Simple cell response properties imply receptive field structure: balanced Gabor and/or bandlimited field functions

Davis Cope; Barbara Blakeslee; Mark E. McCourt

doi:10.1364/JOSAA.26.002067

Journal of the Optical Society of America A
Vol. 26,
Issue 9,
pp. 2067-2092
(2009)
•https://doi.org/10.1364/JOSAA.26.002067

Simple cell response properties imply receptive field structure: balanced Gabor and/or bandlimited field functions

Davis Cope, Barbara Blakeslee, and Mark E. McCourt

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Davis Cope, Barbara Blakeslee, and Mark E. McCourt, "Simple cell response properties imply receptive field structure: balanced Gabor and/or bandlimited field functions," J. Opt. Soc. Am. A 26, 2067-2092 (2009)

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- Vision, Color, and Visual Optics
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About this Article
History
- Original Manuscript: October 10, 2008
- Revised Manuscript: March 23, 2009
- Manuscript Accepted: April 22, 2009
- Published: August 24, 2009
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Virtual Journal for Biomedical Optics Vol. 4, Iss. 11

September 11, 2009 Spotlight on Optics

Abstract

The classical receptive fields of simple cells in mammalian primary visual cortex demonstrate three cardinal response properties: (1) they do not respond to stimuli that are spatially homogeneous; (2) they respond best to stimuli in a preferred orientation (direction); and (3) they do not respond to stimuli in other, nonpreferred orientations (directions). We refer to these as the balanced field property, the maximum response direction property, and the zero response direction property, respectively. These empirically determined response properties are used to derive a complete characterization of elementary receptive field functions defined as products of a circularly symmetric weight function and a simple periodic carrier. Two disjoint classes of elementary receptive field functions result: the balanced Gabor class, a generalization of the traditional Gabor filter, and a bandlimited class whose Fourier transforms have compact support (i.e., are zero valued outside of a bounded range). The detailed specification of these two classes of receptive field functions from empirically based postulates may prove useful to neurophysiologists seeking to test alternative theories of simple cell receptive field structure and to computational neuroscientists seeking basis functions with which to model human vision.

1. INTRODUCTION

The pioneering work of Hubel and Weisel [1, 2] established that neurons in the mammalian primary visual cortex possessed receptive fields that, unlike the circularly concentric receptive fields of subcortical visual neurons, were spatially oriented. The receptive field of one type of cortical neuron, the simple cell, possessed discrete elongated regions within which stimulation by light evoked either excitation or inhibition. This geometry suggested that simple cells might be especially responsive to visual stimuli consisting of bars or edges. Reinterpreting the presumptive role of simple cells as bar and edge detectors, DeValois and colleagues [3, 4] demonstrated that the periodic structure of simple cell receptive fields rendered them highly selective for stimulus spatial frequency and suggested that they might therefore constitute the basis functions for a piecewise Fourier (or wavelet) encoding of the retinal intensity distribution. Much attention was concurrently devoted to developing a mathematical description of the simple cell receptive field function. The first such descriptions, formulated by Marcelja [5] and Daugman [6, 7], inspired by the earlier work of Gabor [8], showed that the spatial sensitivity of simple cells was well described by a 2D Gaussian-damped sine-wave carrier signal, a so-called Gabor function. Owing to some undesirable features of the Gabor function, in particular the fact that it does not integrate to zero for all carrier phases, a variety of competing mathematical formulations have been proposed, which include the difference-of-Gaussians [9, 10], the Laplacian-of-a-Gaussian [11], the log-Gabor [12] and the Cauchy function [13]; see Wallis [14] for a critical review. However, the Gabor model has withstood the test of repeated physiological [15, 16, 4, 17], and psychophysical [18, 19] verification and has therefore emerged as the most commonly accepted mathematical description of cortical simple cell receptive field structure. Theorists interested in developing mathematical descriptions and computational models of various aspects of human spatial vision therefore frequently employ Gabor filters as basis functions [20, 21, 22, 23, 24].

Experimental studies have established that simple cells show a variety of well-defined behaviors [17]. We are concerned here with three essential properties of simple cell receptive fields: (1) the balanced field property (i.e., spatially homogeneous patterns produce a zero response); (2) the zero response direction (ZRD) property [i.e., there is a direction (viz, stimulus orientation) that elicits a zero response to sinusoidal gratings]; and (3) the maximum response direction (MRD) property [the maximum response decreases monotonically to zero as direction (i.e., stimulus orientation) changes from the optimal direction to a direction perpendicular to the optimal]. Note that we use the term “direction” to describe what vision scientists commonly refer to as “orientation,” where grating direction is orthogonal to grating orientation.

In this paper we use these three empirical response properties of simple cells as postulates, which are themselves independent of any particular receptive field structure, and apply them to obtain a complete classification of elementary receptive field functions, which are defined as the product of a circularly symmetric weight function and a simple periodic carrier. Thus, we describe all possible types of elementary receptive field functions that satisfy these empirical constraints. Two disjoint classes of receptive field functions result. The first class, designated here as balanced Gabor receptive fields, is found to be a natural generalization of the traditional Gabor receptive field model. It includes as its simplest case the simple balanced Gabor. The simple balanced Gabor has a receptive field structure similar to the traditional Gabor receptive field model but integrates to zero for all spatial phases of the periodic carrier function. Balanced Gabor filters possess all of the desirable features of the traditional Gabor model, viz., spatial frequency and orientation (direction) tuning, etc., but correct what is commonly regarded as the chief deficiency of the traditional Gabor, which is that it possesses a nonzero DC response. The second class, designated here as bandlimited receptive fields [25] also possess well-behaved spatial frequency and orientation response properties but differ sufficiently from balanced Gabor functions in other respects that an empirical determination of which class best describes simple cell receptive field structure may be possible.

This paper presents formulation, terminology, and a complete statement of our results together with examples of elementary receptive field functions. The more extensive mathematical derivations are given as Appendices A, B, C in a separate Supplement together with additional figures [26]. Appendices A and B derive our main results stated in Section 4, and Appendix C contains miscellaneous derivations. Appendices are referenced herein as appropriate.

2. FORMALIZATION OF POSTULATED SIMPLE CELL RECEPTIVE FIELD PROPERTIES

This section states the ZRD and MRD properties and provides a concise mathematical formulation. It should be noted that these three properties are cumulative, not independent; that is, the MRD property mathematically implies the ZRD property, which in turn implies the balanced field property. Although experimental procedure may emphasize the MRD, the ZRD has more immediate theoretical implications, and this cumulative formulation has been found useful for the development of such implications. It is assumed that the integral [Eq. (1) below] describes the interaction of the receptive field with the stimulus, but no assumption about a specific structure of the receptive field is made in this section. Section 3 will make such an assumption in defining an elementary receptive field function and a mathematically rigorous formulation will be given there.

We define the visual field as a plane, described by the Cartesian coordinates $(x_{1}, x_{2})$ (briefly, $\overset{⃑}{x}$ ). We define visual stimuli, or patterns in the visual field, as nonnegative, bounded, and possibly time-dependent functions $p (\overset{⃑}{x}; t)$ on the plane. The receptive field of the simple cell is described with respect to a fixed reference location, its center, taken here to be the origin. The receptive field is modeled by the receptive field function $R (\overset{⃑}{x})$ , an absolutely integrable function that is assumed to interact with the stimulus pattern by

R \circ p (t) ≔ \int_{R \times R} R (\overset{⃑}{x}) p (\overset{⃑}{x}; t) d \overset{⃑}{x} .

Here

R \circ p

is the response of the receptive field, briefly, the response. The response to a spatially homogeneous pattern

p = c

c R_{0}

, where

R_{0} ≔ \int_{R \times R} R (\overset{⃑}{x}) d \overset{⃑}{x} .

The properties below describe the response

R \circ p

to a sinusoidal grating stimulus p, which (normalized to mean value 1) is a pattern of the form

p (\overset{⃑}{x}) ≔ 1 + c_{P} \cos (2 π \frac{\overset{⃑}{d} (α_{P}) \cdot \overset{⃑}{x}}{λ_{P}} - ϕ_{P}),

where

c_{P}

is the grating contrast

(- 1 ⩽ c_{P} ⩽ + 1)

λ_{P} > 0

is the grating wavelength,

\overset{⃑}{d}

is a unit vector defining grating direction

α_{P}

, that is,

\overset{⃑}{d} (α_{P}) \cdot \overset{⃑}{x} = \cos (α_{P}) x_{1} + \sin (α_{P}) x_{2},

and

ϕ_{P}

is a phase shift, which is time dependent in the case of a drifting grating. The subscript P designates pattern parameters. The pattern is a periodic sequence of bright and dark bands perpendicular to

\overset{⃑}{d}

, normalized to have mean value one. Notice that the grating parameter sets

c_{P}, λ_{P}, α_{P}, ϕ_{P}

;

- c_{P}, λ_{P}, α_{P}, ϕ_{P} \pm π

; and

c_{P}, λ_{P}, α_{P} \pm π, - ϕ_{P}

all describe the same grating and hence will produce the same response.

The response to a sinusoidal grating stimulus is closely related to the Fourier transform of the receptive field function. Specifically, the Fourier transform $F_{R} (s_{1}, s_{2})$ of $R (x_{1}, x_{2})$ is defined by

F_{R} (s_{1}, s_{2}) ≔ \int \int_{R \times R} \exp (- 2 π i (s_{1} x_{1} + s_{2} x_{2})) R (x_{1}, x_{2}) d x_{1} d x_{2} .

The response

c R_{0}

to the constant pattern is related to the Fourier transform by

R_{0} = F_{R} (0, 0),

while the response

R \circ p

to a sinusoidal grating pattern p is given by (since R is real-valued)

R \circ p = \int_{R \times R} R (\overset{⃑}{x}) (1 + c_{P} \cos (2 π \frac{\overset{⃑}{d} (α_{P}) \cdot \overset{⃑}{x}}{λ_{P}} - ϕ_{P})) d \overset{⃑}{x} = R_{0} + c_{P} Re [e^{i ϕ_{P}} F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P}))],

which determines the transform (e.g.,

Re [F_{R}], - Im [F_{R}]

) for

ϕ_{P} = 0, π ∕ 2

). The max-response M to a sinusoidal grating stimulus is the maximum over all phases

ϕ_{P}

M (c_{P}, α_{P}, λ_{P}) ≔ \max_{ϕ_{P}} (R \circ p) = \max_{ϕ_{P}} {R_{0} + c_{P} Re [e^{i ϕ_{P}} F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P}))]} = R_{0} + | c_{P} | | F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) | .

Since R is real-valued, the transform satisfies

\bar{F_{R}} (s_{1}, s_{2}) = F_{R} (- s_{1}, - s_{2})

. Consequently,

M (c_{P}, α_{P}, λ_{P}) = M (| c_{P} |, α_{P}, λ_{P}) = M (| c_{P} |, α_{P} \pm π, λ_{P}) .

In particular, M has period π with respect to

α_{P}

Since the response function, that is, the response over all grating directions and wavelengths, determines the Fourier transform of R, sinusoidal grating experiments have fundamental significance for the study of the receptive field. If Eq. (1) completely described the response to a visual stimulus, then the Fourier transform, that is, the response function, would completely determine the receptive field. In practice, the response necessarily includes nonlinear behavior not described by Eq. (1), but it is widely accepted that the equation models a major portion of the response, and the response function determined by sinusoidal grating experiments plays a correspondingly major role. A second reason for the fundamental significance of grating experiments is that simple cells respond in a remarkably well-defined way to sinusoidal grating stimuli, showing a maximum response for an optimum grating direction and optimum grating wavelength, then decreasing steadily as these parameters vary from the optimum with, in particular, a zero response for grating directions perpendicular to the optimum direction. Our postulates are expressed in terms of such observed behavior for the response to sinusoidal gratings.

The first postulated property of simple cells is that spatially homogeneous patterns produce a zero response, i.e., the

Balanced Field Property. The response $R \circ p$ to a spatially homogeneous pattern $p \equiv constant$ is zero.

Such patterns correspond to a zero-contrast grating. The constant scales out, and the mathematical form is

R_{0} = F_{R} (0, 0) = 0 .

The second postulated property of simple cells is that there is a stimulus direction (i.e., grating orientation) that elicits a zero response, the ZRD property:

ZRD Property (Form 1). There exist values $α_{ZR}$ and $c_{ZR} \neq 0$ such that $R \circ p = 0$ for all sinusoidal gratings p with grating direction $α_{ZR}$ and contrast $c_{ZR}$ .

The mathematical form is that, for all grating phases $ϕ_{P}$ and wavelengths $λ_{P} > 0$ ,

R_{0} + c_{ZR} Re [e^{i ϕ_{P}} F_{R} (\frac{1}{λ_{P}} \cos (α_{ZR}), \frac{1}{λ_{P}} \sin (α_{ZR}))] = 0

Notice that the ZRD property implies the balanced field property (consider the phase shifts

ϕ_{P}

ϕ_{P} + π

and add), that is,

R_{0} = 0

. Since Eq. (9) can then hold for all

ϕ_{P}

if and only if

F_{R} = 0

, the ZRD property is equivalent to the following formulation, which drops all reference to grating contrast and phase:

ZRD Property (Form 2). There exists a value $α_{ZR}$ such that, for all $λ_{P} > 0$ ,

F_{R} (\frac{1}{λ_{P}} \cos (α_{ZR}), \frac{1}{λ_{P}} \sin (α_{ZR})) = 0 .

To see that Form 2 implies Form 1, let

λ_{P} \to \infty

in Eq. (10) to show that the receptive field is balanced; then Eq. (9) follows for arbitrary contrast. The direction

α_{ZR}

is a ZRD and is determined only up to an additive multiple of π; values differing by multiples of π are equivalent directions, and multiple, that is, nonequivalent, zero response directions may occur.

Incidentally, in modeling the neurophysiological response to excitation and inhibition, the response is sometimes taken to be the positive part of the calculated numerical value since firing rates cannot be negative. Using that convention, a zero response would mean $Pos (R \circ p) = 0$ , that is, $R \circ p ⩽ 0$ . That convention is not used here: response means $R \circ p$ , and a zero response means $R \circ p = 0$ .

The third postulated property is essentially that the maximum response decreases monotonically to zero as direction (i.e., stimulus orientation) changes from the optimal direction to a direction perpendicular to the optimal, the MRD property:

MRD Property (Form 1). There exists a value $c_{MR} \neq 0$ such that $M (c_{MR}, α_{P}, λ_{P})$ is not identically zero and a value $α_{MR}$ such that, for all sinusoidal gratings p with contrast $c_{MR}$ and at each fixed wavelength $λ_{P}$ , either the response $M (c_{MR}, α_{P}, λ_{P})$ is zero for all grating directions $α_{P}$ or it is strictly decreasing to zero (and remains zero) as the grating direction changes from $α_{MR}$ .

The range $| α_{P} - α_{MR} | ⩽ π ∕ 2$ covers one period of $M (c_{MR}, α_{P}, λ_{P})$ , and the MRD property implies $M (c_{MR}, α_{MR} \pm π ∕ 2, λ_{P}) = 0$ for each wavelength $λ_{P}$ . Thus, the MRD property implies the ZRD property with $α_{ZR} = α_{MR} \pm π ∕ 2$ . The balanced field property therefore holds as well, so that

M (c_{MR}, α_{P}, λ_{P}) = | c_{MR} | | F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) | .

A further implication is that ZRDs necessarily occur as a sector (possibly reducing to a single line). For example, if

α_{ZR} = α_{MR} + ζ_{0}

with

0 < ζ_{0} < π ∕ 2

is a ZRD, then, for each fixed

λ_{P}

, the response will have dropped to zero when the stimulus direction

α_{P}

reaches

α_{ZR}

and then remains zero as

α_{P}

increases. The corresponding range

α_{MR} + ζ_{0} ⩽ α_{P} ⩽ α_{MR} + π ∕ 2

is then a sector of ZRDs.

The MRD Property is equivalent to the following formulation, which drops all reference to grating contrast and phase:

MRD Property (Form 2). For some pair $α_{P}, λ_{P}$ ,

F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) \neq 0

and there exists a value

α_{MR}

such that, at each fixed wavelength

λ_{P}

, either

| F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) |

is zero for all grating directions

α_{P}

or it is strictly decreasing to zero (and remains zero) as

| α_{P} - α_{MR} |

increases.

To see that Form 2 implies Form 1, notice that

| F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) |

has period π with respect to

α_{P}

. Then

| α_{P} - α_{MR} | ⩽ π ∕ 2

covers one period, the argument above can be repeated to obtain the ZRD and balanced field properties, and the behavior for

M (c_{MR}, α_{P}, λ_{P})

stated in Form 1 follows.

The direction $α_{MR}$ is a MRD and is determined only up to an additive multiple of π; values differing by multiples of π are equivalent directions (orientations). It should be emphasized that this property refers not only to the existence of a direction (orientation) of maximum response but also to the monotonic decrease in response with increasing angular distance from the optimal direction. Such monotonic behavior implies that a maximum response direction, if it exists, is unique.

It should be noted that the properties are robust with respect to nonlinear processing. For example, if the response defined by Eq. (1) is followed by a nonlinear operation, then the properties will continue to hold for the new result given some mild conditions on the nonlinear operation, such as monotonicity and having zero as a fixed point.

3. ELEMENTARY RECEPTIVE FIELD FUNCTIONS AND HANKEL TRANSFORMS

Receptive fields are typically modeled as a product of a weight function, providing localization of the response, and an oscillatory carrier function, providing directionality (orientation) to the response. This section defines an elementary receptive field function to be the product of a circularly symmetric weight function and a simple periodic carrier. For such receptive field functions, the response to a sinusoidal grating can be expressed in terms of the Hankel transform of the weight function. This section describes the Hankel transform and its use to provide reduced mathematical formulations of the ZRD and MRD properties for elementary receptive field functions. These reduced formulations are the basis for the results of Section 4.

We define an elementary receptive field function (centered at the origin) to be the product of a circularly symmetric weight function q and a simple periodic carrier function, that is,

R (\overset{⃑}{x}) ≔ \frac{2 π}{λ_{R}^{2}} q (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - b_{R}),

where

λ_{R} > 0

is the spatial wavelength,

α_{R}

is the field orientation,

ϕ_{R}

is the phase shift, and

b_{R}

is the balancing parameter (the subscript R designates receptive field parameters). The weight function

q (r)

may be negative and is assumed to satisfy the mild regularity conditions

q (r) is C (0, + \infty),

It is useful to distinguish the cases

\sin (ϕ_{R}) = 0

\cos (ϕ_{R}) \sin (ϕ_{R}) \neq 0

\cos (ϕ_{R}) = 0

as elementary receptive field functions of cosine type, mixed type, and sine type, respectively. Note that a specific form is assumed for the carrier function, but the weight function

(q)

is only required to be circularly symmetric and is otherwise unconstrained.

Section 2 formulated properties of the receptive field $R (x_{1}, x_{2})$ in terms of the response to sinusoidal gratings, and then gave mathematical formulations in terms of the Fourier transform $F_{R} (s_{1}, s_{2})$ . The form (11) for the receptive field is essentially a modulated circularly symmetric function. The 2D Fourier transform of a circularly symmetric function is also a circularly symmetric function, and the relation between these two functions of a single radial variable is given by the Hankel transform:

If f (x_{1}, x_{2}) = g (r) where r = {(x_{1}^{2} + x_{2}^{2})}^{1 ∕ 2},

where

H_{g}

is the Hankel transform of g [and

F_{f}

is defined by Eq. (4)]. The transform and the inverse transform are given specifically by

H_{f} (ρ) = \int_{0}^{\infty} J_{0} (ρ r) f (r) r d r and f (r) = \int_{0}^{\infty} J_{0} (ρ r) H_{f} (ρ) ρ d ρ .

Here r represents distance from the origin, ρ can be thought of as a radial frequency, and

J_{0} (x)

is the Bessel function of order 0. Bessel functions are damped oscillatory waveforms that constitute the basis for Hankel analysis/synthesis, just as sinusoids form the basis functions for Fourier analysis/synthesis. In particular, conditions (12) imply

H_{q} (ρ) is C [0, + \infty), H_{q} (0) = 1, and H_{q} (+ \infty) = 0 .

Since the elementary receptive field functions R given by Eq. (11) are simple modulations of the circularly symmetric weight functions q, the Fourier transform

F_{R}

can be expressed in terms of translations of the Fourier transform

F_{q}

, which in turn can be expressed in terms of the Hankel transform

H_{q}

(see Appendix C in the Supplement [26]). To express various relations, it will be useful to write

\overset{⃑}{s} = (s_{1}, s_{2})

and

\overset{⃑}{s_{R}} = (\frac{\cos (α_{R})}{λ_{R}}, \frac{\sin (α_{R})}{λ_{R}}), \overset{⃑}{s_{P}} = (\frac{\cos (α_{P})}{λ_{P}}, \frac{\sin (α_{P})}{λ_{P}})

so that, for example,

‖ \overset{⃑}{s} \pm \overset{⃑}{s_{R}} ‖ = \frac{1}{λ_{R}} \sqrt{{(λ_{R} s_{1} \pm \cos (α_{R}))}^{2} + {(λ_{R} s_{2} \pm \sin (α_{R}))}^{2}},

The Fourier transform can then be written as

F_{R} (s_{1}, s_{2}) = \frac{1}{2} (e^{+ i ϕ_{R}} H_{q} (λ_{R} ‖ \overset{⃑}{s} + \overset{⃑}{s_{R}} ‖) + e^{- i ϕ_{R}} H_{q} (λ_{R} ‖ \overset{⃑}{s} - \overset{⃑}{s_{R}} ‖)) - b_{R} H_{q} (λ_{R} ‖ \overset{⃑}{s} ‖) .

Section 3 noted general relations between the Fourier transform

F_{R}

of the receptive field function R and its response to a sinusoidal grating, and the corresponding quantities for an elementary receptive field function therefore reduce to expressions involving the Hankel transform

H_{q}

. Specifically, for constant patterns, Eq. (5) becomes

R_{0} = \cos (ϕ_{R}) H_{q} (1) - b_{R} H_{q} (0) .

The response

R \circ p

of the elementary receptive field function to a sinusoidal grating pattern p, given by Eq. (6), becomes

R \circ p = R_{0} + \frac{c_{P}}{2} \cos (ϕ_{P} + ϕ_{R}) H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) + \frac{c_{P}}{2} \cos (ϕ_{P} - ϕ_{R}) H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) - b_{R} c_{P} \cos (ϕ_{P}) H_{q} (\frac{λ_{R}}{λ_{P}}) .

The max-response M of an elementary receptive field function to a sinusoidal grating pattern, given by Eq. (7), becomes

M (c_{P}, α_{P}, λ_{P}) = \max_{ϕ_{P}} (R \circ p) = R_{0} + \frac{1}{2} | c_{P} | N (α_{P}, λ_{P}),

where the amplitude factor

N (α_{P}, λ_{P}) ≔ 2 | F_{R} (\cos (α_{P}) ∕ λ_{P}, \sin (α_{P}) ∕ λ_{P}) | ⩾ 0

can be obtained by expanding Eq. (20) in terms of

\cos (ϕ_{P})

and

\sin (ϕ_{P})

and is now given by

N {(α_{P}, λ_{P})}^{2} = {[\cos (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) + H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖)) - 2 b_{R} H_{q} (\frac{λ_{R}}{λ_{P}})]}^{2} + {[\sin (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) - H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖))]}^{2} .

Note that Eqs. (19, 20, 23) are completely general formulas describing, in terms of the Hankel transform

H_{q}

, the response of elementary receptive field functions (11) to sinusoidal grating stimuli.

Section 2 formulated the three properties in terms of the Fourier transform $F_{R}$ of the general receptive field function R. Using Eq. (18), those results now become formulations of the properties for an elementary receptive field function (11) in terms of the Hankel transform $H_{q}$ .

The first result follows immediately from Eq. (19) and uniquely determines the balancing parameter $b_{R}$ :

Lemma 1. Let the receptive field function $R (\overset{⃑}{x})$ be given by Eq. (11). Let $q (r)$ satisfy Eq. (12) [in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies Eq. (15)]. Then $R (\overset{⃑}{x})$ has the balanced field property if and only if

b_{R} = \cos (ϕ_{R}) H_{q} (1) .

The second result follows directly from Eq. (10), Eq. (15), and the representation (18) with

α_{P} = α_{ZR}

Lemma 2. Let the receptive field function $R (\overset{⃑}{x})$ be given by Eq. (11). Let $q (r)$ satisfy Eq. (12) [in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies Eq. (15)]. Then $R (\overset{⃑}{x})$ has the ZRD property if and only if there exists a value $α_{ZR}$ such that the following relations hold for all $λ_{P} > 0$ :

\cos (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} + \overset{⃑}{s_{R}} ‖) + H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} - \overset{⃑}{s_{R}} ‖)) = 2 b_{R} H_{q} (\frac{λ_{R}}{λ_{P}}),

\sin (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} + \overset{⃑}{s_{R}} ‖) - H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} - \overset{⃑}{s_{R}} ‖)) = 0,

b_{R} = \cos (ϕ_{R}) H_{q} (1) .

In such a case,

α_{ZR}

is a ZRD.

The third result is a direct restatement of the MRD property (Form 2) in terms of the amplitude factor $N (α_{P}, λ_{P})$ :

Lemma 3. Let the receptive field function $R (\overset{⃑}{x})$ be given by Eq. (11). Let $q (r)$ satisfy Eq. (12) [in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies Eq. (15)]. Then $R (\overset{⃑}{x})$ has the MRD property if and only if $N (α_{P}, λ_{P}) \neq 0$ for some pair $α_{P}, λ_{P}$ and there exists a value $α_{MR}$ such that, at each fixed wavelength $λ_{P}$ , either $N (α_{P}, λ_{P})$ is zero for all grating directions $α_{P}$ or it is strictly decreasing to zero (and remains zero) as $| α_{P} - α_{MR} |$ increases.

Section 2 observed that an MRD, if it exists, is unique. It is evident from Eq. (23) that $N (α_{P}, λ_{P})$ is an even function of $α_{P} - α_{R}$ , which implies $α_{MR} = α_{R}$ . Section 2 also observed that the MRD property implies the ZRD property, specifically, that a ZRD is given by $α_{ZR} = α_{R} \pm π ∕ 2$ .

4. CHARACTERIZATION OF ELEMENTARY RECEPTIVE FIELD FUNCTIONS WITH POSTULATED PROPERTIES

This section first states Theorems A.1 and A.2, which describe elementary receptive field functions with the ZRD property, and then states the main result, Theorem B.1, a complete characterization of elementary receptive field functions with the MRD property for cosine-type and mixed-type receptive fields. Related results for sine-type receptive fields are summarized in the course of discussion as Theorems B.2 and B.3. The proofs for A.1 and A.2 involve a preparatory lemma and are given in Appendix A [26]. The proof for Theorem B.1 builds on Theorems A.1 and A.2 and is given in Appendix B [26].

Theorem A.1 (Cosine-type and mixed-type receptive field functions). Let the receptive field function $R (\overset{⃑}{x})$ be given by Eq. (11). Let $q (r)$ satisfy Eq. (12) [in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies Eq. (15)]. Assume $\cos (ϕ_{R}) \neq 0$ . Then $R (\overset{⃑}{x})$ has the ZRD property if and only if $H_{q} (ρ)$ satisfies one of the following cases:

$0 < H_{q} (1) < 1$ and $H_{q} (ρ) = H_{q} {(1)}^{ρ^{2}} F (ρ^{2})$ where $F (y)$ is $C [0, \infty)$ with $F (0) = 1$ and $F (y + 1) = F (y)$ . In this case, there is exactly one ZRD given by $α_{ZR} - α_{R} = π ∕ 2$ .
$- 1 < H_{q} (1) < 0$ and $H_{q} (ρ) = {(- H_{q} (1))}^{ρ^{2}} F (ρ^{2})$ , where $F (y)$ is $C [0, \infty)$ with $F (0) = 1$ and $F (y + 1) = - F (y)$ . In this case, there is exactly one ZRD given by $α_{ZR} - α_{R} = π ∕ 2$ .
$H_{q} (1) = 0$ and $H_{q} (ρ) = 0$ on the interval $0 < \sin (ζ_{R}) ⩽ ρ < + \infty$ but on no larger interval. In this case, there is a sector of ZRDs and the sector is given by $0 < ζ_{R} ⩽ | α_{ZR} - α_{R} | ⩽ π ∕ 2$ .
$0 < | H_{q} (1) | < 1$ and $\cos (ϕ_{R}) = \pm 1$ and, for some $0 < ζ_{0} < π ∕ 2$ , $H_{q} (ρ)$ satisfies $H_{q} (\sqrt{1 - 2 ρ \cos (ζ_{0}) + ρ^{2}}) + H_{q} (\sqrt{1 + 2 ρ \cos (ζ_{0}) + ρ^{2}}) = 2 H_{q} (1) H_{q} (ρ)$ for $ρ ⩾ 0$ . In this case, there are ZRDs given by $α_{ZR} - α_{R} = \pm ζ_{0}$ .

Note that Theorem A.1(4) is simply a remaining case, where the theorem does not provide a definitive conclusion. We conjecture that no $H_{q} (x)$ exists, that is, that the case is vacuous, but we have been unable to settle the conjecture under the sole condition that a ZRD exists. This case is eliminated at the next stage, where the MRD Property is assumed.

Theorem A.2 (Sine-type receptive field functions). Let the receptive field function $R (\overset{⃑}{x})$ be given by Eq. (11). Let $q (r)$ satisfy Eq. (12) [in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies Eq. (15)]. Assume $\cos (ϕ_{R}) = 0$ . Then

$R (\overset{⃑}{x})$ has a ZRD given by $α_{ZR} - α_{R} = π ∕ 2$ .
$R (\overset{⃑}{x})$ has more than one ZRD if and only if $H_{q} (ρ) = 0$ on an interval $\sin (ζ_{0}) ⩽ ρ < + \infty$ with $0 < \sin (ζ_{R}) < 1$ . In this case, taking [ $\sin (ζ_{R}), + \infty)$ to be the largest such interval on which $H_{q} (ρ)$ vanishes, there is a sector of ZRDs and the sector is given by $0 < ζ_{R} ⩽ | α_{ZR} - α_{R} | ⩽ π ∕ 2 .$ Our main result is the following characterization of the MRD property:

Theorem B.1. Let the receptive field function $R (\overset{⃑}{x})$ be given by Eq. (11). Let $q (r)$ satisfy Eq. (12) [in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies Eq. (15)]. Assume $\cos (ϕ_{R}) \neq 0$ . Then $R (\overset{⃑}{x})$ has the MRD property if and only if $H_{q} (ρ)$ satisfies one of the following cases:

(1) $H_{q} (ρ) = e^{- c ρ^{2}} F (ρ^{2})$ where $c > 0$ , $F (y)$ is continuous and positive on the real line, $F (0) = 1$ , $F (y + 1) = F (y)$ , and

$(^{*})$ if $\cos (ϕ_{R}) = \pm 1$ , then, for each a, the function

f (a; y) = e^{c y} F (a - y) + e^{- c y} F (a + y) - 2 F (a)

is initially zero on $0 ⩽ y ⩽ δ_{a}$ and strictly increasing on $δ_{a} ⩽ y < + \infty$ ;

$(^{* *})$ if $\cos (ϕ_{R}) \neq \pm 1$ , then, for each a, the function

{(\cos (ϕ_{R}))}^{2} {(e^{c y} F (a - y) + e^{- c y} F (a + y) - 2 F (a))}^{2} + {(\sin (ϕ_{R}))}^{2} {(e^{c y} F (a - y) - e^{- c y} F (a + y))}^{2}

is strictly increasing on $0 ⩽ y < + \infty$ .

In $(^{*})$ , $δ_{a} > 0$ (that is, $f (a; y)$ is initially zero on a nontrivial interval) if and only if the value a satisfies

e^{- c y} F (y) = e^{- c a} F (a) - K (a) (y - a) on | y - a | ⩽ δ_{a} for some constant K (a) > 0 .

For both $(^{*})$ and $(^{* *})$ :

(a) the MRD is given by $α_{MR} = α_{R}$ ;

(b) the ZRD is unique and given by $α_{ZR} = α_{R} + π ∕ 2$ ;

(2) $H_{q} (ρ)$ is strictly decreasing on $0 ⩽ ρ < \sin (ζ_{R})$ and $H_{q} (ρ) = 0$ on $\sin (ζ_{R}) ⩽ ρ < + \infty$ . In this case:

(a) the MRD is given by $α_{MR} = α_{R}$ ;

(b) there is a sector of ZRDs given by $0 < ζ_{R} ⩽ | α_{ZR} - α_{R} | ⩽ π ∕ 2$ .

The remainder of this section discusses implications of Theorem B.1, in particular, that it leads to two disjoint classes of receptive field functions. A preliminary analysis of the two classes is given. Subsections 5A, 5B, respectively, provide more detailed results for each class as well as explicit analytic examples illustrating those results.

In result (1) of Theorem B.1, parts $(^{*})$ and $(^{* *})$ can be considered as providing tests for candidate periodic functions $F (y)$ in forming the Hankel transform. Notice it is sufficient to test the expressions for $0 ⩽ a < 1$ since $F (y)$ , and thus the given expressions as well, has period 1. However, it is useful to leave the criterion in terms of arbitrary a. Equation (30) says that the strictly decreasing function $e^{- c y} F (y)$ reduces to a linear function on some intervals. This condition (with $δ_{a} > 0$ ) cannot hold for all a, or even for $0 ⩽ a ⩽ 1$ , because $e^{- c y} F (y)$ would reduce to a strictly linear function over an interval larger than a period. Consequently, expression (28) must be strictly increasing on $0 ⩽ y < + \infty$ for some values of a.

Theorem B.1 completely characterizes elementary receptive field functions $R (\overset{⃑}{x})$ (of cosine-type and mixed-type) possessing the MRD property in terms of the Hankel transform $H_{q} (ρ)$ of the receptive field weight functions $q (r)$ . For the moment, let us refer to weights $q (r)$ occurring under result (1) as Type I weights and those under result (2) as Type II weights. These two classes of weight functions are completely disjoint, since the Hankel transform $H_{q} (ρ)$ for a Type I weight is positive for all ρ, while the transform for a Type II weight is zero outside a finite interval. The characterization theorem thus results in two disjoint classes of receptive field functions $R (\overset{⃑}{x})$ , each possessing the MRD property.

Type I and Type II weights yield quite different formulas for the amplitude response factor $N (α_{P}, λ_{P}) ⩾ 0$ given by Eq. (23), and the formulas may be helpful in understanding the results of Theorem B.1. (The formulas continue to hold for sine-type receptive fields.) For Type I weights with Hankel transform $H_{q} (ρ) = e^{- c ρ^{2}} F (ρ^{2})$ where $F (y)$ has period one, the factor is

N {(α_{P}, λ_{P})}^{2} = \exp (- c (1 + \frac{λ_{R}^{2}}{λ_{P}^{2}})) \cdot T_{F} (ϕ_{R}, c, 1 + \frac{λ_{R}^{2}}{λ_{P}^{2}}; 2 \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}),

where

T_{F} (ϕ_{R}, c, a; y) ≔ {(\cos (ϕ_{R}))}^{2} {(e^{c y} F (a - y) + e^{- c y} F (a + y) - 2 F (a))}^{2} + {(\sin (ϕ_{R}))}^{2} {(e^{c y} F (a - y) - e^{- c y} F (a + y))}^{2} .

Comparison of Eq. (32) with result (1) shows that

(^{*})

(^{* *})

implies that, for fixed

λ_{P}

N (α_{P}, λ_{P})

is strictly decreasing as

| α_{P} - α_{R} |

increases. For Type II weights with Hankel transform

H_{q} (ρ)

vanishing on some interval

\sin (ζ_{R}) ⩽ ρ < + \infty

, the factor is simply

N (α_{P}, λ_{P}) = H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖),

and result (2), that

H_{q} (ρ)

is strictly decreasing, implies that

N (α_{P}, λ_{P})

is strictly decreasing as

| α_{P} - α_{R} |

increases.

Type II weights are well defined by result (2); in particular, the definition is independent of the receptive field parameters. In contrast, the definition of Type I weights in result (1) depends on the receptive field phase $ϕ_{R}$ . That is, $q (r)$ is a Type I weight if its Hankel transform $H_{q} (ρ)$ satisfies the conditions of result (1) for some $ϕ_{R}$ , and it may be that $q (r)$ satisfies the conditions $(^{*})$ or $(^{* *})$ for some values of $ϕ_{R}$ and not for others. Roughly speaking, test $(^{* *})$ becomes less restrictive as $\cos (ϕ_{R})$ approaches 0, that is, as the receptive field approaches a sine-type receptive field. This observation suggests that, within the Type I class, cosine-type receptive fields are the most basic type. The following theorem illustrates this idea by showing that cosine-type receptive field functions can be used to construct receptive field functions for both mixed-type and sine-type receptive fields. This theorem is the basis for the definition of balanced Gabor weights in Subsection 5A, the subclass of Type I weights that can be used to define elementary receptive field functions with the MRD Property for arbitrary receptive field phase. Type II weights are precisely the bandlimited weights of Subsection 5B. Notice that Theorem B.2 yields sine-type receptive field functions.

Theorem B.2. Let $q (r)$ satisfy Eq. (12) [in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies (15)]. Assume:

$H_{q} (ρ) = e^{- c ρ^{2}} F (ρ^{2})$ where $c > 0$ , $F (y)$ is continuous and positive on the real line, $F (0) = 1$ , $F (y + 1) = F (y)$ .
$e^{- c y} F (y)$ is positive and strictly decreasing for all y.
For each a, the function $f (a; y) = e^{c y} F (a - y) + e^{- c y} F (a + y) - 2 F (a)$ is initially zero on $0 ⩽ y ⩽ δ_{a}$ and strictly increasing on $δ_{a} ⩽ y < + \infty$ .
In (c), $δ_{a} > 0$ if and only if the value a satisfies $e^{- c y} F (y) = e^{- c a} F (a) - K (a) (y - a)$ on $| y - a | ⩽ δ_{a}$ for some constant $K (a) > 0$ .

Then the elementary receptive field functions

R (\overset{⃑}{x})

defined by (11) with

b_{R} = \cos (ϕ_{R}) e^{- c}

have the MRD Property for all values of the receptive field parameters, in particular, all values of the receptive field phase

ϕ_{R}

Theorem B.2 follows directly from result (1) of Theorem B.1:

For cosine-type receptive fields, $R (\overset{⃑}{x})$ has the MRD Property because the conditions are simply a restatement of $(^{*})$ .
For mixed-type receptive fields, $R (\overset{⃑}{x})$ has the property because $(^{* *})$ holds. Conditions (c) and (b) respectively imply that the $\cos {(ϕ_{R})}^{2}$ component of Eq. (29) is increasing and the $\sin {(ϕ_{R})}^{2}$ component is strictly increasing because $e^{+ c y} F (a - y) - e^{- c y} F (a + y)$ is strictly increasing.
For sine-type receptive fields, the preceding observation applied to Eqs. (31, 32) ensures that, for fixed $λ_{P}$ , $N (α_{P}, λ_{P})$ is strictly decreasing as $| α_{P} - α_{R} |$ increases.

Theorem A.2 shows that sine-type elementary receptive field functions possessing a ZRD form a much broader class than the corresponding cosine-type and mixed-type receptive field functions described by Theorem A.1. Similarly, sine-type elementary receptive field functions with the MRD property form a broader class than the cosine-type and mixed-type receptive field functions described by Theorem B.1. The following theorem gives partial results for sine-type receptive fields.

Theorem B.3. Let the receptive field function $R (\overset{⃑}{x})$ be given by Eq. (11). Let $q (r)$ satisfy Eq. (12) (in which case the Hankel transform $H_{q} (ρ)$ exists and satisfies (15)). Assume $\cos (ϕ_{R}) = 0$ .

If $H_{q} (ρ) = e^{- c ρ^{2}} F (ρ^{2})$ with $c > 0$ and $e^{- c y} F (y)$ is positive and strictly decreasing for all y, then $R (\overset{⃑}{x})$ with $b_{R} = \cos (ϕ_{R}) e^{- c} F (1)$ has the MRD property. In this case, there is a unique ZRD given by $α_{ZR} = α_{MR} + π ∕ 2$ .
If $H_{q} (ρ)$ is strictly decreasing on $0 ⩽ ρ < \sin (ζ_{R})$ and $H_{q} (ρ) = 0$ on $\sin (ζ_{R}) ⩽ ρ < + \infty$ , then $R (\overset{⃑}{x})$ with $b_{R} = 0$ has the MRD property. In this case, there is a sector of ZRDs and the sector is given by $0 < ζ_{R} ⩽ | α_{ZR} - α_{MR} | ⩽ π ∕ 2$ .
$R (\overset{⃑}{x})$ has the MRD property and more than one ZRD if and only if $H_{q} (ρ)$ satisfies (2) with $0 < \sin (ζ_{R}) < 1$ . In this case, $b_{R} = 0$ and there is a sector of ZRDs.

Result (1) follows by a direct check that, for fixed

λ_{P}

N (α_{P}, λ_{P}) = | H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) - H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) |

is strictly decreasing as

| α_{P} - α_{R} |

increases. Result (2) follows by the same check after noting that Eq. (34) reduces to a single term since one argument is necessarily

⩾ 1

. Result (3) follows from Theorem A.2, which characterizes the condition under which a sector of ZRDs can occur.

5. RECEPTIVE FIELD FUNCTIONS

5A. Balanced Gabor Receptive Field Functions: Discussion and Examples

This subsection defines and discusses the balanced Gabor class of weights and elementary receptive field functions mentioned in Section 4. Two cases are analyzed: the simple balanced Gabor receptive field function, corresponding to the traditional Gabor filter model of the receptive field, and a more general class of balanced Gabor weights, corresponding to elementary receptive fields with weight functions that are not simple Gaussians but that are products of Gaussians and oscillatory components.

5A1. Balanced Gabor Weights and Receptive Fields

Balanced Gabor weights (with exponent $γ_{R} > 0$ ) are weight functions $g (r)$ satisfying

g (r) is C (0, + \infty),

in which case the Hankel transform

H_{g} (ρ)

satisfies

H_{g} (ρ) is C [0, + \infty), H_{g} (0) = 1, and H_{g} (+ \infty) = 0

and such that the following properties hold:

$H_{g} (ρ) = e^{- γ_{R} ρ^{2}} G (ρ^{2})$ , where $G (y)$ is continuous and positive on the real line, $G (0) = 1$ , $G (y + 1) = G (y)$ ;
$e^{- γ_{R} y} G (y)$ is strictly decreasing on the real line;
for each real a, $T_{G} (γ_{R}, a; y) ≔ e^{γ_{R} y} G (a - y) + e^{- γ_{R} y} G (a + y) - 2 G (a)$ is initially zero on $0 ⩽ y ⩽ δ_{a}$ and strictly increasing on $δ_{a} ⩽ y < + \infty$ ;
in (c), $δ_{a} > 0$ if and only if the value a satisfies $e^{- c y} F (y) = e^{- c a} F (a) - K (a) (y - a) on | y - a | ⩽ δ_{a}$ for some constant $K (a) > 0$ .

All parameters and independent variables are dimensionless. For each balanced Gabor Hankel transform (and thus for the corresponding weight function), the exponent is uniquely determined. The exponents thus partition or classify both the transforms and the weights. For balanced Gabor weights $g (r)$ , $g (0)$ is necessarily positive because Eq. (14) reduces to a positive integrand for $r = 0$ . Simple balanced Gabor weights refer to the special case $G (y) \equiv 1$ . The following subsection discusses these weights and compares them with the traditional Gabor filter model for receptive fields.

Some points should be mentioned with regard to checking conditions (b), (c), and (d) of the definition. In checking (b) for a candidate periodic function $G (y)$ , it is necessary to check only that $e^{- γ_{R} y} G (y)$ is strictly decreasing over an interval $[y_{0}, y_{0} + 1]$ of length one, since shifting this function by a period multiplies it by a constant:

e^{- γ_{R} (y + 1)} G (y + 1) = e^{- γ_{R}} e^{- γ_{R} y} G (y) .

In checking (c) for a candidate function

G (y)

, it is necessary to check only

the condition on $T_{G} (γ_{R}, a; y)$ for some interval $a_{0} ⩽ a < a_{0} + 1$ since the expression is 1-periodic in a;
for given a, that $T_{G} (γ_{R}, a; y)$ is strictly increasing on $0 ⩽ y < 1$ , since strictly increasing behavior on any unit interval implies a continued strict increase due to (b), which implies that $e^{γ_{R} y} G (- y)$ is strictly increasing, and to the relation $T_{G} (γ_{R}, a; y + 1) = e^{- γ_{R}} T_{G} (γ_{R}, a; y) + (e^{γ_{R}} - e^{- γ_{R}}) e^{γ_{R} y} G (- y) .$

Condition (d) says that the strictly decreasing function

e^{- c y} F (y)

reduces to a linear function on some intervals. This condition (with

δ_{a} > 0

) cannot hold for all a, or even for

0 ⩽ a ⩽ 1

, because

e^{- c y} F (y)

would reduce to a strictly linear function over an interval larger than a period. Thus,

T_{G} (γ_{R}, a; y)

must be strictly increasing on

0 ⩽ y < + \infty

for some values of a.

While the sets of balanced Gabor Hankel transforms for different exponents are disjoint, the corresponding sets of periodic components are related:

Closure Lemma. Let

Γ (γ_{R}) ≔ {G (y) : G (y) satisfies conditions (a), (b), (c) for balanced Gabor weights with exponent γ_{R}}

Then

$Γ (γ_{R})$ is closed under convex combinations.
If $γ_{1} < γ_{2}$ , then $Γ (γ_{1}) \subset_{\neq} Γ (γ_{2})$ .
$\cup_{0 < γ < \infty} Γ (γ)$ is closed under convex combinations.

Result (1) follows by a straightforward check of the conditions of the balanced Gabor definition when applied to convex combinations. Result (2) requires some manipulation, and the proof is given in Appendix C [26]. Result (3) follows directly from (1) and (2).

Two aspects of the Closure Lemma should be noted. First, the sets of periodic components for different exponents $γ_{R}$ simply increase in extent as $γ_{R}$ increases. Second, new weights can be formed by appropriately interpreted convex combinations of old weights. For example, let $g_{1} (r)$ , $g_{2} (r)$ be balanced Gabor weights with corresponding Hankel transforms $e^{- γ_{1} ρ^{2}} G_{1} (ρ^{2})$ , $e^{- γ_{2} ρ^{2}} G_{2} (ρ^{2})$ , where $γ_{1} < γ_{2}$ . Then, by result (2), convex combinations of the form

H_{3} (ρ) = e^{- γ_{2} ρ^{2}} (α_{1} G_{1} (ρ^{2}) + α_{2} G_{2} (ρ^{2}))

with

α_{1}, α_{2} ⩾ 0

and

α_{1} + α_{2} = 1

are Hankel transforms of new balanced Gabor weight functions

g_{3} (r)

A balanced Gabor receptive field function $R_{BG} (\overset{⃑}{x})$ (with exponent $γ_{R}$ ) is an elementary receptive field function where the receptive field weight is a balanced Gabor function, that is,

R_{BG} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} g (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - b_{R}),

where

g (r)

is a balanced Gabor weight with associated exponent

γ_{R}

and

b_{R} = \cos (ϕ_{R}) H_{g} (1) = \cos (ϕ_{R}) e^{- γ_{R}}

. Notice this formula for the balance parameter

b_{R}

holds for all balanced Gabor receptive fields. Here

λ_{R} > 0

is the spatial wavelength,

α_{R}

is the field orientation, and

ϕ_{R}

is the phase shift. The function

R_{BG} (\overset{⃑}{x})

is an elementary receptive field function with the MRD Property (and hence with the ZRD and balanced field properties) for all values of the parameters, in particular, all values of the phase shift

ϕ_{R}

(Theorem B.2). Within the classes of Type I weights discussed in Section 4, balanced Gabor weights are precisely the weights that are phase independent in the sense that elementary receptive fields based on these weights have the MRD property for all receptive field phase values

ϕ_{R}

Balanced Gabor receptive field functions have a unique ZRD given by $ζ_{R} = α_{R} + π ∕ 2$ (Theorem B.1).

The Fourier transform of a balanced Gabor receptive field function $R_{BG} (\overset{⃑}{x})$ , where the weight $g (r)$ has Hankel transform $H_{g} (ρ) = e^{- γ_{R} ρ^{2}} G (ρ^{2})$ , follows from Eq. (18) and is given by

F_{R} (s_{1}, s_{2}) = \frac{1}{2} \exp (- γ_{R} (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2} + 1)) \cdot [e^{+ i ϕ_{R}} \exp (- 2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) G (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2} + 2 λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) + e^{- i ϕ_{R}} \exp (+ 2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) G (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2} - 2 λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) - 2 \cos (ϕ_{R}) G (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2})] .

It will be useful to introduce

C_{\pm} (α_{P} - α_{R}) ≔ \exp (\pm 2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s_{P}}) G (λ_{R}^{2} {‖ \overset{⃑}{s_{P}} ‖}^{2} \mp 2 λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s_{P}}) = \exp (\pm 2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) G (\frac{λ_{R}^{2}}{λ_{P}^{2}} \mp 2 \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) .

The response

R \circ p

of a balanced Gabor receptive field function to a sinusoidal grating pattern p, where the weight

g (r)

has Hankel transform

H_{g} (ρ) = e^{- γ_{R} ρ^{2}} G (ρ^{2})

, follows from Eq. (20) and is given by

R \circ p = \frac{c_{P}}{2} \exp (- γ_{R} (1 + \frac{λ_{R}^{2}}{λ_{P}^{2}})) \cdot [\cos (ϕ_{P} - ϕ_{R}) (C_{+} (α_{P} - α_{R}) - G (\frac{λ_{R}^{2}}{λ_{P}^{2}})) + \cos (ϕ_{P} + ϕ_{R}) (C_{-} (α_{P} - α_{R}) - G (\frac{λ_{R}^{2}}{λ_{P}^{2}}))] .

Notice the response

R \circ p

does not reduce to a simple multiple of

\cos (ϕ_{P} - ϕ_{R})

, in contrast to the bandlimited case (Subsection 5B), although this term is the exponentially dominant component. Consequently, for fixed grating orientation

α_{P}

, grating wavelength

λ_{P}

, and grating contrast

c_{P}

, the maximum of the response

R \circ p

does not occur at a grating phase

ϕ_{P}

that exactly lines up with the receptive field phase

ϕ_{R}

The max-response M of a balanced Gabor receptive field function, where the weight $g (r)$ has Hankel transform $H_{g} (ρ) = e^{- γ_{R} ρ^{2}} G (ρ^{2})$ , to a sinusoidal grating pattern p follows from Eqs. (21, 23) and is given by

M (c_{P}, α_{P}, λ_{P}) = \max_{ϕ_{P}} (R \circ p) = \frac{1}{2} | c_{P} | N (α_{P}, λ_{P}),

where

N {(α_{P}, λ_{P})}^{2} = \exp (- 2 γ_{R} (\frac{λ_{R}^{2}}{λ_{P}^{2}} + 1)) [\cos {(ϕ_{R})}^{2} N_{c}^{2} + \sin {(ϕ_{R})}^{2} N_{s}^{2}],

Notice that the max-response depends on the receptive field phase

ϕ_{R}

, in contrast to the bandlimited case (Subsection 5B). The maximum of the max-response, that is, the max of

N (α_{P}, λ_{P})

, occurs at

α_{P} = α_{R}

for fixed

λ_{P}

; that is, the optimal grating orientation lines up with the receptive field orientation

α_{R}

. This is a general consequence of the MRD property, but it can also be seen in the expression for

N {(α_{P}, λ_{P})}^{2}

, since the

\sin {(ϕ_{R})}^{2}

portion is strictly decreasing as

| α_{P} - α_{R} |

increases by part (b) of the definition of balanced Gabor weight and the

\cos {(ϕ_{R})}^{2}

portion is strictly decreasing by part (c). [Cosine-type filters may decrease to intervals where

N = 0

due to condition (d).] However, examples show that the maximum, for fixed grating orientation

α_{P}

, does not necessarily occur at

λ_{P} = λ_{R}

; that is, the optimal grating spatial wavelength does not necessarily occur at the receptive field spatial wavelength, another difference between the balanced Gabor and bandlimited cases (see examples below and Subsection 5B). Notice that

N_{s}^{2} ⩾ N_{c}^{2}

(for

\cos (α_{P} - α_{R}) ⩾ 0

) because

N_{s}^{2} - N_{c}^{2} = 4 (- C_{-} (α_{P} - α_{R}) + G (\frac{λ_{R}^{2}}{λ_{P}^{2}})) (C_{+} (α_{P} - α_{R}) + G (\frac{λ_{R}^{2}}{λ_{P}^{2}})),

and each factor is positive because

\exp (- γ_{R} z) G (z + a) and \exp (+ γ_{R} z) G (- z + a)

are strictly decreasing and strictly increasing functions of z by part (b) of the definition. In particular, the response of a pure sine-type receptive field will always be larger than the response of a pure cosine-type with an intermediate response for a mixed-type receptive field.

An explicit transform pair for the Hankel transform is

z (r) = \frac{1}{2} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}} \exp (- \frac{r^{2}}{4} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}}),

Consequently, if the Hankel transform

H_{q} (ρ) = e^{- c ρ^{2}} G (ρ^{2})

has its periodic factor

G (y)

expanded as a Fourier series, there is a corresponding explicit inversion:

q (r) = \sum_{k = - \infty}^{+ \infty} \frac{g_{k}}{2} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}} \exp (- \frac{r^{2}}{4} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}}),

The most basic instance of such explicit solutions, the case of a single harmonic for

G (y)

, is analyzed in Subsection 5.A.3 below.

5A2. Example: Simple Balanced Gabor Receptive Field Functions (and Traditional Gabor Filters)

The traditional Gabor filter model for a receptive field function is

R (\overset{⃑}{x}) = \frac{1}{2 π σ_{R}^{2}} \exp (- \frac{{‖ \overset{⃑}{x} ‖}^{2}}{2 σ_{R}^{2}}) \cos (2 π \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x}}{λ_{R}} - ϕ_{R}),

where

σ_{R} > 0

is the spatial constant and the other parameters correspond to Eq. (11). This is a typical normalization of the Gabor filter, where the Gaussian factor has unit volume. Its Fourier transform is

F_{R} (\overset{⃑}{s}) = \exp (- 2 π^{2} σ_{R}^{2} ({‖ \overset{⃑}{s} ‖}^{2} + \frac{1}{λ_{R}^{2}})) \cdot [\cos (ϕ_{R}) \cosh (4 π^{2} σ_{R}^{2} \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}}{λ_{R}}) - i \sin (ϕ_{R}) \sinh (4 π^{2} σ_{R}^{2} \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}}{λ_{R}})] .

The traditional Gabor filter is not in general balanced. The exact value for the integral (2) is

\int_{R \times R} R (\overset{⃑}{x}) d \overset{⃑}{x} = \cos (ϕ_{R}) \exp (- 2 π^{2} \frac{σ_{R}^{2}}{λ_{R}^{2}}) .

Consequently, it is balanced for sine-type filters

(\cos (ϕ_{R}) = 0)

and is nearly zero when the dimensionless ratio

λ_{R} ∕ σ_{R}

is sufficiently small. In the context of the Gabor filter, the ratio

λ_{R} ∕ σ_{R}

is often interpreted as a measure of “bandwidth.” The value

λ_{R} ∕ σ_{R} ≃ 5 ∕ 2

is considered a typical midrange value for simple cells corresponding to a spatial frequency tuning bandwidth of approximately 1.7 octaves [4], giving 0.0425 for the exponential term in Eq. (47). That is, for this ratio, the imbalance of a cosine-type filter is about 4% of the Gaussian component’s unit volume and becomes worse as

λ_{R} ∕ σ_{R}

increases, that is, at large bandwidth.

The simple balanced Gabor weight $g (r)$ is determined by an exponent $γ_{R} > 0$ . The weight and its Hankel transform $H_{g} (ρ)$ are given by

g (r) = \frac{1}{2 γ_{R}} \exp (- \frac{r^{2}}{4 γ_{R}}) with H_{g} (ρ) = \exp (- γ_{R} ρ^{2}) .

The simple balanced Gabor receptive field function

R_{SBG} (\overset{⃑}{x})

and its Fourier transform are special cases of Eqs. (37, 38):

R_{SBG} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} g (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - \cos (ϕ_{R}) e^{- γ_{R}}),

F_{R} (\overset{⃑}{s}) = \exp (- γ_{R} λ_{R}^{2} ({‖ \overset{⃑}{s} ‖}^{2} + \frac{1}{λ_{R}^{2}})) [\cos (ϕ_{R}) (\cosh (2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) - 1) - i \sin (ϕ_{R}) \sinh (2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s})] .

Comparison of Eqs. (45, 49) shows that the receptive field function

R_{SBG} (\overset{⃑}{x})

is essentially the Gabor filter corrected to achieve balance. This is, of course, the motivation for the name of this class of weights. The exponent

γ_{R}

can be considered a dimensionless “shape” parameter that arises naturally in the derivation of these functions and unifies all cases of balanced Gabor weights, or, since matching the Gaussian components of Eqs. (45, 49) gives the relation

γ_{R} = 2 π^{2} {(σ_{R} ∕ λ_{R})}^{2}

γ_{R}

can be considered a measure of “reciprocal bandwidth.” Figure 1 illustrates weights

g (r)

and corresponding Hankel transforms

H_{g} (ρ)

for a range of values of

γ_{R}

. Figure 2 compares the traditional Gabor filter and the corresponding simple balanced Gabor receptive field function with identical Gaussian components, that is, with identical weight functions (“envelopes”). Figure 2 also shows the corresponding Fourier transforms. The imbalance of the traditional Gabor receptive field is shown by the nonzero values of its Fourier transform at the origin.

It will be useful to express our results in terms of the exponent $γ_{R}$ since it provides a common unifying parameter for the entire class of balanced Gabor receptive field functions, both simple and nonsimple (see the following subsection). The typical value $λ_{R} ∕ σ_{R} ≃ 2.5$ gives $γ_{R} ≃ 3.2$ . Consequently, typical trial values for our plots in this subsection will be $γ_{R} = 6.0, 3.0, 1.5$ (respectively, $λ_{R} ∕ σ_{R} = 1.8, 2.6, 3.6$ ) with the value $γ_{R} = 0.75$ $(λ_{R} ∕ σ_{R} = 5.1)$ included to indicate trends at more extreme values. Figure 2 shows that the two types of receptive fields essentially merge for large $γ_{R}$ $(γ_{R} > 3)$ and increasingly diverge for small $γ_{R}$ $(γ_{R} < 3)$ with $γ_{R} ≃ 3$ as a nominal boundary between the two regimes. The same values will be used in the following subsection for nonsimple balanced Gabor receptive field functions for comparison.

The response $R \circ p$ of a simple balanced Gabor receptive field function to a sinusoidal grating pattern p is a special case of Eq. (40) and is given by

R \circ p = \frac{c_{P}}{2} \exp (- γ_{R} (1 + \frac{λ_{R}^{2}}{λ_{P}^{2}})) \cdot [\cos (ϕ_{P} - ϕ_{R}) (\exp (+ 2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) - 1) + \cos (ϕ_{P} + ϕ_{R}) (\exp (- 2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) - 1)] .

The behavior of this special case is the same as noted for nonsimple balanced Gabor behavior: the

\cos (ϕ_{P} - ϕ_{R})

component is exponentially dominant at

ϕ_{P} = ϕ_{R}

, that is, where the grating phase matches the receptive field phase, but the maximum does not occur at exactly this value due to the

\cos (ϕ_{P} + ϕ_{R})

component.

Similarly, the max-response function amplitude factor $N (α_{P}, λ_{P}) > 0$ is a special case of Eq. (42) and is given by

N {(α_{P}, λ_{P})}^{2} = 4 \exp (- 2 γ_{R} (\frac{λ_{R}^{2}}{λ_{P}^{2}} + 1)) \cdot [\cos {(ϕ_{R})}^{2} {(\cosh (2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) - 1)}^{2} + \sin {(ϕ_{R})}^{2} {(\sinh (2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}))}^{2}] .

For fixed grating orientation

α_{P}

, the maximum for

N (α_{P}, λ_{P})

does not necessarily occur at

λ_{P} = λ_{R}

, that is, where the grating wavelength matches the receptive field wavelength. This behavior is illustrated in Fig. 3 . Plots of

N (α_{P}, λ_{P})

vs.

λ_{P}

are given for the optimal orientation

α_{P} = α_{R}

. These plots are essentially the same for all receptive field phases at

γ_{R} = 6

and with maxima essentially at

λ_{P} = λ_{R}

, but they show increasing dependence on the receptive field phase

ϕ_{R}

and increasing deviation of the optimal grating wavelength from the receptive field wavelength

λ_{R}

γ_{R}

decreases. This behavior is quite different from the bandlimited case, where the max-response is completely independent of

ϕ_{R}

(Subsection 5B).

For fixed grating wavelength $λ_{P}$ , $N (α_{P}, λ_{P})$ has a maximum at $α_{P} = α_{R}$ , that is, where the grating orientation matches the field orientation, due to the MRD property. Figure 3 also shows such plots of response versus grating orientation at optimal grating wavelengths. The increasing dependence of the response on the receptive field phase $ϕ_{R}$ as $γ_{R}$ decreases is again evident.

5A3. Example: a Class of Nonsimple Balanced Gabor Receptive Field Functions

We define general balanced Gabor weights of order 1 (with exponent $γ_{R}$ ) as those weights given by

g (r) = \frac{1}{2 γ_{R} (1 + c_{R} \cos (ψ_{R}))} \exp (- \frac{r^{2}}{4 γ_{R}}) + \frac{c_{R} γ_{R}}{2 (1 + c_{R} \cos (ψ_{R})) (γ_{R}^{2} + 4 π^{2})} \exp (\frac{- γ_{R} r^{2}}{4 (γ_{R}^{2} + 4 π^{2})}) \cdot (\cos (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R}) + \frac{2 π}{γ_{R}} \sin (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R})),

with Hankel transform given by

H_{g} (ρ) = e^{- γ_{R} ρ^{2}} G (ρ^{2}) where G (y) = \frac{1 + c_{R} \cos (2 π y - ψ_{R})}{1 + c_{R} \cos (ψ_{R})},

where the coefficient

c_{R}

satisfies the constraint

| c_{R} | ⩽ \frac{γ_{R}^{2}}{γ_{R}^{2} + 4 π^{2}} .

In the Hankel transform, notice that $G (y)$ is the most general form of periodic function with a single harmonic term that satisfies the definition for a balanced Gabor weight, and it can be shown that the bound on $| c_{R} |$ is a necessary and sufficient condition to satisfy parts (a), (b), (c) of that definition (Appendix C [26]). Part (d) is vacuous for this case. The simple balanced Gabor occurs for $c_{R} = 0$ and can be considered the “order 0” form. With the Hankel transform $H_{g} (ρ)$ determined, the inverse transform $g (r)$ follows by the explicit inversion formula (43).

In $g (r)$ , notice the oscillatory component is the more slowly decaying component:

g (r) = \frac{1}{2 γ_{R} (1 + c_{R} \cos (ψ_{R}))} \exp (\frac{- γ_{R} r^{2}}{4 (γ_{R}^{2} + 4 π^{2})}) \cdot [\exp (- \frac{π^{2} r^{2}}{γ_{R} (γ_{R}^{2} + 4 π^{2})}) + \frac{c_{R} γ_{R}^{2}}{γ_{R}^{2} + 4 π^{2}} (\cos (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R}) + \frac{2 π}{γ_{R}} \sin (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R}))] .

The balanced Gabor receptive field function (order 1) $R_{BG} (\overset{⃑}{x})$ is the special case of Eq. (37) given by

R_{BG} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} g (\frac{2 π}{λ_{R}} | \overset{⃑}{x} |) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - \cos (ϕ_{R}) e^{- γ_{R}}),

where

g (r)

is given by Eq. (53). In the same way, the Fourier transform

F_{R} (s_{1}, s_{2})

, the response function

R \circ p

for sinusoidal grating patterns p, and the max-response amplitude factor

N (α_{P}, λ_{P}) > 0

are given by formulas combining Eq. (53) with the general formulas (38), (40), and (42), respectively.

The figures show examples of these nonsimple balanced Gabor receptive field functions and their behavior. There are two sets: Figures 4, 5, 6 have $ψ_{R} = 0 (rad)$ and Figs. 7, 8, 9 have $π ∕ 2 (rad)$ . Each plot shows behavior for exponents $γ_{R} = 0.75, 1.5, 3.0, 6.0$ and for cases of $G (y)$ with the coefficient $c_{R}$ taking the value 0 (that is, the simple balanced Gabor case) and the two most extreme values permitted by the bound (55). Since this bound is small for small $γ_{R}$ , the plots for $γ_{R} = 0.75$ essentially reduce to the simple balanced Gabor case. As $γ_{R}$ increases, however, the bound also increases, and substantial differences from the simple balanced Gabor are evident at $γ_{R} = 6.0$ . Specifically, Figs. 4, 7 show the Hankel transforms $H_{g} (ρ)$ and weight functions $g (r)$ for these parameter values. Notice that the weight functions $g (r)$ can become negative, in contrast to the simple balanced Gabor, where the weights are strictly positive. Figures 5, 8 show the corresponding receptive field functions (with the weight functions as envelopes) and Fourier transforms for the receptive field functions (cosine-type receptive fields). Figures 6, 9 show plots of $N (α_{P}, λ_{P})$ versus $λ_{P}$ at the optimal grating orientation $(α_{P} = α_{R})$ . The optimal grating wavelength is quite different from the receptive field wavelength $λ_{R}$ for small $γ_{R}$ but begins to match it as $γ_{R}$ increases. At the same time, the variation in these curves increases with $γ_{R}$ . That is, as bandwidth decreases and the simple balanced Gabor and traditional Gabor receptive fields become identical (see Fig. 2), the nonsimple balanced Gabor weights and receptive fields show a widening range of behavior. The same figures show plots of $N (α_{P}, λ_{P})$ versus $α_{P}$ at optimal wavelengths $λ_{P}$ . The maximum response occurs at $α_{P} = α_{R}$ , that is, when the grating orientation matches the receptive field orientation, in agreement with the MRD property, and the variation in the response increases with $γ_{R}$ .

5B. Bandlimited Receptive Field Functions: Discussion and Examples

This section defines and discusses the bandlimited class of weights and elementary receptive field functions mentioned in Section 4. A bandlimited class with explicit analytic solutions (Bessel receptive fields) is given as an example.

5B1. Bandlimited Weights and Receptive Fields

Bandlimited weights (with support $0 < s_{R} ⩽ 1$ ) are weight functions $b (r)$ satisfying

b (r) is C (0, + \infty),

in which case the Hankel transform

H_{b} (ρ)

satisfies

H_{b} (ρ) is C [0, + \infty), H_{b} (0) = 1, and H_{b} (+ \infty) = 0

and such that there exists a value

0 < s_{R} ⩽ 1

with the properties

(a) $H_{b} (ρ) = 0$ on $s_{R} ⩽ x < + \infty$ ;

(b) $H_{b} (ρ)$ is strictly decreasing on $0 ⩽ x < s_{R}$ with $H_{b} (0) = 1$ .

All parameters and independent variables are dimensionless. For each bandlimited Hankel transform (and thus for the corresponding weight function), the support value $s_{R}$ is uniquely determined. The support values thus partition or classify both the transforms and the weights. Since the support $s_{R}$ is a measure of the width of the Hankel transform and thus of the Fourier transform for the receptive field (see below), it corresponds to a measure of the bandwidth of the bandlimited receptive field function. For bandlimited weights $b (r)$ , $b (0)$ is necessarily positive [because Eq. (14) reduces to a positive integrand for $r = 0$ ]. In contrast to balanced Gabor weights, no explicit analytic structure occurs naturally for bandlimited weights, and no analog for “simple” balanced Gabor weights occurs.

A closure result can be stated for the bandlimited case and takes a simpler form than for the balanced Gabor case:

Closure Lemma.

The set of bandlimited Hankel transforms and the corresponding set of bandlimited weights for a fixed support value $s_{R}$ are closed under convex combinations.
The set of all bandlimited Hankel transforms and the corresponding set of all bandlimited weights are closed under convex combinations.

Both parts follow by a straightforward check of the conditions of the bandlimited definition when applied to convex combinations.

If $u (r)$ is a bandlimited weight function with support $s_{R} = 1$ , it follows directly from the definition that, for an arbitrary c, $0 < c ⩽ 1$ , the function $b (r) = c^{2} u (c r)$ is a bandlimited weight with support $s_{R} = c$ and Hankel transform $H_{b} (ρ) = H_{u} (ρ ∕ c)$ . That is, scaling maps bandlimited weights with a given support value to weights with other support values. In particular, the equivalence class of weights for a given support value $s_{R}$ is a scaled copy of any other equivalence class.

A bandlimited receptive field function $R_{BL} (\overset{⃑}{x})$ is an elementary receptive field function where the receptive field weight is a bandlimited function; that is, $q (r)$ has been replaced by $b (r)$ ,

R_{BL} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} b (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) \cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) .

The balance parameter vanishes because

b_{R} = \cos (ϕ_{R}) H_{g} (1) = 0

for all bandlimited receptive fields. Bandlimited receptive field functions have the MRD property (and hence the ZRD and balanced field properties) because bandlimited weights are identical to the Type II class of weights for elementary receptive field functions defined by Theorem B.1.

The support value $s_{R}$ determines the corresponding sector of ZRDs (possibly reducing to a single line) by $s_{R} = \sin (ζ_{R})$ . The sector is centered on $α_{R} + π ∕ 2$ and is bounded by $ζ_{R}$ ; that is, the ZRD sector consists of $ζ_{R} ⩽ α_{P} - α_{R} ⩽ π - ζ_{R}$ .

The Fourier transform of a bandlimited receptive field function reduces to

F_{R} (s_{1}, s_{2}) = \frac{1}{2} (e^{+ i ϕ_{R}} H_{b} (λ_{R} ‖ \overset{⃑}{s} + \overset{⃑}{s_{R}} ‖) + e^{- i ϕ_{R}} H_{b} (λ_{R} ‖ \overset{⃑}{s} - \overset{⃑}{s_{R}} ‖)) .

For any point

(s_{1}, s_{2})

, at least one of the arguments is

⩾ 1

, so the sum always reduces to a single term. The motivation for the name “bandlimited” comes from the fact that the Fourier transforms of the weight and receptive field functions have compact support.

The response $R \circ p$ of a bandlimited receptive field function to a sinusoidal grating pattern p is given by (assuming $| α_{P} - α_{R} | ⩽ π ∕ 2$ )

R \circ p = \frac{c_{P}}{2} \cos (ϕ_{P} - ϕ_{R}) H_{b} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) .

In contrast to the balanced Gabor case, the response

R \circ p

reduces to a simple multiple of

\cos (ϕ_{P} - ϕ_{R})

. Thus, for fixed grating orientation

α_{P}

, grating wavelength

λ_{P}

, and grating contrast

c_{P}

, the maximum of the response

R \circ p

necessarily occurs when the grating phase

ϕ_{P}

matches the receptive field phase

ϕ_{R}

The max-response of a bandlimited receptive field function to a sinusoidal grating pattern is given by

M (c_{P}, α_{P}, λ_{P}) = \max_{ϕ_{P}} (R \circ p) = \frac{1}{2} | c_{P} | N (α_{P}, λ_{P}),

where the amplitude factor

N (α_{P}, λ_{P})

for the max-response is given by

N (α_{P}, λ_{P}) = H_{b} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) .

The simplicity of the formula for $N (α_{P}, λ_{P})$ provides several results:

The maximum of the max-response amplitude factor is $N = 1$ and occurs for exactly one pair of values, $(α_{P}, λ_{P}) = (α_{R}, λ_{R})$ .
The response function is independent of the receptive field phase $ϕ_{R}$ .
If the grating direction $α_{P}$ is fixed, then $| α_{P} - α_{R} | < ζ_{R}$ , where $s_{R} = \sin (ζ_{R})$ , is required for a nonzero response. A nonzero response occurs only for wavelengths $λ_{P}$ in the interval $(δ ≔ {[\cos {(α_{P} - α_{R})}^{2} - \cos {(ζ_{R})}^{2}]}^{1 ∕ 2})$ $\cos (α_{P} - α_{R}) - δ ⩽ \frac{λ_{R}}{λ_{P}} ⩽ \cos (α_{P} - α_{R}) + δ,$ which narrows to a single point when $α_{P} - α_{R} = ζ_{R}$ . The max over this interval occurs at the midpoint $λ_{R} ∕ λ_{P} = \cos (α_{P} - α_{R})$ , and that max-value in this fixed direction is given by $N = H_{b} (\sin (α_{P} - α_{R}))$ .
If the grating wavelength $λ_{P}$ is fixed, then $| λ_{R} ∕ λ_{P} - 1 | < s_{R}$ , where $s_{R} = \sin (ζ_{R})$ , is required for a nonzero response. A nonzero response occurs only for orientations $α_{P}$ in the range
$\cos (α_{P} - α_{R}) ⩾ \frac{1}{2} (\frac{λ_{R}}{λ_{P}} + \cos {(ζ_{R})}^{2} \frac{λ_{P}}{λ_{R}}),$
and the maximum response within this interval is at the midpoint $α_{P} = α_{R}$ giving $N = H_{b} (| λ_{R} ∕ λ_{P} - 1 |)$ .
1. The broadest interval of nonzero response occurs when $\frac{λ_{R}}{λ_{P}} = \cos (ζ_{R}) < 1,$ in which case $| α_{P} - α_{R} | < ζ_{R}$ (that is, extends to the ZRD sector) with maximum value $H_{b} (1 - \cos (ζ_{R}))$ .
2. The interval of nonzero response for the optimal wavelength $λ_{R} ∕ λ_{P} = 1$ is $| \sin (\frac{α_{P} - α_{R}}{2}) | ⩽ \frac{1}{2} \sin (ζ_{R})$ with maximum value $H_{b} (0) = 1$ .

Notice that the broadest interval in (a) is larger than the optimal wavelength interval in (b).

The synthesis/analysis formulas for the Hankel transform and the fact that $H_{b} (ρ)$ vanishes outside the interval $[0, s_{R}]$ with $0 < s_{R} ⩽ 1$ for the bandlimited class give a general formula for the weights:

b (r) = \int_{0}^{1} J_{0} (ρ r) H_{b} (ρ) ρ d ρ = \sum_{k = 0}^{\infty} \frac{{(- r^{2} 4)}^{k}}{{(k!)}^{2}} \int_{0}^{s_{R}} H_{b} (ρ) ρ^{2 k + 1} d ρ

5B2. Example: Bessel Receptive Field Functions

An example of the explicit inversion (65) is the following Hankel transform and inverse transform. These functions satisfy the conditions for a bandlimited weight function with support parameter $s_{R} = 1$ : for $ν_{R} > 3 ∕ 2$ ,

H_{b} (ν_{R}; ρ) ≔ (\begin{matrix} {(1 - ρ^{2})}^{ν_{R} - 1} & for 0 ⩽ ρ < 1 \\ 0 & for 1 ⩽ ρ \end{matrix}),

b (ν_{R}; r) ≔ 2^{ν_{R} - 1} Γ (ν_{R}) r^{- ν_{R}} J_{ν_{R}} (r) = \int_{0}^{1} J_{0} (ρ r) {(1 - ρ^{2})}^{ν_{R} - 1} ρ d ρ .

ν_{R}

increases, the transform

H_{b}

decreases more rapidly to zero and has a smoother transition at

ρ = 1

. The weight function

b (ν_{R}; r)

is positive at the origin with

b (ν_{R}; 0) = {(2 ν_{R})}^{- 1}

, has infinitely many positive roots and has a first root at

r = j_{ν_{R}, 1} \sim ν_{R} + 1.8557571 ν_{R}^{1 ∕ 3} + O (ν_{R}^{- 1 ∕ 3})

ν_{R} \to \infty

[27]. The function satisfies

b (ν_{R}; r) = O (r^{- ν_{R} - 1 ∕ 2})

r \to \infty

, and the condition

ν_{R} > 3 ∕ 2

ensures the absolute integrability of

b (r) r

as well as continuity of the Hankel transform.

Scaling gives weight functions with arbitrary support $0 < s_{R} ⩽ 1$ :

b (r) = s_{R}^{2} b (ν_{R}; s_{R} r) and H_{b} (ρ) = H_{b} (ν_{R}; \frac{ρ}{s_{R}}) .

This set of scaled functions $b (r)$ will be called Bessel weights of order $ν_{R}$ , referring to the order of the corresponding Bessel function. The support parameter $s_{R}$ determines an equivalence class of weights.

The Bessel receptive field functions $R_{B} (\overset{⃑}{x})$ are the elementary receptive field functions given by

R_{B} (\overset{⃑}{x}) = 2 π \frac{s_{R}^{2}}{λ_{R}^{2}} b (ν_{R}; 2 π \frac{s_{R}}{λ_{R}} ‖ \overset{⃑}{x} ‖) \cos (2 π \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x}}{λ_{R}} - ϕ_{R}),

where

0 < s_{R} ⩽ 1

is the support parameter. Increasing the order

ν_{R}

broadens and flattens the weight factor b. As noted under the general discussion of bandlimited receptive fields, these receptive fields have the MRD property (and hence the ZRD and balanced field properties). The boundary angle

ζ_{R}

between the sector of nonzero response and the sector of ZRDs is given by

s_{R} = \sin (ζ_{R})

Formulas and properties for the Fourier transform, response function, and max-response for Bessel receptive fields carry over directly from the general discussion. We note a couple of simplifications. The response function $R \circ p$ for a sinusoidal grating pattern p reduces to (assuming $| α_{P} - α_{R} | ⩽ π ∕ 2$ )

R \circ p = \frac{c_{P}}{2} \cos (ϕ_{P} - ϕ_{R}) H_{b} (ν_{R}; \frac{λ_{R}}{s_{R}} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖)

= \frac{c_{P}}{2} \frac{\cos (ϕ_{P} - ϕ_{R})}{s_{R}^{2 ν_{R} - 2}} D^{ν_{R} - 1}

when D ≔ s_{R}^{2} - \sin {(α_{P} - α_{R})}^{2} - {(\frac{λ_{R}}{λ_{P}} - \cos (α_{P} - α_{R}))}^{2} > 0, = 0 otherwise,

and the max-response to a sinusoidal grating pattern has amplitude factor

N (α_{P}, λ_{P})

given explicitly by

N (α_{P}, λ_{P}) = H_{b} (ν_{R}; \frac{λ_{R}}{s_{R}} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) = \frac{1}{s_{R}^{2 ν_{R} - 2}} D^{ν_{R} - 1} when D > 0 = 0 otherwise .

Figures 10, 11, 12 show examples of Bessel receptive field functions and their behavior. Each plot shows curves for orders $ν_{R} = 2.0, 3.5, 5.0$ . Each figure shows plots for support values $s_{R} = 0.5, 0.7, 0.85, 1.0$ , corresponding to scaled versions of each other. Figure 10 shows Hankel transforms and corresponding weight functions. Figure 11 shows Fourier transforms and corresponding receptive field functions. Since the balance parameter $b_{R} = 0$ for the bandlimited case, the receptive field curves always match up with their envelopes and the nodes always match exactly with the zeros of the periodic carrier. Figure 12 shows amplitude factors $N (α_{P}, λ_{P})$ at optimal orientations $(α_{P} = α_{R})$ and optimal wavelengths $(λ_{P} = λ_{R})$ . Notice, in contrast to the balanced Gabor case, the max-response is independent of the receptive field phase $ϕ_{R}$ .

6. DISCUSSION

First, we suggest three properties as theoretical postulates for simple-cell receptive fields (Section 2). Experimental studies have established that simple cells show three well-defined properties, formulated here as the balanced field property (spatially homogeneous patterns produce a zero response); the zero response direction (ZRD) property (there is a direction, i.e., stimulus orientation, which elicits a zero response to sinusoidal gratings); and the maximum response direction (MRD) property (the maximum response to sinusoidal gratings decreases monotonically to zero as direction, i.e., stimulus orientation, changes from the optimal direction to a direction perpendicular to the optimal). These properties are directly motivated by well-established experimentally observed behavior [1, 2, 4, 17]. Our analysis complements that of Olshausen and Field [28, 29], who numerically derived spatial structures for linear V1 neurons from natural scenes coupled with a sparse coding constraint. Since our analysis is based on responses to sinusoidal grating stimuli, these properties can be restated in terms of the Fourier transform of the simple cell receptive field function. For the Fourier transform $F_{R} (s_{1}, s_{2})$ of the receptive field function $R (x_{1}, x_{2})$ , the properties are

Balanced field property: $F_{R} (0, 0) = 0$ .
ZRD Property: There exists a direction $α_{ZR}$ such that $F_{R} (s \cos (α_{ZR}), s \sin (α_{ZR})) = 0$ for all real s.
MRD Property: There exists a direction $α_{MR}$ such that, for each fixed s, $F_{R} (s \cos (α), s \sin (α))$ either is zero for all α or strictly decreases to zero (and remains zero) as $| α - α_{MR} |$ increases.

Second, we demonstrate that these properties can serve as a productive basis for theoretical development. A simple role for such postulates is to provide definite criteria that any proposed receptive field model should satisfy, where such models either should satisfy the criteria or, if not, should possess compensating advantage(s). For example, the traditional Gabor filter is not balanced, but the imbalance is negligible over a certain parameter range, and the filter provides a useful explicit approximation to observed receptive field functions within that range. A deeper role is to provide a catalyst. We have demonstrated such a role by using the properties to completely characterize elementary receptive field functions, that is, to characterize the weight functions q defining such receptive fields for cosine-type and mixed-type receptive fields (Sections 3, 4). This derivation began by obtaining a description of receptive field functions with the ZRD property (Appendix A [26]), then refining that description to obtain receptive field functions with the MRD property (Appendix B [26]). The usefulness of this stepwise approach motivates the above formulation of three cumulative properties.

This characterization yields two disjoint classes of elementary receptive field functions, the balanced Gabor class (Subsection 5A) and the bandlimited class (Subsection 5B). The balanced Gabor class appears to be an essentially new class of receptive field models, although its most basic case, the simple balanced Gabor receptive field function, is a straightforward modification of the traditional Gabor filter. Bandlimited models for receptive fields have been proposed [25], and the bandlimited class derived here is an unexpected connection with that literature. One difference between the classes is that the balanced Gabor class has a unique ZRD (orthogonal to the MRD), while the bandlimited class can have a sector of zero response directions/orientations (centered on an orientation orthogonal to the maximum response orientation, but possibly reducing to a single orientation). Another difference is that the Hankel transform $H_{q} (ρ)$ of the weight function q is strictly positive for the balanced Gabor class (although it decays exponentially fast owing to a characteristic Gaussian factor), while $H_{q} (ρ)$ for the bandlimited case is nonnegative and is always zero outside some bounded region (i.e., has compact support).

Third, we provide explicit examples and short studies of these two classes of elementary receptive field functions (Subsections 5A, 5B). It should be noticed that the balanced Gabor class has a partially determined analytic structure [the Hankel transform $H_{q} (ρ)$ of the weight function is the product of a gaussian and a periodic function]; while equally well defined, the bandlimited class does not imply a comparable analytic form for the Hankel transform.

With regard to further work, the fact that a set of three experimentally based properties leads to two disjoint classes of elementary receptive field functions raises the obvious question of which class better describes the simple cell receptive field. Consequently, the most immediate question raised by this paper is whether experimental study might lead to a decision between these two classes. In addition, there are natural directions for generalizing both the results obtained here and the approach. We set these out under the following suggestions.

Experimental diagnostics for receptive field functions. The two classes of receptive field functions resulting from our analysis necessarily exhibit similar behavior since they both satisfy the three imposed experimentally based response properties to sinusoidal grating stimuli. Not surprisingly, therefore, identifying diagnostic differences in the responses of the two classes of elementary receptive field functions that are both experimentally detectable and empirically conclusive may prove challenging. The construction of explicit examples of receptive field functions of both types, based on the selection of receptive field parameter values that result in physiologically plausible behavior, has suggested some differences which may have experimental significance. We list some of these differences by way of example but note that further study is required:

(a) Bandlimited receptive field functions can have a sector of ZRDs, whereas balanced Gabor receptive fields always possess a unique ZRD. While a narrow range of ZRDs (orientations) may be experimentally indistinguishable from a single ZRD, a broad range of grating orientations yielding zero response points to bandlimited receptive field structure.

(b) Bandlimited receptive field functions have max-response (i.e., tuning) functions with respect to both spatial frequency and orientation that are completely independent of the receptive field phase shift $ϕ_{R}$ . By contrast, balanced Gabor max-response functions depend on receptive field phase, and max-response values for sine-type receptive field functions are always larger than those for cosine-type receptive field functions. In addition, initial numerical calculations suggest that, for balanced Gabor types, cosine phase receptive field functions are more narrowly tuned for both spatial frequency and orientation than are mixed or sine phase receptive fields (see Figs. 3, 6, 9). Population level surveys of simple cell behavior might be diagnostic if, as a group, simple cell max-responses, orientation, or spatial frequency tuning were found to systematically vary with receptive field phase, since this would contraindicate the bandlimited class of receptive field model for which these response properties are invariant with receptive field phase.

(c) The weight function, q, of the simple balanced Gabor receptive field function is always nonnegative, whereas the weight functions of nonsimple balanced Gabor and bandlimited types may assume negative values. When the weight function changes sign, the phase of the periodic carrier function undergoes an abrupt reversal. Such abrupt carrier phase reversals may be of diagnostic value since discovering simple cells with such receptive field structure would disqualify the simple balanced Gabor receptive field type.

Theoretical diagnostics for evaluating receptive field functions. There may be theoretical grounds, such as minimal energy or other optimality arguments, for preferring one class of elementary receptive field function over the other. We have not extended our analysis to include such considerations, but this may be a fruitful direction for further study.

Further developments of the present analysis. The present analysis has assumed that the weight function, q, of the elementary receptive field function is circularly symmetric and satisfies the regularity conditions described in Eq. (12), in particular that $\int_{0}^{\infty} q (r) r d r = 1$ (i.e., that the weight function possesses unit volume), which in turn implies that the Hankel transform satisfies $H_{q} (ρ) = 1$ . This is a broad normalization condition since a simple scaling of the weight function can transform any nonzero weight integral to unit value. Several questions regarding further generalizations naturally arise:

(a) What receptive field structures might satisfy the balanced field, ZRD, and MRD constraints if our assumed regularity condition $\int_{0}^{\infty} q (r) r d r = 1$ is modified to be the singular condition $\int_{0}^{\infty} q (r) r d r = 0$ , in which case the weight function itself must integrate to zero? If valid receptive field functions result, that is, if nontrivial weight functions, q, are discovered that satisfy the constraints imposed by the three postulated response properties, then they would necessarily form a new class of receptive field structures distinct from the two classes described in this paper, since no scaling of independent or dependent variables can convert the singular to the regular condition.

(b) What are the implications for our analysis of receptive field functions if the condition of a circularly symmetric weight function, q, is relaxed to allow for elliptically symmetric weights? Both psychophysical [19, 30, 31] (but see [18] and physiological observations [17, 32]) indicate that simple cell receptive fields with elliptically symmetric weight functions may commonly occur. This suggests that extending the current analysis to include such receptive field variations would be profitable. This extension of our analysis is, however, beyond the scope of the present paper.

(c) We have defined elementary receptive field functions in Eq. (11) to be the product of a circularly symmetric weight and a simple periodic carrier. The simple periodic carrier corresponds to the first two terms of a Fourier series expansion. Can receptive field functions with a general periodic carrier be characterized? That is, can nonelementary receptive field functions that satisfy our postulated response behaviors be characterized, where the functions have the following form, and $p (y)$ is an arbitrary 1-periodic function?

R (\overset{⃑}{x}) ≔ \frac{2 π}{λ_{R}^{2}} q (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) p (\frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x}}{λ_{R}}) .

(d) Our approach has been to deduce the mathematical implications of three simple, but generally accepted, properties of the response of simple cells to sinusoidal gratings. Our analysis relies, in part, on the fact that statements about such responses convert directly to statements about the Fourier transform of elementary receptive field functions and on the leverage afforded by the intersection of constraints imposed by these three response properties. It is therefore of interest to ask whether there might be additional properties of simple cells, well defined but perhaps less widely recognized (and not necessarily restricted to sinusoidal grating stimuli) that could be incorporated to supplement the three postulated properties. It should be kept in mind that such properties might be considered of minor or secondary importance from the viewpoint of experimental significance yet might have unexpected power in mathematical terms. For example, the ZRD property might easily be regarded as of secondary importance in comparison with the MRD property, yet studying the implications of the existence of a ZRD provided important initial results for this work. The incorporation of such an additional response property to the present analysis would quite probably determine which class of elementary receptive field function best describes simple cell receptive fields.

Classical versus extraclassical receptive fields. A final caveat is that our analysis is motivated by considering the response properties of the so-called classical receptive field of V1 neurons, which typically demonstrate the balanced field property. A large body of literature suggests that V1 neurons also possess “extraclassical” receptive fields [33, 34], and a sizable proportion of V1 neurons respond to homogeneous luminance modulations if these stimuli extend into the extraclassical region, which may subtend many degrees of visual angle beyond the classical receptive field [35, 36, 37, 38, 39, 40, 41, 42, 43].

ACKNOWLEDGMENTS

This work was supported by grants R01EY014015 from the National Eye Institute (NEI) and NIH P20 RR020151 from the National Center for Research Resources (NCRR). The NEI and the NCRR are components of the National Institutes of Health (NIH). The contents of this report are solely the responsibility of the authors and do not necessarily reflect the official views of the NIH, NCRR, or NEI. Commercial relationships: none.

Fig. 1 Simple balanced Gabor weight functions, $g (r) ∕ g (0)$ vs. r, (left) and Hankel transforms, $H_{g} (ρ)$ vs. ρ, (right) for four exponents $γ_{R} = 0.75$ (dashed), 1.5 (dotted), 3.0 (solid), 6.0 (near-solid). As the exponent $γ_{R}$ increases, the weight $g (r)$ broadens and the Hankel transform $H_{g} (ρ)$ narrows. Note that the weight functions are strictly positive and monotonic and are not oscillatory.

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Fig. 2 Left panels: comparison of traditional cosine-type Gabor receptive field function (near-solid) with the corresponding simple balanced Gabor receptive field function (dashed) with identical Gaussian components, that is, with identical weight functions (envelopes, solid). Right panels: corresponding Fourier transforms. All receptive field functions are cosine-type. The imbalance of the traditional Gabor is indicated by the nonzero values of its Fourier transform at the origin. Comparisons are for $γ_{R} = 0.75, 1.5, 3.0, 6.0$ . Note that the traditional and simple balanced Gabor receptive field functions become increasingly similar (i.e., the balancing parameter approaches 0) as $γ_{R}$ increases (i.e., as effective spatial frequency bandwidth decreases), becoming virtually identical for $γ_{R} > 3$ .

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Fig. 3 Left panels: orientation max-response functions (in degrees). Right panels: spatial frequency max-response functions for simple balanced Gabor receptive field functions as they vary with carrier spatial phase: $0 \deg$ (cosine-type, dashed), $45 \deg$ (mixed-type, dotted) and $90 \deg$ (sine-type, solid). Response curves are for $γ_{R} = 0.75, 1.5, 3.0, 6.0$ . Note that as exponent $γ_{R}$ increases, max-response curves become independent of carrier spatial phase $ϕ_{R}$ and that sine-type receptive fields always give larger responses than cosine-type receptive fields with corresponding parameters. The cosine-type spatial frequency max-response matches the Fourier transform of Fig. 2 (differing only by a scale factor).

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Fig. 4 Nonsimple balanced Gabor weight functions, $g (r) ∕ g (0)$ vs. r, (left) and Hankel transforms, $H_{g} (ρ)$ vs. ρ, (right) for four exponents $γ_{R} = 0.75, 1.5, 3.0, 6.0$ (with $ψ_{R} = 0$ ). Each plot shows three curves. The curves plot weight functions when $c_{R}$ is set to a positive (solid) or negative (dotted) boundary value. These extreme curves (solid and dotted) are interchanged relative to the corresponding extreme curves for $ψ_{R} = π ∕ 2$ (see Fig. 7 below). The dashed curve replots the weight function of the simple balanced Gabor $(c_{R} = 0)$ for comparison. Note that nonsimple Gabor weight functions exhibit an oscillatory behavior and can take on negative values. The departure from the simple balanced Gabor weight function is most pronounced for large values of $γ_{R}$ . As the exponent $γ_{R}$ increases, the weight $g (r)$ broadens and the Hankel transform $H_{g} (ρ)$ narrows.

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Fig. 5 Normalized balanced Gabor receptive field functions (cosine-type), $λ_{R}^{2} R (x_{1}, 0) ∕ 2 π g (0)$ vs. $x_{1} ∕ λ_{R}$ , and associated weight functions (left) for $γ_{R} = 0.75, 1.5, 3.0, 6.0$ $(ψ_{R} = 0)$ , together with their Fourier transforms (right). Weight functions (left) are in thinner gray curves. Each plot shows $c_{R} = 0$ (simple balanced Gabor, dashed) and $c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})$ for max (solid) and min (dotted) bounds of this cofficient. For the Fourier transforms, note that as $γ_{R}$ increases, (1) the transform narrows (consistent with decreasing bandwidth) and (2) the range of variation of the curves increases (consistent with the increased bounds for $c_{R}$ ). Note the variation in the Fourier transforms for small values of $s_{1}$ (low spatial frequency). There is no apparent corresponding variation in the receptive field functions for small $γ_{R}$ , indicating the importance of the large-scale receptive field structure.

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Fig. 6 Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for nonsimple balanced Gabor receptive field functions for $γ_{R} = 0.75, 1.5, 3.0, 6.0$ $(ψ_{R} = 0)$ . Each plot shows $c_{R} = 0$ (simple balanced Gabor, dashed) and $c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})$ for max (solid) and min (dotted) bounds of this coefficient. Three curves are plotted for each $c_{R}$ corresponding to receptive field phase $ϕ_{R} = 0, 45, 90 \deg$ , respectively, cosine-type, mixed-type, sine-type. As $γ_{R}$ increases, the dependence of both orientation and spatial frequency response on field phase $ϕ_{R}$ ( $c_{R}$ fixed) decreases, becoming virtually independent at $γ_{R} = 6.0$ . The cosine-type spatial frequency max-response function matches the Fourier transform of Fig. 5 (up to scaling).

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Fig. 7 Nonsimple balanced Gabor weight functions, $g (r) ∕ g (0)$ vs. r, (left) and Hankel transforms, $H_{g} (ρ)$ vs. ρ, (right) for four exponents $γ_{R} = 0.75, 1.5, 3.0, 6.0$ (with $ψ_{R} = π ∕ 2$ ). Each plot shows three curves. The curves plot weight functions when $c_{R}$ is set to a positive (solid) or negative (dotted) boundary value. These extreme curves (solid and dotted) are interchanged relative to the corresponding extreme curves for $ψ_{R} = 0$ (Fig. 4). The dashed curve replots the weight function of the simple balanced Gabor $(c_{R} = 0)$ for comparison. Note that nonsimple Gabor weight functions exhibit an oscillatory behavior and can take on negative values. The departure from the simple balanced Gabor weight function is most pronounced for large values of $γ_{R}$ . As the exponent $γ_{R}$ increases, the weight $g (r)$ broadens and the Hankel transform $H_{g} (ρ)$ narrows.

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Fig. 8 Four pairs of plots showing normalized balanced Gabor receptive field functions (cosine-type), $λ_{R}^{2} R (x_{1}, 0) ∕ 2 π g (0)$ vs. $x_{1} ∕ λ_{R}$ , and associated weight functions (left) with their Fourier transforms (right) for $γ_{R} = 0.75, 1.5, 3.0, 6.0$ $(ψ_{R} = π ∕ 2)$ . Weight functions (left) are in thinner gray curves. Each plot shows $c_{R} = 0$ (simple balanced Gabor, dashed) and $c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})$ for max (solid) and min (dotted) bounds of this coefficient. For the Fourier tranforms, note that as $γ_{R}$ increases, (1) the transform narrows (consistent with decreasing bandwidth) and (2) the range of variation of the curves increases (consistent with the increased bounds for $c_{R}$ ). Note the variation in the Fourier transforms for small values of $s_{1}$ seen in Fig. 5 $(ψ_{R} = 0)$ is here most prominent for $s_{1}$ slightly less than 1.0. Note the field phase reversal for $γ_{R} = 6.0$ (solid).

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Fig. 9 Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for nonsimple balanced Gabor receptive field functions for $γ_{R} = 0.75$ , 1.5, 3.0, 6.0 $(ψ_{R} = π ∕ 2)$ . Each plot shows $c_{R} = 0$ (simple balanced Gabor, dashed) and $c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})$ for max (solid) and min (dotted) bounds of this coefficient. Three curves are plotted for each $c_{R}$ corresponding to receptive field phase $ϕ_{R} = 0, 45, 90 \deg$ , respectively cosine-type, mixed-type, sine-type. As $γ_{R}$ increases, the dependence of both orientation and spatial frequency response on receptive field phase $ϕ_{R}$ ( $c_{R}$ fixed) decreases, becoming virtually independent at $γ_{R} = 6.0$ . The cosine-type spatial frequency max-response function matches the Fourier transform of Fig. 7 (up to scaling).

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Fig. 10 Bandlimited weight functions for $b (r) ∕ b (0)$ vs. r, (left) and Hankel transforms, $H_{b} (ρ)$ vs. ρ, (right) for support parameter values $s_{R} = 1.0$ , 0.85,0.7,0.5. Each plot shows three curves that plot weight functions when the order of the Bessel weight $ν_{R}$ is set to 2.0 (dotted), 3.5 (dashed), and 5.0 (solid). Note that Bessel weight functions exhibit oscillatory behavior and take on negative values. As the support parameter $s_{R}$ decreases, the weight $b (r)$ broadens and the Hankel transform $H_{b} (ρ)$ narrows.

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Fig. 11 Bandlimited Bessel receptive field functions, $λ_{R}^{2} R (x_{1}, 0) ∕ 2 π b (0)$ vs. $x_{1} ∕ λ_{R}$ , (left), and their Fourier transforms, $F_{R} (s_{1}, 0)$ vs. $\log_{10} (λ_{R} s_{1})$ (right), for support parameter values $s_{R} = 1.0$ , 0.85, 0.7, 0.5. Each plot shows three curves corresponding to Bessel weights of order $ν_{R} = 2.0$ (dotted), 3.5 (dashed), and 5.0 (solid). Unlike balanced Gabor functions, the bandlimited receptive field function and its weight function are equal at the origin and, for the Fourier transform, the transform max always occurs for $λ_{P} = λ_{R}$ . Like balanced Gabor receptive field functions, the bandlimited receptive field function can show phase reversal (note $s_{R} = 0.7$ ). Note that the bandwidth narrows as $s_{R}$ decreases and also narrows for fixed $s_{R}$ and increasing order $ν_{R}$ . of the Bessel weight.

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Fig. 12 Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for bandlimited Bessel receptive field functions as they vary with $s_{R} = 1.0$ , 0.85, 0.7, 0.5, and $ν_{R} = 2.0$ (dotted), 3.5 (dashed), and 5.0 (solid). Unlike balanced Gabor receptive field functions, the max-response of bandlimited receptive field functions is independent of the receptive field phase $ϕ_{R}$ . As support parameter $s_{R}$ values decrease, the receptive field function narrows in effective bandwidth. For a particular support parameter value, as the order of the Bessel weight $ν_{R}$ increases, the receptive field function narrows in effective bandwidth.

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Fig. 1 Simple balanced Gabor weight functions,

g (r) ∕ g (0)

vs. r, (left) and Hankel transforms,

H_{g} (ρ)

vs. ρ, (right) for four exponents

γ_{R} = 0.75

(dashed), 1.5 (dotted), 3.0 (solid), 6.0 (near-solid). As the exponent

γ_{R}

increases, the weight

g (r)

broadens and the Hankel transform

H_{g} (ρ)

narrows. Note that the weight functions are strictly positive and monotonic and are not oscillatory.

View in Article | Download Full Size | PDF

γ_{R} = 0.75, 1.5, 3.0, 6.0

. Note that the traditional and simple balanced Gabor receptive field functions become increasingly similar (i.e., the balancing parameter approaches 0) as

γ_{R}

increases (i.e., as effective spatial frequency bandwidth decreases), becoming virtually identical for

γ_{R} > 3

View in Article | Download Full Size | PDF

0 \deg

(cosine-type, dashed),

45 \deg

(mixed-type, dotted) and

90 \deg

(sine-type, solid). Response curves are for

γ_{R} = 0.75, 1.5, 3.0, 6.0

. Note that as exponent

γ_{R}

increases, max-response curves become independent of carrier spatial phase

ϕ_{R}

and that sine-type receptive fields always give larger responses than cosine-type receptive fields with corresponding parameters. The cosine-type spatial frequency max-response matches the Fourier transform of Fig. 2 (differing only by a scale factor).

View in Article | Download Full Size | PDF

Fig. 4 Nonsimple balanced Gabor weight functions,

g (r) ∕ g (0)

vs. r, (left) and Hankel transforms,

H_{g} (ρ)

vs. ρ, (right) for four exponents

γ_{R} = 0.75, 1.5, 3.0, 6.0

(with

ψ_{R} = 0

). Each plot shows three curves. The curves plot weight functions when

c_{R}

is set to a positive (solid) or negative (dotted) boundary value. These extreme curves (solid and dotted) are interchanged relative to the corresponding extreme curves for

ψ_{R} = π ∕ 2

(see Fig. 7 below). The dashed curve replots the weight function of the simple balanced Gabor

(c_{R} = 0)

for comparison. Note that nonsimple Gabor weight functions exhibit an oscillatory behavior and can take on negative values. The departure from the simple balanced Gabor weight function is most pronounced for large values of

γ_{R}

. As the exponent

γ_{R}

increases, the weight

g (r)

broadens and the Hankel transform

H_{g} (ρ)

narrows.

View in Article | Download Full Size | PDF

Fig. 5 Normalized balanced Gabor receptive field functions (cosine-type),

λ_{R}^{2} R (x_{1}, 0) ∕ 2 π g (0)

vs.

x_{1} ∕ λ_{R}

, and associated weight functions (left) for

γ_{R} = 0.75, 1.5, 3.0, 6.0

(ψ_{R} = 0)

, together with their Fourier transforms (right). Weight functions (left) are in thinner gray curves. Each plot shows

c_{R} = 0

(simple balanced Gabor, dashed) and

c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})

for max (solid) and min (dotted) bounds of this cofficient. For the Fourier transforms, note that as

γ_{R}

increases, (1) the transform narrows (consistent with decreasing bandwidth) and (2) the range of variation of the curves increases (consistent with the increased bounds for

c_{R}

). Note the variation in the Fourier transforms for small values of

s_{1}

(low spatial frequency). There is no apparent corresponding variation in the receptive field functions for small

γ_{R}

, indicating the importance of the large-scale receptive field structure.

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Fig. 6 Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for nonsimple balanced Gabor receptive field functions for

γ_{R} = 0.75, 1.5, 3.0, 6.0

(ψ_{R} = 0)

. Each plot shows

c_{R} = 0

(simple balanced Gabor, dashed) and

c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})

for max (solid) and min (dotted) bounds of this coefficient. Three curves are plotted for each

c_{R}

corresponding to receptive field phase

ϕ_{R} = 0, 45, 90 \deg

, respectively, cosine-type, mixed-type, sine-type. As

γ_{R}

increases, the dependence of both orientation and spatial frequency response on field phase

ϕ_{R}

(

c_{R}

fixed) decreases, becoming virtually independent at

γ_{R} = 6.0

. The cosine-type spatial frequency max-response function matches the Fourier transform of Fig. 5 (up to scaling).

View in Article | Download Full Size | PDF

Fig. 7 Nonsimple balanced Gabor weight functions,

g (r) ∕ g (0)

vs. r, (left) and Hankel transforms,

H_{g} (ρ)

vs. ρ, (right) for four exponents

γ_{R} = 0.75, 1.5, 3.0, 6.0

(with

ψ_{R} = π ∕ 2

). Each plot shows three curves. The curves plot weight functions when

c_{R}

is set to a positive (solid) or negative (dotted) boundary value. These extreme curves (solid and dotted) are interchanged relative to the corresponding extreme curves for

ψ_{R} = 0

(Fig. 4). The dashed curve replots the weight function of the simple balanced Gabor

(c_{R} = 0)

γ_{R}

. As the exponent

γ_{R}

increases, the weight

g (r)

broadens and the Hankel transform

H_{g} (ρ)

narrows.

View in Article | Download Full Size | PDF

Fig. 8 Four pairs of plots showing normalized balanced Gabor receptive field functions (cosine-type),

λ_{R}^{2} R (x_{1}, 0) ∕ 2 π g (0)

vs.

x_{1} ∕ λ_{R}

, and associated weight functions (left) with their Fourier transforms (right) for

γ_{R} = 0.75, 1.5, 3.0, 6.0

(ψ_{R} = π ∕ 2)

. Weight functions (left) are in thinner gray curves. Each plot shows

c_{R} = 0

(simple balanced Gabor, dashed) and

c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})

for max (solid) and min (dotted) bounds of this coefficient. For the Fourier tranforms, note that as

γ_{R}

increases, (1) the transform narrows (consistent with decreasing bandwidth) and (2) the range of variation of the curves increases (consistent with the increased bounds for

c_{R}

). Note the variation in the Fourier transforms for small values of

s_{1}

seen in Fig. 5

(ψ_{R} = 0)

is here most prominent for

s_{1}

slightly less than 1.0. Note the field phase reversal for

γ_{R} = 6.0

(solid).

View in Article | Download Full Size | PDF

Fig. 9 Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for nonsimple balanced Gabor receptive field functions for

γ_{R} = 0.75

, 1.5, 3.0, 6.0

(ψ_{R} = π ∕ 2)

. Each plot shows

c_{R} = 0

(simple balanced Gabor, dashed) and

c_{R} = \pm γ_{R}^{2} ∕ (γ_{R}^{2} + 4 π^{2})

for max (solid) and min (dotted) bounds of this coefficient. Three curves are plotted for each

c_{R}

corresponding to receptive field phase

ϕ_{R} = 0, 45, 90 \deg

, respectively cosine-type, mixed-type, sine-type. As

γ_{R}

increases, the dependence of both orientation and spatial frequency response on receptive field phase

ϕ_{R}

(

c_{R}

fixed) decreases, becoming virtually independent at

γ_{R} = 6.0

. The cosine-type spatial frequency max-response function matches the Fourier transform of Fig. 7 (up to scaling).

View in Article | Download Full Size | PDF

Fig. 10 Bandlimited weight functions for

b (r) ∕ b (0)

vs. r, (left) and Hankel transforms,

H_{b} (ρ)

vs. ρ, (right) for support parameter values

s_{R} = 1.0

, 0.85,0.7,0.5. Each plot shows three curves that plot weight functions when the order of the Bessel weight

ν_{R}

is set to 2.0 (dotted), 3.5 (dashed), and 5.0 (solid). Note that Bessel weight functions exhibit oscillatory behavior and take on negative values. As the support parameter

s_{R}

decreases, the weight

b (r)

broadens and the Hankel transform

H_{b} (ρ)

narrows.

View in Article | Download Full Size | PDF

Fig. 11 Bandlimited Bessel receptive field functions,

λ_{R}^{2} R (x_{1}, 0) ∕ 2 π b (0)

vs.

x_{1} ∕ λ_{R}

, (left), and their Fourier transforms,

F_{R} (s_{1}, 0)

vs.

\log_{10} (λ_{R} s_{1})

(right), for support parameter values

s_{R} = 1.0

, 0.85, 0.7, 0.5. Each plot shows three curves corresponding to Bessel weights of order

ν_{R} = 2.0

(dotted), 3.5 (dashed), and 5.0 (solid). Unlike balanced Gabor functions, the bandlimited receptive field function and its weight function are equal at the origin and, for the Fourier transform, the transform max always occurs for

λ_{P} = λ_{R}

. Like balanced Gabor receptive field functions, the bandlimited receptive field function can show phase reversal (note

s_{R} = 0.7

). Note that the bandwidth narrows as

s_{R}

decreases and also narrows for fixed

s_{R}

and increasing order

ν_{R}

. of the Bessel weight.

View in Article | Download Full Size | PDF

Fig. 12 Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for bandlimited Bessel receptive field functions as they vary with

s_{R} = 1.0

, 0.85, 0.7, 0.5, and

ν_{R} = 2.0

(dotted), 3.5 (dashed), and 5.0 (solid). Unlike balanced Gabor receptive field functions, the max-response of bandlimited receptive field functions is independent of the receptive field phase

ϕ_{R}

. As support parameter

s_{R}

values decrease, the receptive field function narrows in effective bandwidth. For a particular support parameter value, as the order of the Bessel weight

ν_{R}

increases, the receptive field function narrows in effective bandwidth.

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Equations (107)

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R \circ p (t) ≔ \int_{R \times R} R (\overset{⃑}{x}) p (\overset{⃑}{x}; t) d \overset{⃑}{x} .

R_{0} ≔ \int_{R \times R} R (\overset{⃑}{x}) d \overset{⃑}{x} .

p (\overset{⃑}{x}) ≔ 1 + c_{P} \cos (2 π \frac{\overset{⃑}{d} (α_{P}) \cdot \overset{⃑}{x}}{λ_{P}} - ϕ_{P}),

\overset{⃑}{d} (α_{P}) \cdot \overset{⃑}{x} = \cos (α_{P}) x_{1} + \sin (α_{P}) x_{2},

F_{R} (s_{1}, s_{2}) ≔ \int \int_{R \times R} \exp (- 2 π i (s_{1} x_{1} + s_{2} x_{2})) R (x_{1}, x_{2}) d x_{1} d x_{2} .

R_{0} = F_{R} (0, 0),

R \circ p = \int_{R \times R} R (\overset{⃑}{x}) (1 + c_{P} \cos (2 π \frac{\overset{⃑}{d} (α_{P}) \cdot \overset{⃑}{x}}{λ_{P}} - ϕ_{P})) d \overset{⃑}{x} = R_{0} + c_{P} Re [e^{i ϕ_{P}} F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P}))],

M (c_{P}, α_{P}, λ_{P}) ≔ \max_{ϕ_{P}} (R \circ p) = \max_{ϕ_{P}} {R_{0} + c_{P} Re [e^{i ϕ_{P}} F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P}))]} = R_{0} + | c_{P} | | F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) | .

M (c_{P}, α_{P}, λ_{P}) = M (| c_{P} |, α_{P}, λ_{P}) = M (| c_{P} |, α_{P} \pm π, λ_{P}) .

R_{0} = F_{R} (0, 0) = 0 .

R_{0} + c_{ZR} Re [e^{i ϕ_{P}} F_{R} (\frac{1}{λ_{P}} \cos (α_{ZR}), \frac{1}{λ_{P}} \sin (α_{ZR}))] = 0

F_{R} (\frac{1}{λ_{P}} \cos (α_{ZR}), \frac{1}{λ_{P}} \sin (α_{ZR})) = 0 .

M (c_{MR}, α_{P}, λ_{P}) = | c_{MR} | | F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) | .

α_{MR} + ζ_{0} ⩽ α_{P} ⩽ α_{MR} + π ∕ 2

F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) \neq 0

| F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) |

| F_{R} (\frac{1}{λ_{P}} \cos (α_{P}), \frac{1}{λ_{P}} \sin (α_{P})) |

R (\overset{⃑}{x}) ≔ \frac{2 π}{λ_{R}^{2}} q (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - b_{R}),

q (r) is C (0, + \infty),

q (r) r is absolutely integrable on (0, + \infty),

and \int_{0}^{\infty} q (r) r d r = 1 .

If f (x_{1}, x_{2}) = g (r) where r = {(x_{1}^{2} + x_{2}^{2})}^{1 ∕ 2},

then F_{f} (s_{1}, s_{2}) = 2 π H_{g} (2 π ρ) where ρ = {(s_{1}^{2} + s_{2}^{2})}^{1 ∕ 2},

H_{f} (ρ) = \int_{0}^{\infty} J_{0} (ρ r) f (r) r d r and f (r) = \int_{0}^{\infty} J_{0} (ρ r) H_{f} (ρ) ρ d ρ .

H_{q} (ρ) is C [0, + \infty), H_{q} (0) = 1, and H_{q} (+ \infty) = 0 .

\overset{⃑}{s_{R}} = (\frac{\cos (α_{R})}{λ_{R}}, \frac{\sin (α_{R})}{λ_{R}}), \overset{⃑}{s_{P}} = (\frac{\cos (α_{P})}{λ_{P}}, \frac{\sin (α_{P})}{λ_{P}})

‖ \overset{⃑}{s} \pm \overset{⃑}{s_{R}} ‖ = \frac{1}{λ_{R}} \sqrt{{(λ_{R} s_{1} \pm \cos (α_{R}))}^{2} + {(λ_{R} s_{2} \pm \sin (α_{R}))}^{2}},

‖ \overset{⃑}{s_{P}} \pm \overset{⃑}{s_{R}} ‖ = \frac{1}{λ_{R}} \sqrt{1 \pm 2 \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}} + \frac{λ_{R}^{2}}{λ_{P}^{2}}} .

F_{R} (s_{1}, s_{2}) = \frac{1}{2} (e^{+ i ϕ_{R}} H_{q} (λ_{R} ‖ \overset{⃑}{s} + \overset{⃑}{s_{R}} ‖) + e^{- i ϕ_{R}} H_{q} (λ_{R} ‖ \overset{⃑}{s} - \overset{⃑}{s_{R}} ‖)) - b_{R} H_{q} (λ_{R} ‖ \overset{⃑}{s} ‖) .

R_{0} = \cos (ϕ_{R}) H_{q} (1) - b_{R} H_{q} (0) .

R \circ p = R_{0} + \frac{c_{P}}{2} \cos (ϕ_{P} + ϕ_{R}) H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) + \frac{c_{P}}{2} \cos (ϕ_{P} - ϕ_{R}) H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) - b_{R} c_{P} \cos (ϕ_{P}) H_{q} (\frac{λ_{R}}{λ_{P}}) .

M (c_{P}, α_{P}, λ_{P}) = \max_{ϕ_{P}} (R \circ p) = R_{0} + \frac{1}{2} | c_{P} | N (α_{P}, λ_{P}),

N (α_{P}, λ_{P}) ≔ 2 | F_{R} (\cos (α_{P}) ∕ λ_{P}, \sin (α_{P}) ∕ λ_{P}) | ⩾ 0

N {(α_{P}, λ_{P})}^{2} = {[\cos (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) + H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖)) - 2 b_{R} H_{q} (\frac{λ_{R}}{λ_{P}})]}^{2} + {[\sin (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) - H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖))]}^{2} .

b_{R} = \cos (ϕ_{R}) H_{q} (1) .

\cos (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} + \overset{⃑}{s_{R}} ‖) + H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} - \overset{⃑}{s_{R}} ‖)) = 2 b_{R} H_{q} (\frac{λ_{R}}{λ_{P}}),

\sin (ϕ_{R}) (H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} + \overset{⃑}{s_{R}} ‖) - H_{q} (λ_{R} ‖ \overset{⃑}{s_{ZR}} - \overset{⃑}{s_{R}} ‖)) = 0,

b_{R} = \cos (ϕ_{R}) H_{q} (1) .

H_{q} (\sqrt{1 - 2 ρ \cos (ζ_{0}) + ρ^{2}}) + H_{q} (\sqrt{1 + 2 ρ \cos (ζ_{0}) + ρ^{2}}) = 2 H_{q} (1) H_{q} (ρ)

0 < ζ_{R} ⩽ | α_{ZR} - α_{R} | ⩽ π ∕ 2 .

f (a; y) = e^{c y} F (a - y) + e^{- c y} F (a + y) - 2 F (a)

{(\cos (ϕ_{R}))}^{2} {(e^{c y} F (a - y) + e^{- c y} F (a + y) - 2 F (a))}^{2} + {(\sin (ϕ_{R}))}^{2} {(e^{c y} F (a - y) - e^{- c y} F (a + y))}^{2}

e^{- c y} F (y) = e^{- c a} F (a) - K (a) (y - a) on | y - a | ⩽ δ_{a} for some constant K (a) > 0 .

N {(α_{P}, λ_{P})}^{2} = \exp (- c (1 + \frac{λ_{R}^{2}}{λ_{P}^{2}})) \cdot T_{F} (ϕ_{R}, c, 1 + \frac{λ_{R}^{2}}{λ_{P}^{2}}; 2 \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}),

T_{F} (ϕ_{R}, c, a; y) ≔ {(\cos (ϕ_{R}))}^{2} {(e^{c y} F (a - y) + e^{- c y} F (a + y) - 2 F (a))}^{2} + {(\sin (ϕ_{R}))}^{2} {(e^{c y} F (a - y) - e^{- c y} F (a + y))}^{2} .

N (α_{P}, λ_{P}) = H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖),

N (α_{P}, λ_{P}) = | H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) - H_{q} (λ_{R} ‖ \overset{⃑}{s_{P}} + \overset{⃑}{s_{R}} ‖) |

g (r) is C (0, + \infty),

g (r) r is absolutely integrable on (0, + \infty),

and \int_{0}^{\infty} g (r) r d r = 1,

H_{g} (ρ) is C [0, + \infty), H_{g} (0) = 1, and H_{g} (+ \infty) = 0

T_{G} (γ_{R}, a; y) ≔ e^{γ_{R} y} G (a - y) + e^{- γ_{R} y} G (a + y) - 2 G (a)

e^{- c y} F (y) = e^{- c a} F (a) - K (a) (y - a) on | y - a | ⩽ δ_{a}

e^{- γ_{R} (y + 1)} G (y + 1) = e^{- γ_{R}} e^{- γ_{R} y} G (y) .

T_{G} (γ_{R}, a; y + 1) = e^{- γ_{R}} T_{G} (γ_{R}, a; y) + (e^{γ_{R}} - e^{- γ_{R}}) e^{γ_{R} y} G (- y) .

Γ (γ_{R}) ≔ {G (y) : G (y) satisfies conditions (a), (b), (c) for balanced Gabor weights with exponent γ_{R}}

H_{3} (ρ) = e^{- γ_{2} ρ^{2}} (α_{1} G_{1} (ρ^{2}) + α_{2} G_{2} (ρ^{2}))

R_{BG} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} g (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - b_{R}),

F_{R} (s_{1}, s_{2}) = \frac{1}{2} \exp (- γ_{R} (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2} + 1)) \cdot [e^{+ i ϕ_{R}} \exp (- 2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) G (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2} + 2 λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) + e^{- i ϕ_{R}} \exp (+ 2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) G (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2} - 2 λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) - 2 \cos (ϕ_{R}) G (λ_{R}^{2} {‖ \overset{⃑}{s} ‖}^{2})] .

C_{\pm} (α_{P} - α_{R}) ≔ \exp (\pm 2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s_{P}}) G (λ_{R}^{2} {‖ \overset{⃑}{s_{P}} ‖}^{2} \mp 2 λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s_{P}}) = \exp (\pm 2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) G (\frac{λ_{R}^{2}}{λ_{P}^{2}} \mp 2 \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) .

R \circ p = \frac{c_{P}}{2} \exp (- γ_{R} (1 + \frac{λ_{R}^{2}}{λ_{P}^{2}})) \cdot [\cos (ϕ_{P} - ϕ_{R}) (C_{+} (α_{P} - α_{R}) - G (\frac{λ_{R}^{2}}{λ_{P}^{2}})) + \cos (ϕ_{P} + ϕ_{R}) (C_{-} (α_{P} - α_{R}) - G (\frac{λ_{R}^{2}}{λ_{P}^{2}}))] .

M (c_{P}, α_{P}, λ_{P}) = \max_{ϕ_{P}} (R \circ p) = \frac{1}{2} | c_{P} | N (α_{P}, λ_{P}),

N {(α_{P}, λ_{P})}^{2} = \exp (- 2 γ_{R} (\frac{λ_{R}^{2}}{λ_{P}^{2}} + 1)) [\cos {(ϕ_{R})}^{2} N_{c}^{2} + \sin {(ϕ_{R})}^{2} N_{s}^{2}],

N_{c} ≔ C_{+} (α_{P} - α_{R}) + C_{-} (α_{P} - α_{R}) - 2 G (\frac{λ_{R}^{2}}{λ_{P}^{2}}),

N_{s} ≔ C_{+} (α_{P} - α_{R}) - C_{-} (α_{P} - α_{R}) .

N_{s}^{2} - N_{c}^{2} = 4 (- C_{-} (α_{P} - α_{R}) + G (\frac{λ_{R}^{2}}{λ_{P}^{2}})) (C_{+} (α_{P} - α_{R}) + G (\frac{λ_{R}^{2}}{λ_{P}^{2}})),

\exp (- γ_{R} z) G (z + a) and \exp (+ γ_{R} z) G (- z + a)

z (r) = \frac{1}{2} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}} \exp (- \frac{r^{2}}{4} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}}),

H_{z} (ρ) = e^{- c ρ^{2} + 2 π i k ρ^{2}} .

q (r) = \sum_{k = - \infty}^{+ \infty} \frac{g_{k}}{2} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}} \exp (- \frac{r^{2}}{4} \frac{c + 2 π i k}{c^{2} + {(2 π k)}^{2}}),

H_{q} (ρ) = e^{- c ρ^{2}} \sum_{k = - \infty}^{+ \infty} g_{k} e^{+ 2 π i k ρ^{2}} .

R (\overset{⃑}{x}) = \frac{1}{2 π σ_{R}^{2}} \exp (- \frac{{‖ \overset{⃑}{x} ‖}^{2}}{2 σ_{R}^{2}}) \cos (2 π \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x}}{λ_{R}} - ϕ_{R}),

F_{R} (\overset{⃑}{s}) = \exp (- 2 π^{2} σ_{R}^{2} ({‖ \overset{⃑}{s} ‖}^{2} + \frac{1}{λ_{R}^{2}})) \cdot [\cos (ϕ_{R}) \cosh (4 π^{2} σ_{R}^{2} \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}}{λ_{R}}) - i \sin (ϕ_{R}) \sinh (4 π^{2} σ_{R}^{2} \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}}{λ_{R}})] .

\int_{R \times R} R (\overset{⃑}{x}) d \overset{⃑}{x} = \cos (ϕ_{R}) \exp (- 2 π^{2} \frac{σ_{R}^{2}}{λ_{R}^{2}}) .

g (r) = \frac{1}{2 γ_{R}} \exp (- \frac{r^{2}}{4 γ_{R}}) with H_{g} (ρ) = \exp (- γ_{R} ρ^{2}) .

R_{SBG} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} g (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - \cos (ϕ_{R}) e^{- γ_{R}}),

F_{R} (\overset{⃑}{s}) = \exp (- γ_{R} λ_{R}^{2} ({‖ \overset{⃑}{s} ‖}^{2} + \frac{1}{λ_{R}^{2}})) [\cos (ϕ_{R}) (\cosh (2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s}) - 1) - i \sin (ϕ_{R}) \sinh (2 γ_{R} λ_{R} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{s})] .

R \circ p = \frac{c_{P}}{2} \exp (- γ_{R} (1 + \frac{λ_{R}^{2}}{λ_{P}^{2}})) \cdot [\cos (ϕ_{P} - ϕ_{R}) (\exp (+ 2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) - 1) + \cos (ϕ_{P} + ϕ_{R}) (\exp (- 2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) - 1)] .

N {(α_{P}, λ_{P})}^{2} = 4 \exp (- 2 γ_{R} (\frac{λ_{R}^{2}}{λ_{P}^{2}} + 1)) \cdot [\cos {(ϕ_{R})}^{2} {(\cosh (2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}) - 1)}^{2} + \sin {(ϕ_{R})}^{2} {(\sinh (2 γ_{R} \cos (α_{P} - α_{R}) \frac{λ_{R}}{λ_{P}}))}^{2}] .

g (r) = \frac{1}{2 γ_{R} (1 + c_{R} \cos (ψ_{R}))} \exp (- \frac{r^{2}}{4 γ_{R}}) + \frac{c_{R} γ_{R}}{2 (1 + c_{R} \cos (ψ_{R})) (γ_{R}^{2} + 4 π^{2})} \exp (\frac{- γ_{R} r^{2}}{4 (γ_{R}^{2} + 4 π^{2})}) \cdot (\cos (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R}) + \frac{2 π}{γ_{R}} \sin (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R})),

H_{g} (ρ) = e^{- γ_{R} ρ^{2}} G (ρ^{2}) where G (y) = \frac{1 + c_{R} \cos (2 π y - ψ_{R})}{1 + c_{R} \cos (ψ_{R})},

| c_{R} | ⩽ \frac{γ_{R}^{2}}{γ_{R}^{2} + 4 π^{2}} .

g (r) = \frac{1}{2 γ_{R} (1 + c_{R} \cos (ψ_{R}))} \exp (\frac{- γ_{R} r^{2}}{4 (γ_{R}^{2} + 4 π^{2})}) \cdot [\exp (- \frac{π^{2} r^{2}}{γ_{R} (γ_{R}^{2} + 4 π^{2})}) + \frac{c_{R} γ_{R}^{2}}{γ_{R}^{2} + 4 π^{2}} (\cos (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R}) + \frac{2 π}{γ_{R}} \sin (\frac{π r^{2}}{2 (γ_{R}^{2} + 4 π^{2})} + ψ_{R}))] .

R_{BG} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} g (\frac{2 π}{λ_{R}} | \overset{⃑}{x} |) (\cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) - \cos (ϕ_{R}) e^{- γ_{R}}),

b (r) is C (0, + \infty),

b (r) r is absolutely integrable on (0, + \infty),

and \int_{0}^{\infty} b (r) r d r = 1,

H_{b} (ρ) is C [0, + \infty), H_{b} (0) = 1, and H_{b} (+ \infty) = 0

R_{BL} (\overset{⃑}{x}) = \frac{2 π}{λ_{R}^{2}} b (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) \cos (\frac{2 π}{λ_{R}} \overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x} - ϕ_{R}) .

F_{R} (s_{1}, s_{2}) = \frac{1}{2} (e^{+ i ϕ_{R}} H_{b} (λ_{R} ‖ \overset{⃑}{s} + \overset{⃑}{s_{R}} ‖) + e^{- i ϕ_{R}} H_{b} (λ_{R} ‖ \overset{⃑}{s} - \overset{⃑}{s_{R}} ‖)) .

R \circ p = \frac{c_{P}}{2} \cos (ϕ_{P} - ϕ_{R}) H_{b} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) .

M (c_{P}, α_{P}, λ_{P}) = \max_{ϕ_{P}} (R \circ p) = \frac{1}{2} | c_{P} | N (α_{P}, λ_{P}),

N (α_{P}, λ_{P}) = H_{b} (λ_{R} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) .

\cos (α_{P} - α_{R}) - δ ⩽ \frac{λ_{R}}{λ_{P}} ⩽ \cos (α_{P} - α_{R}) + δ,

\cos (α_{P} - α_{R}) ⩾ \frac{1}{2} (\frac{λ_{R}}{λ_{P}} + \cos {(ζ_{R})}^{2} \frac{λ_{P}}{λ_{R}}),

\frac{λ_{R}}{λ_{P}} = \cos (ζ_{R}) < 1,

| \sin (\frac{α_{P} - α_{R}}{2}) | ⩽ \frac{1}{2} \sin (ζ_{R})

b (r) = \int_{0}^{1} J_{0} (ρ r) H_{b} (ρ) ρ d ρ = \sum_{k = 0}^{\infty} \frac{{(- r^{2} 4)}^{k}}{{(k!)}^{2}} \int_{0}^{s_{R}} H_{b} (ρ) ρ^{2 k + 1} d ρ

H_{b} (ν_{R}; ρ) ≔ (\begin{matrix} {(1 - ρ^{2})}^{ν_{R} - 1} & for 0 ⩽ ρ < 1 \\ 0 & for 1 ⩽ ρ \end{matrix}),

b (ν_{R}; r) ≔ 2^{ν_{R} - 1} Γ (ν_{R}) r^{- ν_{R}} J_{ν_{R}} (r) = \int_{0}^{1} J_{0} (ρ r) {(1 - ρ^{2})}^{ν_{R} - 1} ρ d ρ .

b (r) = s_{R}^{2} b (ν_{R}; s_{R} r) and H_{b} (ρ) = H_{b} (ν_{R}; \frac{ρ}{s_{R}}) .

R_{B} (\overset{⃑}{x}) = 2 π \frac{s_{R}^{2}}{λ_{R}^{2}} b (ν_{R}; 2 π \frac{s_{R}}{λ_{R}} ‖ \overset{⃑}{x} ‖) \cos (2 π \frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x}}{λ_{R}} - ϕ_{R}),

R \circ p = \frac{c_{P}}{2} \cos (ϕ_{P} - ϕ_{R}) H_{b} (ν_{R}; \frac{λ_{R}}{s_{R}} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖)

= \frac{c_{P}}{2} \frac{\cos (ϕ_{P} - ϕ_{R})}{s_{R}^{2 ν_{R} - 2}} D^{ν_{R} - 1}

when D ≔ s_{R}^{2} - \sin {(α_{P} - α_{R})}^{2} - {(\frac{λ_{R}}{λ_{P}} - \cos (α_{P} - α_{R}))}^{2} > 0, = 0 otherwise,

N (α_{P}, λ_{P}) = H_{b} (ν_{R}; \frac{λ_{R}}{s_{R}} ‖ \overset{⃑}{s_{P}} - \overset{⃑}{s_{R}} ‖) = \frac{1}{s_{R}^{2 ν_{R} - 2}} D^{ν_{R} - 1} when D > 0 = 0 otherwise .

R (\overset{⃑}{x}) ≔ \frac{2 π}{λ_{R}^{2}} q (\frac{2 π}{λ_{R}} ‖ \overset{⃑}{x} ‖) p (\frac{\overset{⃑}{d} (α_{R}) \cdot \overset{⃑}{x}}{λ_{R}}) .

Abstract

1. INTRODUCTION

2. FORMALIZATION OF POSTULATED SIMPLE CELL RECEPTIVE FIELD PROPERTIES

3. ELEMENTARY RECEPTIVE FIELD FUNCTIONS AND HANKEL TRANSFORMS

4. CHARACTERIZATION OF ELEMENTARY RECEPTIVE FIELD FUNCTIONS WITH POSTULATED PROPERTIES

5. RECEPTIVE FIELD FUNCTIONS

5A. Balanced Gabor Receptive Field Functions: Discussion and Examples

5A1. Balanced Gabor Weights and Receptive Fields

5A2. Example: Simple Balanced Gabor Receptive Field Functions (and Traditional Gabor Filters)

5A3. Example: a Class of Nonsimple Balanced Gabor Receptive Field Functions

5B. Bandlimited Receptive Field Functions: Discussion and Examples

5B1. Bandlimited Weights and Receptive Fields

5B2. Example: Bessel Receptive Field Functions

6. DISCUSSION

ACKNOWLEDGMENTS

Cited By

Figures (12)

Equations (107)

Journal of the Optical Society of America A