Abstract

Recent work has shown that salient perceptual singularities occur in visual textures even in the absence of feature gradients. In smoothly varying orientation-defined textures, these striking non-smooth percepts can be predicted from two texture curvatures, one tangential and one normal [Proc. Natl. Acad. Sci. USA 103, 15704 (2006) ]. We address the issue of detecting these perceptual singularities in a biologically plausible manner and present three different models to compute the tangential and normal curvatures using early cortical mechanisms. The first model relies on the response summation of similarly scaled even-symmetric simple cells at different positions by utilizing intercolumnar interactions in the primary visual cortex (V1). The second model is based on intracolumnar interactions in a two-layer mechanism of simple cells having the same orientation tuning but significantly different scales. Our third model uses a three-layer circuit in which both even-symmetric and odd-symmetric receptive fields (RFs) are used to compute all possible directional derivatives of the dominant orientation, from which the tangential and normal curvatures at each spatial position are selected using nonlinear shunting inhibition. We show experimental results of all three models, we outline an extension to oriented textures with multiple dominant orientations at each point, and we discuss how our results may be relevant to the processing of general textures.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  65. K. Schmidt, R. Goebel, S. Löwel, and W. Singer, “The perceptual grouping criterion of colinearity is reflected by anisotropies in the primary visual cortex,” Eur. J. Neurosci. 9, 1083-1089 (1997).
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    [CrossRef]

2007 (1)

A. Johnson, N. Prins, F. Kingdom, and C. Baker, “Ecologically valid combinations of first- and second-order surface markings facilitate texture discrimination,” Vision Res. 47, 2281-2290 (2007).
[CrossRef] [PubMed]

2006 (3)

L. Strother and M. Kubovy, “On the surprising salience of curvature in grouping by proximity,” J. Exp. Psychol. 32, 226-234 (2006).

O. Ben-Shahar, “Visual saliency and texture segregation without feature gradient,” Proc. Natl. Acad. Sci. U.S.A. 103, 15704-15709 (2006).
[CrossRef] [PubMed]

O. Ben-Shahar, “Saliency and segregation without feature gradient: New insights for segmentation from orientation-defined textures,” in The Fifth IEEE Computer Society Workshop on Perceptual Organization in Computer Vision (IEEE, 2006), pp. 175-182.

2005 (1)

R. Oliveira, L. da Fondtoura Costa, and A. Roque, “A possible mechanism of curvature coding in early vision,” Neurocomputing 65-66, 117-124 (2005).
[CrossRef]

2004 (4)

O. Ben-Shahar and S. Zucker, “Sensitivity to curvatures in orientation-based texture segmentation,” Vision Res. 44, 257-277 (2004).
[CrossRef]

M. S. Landy and N. Graham, “Visual perception of texture,” in The Visual Neurosciences, L.M.Chalupa and J.S.Werner, eds. (MIT, 2004), pp. 1106-1118.

O. Ben-Shahar and S. Zucker, “Geometrical computations explain projection patterns of long range horizontal connections in visual cortex,” Neural Comput. 16, 445-476 (2004).
[CrossRef] [PubMed]

I. Lampl, D. Ferster, T. Poggio, and M. Riesenhuber, “Intracellular measurements of spatial integration and the MAX operator in complex cells of the cat primary visual cortex,” J. Neurophysiol. 92, 2704-2713 (2004).
[CrossRef] [PubMed]

2003 (5)

Y. Frégnac, C. Monier, F. Chavane, P. Baudot, and L. Graham, “Shunting inhibition, a silent step in visual cortical computation,” J. Physiol. (Paris) 97, 441-451 (2003).
[CrossRef]

O. Ben-Shahar and S. Zucker, “The perceptual organization of texture flows: A contextual inference approach,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 401-417 (2003).
[CrossRef]

F. Kingdom, N. Prins, and A. Hayes, “Mechanism independence for texture-modulations detection is consistent with filter-rectify-filter mechanism,” Visual Neurosci. 20, 65-76 (2003).
[CrossRef]

N. Prins and F. Kingdom, “Detection and discrimination of texture modulations defined by orientation, spatial frequency, and contrast,” J. Opt. Soc. Am. A 20, 401-410 (2003).
[CrossRef]

A. Angelucci and J. Bullier, “Reaching beyond the classical receptive field of V1 neurons: Horizontal or feedback axons?” J. Physiol. (Paris) 97, 141-154 (2003).
[CrossRef]

2002 (1)

A. Yu, M. Gisse, and T. Poggio, “Biophysiologically plausible implementations of the maximum operation,” Neural Comput. 14, 2857-2881 (2002).
[CrossRef] [PubMed]

2001 (2)

2000 (2)

H. Nothdurft, J. Gallant, and D. Van Essen, “Response profiles to texture border patterns in area V1,” Visual Neurosci. 17, 421-436 (2000).
[CrossRef]

J. Serrat, A. López, and D. Lloret, “On ridges and valleys,” in Proceedings of the 15th IEEE International Conference on Pattern Recognition (IEEE, 2000), pp. 59-66.
[CrossRef]

1999 (3)

S. Dakin, C. Williams, and R. Hess, “The interaction of first- and second-order cues to orientation,” Vision Res. 39, 2867-2884 (1999).
[CrossRef] [PubMed]

M. Riesenhuber and T. Poggio, “Hierarchical models of object recognition in cortex,” Nat. Neurosci. 2, 1019-1025 (1999).
[CrossRef] [PubMed]

A. Mussap and D. Levi, “Orientation-based texture segmentation in strabismic amblyopia,” Vision Res. 39, 411-418 (1999).
[CrossRef] [PubMed]

1998 (3)

L. Borg-Graham, C. Monier, and Y. Frégnac, “Visual input evokes transient and strong shunting inhibition in visual cortical neurons,” Nature 393, 369-373 (1998).
[CrossRef] [PubMed]

H. Wilson and F. Wilkinson, “Detection of global structure in glass patterns: Implications for form vision,” Vision Res. 38, 2933-2947 (1998).
[CrossRef] [PubMed]

I. Mareschal and C. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150-154 (1998).
[CrossRef]

1997 (3)

K. Schmidt, R. Goebel, S. Löwel, and W. Singer, “The perceptual grouping criterion of colinearity is reflected by anisotropies in the primary visual cortex,” Eur. J. Neurosci. 9, 1083-1089 (1997).
[CrossRef] [PubMed]

W. Bosking, Y. Zhang, B. Schofield, and D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in the tree shrew striate cortex,” J. Neurosci. 17, 2112-2127 (1997).
[PubMed]

J. Rieger, “Topographical properties of generic images,” Int. J. Comput. Vis. 23, 79-92 (1997).
[CrossRef]

1996 (2)

D. Eberly, Ridges in Image and Data Analysis (Kluwer Academic, 1996).

A. López and J. Serrat, “Tracing crease curves by solving a system of differential equations,” in Proceedings of the European Conference on Computer Vision, Vol. 1064 of Lecture Notes in Computer Science (Springer-Verlag, 1996), pp. 241-250.

1995 (2)

S. Wolfson and M. Landy, “Discrimination of oriention-defined texture edges,” Vision Res. 35, 2863-2877 (1995).
[CrossRef] [PubMed]

D. Sagi, “The psychophysics of texture segmentation,” in Early Vision and Beyond, T.Papathomas, C.Chubb, A.Gorea, and E.Kowler, eds. (MIT, 1995), pp. 69-78.

1993 (6)

B. Carbal and L. Leedom, “Imaging vector fields using line integral convolution,” in Proceedings of SIGGRAPH (ACMSIGGRAPH, 1993), pp. 263-270.

H. Nothdurft, “The role of features in preattentive vision: Comparison of orientation, motion, and color cues,” Vision Res. 33, 1937-1958 (1993).
[CrossRef] [PubMed]

N. Fisher, Statistical Analysis of Circular Data (Cambridge U. Press, 1993).
[CrossRef]

D. Field, A. Hayes, and R. Hess, “Contour integration in the human visual system: Evidence for a local 'association' field,” Vision Res. 33, 173-193 (1993).
[CrossRef] [PubMed]

R. Malach, Y. Amir, M. Harel, and A. Grinvald, “Relationship between intrinsic connections and functional architecture revealed by optical imaging and in vivo targeted biocytin injections in primate striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 90, 10469-10473 (1993).
[CrossRef] [PubMed]

J. Koenderink and A. van Doorn, “Local features of smooth shapes: Ridges and courses,” Proc. SPIE 2031, 2-13 (1993).
[CrossRef]

1992 (2)

A. Rao and R. Jain, “Computerized flow field analysis: Oriented texture fields,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 693-709 (1992).
[CrossRef]

C. Gilbert, “Horizontal integration and cortical dynamics,” Neuron 9, 1-13 (1992).
[CrossRef] [PubMed]

1991 (3)

J. Bergen and M. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M.Landy and J.Movshon, eds. (MIT, 1991), pp. 253-271.

H. Nothdurft, “Texture segmentation and pop-out from orientation contrast,” Vision Res. 31, 1073-1078 (1991).
[CrossRef] [PubMed]

M. Landy and J. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679-691 (1991).
[CrossRef] [PubMed]

1990 (3)

J. Todd and F. Reichel, “Visual perception of smoothly curved surfaces from double-projected contour patterns,” J. Exp. Psychol. 16, 665-674 (1990).

J. Malik and P. Perona, “Preattentive texture discrimination with early vision mechanisms,” J. Opt. Soc. Am. A 7, 923-932 (1990).
[CrossRef] [PubMed]

M. Versavel, G. Orban, and L. Lagae, “Responses of visual cortical neurons to curved stimuli and chevrons,” Vision Res. 30, 235-248 (1990).
[CrossRef] [PubMed]

1989 (1)

1988 (2)

A. Sha'ashua and S. Ullman, “Structural saliency: The detection of globally salient structures using a locally connected network,” in Proceedings of the Second IEEE International Conference on Computer Vision (IEEE, 1988), pp. 321-327.
[CrossRef]

M. Concetta Morrone and D. Burr, “Feature detection in human vision: A phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

1987 (3)

A. Dobbins, S. Zucker, and M. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438-441 (1987).
[CrossRef] [PubMed]

M. Kass and A. Witkin, “Analyzing oriented patterns,” Comput. Vis. Graph. Image Process. 37, 362-385 (1987).
[CrossRef]

W. Richards, J. Keonderink, and D. Hoffman, “Inferring 3D shapes from 2D silhouettes,” J. Opt. Soc. Am. A 4, 1168-1175 (1987).
[CrossRef]

1985 (1)

H. Nothdurft, “Orientation sensitivity and texture segmentation in patterns with different line orientation,” Vision Res. 25, 551-560 (1985).
[CrossRef] [PubMed]

1983 (2)

C. Koch, T. Poggio, and V. Torre, “Nonlinear interactions in a dendritic tree: Localization, timing, and role in information processing,” Proc. Natl. Acad. Sci. U.S.A. 80, 2799-2802 (1983).
[CrossRef] [PubMed]

R. Haralik, “Ridges and valleys on digital images,” Comput. Vis. Graph. Image Process. 22, 28-38 (1983).
[CrossRef]

1982 (1)

K. Rockland and J. Lund, “Widespread periodic intrinsic connections in the tree shrew visual cortex,” Science 215, 1532-1534 (1982).
[CrossRef] [PubMed]

1981 (2)

K. Stevens, “The visual interpretation of surface contours,” Artif. Intell. 17, 47-73 (1981).
[CrossRef]

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91-97 (1981).
[CrossRef] [PubMed]

1979 (1)

G. Kanizsa, Organization in Vision: Essays on Gestalt Perception (Praeger, 1979).

1978 (1)

V. Torre and T. Poggio, “A synaptic mechanism possibly underlying directional selectivity to motion,” Proc. R. Soc. London, Ser. B 202, 409-416 (1978).
[CrossRef]

1977 (1)

D. Hubel and T. Wiesel, “Functional architecture of macaque monkey visual cortex,” in Proc. R. Soc. London, Ser. B 198, 1-59 (1977).
[CrossRef]

1976 (1)

M. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, 1976).

1970 (1)

R. Olson and F. Attneave, “What variables produce similarity grouping?” Am. J. Psychol. 83, 1-21 (1970).
[CrossRef]

1966 (2)

J. Beck, “Effect of orientation and the shape similarity on perceptual grouping,” Percept. Psychophys. 1, 300-302 (1966).

B. O'Neill, Elementary Differential Geometry (Academic, 1966).

1965 (1)

H. Barlow and W. Levick, “The mechanism of directionally selective units in rabbit's retina,” J. Physiol. (London) 178, 477-504 (1965).

Amir, Y.

R. Malach, Y. Amir, M. Harel, and A. Grinvald, “Relationship between intrinsic connections and functional architecture revealed by optical imaging and in vivo targeted biocytin injections in primate striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 90, 10469-10473 (1993).
[CrossRef] [PubMed]

Angelucci, A.

A. Angelucci and J. Bullier, “Reaching beyond the classical receptive field of V1 neurons: Horizontal or feedback axons?” J. Physiol. (Paris) 97, 141-154 (2003).
[CrossRef]

Attneave, F.

R. Olson and F. Attneave, “What variables produce similarity grouping?” Am. J. Psychol. 83, 1-21 (1970).
[CrossRef]

Baker, C.

A. Johnson, N. Prins, F. Kingdom, and C. Baker, “Ecologically valid combinations of first- and second-order surface markings facilitate texture discrimination,” Vision Res. 47, 2281-2290 (2007).
[CrossRef] [PubMed]

C. Baker and I. Mareschal, “Processing of second-order stimuli in the visual cortex,” Prog. Brain Res. 134, 171-191 (2001).
[CrossRef] [PubMed]

I. Mareschal and C. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150-154 (1998).
[CrossRef]

Barlow, H.

H. Barlow and W. Levick, “The mechanism of directionally selective units in rabbit's retina,” J. Physiol. (London) 178, 477-504 (1965).

Baudot, P.

Y. Frégnac, C. Monier, F. Chavane, P. Baudot, and L. Graham, “Shunting inhibition, a silent step in visual cortical computation,” J. Physiol. (Paris) 97, 441-451 (2003).
[CrossRef]

Beck, J.

J. Beck, “Effect of orientation and the shape similarity on perceptual grouping,” Percept. Psychophys. 1, 300-302 (1966).

Ben-Shahar, O.

O. Ben-Shahar, “Saliency and segregation without feature gradient: New insights for segmentation from orientation-defined textures,” in The Fifth IEEE Computer Society Workshop on Perceptual Organization in Computer Vision (IEEE, 2006), pp. 175-182.

O. Ben-Shahar, “Visual saliency and texture segregation without feature gradient,” Proc. Natl. Acad. Sci. U.S.A. 103, 15704-15709 (2006).
[CrossRef] [PubMed]

O. Ben-Shahar and S. Zucker, “Geometrical computations explain projection patterns of long range horizontal connections in visual cortex,” Neural Comput. 16, 445-476 (2004).
[CrossRef] [PubMed]

O. Ben-Shahar and S. Zucker, “Sensitivity to curvatures in orientation-based texture segmentation,” Vision Res. 44, 257-277 (2004).
[CrossRef]

O. Ben-Shahar and S. Zucker, “The perceptual organization of texture flows: A contextual inference approach,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 401-417 (2003).
[CrossRef]

Bergen, J.

J. Bergen and M. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M.Landy and J.Movshon, eds. (MIT, 1991), pp. 253-271.

M. Landy and J. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679-691 (1991).
[CrossRef] [PubMed]

Borg-Graham, L.

L. Borg-Graham, C. Monier, and Y. Frégnac, “Visual input evokes transient and strong shunting inhibition in visual cortical neurons,” Nature 393, 369-373 (1998).
[CrossRef] [PubMed]

Bosking, W.

W. Bosking, Y. Zhang, B. Schofield, and D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in the tree shrew striate cortex,” J. Neurosci. 17, 2112-2127 (1997).
[PubMed]

Bullier, J.

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J. Neurophysiol. (1)

I. Lampl, D. Ferster, T. Poggio, and M. Riesenhuber, “Intracellular measurements of spatial integration and the MAX operator in complex cells of the cat primary visual cortex,” J. Neurophysiol. 92, 2704-2713 (2004).
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J. Physiol. (Paris) (2)

Y. Frégnac, C. Monier, F. Chavane, P. Baudot, and L. Graham, “Shunting inhibition, a silent step in visual cortical computation,” J. Physiol. (Paris) 97, 441-451 (2003).
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I. Mareschal and C. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150-154 (1998).
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Nature (3)

L. Borg-Graham, C. Monier, and Y. Frégnac, “Visual input evokes transient and strong shunting inhibition in visual cortical neurons,” Nature 393, 369-373 (1998).
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B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91-97 (1981).
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Neural Comput. (2)

O. Ben-Shahar and S. Zucker, “Geometrical computations explain projection patterns of long range horizontal connections in visual cortex,” Neural Comput. 16, 445-476 (2004).
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R. Oliveira, L. da Fondtoura Costa, and A. Roque, “A possible mechanism of curvature coding in early vision,” Neurocomputing 65-66, 117-124 (2005).
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C. Gilbert, “Horizontal integration and cortical dynamics,” Neuron 9, 1-13 (1992).
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Figures (21)

Fig. 1
Fig. 1

Smoothly varying ODTs exhibit salient perceptual singularities that are poorly predicted by orientation gradients. (a) A smoothly varying ODT defined by the function θ ( x , y ) = x + y and its orientation gradient magnitude (as a graph). The fundamental gap between the (inhomogeneous) perceptual outcome and the (constant) orientation gradient is evident. Here and throughout the paper, dense ODTs are depicted as line integral convolution (LIC) patterns [62] of constant local luminance. (b) Another example of an ODT whose perceptual structure is a (double) spiral, while its orientation gradient is concentric in nature. (c) Phased pair of ODTs are two ODTs that are different only by a constant phase shift of 90 ° , i.e., θ l o w e r ( x , y ) = θ u p p e r ( x , y ) + π 2 . Despite having an identical orientation gradient across the pattern, the perceptual outcome is drastically different. In particular, note the strong segregation to concentric rings in the lower pattern but not in the upper one.

Fig. 2
Fig. 2

Perceptual singularities in ODTs are robust to the specific visualization method and/or to the presence of multiple dominant orientations at a point. (a) A dense ODT of the sort shown in Fig. 1. (b) The same ODT depicted now as a sparse array of oriented segments (texels). Note that the same global structure mediated by the perceptual singularities is equally salient in both patterns. Note further that collinear structure, which classically is predicted to be perceptually dominant [63, 64, 65], could in fact be the least salient in such displays (as is between the black dots). (c) A dense multioriented ODT with two dominant orientations at each point. Note that the global perceptual structure of a mosaic of diamonds although no orientation discontinuities are present.

Fig. 3
Fig. 3

Perceptual singularities in smoothly varying ODTs are not related to, and cannot be explained by, biases in the distribution of energy in the frequency domain. (a) A dense ODT of the sort shown in Fig. 1a. The marked region of interest (ROI) that includes a singularity is used for the local analysis in the second row of this figure. (b) The global amplitude spectrum of this stimulus shows no special bias upon casual inspection. (c) A map of four 45 ° sectors (centered about the directions 0 ° , 45 ° , 90 ° , and 135 ° in the frequency domain) in which energy was accumulated for testing whether the spectrum nevertheless contains any bias in a direction corresponding to the perceptual singularities (i.e., 135 ° ). (d) The distribution of energy in the spectrum of the pattern shows no clear bias that can explain the existence of the perceptual singularities in the ODT. (e) Another ODT and a ROI that does not include a perceptual singularity. (f) The spectrum of the ROI in panel (e). (g) The spectrum of the ROI in panel (a). Note how it is virtually identical to the previous spectrum, although one patch includes a singularity while the other does not. (h) The orientation analysis of the distribution of energy in the spectra from panels (f) and (g). Shown is only one graph because the two spectra indeed have identical distributions. Hence, note how the same spectra can emerge from a patch that contains a singularity and from a patch that does not include such a singularity.

Fig. 4
Fig. 4

Intrinsic local geometry of smooth ODTs is best captured by its representation as a differentiable frame field, which is everywhere tangent and normal to the direction of the flow. An infinitesimal translation of the frame in some direction V rotates it by some angle determined by the connection form of the frame field. Since the connection form is a linear operator, it is fully characterized by two numbers obtained by orthogonal expansion. The natural expansion based on the frame itself yields the two curvatures κ T and κ N . This figure exemplifies all these notions on a blown-up (and brightened) section of the spiral-like stimuli from Fig. 1b. Tangent and normal vectors are shown in red and green, respectively. (a) An infinitesimal translation of the frame from point q 1 along an arbitrary direction V (blue vector) rotates it (counterclockwise in this case) by an amount related to the covariant derivatives V E T and V E N . The initial rate of rotation along the directions E T and E N themselves defines the two curvatures κ T and κ N . Here a “tangential translation” induces a small counterclockwise rotation Δ θ T relative to the original pose (dashed), while the “normal translation” induces a large clockwise rotation Δ θ N . The corresponding (small) κ T and (large) κ N translate to a high perceptual singularity measure [PSM in Eq. (3)], which peaks at the perceptual singularity that passes around q 1 . (b) Similar analysis around q 2 shows a large “tangential rotation” Δ θ T and a small “normal rotation” Δ θ N . The corresponding curvature values translate to a small PSM, which predicts correctly the lack of saliency around q 2 .

Fig. 5
Fig. 5

Structure of the tangential and normal curvature maps is intimately related to the perceptual structure in ODTs. (a) ODT of constant orientation gradient [same as Fig. 1a] with one perceptual singularity highlighted. (b) The κ T ( x , y ) of the ODT from panel (a) depicted as intensity. Note how the perceptual singularities coincide with κ T ( x , y ) ’s zeros. (c) The κ N ( x , y ) of the same ODT. Note the correspondence of the singularities with κ N ( x , y ) ’s maxima.

Fig. 6
Fig. 6

Application of the PSM to selected ODTs whose orientation function is given analytically. (a) The smoothly varying ODT from Fig. 1b and its PSM-predicted perceptual singularities. (b) Another smoothly varying ODT with its PSM prediction. (c) A piecewise-constant ODT and its predicted perceptual borders. Note that in order to apply the PSM on discontinuous ODTs, their nondifferentiable orientation function is first shifted an infinitesimal amount in scale space (i.e., blurred a little bit). (d) Same as panel (c) for a piecewise-smooth ODT. Note the simultaneous detection of the perceptual singularities both within and between smoothly varying regions.

Fig. 7
Fig. 7

Biologically plausible way to compute the oriented structure of an ODT stimulus. Presented are 4 different channels (out of 18 channels), each computing the oriented structure in a different orientation. The visual signal is being convolved with a tuned even-symmetric RF and with its contrast-reversed version, both followed by the intrinsic neural half-wave rectification (shown in red in the online version). This yields the neural responses max { 0 , r θ } and max { 0 , r θ ¯ } , respectively, in each channel. Taking their sum gives max { 0 , r θ } + max { 0 , r θ ¯ } = r θ = R θ , which is the estimated oriented structure of the stimulus in orientation θ. Output examples are shown in Figs. 12b, 13a.

Fig. 8
Fig. 8

Biologically plausible way for computing directional derivatives. A map of some spatial signal (represented here as an image) is being convolved with an odd-symmetric cell (upper filter tuned to 90 ° , lower filter tuned to 225 ° ). The filter responses represent the rate of change of the image intensity along the 0 ° (upper) and 135 ° (lower) directions. Note the graded response along the disk boundary. Importantly, this operation could apply not only to the retinotopic signal but to any intermediate representation along the early visual process. In fact, in some of our models, we apply such a process on the map of dominant orientations in order to estimate the directional derivative of θ ( x , y ) and the ODT curvatures.

Fig. 9
Fig. 9

Schematics of a gating circuit. The input signal S is passed to the output unless the gating signal is nonzero, in which case, the output is zero.

Fig. 10
Fig. 10

Measurement and modeling of a MAX-like operator in the visual cortex. (a) Neurons with a MAX-like behavior in the visual cortex of a cat obtained from bar pairs that were presented both separately and simultaneously (data reproduced from Lampl et al. [33] with permission by the author). Shown are the mean responses of the two simple cells to each bar individually (the smaller response in blue and the larger response in red online) and the actual MAX-like response of a complex cell stimulated by the two bars simultaneously (in green online). The predicted linear response from the summation of the individual responses is shown in black. (b) A MAX operator can be constructed in a biologically plausible fashion by shunting each input with a control signal obtained by the rectified difference of the two input signals. Here the triangle represents a neuron with one excitatory input and one inhibitory input, and its output represents the signed summation of these values up to the usual rectification (red function drawn inside the triangles online). Hence, the output of these units is zero unless the excitatory input is larger than the inhibitory one. These control signals are then fed to gating circuits (Subsection 3C and Fig. 9), which effectively feed the final summation unit a zero value in one input and max ( S 1 , S 2 ) in the other.

Fig. 11
Fig. 11

Variation on the MAX operator gate that blocks its input unless it is bigger than its gating signal. (a) A construction of a MAX-GATING operator using nonlinear shunting inhibition. Symbols represent atomic gates and circuits as discussed in Figs. 9, 10b. (b) Extending the basic MAX-GATING to multiple controls requires only serial concatenation, a circuit whose operation can be written formally as O u t = S ( ( S G 1 ) ( S G 2 ) ( S G n ) ) . While a “parallel” construction is possible also, it is omitted here for space considerations.

Fig. 12
Fig. 12

Sketch and example of Model 1 with emphasis on the estimation of the two curvatures. (a), (b) Simple cell RFs are used to estimate oriented structure (as described in Subsection 3A). High filter response indicates oriented structure in the corresponding orientation. (b) At each point of each R θ map, the amount of response in the tangential and normal directions (green arrows) is evaluated by summation of similarly oriented filters along a “θ-tangential neighborhood” (cyan ellipses) and “θ-normal neighborhoods” (red ellipses), respectively (see inset). The corresponding summation maps of the selected channels are denoted by S T θ and S N θ as discussed in the text. (c) At each point ( x , y ) , the orientation that yields the maximum summation value is selected. By construction, these maps represent complimentary curvature values as discussed in the text. (d) Final PSM ridges results (in red) superimposed on the original stimulus.

Fig. 13
Fig. 13

Sketch and example of Model 2. (a) The filter response of the first layer (i.e., oriented structure estimation as described in Subsection 3A) shown here for two selected channels ( θ = 90 ° and θ = 120 ° ). (b) The second layer filter response for the same two channels. Note that these two second layer filters in channel θ are tuned to the θ direction and the θ + 90 direction. (c) The κ T ¯ and κ N ¯ maps obtained by applying maximum selection among all corresponding second layer outputs. The higher the value in the final maps, the slower the change of orientation along its own (i.e., tangential) or perpendicular (i.e., normal) directions, respectively.

Fig. 14
Fig. 14

Intercolumnar and intracolumnar interactions in V1 [24, 36]. Horizontal connections (dashed/red online) are intercolumnar interactions between cells having the same scale but different position tuning. Vertical connections (solid/blue online) are intracolumnar interactions between cells having a similar orientation preference but possibly different scale/size.

Fig. 15
Fig. 15

Sketch of Model 3. (a) The dominant orientation map is computed by passing each R θ map at each point through a MAX GATING circuit, which is gated (in red) by all the other R θ channels. (b) The dominant orientation map is then being convolved with tuned odd-symmetric simple cells, creating orientation directional derivative maps in all directions (in a 10 ° resolution). Each of these directional derivatives is then being gated (in red) by the dominant orientation in order to preserve only those directional derivatives along and perpendicular to the dominant orientation. The end result are two curvature maps used for the PSM final computation.

Fig. 16
Fig. 16

First possibility for computing the directional derivative on implicit and sparse representation of the dominant orientation map. Here a “directional derivative column” of neurons has a cell for each possible value of the directional derivative. Not unlike the representation of the orientation map itself by orientation-tuned RFs, here an active neuron N d θ , ϕ , x , y implies a directional derivative of magnitude dθ in direction ϕ at position ( x , y ) . Such a neuron should fire if and only if nearby neurons of particular position and orientation tuning fire also. The input afferents of N d θ , ϕ , x , y should therefore come from these particular neurons (shown in dashed red online) and summed up (or combined nonlinearly) to produce the desired output signal. (a) The discussed circuit shown with orientation hypercolumn structures where each hypercolumn is positioned according to the spatial position in the visual field to which it is tuned. Here we use a hexagonal neighborhood structure, but any other structure could be used also as long as the connections are wired up according to the correct spatial neighbors dictated by the direction of derivation. (b) The same circuit shown from a “top view” of the orientation hypercolumn. The same neighborhood orientations that need to interact in order to trigger the directional derivative neuron N d θ , ϕ , x , y are now shown explicitly within the column.

Fig. 17
Fig. 17

Second possibility for computing directional derivatives on an implicit and sparse representation of the dominant orientation map. Now, a single unit N ϕ , x , y signals all possible directional derivative values via different firing rates, and the computation is based on subtracting (via excitatory and inhibitory inputs) the orientations at the appropriate nearby spatial positions by encoding these orientations as synaptic weights.

Fig. 18
Fig. 18

Performance of the three biologically plausible models in predicting perceptual singularities in smoothly varying dense ODT stimuli. Model 1 uses σ = 1 and a = 3 (see Section 4). Model 2 uses σ = 1 , a = 3 , b = 3 , and k = 7 (see Section 5). Model 3 uses σ = 1 and a = 3 for the even-symmetric cells and σ = 2 and a = 1.5 for the odd-symmetric cells (see Section 6). Up to some image margin artifacts, these results compare very well with the analytical results (cf. Section 2) on the right. (If needed, please zoom in using the electronic version of the paper.)

Fig. 19
Fig. 19

Predicted perceptual singularities by Model 3 on piecewise-constant and piecewise-smooth dense ODTs (left) and on sparse, texel-based ODT stimuli (right). Note that exactly the same algorithm and the same implementation parameters (as specified in Fig. 18 for Model 3) were used on all types of stimuli.  

Fig. 20
Fig. 20

Comparison of predicted perceptual singularities by our Model 3 to those made by the classical FRF model due to Malik and Perona [6]. (a) Original ODT with perceptual singularities (in red) as predicted by Model 3. (b) Phased ODT with perceptual singularities (in green) as predicted by Model 3. Note that the result is different from the one in panel (a), in accordance with the perceptual result. (c) Original ODT with perceptual singularities (in red) predicted by Malik and Perona [6]. (d) Phased ODT with perceptual singularities (in green) predicted by Malik and Perona [6]. (e) Both sets of predicted perceptual singularities by Malik and Perona [6] superimposed on the original ODT. Note the striking degree of overlap. (f) A close-up of the regions marked in panel (e).

Fig. 21
Fig. 21

Computational detection of perceptual singularities in multioriented ODTs. (a) Example of a dense multioriented ODT. The perceptual singularities in this pattern trigger the perception of diamonds. (b) Estimation of multiple dominant orientations at the marked point [see inset, taken from Fig. 2c] via fitting a von Mises mixture model [66] to the response of the filter bank at that position. The peaks of each mode in the fitted mixture at each point represent the dominant orientations at that point. (c) Grouping the estimated dominant orientations to manifolds in R 2 × S 1 can lead to the combined set of perceptual singularities. (d) Computational detection (in red online) of perceptual singularities and global perceptual structure.

Equations (24)

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( V E T V E N ) = [ 0 w 12 ( V ) w 12 ( V ) 0 ] ( E T E N ) .
w 12 ( V ) = w 12 ( a E 1 + b E 2 ) = a w 12 ( E 1 ) + b w 12 ( E 2 ) .
κ T w 12 ( E T ) ,
κ N w 12 ( E N ) .
κ T = d θ ( E T ) = θ ( cos θ , sin θ ) ,
κ N = d θ ( E N ) = θ ( sin θ , cos θ ) .
P S M ( x , y ) = κ T 2 + κ N 2 > τ ( R i d g e s [ κ N ( x , y ) 2 κ T ( x , y ) 2 + κ N ( x , y ) 2 ] ) ,
F θ = 0 e ( σ , a ) = e G ( 0 , 0 ; σ x , σ y ) i [ G ( 0 , y i ; σ x , σ y ) + G ( 0 , y i ; σ x , σ y ) ] ,
F θ = 0 o ( σ , a ) = e G ( 0 , y i 2 ; σ x , σ y ) i G ( 0 , y i 2 ; σ x , σ y ) ,
d I ( e θ ) = I * F θ + 90 o : e θ = ( cos θ , sin θ ) ,
S T θ ( q ) = ( x , y ) N T ( θ ) ( q ) R θ ( x , y ) ,
S N θ ( q ) = ( x , y ) N N ( θ ) ( q ) R θ ( x , y ) ,
κ T ¯ ( q ) = max θ { S T θ ( q ) : θ Θ e } ,
κ N ¯ ( q ) = max θ { S N θ ( q ) : θ Θ e } ,
κ T ¯ ( q ) = max θ { ( F θ e ( k σ , a ) * F θ e ( σ , b ) * I ) ( q ) : θ Θ e } ,
κ N ¯ ( q ) = max θ { ( F θ + 90 e ( k σ , a ) * F θ e ( σ , b ) * I ) ( q ) : θ Θ e } ,
θ ( x , y ) = arg max θ { R θ ( x , y ) : θ Θ e } ,
d θ ( e ϕ ) = F ϕ + 90 o * θ .
κ T = ϕ Θ e d θ ( e ϕ ) δ ϕ , θ ,
κ N = ϕ Θ e d θ ( e ϕ ) δ ϕ , θ + 90 ,
δ ϕ , θ = { 1 ϕ = θ 0 ϕ θ .
Model 1 O ( N θ n 2 f 2 2 ) ,
Model 2 O ( N θ n 2 f 2 2 ) ,
Model 3 O ( N θ n 2 f 1 2 ) ,

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