1. Introduction

Burgeoning developments in the polarimetric sensing community have resulted in the utilization of linear systems theory to describe polarimetric snapshot systems in a general way [1,2]. The demosaicing community has utilized the linear systems framework for some time to describe color filter arrays (CFAs) [3–10] and recently to design some spatially multiplexed filter arrays [10]. Much of the work in the demosaicing community has focused on tri-color reconstruction, due to the requirements of the human visual system [3–9], while the polarimetric community has developed some designs which increase spatial bandwidth [11–13], with a systematic approach being taken by our group [14]. Hirakawa and Wolfe [8] described a specific lattice formalism for color arrays which is general assuming no filter array subsampling; and Hao, Li, Zin, and Dubois [9] used this formalism to develop an optimization problem in order to select optimal tri-color CFAs. Yadid-Pecht did some analysis on the modulation transfer function of arbitrary pixel shapes [15]. Recent work by Jia, Barnard, and Hirakawa and by Jia, Ni, Sarangan, and Hirakawa showed spatially multiplexed channels which help to mitigate crosstalk issues for multi-spectral imaging [10,16]. We recently derived a specific physical channel structure for a conventional micropolarizer array (MPA) [17].

The tri-color constraint is typically not useful for remote sensing instruments which sense ocean color and ocean radiometry; often 5 or more spectral bands [18,19] are required. Moreover, conventional polarimetric sensing requires at least 3 filter types, and a full 3-dimensional 2nd order coherence measurement requires at least 5 filter types [20]. In this communication we generalize the results of Hirakawa and Wolfe, Jia et al, and Yadid-Pecht to any finite unit cell pattern of optical filters (on a rectangular lattice), while extending our work [14,17] addressing the physical measurement properties in order to facilitate sub-sampling and filter element shape engineering to be implemented for focal plane filter array (FPFA) design. This work is applicable to any array of filters which can measure a physical property of light as an irradiance.

The article is organized as follows: In Section 2 the physical form of a single filter element is analyzed in the Fourier (channel) domain for an element of arbitrary shape. In Section 3 this physical form is then combined into a finite array which reveals the lattice structure and Section 3.1 demonstrates that the complete physical channel structure available for engineering is a function of easily manipulated parameters.

2. Physical description

A typical tiled filter array will usually adhere to the rectangular pixel grid focal plane array (FPA) geometry. Other grid geometries, e.g. hexagonal lattices, have been shown to be more efficient in terms of bandwidth [21], however there are currently very few commercially available FPAs which offer a hexagonal lattice. In this section we derive an ideal physical model of the channel structure given the assumptions: 1) spatial integrations are disjoint, 2) filter parameters are constant over the filter element, 3) temporal integration time is ignored, and 4) the FPFA is on a regular rectangular lattice. A channel in this manuscript refers to a Dirac δ-function or an approximation to a Dirac δ-function which arises from a periodic spatial carrier structure due to the unit cell tiling of an FPA. The primary physical parameters which affect the channel structure are the filter shape, the spatial functions of the filters, and the array size.

2.1. Single filter

We make the assumption that each filter array element has the same shape and spatial attenuation function, and that the each filter element shape is compact and restricted to the filter element pitch size. This function can be represented by h(x, y) as shown in Fig. 1. A single element on the rectangular grid can be represented by h(x − x_j, y − y_ℓ) multiplied by the sum of periodic spatial filter functions [17]. Periodic spatial filter functions denote the set of filter projections over each filter type, at a specific location (x_j, y_ℓ). Coherence properties (e.g., polarimetric measurements) may have projections of a specific property for a specific filter type. Spectral filters can be designed so that there is only a single scalar transmission factor for each filter type. Mathematically, we define the set of possible projective measurement vectors as {a(x, y), b(x, y), · · · , v(x, y), · · · }. Given a physical irradiance property vector 𝔡(x, y) = [d₀(x, y) d₁(x, y) · · · d_k (x, y)]^T, the irradiance available to the sensor element after the filter can be defined as the inner product between the sum of the measurement vectors and the irradiance property vector (extending Hirakawa and Wolfe [8]) multiplied by the filter element transmission function:

 

Fig. 1 Arbitrary filter shape represented by the function h(x, y), which is assumed to be compact, but not necessarily an indicator (binary) function. Notice that it is centered on (0, 0). Adapted from [17].

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As a general example of the scheme, a 2-band spectral measurement could be taken together with a Stokes measurement by adding 2 additional vectors and extending the length of all of the vectors. We assume disjoint spectral measurements. We can re-define the Stokes filter measurement vector as a₀(x, y) = [a′_0,0(x, y) a′_0,1(x, y) a′_0,2(x, y) 0 0]^T The first color band can be defined by a₁(x, y) = [0 0 0 a_1,3(x, y) 0]^T and the second as a₂(x, y) = [0 0 0 0 a_2,4(x, y)]^T, totaling 5 filters types and yielding a hybrid polarization/color sensor.

Generally, a single element at (x_j, y_ℓ) has an irradiance transmission of

 

Fig. 2 An example of a 5-band transmission filter element for wavelengths 443nm, 520nm, 550nm, 600nm, 660nm. This 5-band transmission is for a single filter element, corrected for the spectral integration and the quantum efficiency at the detector assuming the camera is an Andor Alta F6. This type of multi-band filter is currently being manufactured by companies like PIXELTEQ and Alluxa.

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Note that the multi-band filter shown in Figure 2 is a panchromatic filter. A panchromatic filter passes multiple spectral bands in some specific ratio to one another. Furthermore, we do not account for quantum efficiency, etc. in our mathematical framework, but detector phenomenology would need to be taken into account when designing the filters since the final ratio measurements are what determine the channels in the Fourier domain. The manufacture of such multi-band color filters has recently commenced (e.g., PIXELTEQ [22]).

3. Array of filter elements

The assumption that h(x, y) is compact within a bounding rectangle defined by the filter element pitch implies that we can add an array of the elements together. We can then take the Fourier transform of this sum, and interchange the summation and Fourier transform to obtain

3.1. Unit cell tiling

We now assume that the filter element tiling is some integer number of unit cells of size T_y × T_x, i.e., a_0,p (ñP_x, m̃P_y) , · · · , a_n,p (ñP_x, m̃P_y), · · · are periodic with period (T_x, T_y) when T_x, T_y ∈ ℤ, for each respective p. For this analysis we manipulate the inner part of the generic sum

 

Fig. 3 An example of sub-sum selection from a 6-band spectral CFA. The green sub-sum is denoted by the black circles. Each element type is periodic over the array. a_n,p (0, 0) through a_n,p (2, 1) represent the 6 different periodic sets over the FPA, while a_0,0(−3, 1), a_0,0(0, 1), a_0,0(3, 1), a_0,0(0, 3), a_0,0(3, 3), · · · represent the green sub-sum.

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4. Discussion

Equation (15) completely describes the physical channel structure available to a sensor, subject to sampling in the spatial domain. A filter designer may utilize the following parameters to adjust for a filter design; 1) the number of unit cells, 2) the filter element shape function, 3) the unit cell geometry, and 4) the unit cell element weights. The unit cell geometry specifies the set of available channels and the possible channel configurations in the spatial frequency domain. We define the unit cell geometry as size M × N, where M is the number of rows and N is the number of columns. Note that N corresponds to T_x and M corresponds to T_y.

A designer can choose a set of desired channels, and then invert a matrix equation to obtain the filter weights needed to realize the specified set of channels, but a solution does not necessarily exist. The available channels can be enumerated and the values can be stored in a vector m_k for each filter element of the unit cell. The channel centers are located at positions $\left(\frac{n}{M},\frac{m}{M}\right)$ where n, m ∈ ℤ are constrained such that $\frac{n}{N},\frac{m}{M}\in \left[-\frac{1}{2},\frac{1}{2}\right)$, i.e. at steps of $\frac{1}{N}$, $\frac{1}{M}$ and inside the Nyquist boundary when the sampling rate is maximal and we assume without loss of generality that P_x = P_y = 1 in cycles per pixel. We read off the entries of m_k starting at the lower-leftmost channel location. Figure 4 demonstrates the possible channels for a 2 × 3 unit cell at the 9 locations in an intuitive way, while Fig. 5 shows the physical channel structure locations to a sensor for a variety of unit cell geometries corresponding to the (0, 0) unit cell location.

 

Fig. 4 The set of possible channel locations and values available for a 2 × 3 unit cell. The ξ, η-axes (horizontal and vertical respectively) have ranges in $\left[-\frac{1}{2},-\frac{1}{2}\right)$ in cycles per pixel showing the Nyquist bandwidth square. The circles are located at the channel centers, the magnitude of each channel is 1, and the clocking angle and color indicate the angle of the complex value of each channel.

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Fig. 5 Examples of approximate δ-functions (channels) for different unit cell sizes for the (0, 0) element assuming maximal sampling. Note the 3 × 3 unit cell has approximate δ-functions which lie completely within the Nyquist square. This representation conveys the physical structure available in the Fourier domain, whereas the notation used in Fig. 4 is conceptual.

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A matrix equation can be formed by concatenating all of the m_k vectors column wise,

Once the channel values are chosen, we obtain a set of filter values for each band. When an object is imaged, the Fourier transform of that data is convolved with each channel, for each specific band, then all of the convolved data is added together linearly, or “mixed.” In order to reconstruct the data for each band the data must be “unmixed” first. This can be accomplished directly in the Fourier domain via a matrix, which we denote the Q matrix [2,14,24,25].

This framework can be applied to any filter scheme where an irradiance measurement is being made. If a filter array is sub-sampled, i.e., each filter element has multiple measurement pixels per filter element, then the filter transmission function can also be used to shape the channel structure, however the details of filter shape engineering are beyond the scope of this article.

5. Examples

In this section we use our design methodology to briefly analyze tri-color FPFA systems. We introduce 2 new designs and compare them to designs by Hirakawa and Wolfe [8]; Condat [26]; and Lukac, Rastislav and Plataniotis [27]. The unit cells are shown in Fig. 6 and the associated channel descriptions are given in Fig. 7. Using our framework and the associated software, FiltArrayDesigner.m, (Code 1 [28]), these configurations are easily computed. The Hirakawa, Condat, and Lukac designs have been compared in the past using the standard lighthouse image, however reasonable reconstruction resulted for all designs by using low-pass filters in the Fourier domain for the color sidebands, indicating that the region of interest often shown in the literature has low spatial frequency color content. We also assess the performance of the different patterns by using a synthetic resolution target, which includes spatial frequency content which violates the bandlimit criteria for all filter array designs. The standard lighthouse images and a synthetic resolution target were processed with no noise, using the specified color filter arrays, and reconstructed by linear filtering in the Fourier domain. The synthetic image is 1680 × 1680 pixels because this size is divisible by 2, · · · , 8, facilitating the testing of different unit cell sizes. The mean squared error (MSE) was them computed for each color component to quantitatively compare the different CFA designs. See Fig. 8 for the synthetic image used and the region of interest (ROI) shown in the comparison.

 

Fig. 6 Human perception of the unit cells for the various CFA designs. Generated via our software (Ref. [28])

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Fig. 7 Red (top row), green (center row), and blue (bottom row) channel descriptions for the Hirakawa, Condat, Lukac, and Vaughn 4 × 2 and 8 × 2 color filter array designs.

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Fig. 8 Synthetic resolution target truth image. This image varies between 0 and 1 for each color and is processed as if being incident on an ideal sensor. The white square indicates the region of interest.

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The linear filters in the Fourier domain used for the lighthouse image reconstruction are circular, with a radius of 0.05 cycles per pixel for the sideband color channels and 0.5 cycles per pixel for the luminance channel, except for the Lukac design, where the luminance channel filter had a radius of 0.45 cycles per pixel due to channel locations at the top and bottom of the Nyquist square. The synthetic images required Fourier domain filters which pass higher frequency content in order to achieve reasonable reconstruction. The Fourier filters for the synthetic images were chosen in an ad-hoc manner, but we attempted to minimize the MSE of each design without resorting to extensive optimization techniques, see Fig. 9. No matter which linear reconstruction methods are used, channel crosstalk [29] causes errors, particularly for the 8 × 2 and Hirakawa designs due to the frequency content wrapping around the Nyquist boundary. Optimal Fourier domain filter strategies and filter design are not the primary objective of this communication, and we plan to address this specific topic in greater detail in a subsequent publication.

 

Fig. 9 Fourier domain filters used for the synthetic image reconstruction. Data shown is for the synthetic image and is log scaled on the magnitude of the Fourier domain data. Note that the Hirakawa filters overlap, resulting in the high filter display values there, however, Fourier domain filter sets are used individually for the reconstruction.

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The lighthouse reconstructions all had fairly low MSE, see Fig. 10. The means of the red, green, and blue channel for the lighthouse image are 122.5, 115.5, and 98.2 respectively, with corresponding total MSEs shown in Table 1. The synthetic images varied from 0 to 1, with each channel having the same mean of 0.5. Interestingly, our 4 × 2 design and the Lukac perform the best with the high frequency content of the synthetic image, see Fig. 11 and Table 1. The MSE values shown in Table 1 are the total MSEs, i.e., the sums of the MSEs of each color. MSE for each color is calculated as follows:

 

Fig. 10 Reconstructions of the lighthouse image.

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Fig. 11 Reconstructions of the ROI of the synthetic image.

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Table 1. Table of total MSE values for various unit cells. Note that the lighthouse and synthetic MSEs are not comparable due to the original images having different image values for each color, from 0 to 255 and from 0 to 1 respectively.

View Table

6. Conclusions

Focal plane filter engineering has a formal mathematical structure which applies to all periodic filter array layouts on a rectangular lattice. We have shown the underlying mechanism for filter array design, including the physical channel structure available from the irradiance. We have outlined a procedure in this communication for the engineering of any periodic filter array, given an irradiance measuring sensor and filter elements lying on a rectangular lattice. This procedure is general, and may need optimization over some cost or utility function which quantifies aspects of the final sensor requirements. The procedure follows:

Our design methodology encompasses all filter array designs for irradiance sensors with pixels located on a periodic rectangular lattice and allows for generic engineering of filter arrays to specific specifications. We showed our method applied to previous color filter designs presented in the literature, and presented a new 4 × 2 design which performs well on both the standard lighthouse image and on a synthetic image which violates the band limits of all designs.

Appendix A: unit cells are rectangular

In general, there are 17 different ways to tile the plane in an abstract manner [30], dependent on the symmetries of the primitive cells, denoted the wallpaper groups. There exist an infinite number of specific implementations of the wallpaper groups, and here we show that a unit cell on an FPA with a rectangular lattice can be defined via a rectangular unit cell in general.

Suppose we have some arbitrary (non-rectangular) unit cell shape, which periodically tiles the FPA. The unit cell element is fixed to the rectangular lattice, and is periodic in some direction. The fixation means that any rotational symmetries but be integer rotations of 90°. Generally, there will exist a maximum period in the x-direction, T_x ∈ ℤ and a maximum period in the y-direction, T_y ∈ ℤ in the number of filters for the final tiled array. We can then re-define the unit cell to be the set of filter elements from (0, 0) to (T_x − 1, T_y − 1), which is rectangular.

Appendix B: unit cell channel magnitudes

The magnitudes of the channels from each unit cell filter element can be computed from Eqs. (12) and (15) when

(19)$$-0.5\le \xi =\frac{k_N}{N}\le 0.5,\hspace{0.17em}\hspace{0.17em}-0.5\le \eta =\frac{k_M}{M}\le 0.5,\hspace{0.17em}\hspace{0.17em}{P}_x=P_y=1,\hspace{0.17em}\hspace{0.17em}{a}_{n,k}(j,\ell )=1.$$

Then from Eq. (12) we obtain

(20)$$\begin{array}{r}\hfill D_{n,k,\xi ,\eta }(j,\ell )=\text{exp}\left(\pi i\left(N_{j,\ell }^{\prime }-N_{j,\ell }^{″}-\frac{2O_j}{T_x}\right)T_x\xi \right)\cdot \text{exp}\left(\pi i\left(M_{j,\ell }^{\prime }-M_{j,\ell }^{″}-\frac{2O_{\ell }}{T_y}\right)T_y\eta \right)\\ \hfill \cdot \left\{\frac{\text{sin}\left[T_x\left(N_{j,\ell }^{\prime }+N_{j,\ell }^{″}+1\right)\pi \xi \right]}{\text{sin}(T_x\pi \xi )}\right\}\cdot \left\{\frac{\text{sin}\left[T_y\left(M_{j,\ell }^{\prime }+M_{j,\ell }^{″}+1\right)\pi \xi \right]}{\text{sin}(T_y\pi \eta )}\right\}\end{array}$$

since the exponential terms have magnitude 1. For a unit cell of size M × N, T_x = N and T_y = M will have a step size of M. The U_x = N′_j,ℓ + N″_j,ℓ + 1 and U_y = M′_j,ℓ + M″_j,ℓ + 1 terms are integers depending on the number of unit cells used. Combining the above yields

(21)

where the sin terms can be solved using L’Hôpitals rule due to an indeterminate form of 0/0. If the number of unit cells in the x-direction is odd, then N′_j,ℓ − N″_j,ℓ = 0 and if number of unit cells in the x-direction is even then N′_j,ℓ − N″_j,ℓ = −1. Similarly, if the number of unit cells in the y-direction is odd, then M′_j,ℓ − M″_j,ℓ = 0 and if number of unit cells in the y-direction is even then M′_j,ℓ − M″_j,ℓ = −1. This results in 2 cases:

(22)$$D_{n,k,\xi ,\eta }\left(j,\ell \right)=U_x{U}_y{e}^{\left(-2\pi i\frac{O_j{k}_N}{N}\right)}\cdot e^{\left(-2\pi i\frac{O_{\ell }{k}_M}{M}\right)},$$

or

(23)$$D_{n,k,\xi ,\eta }\left(j,\ell \right)=-U_x{U}_y{e}^{\left(-2\pi i\frac{O_j{k}_N}{N}\right)}\cdot e^{\left(-2\pi i\frac{O_{\ell }{k}_M}{M}\right)}.$$

For a specific sub-sum of a specific filter type we have a fixed (j, ℓ), corresponding to a specific unit cell location, and O_j, O_ℓ are also fixed. Eq. (15) for a specific sub-sum with the above constraints and H(ξ, η) ≡ 1 results in a magnitude of U_xU_y from Eqs. (22) and (23) for the channels available.

The channel value matrix for the 2 × 3 unit cell shown in Fig. 4 is

(24)$$\mathbf{M}=C\left[\begin{array}{cccccc}1& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& -1& \frac{1}{2}+i\text{cos}\frac{\pi }{6}& \frac{1}{2}-i\text{cos}\frac{\pi }{6}\\ 1& 1& 1& -1& -1& -1\\ 1& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -1& \frac{1}{2}-i\text{cos}\frac{\pi }{6}& \frac{1}{2}+i\text{cos}\frac{\pi }{6}\\ 1& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& 1& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}\\ 1& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& 1& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}\\ 1& 1& 1& 1& 1& 1\\ 1& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& 1& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}\\ 1& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& -1& \frac{1}{2}+i\text{cos}\frac{\pi }{6}& \frac{1}{2}-i\text{cos}\frac{\pi }{6}\\ 1& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& -1& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& \frac{1}{2}-i\text{cos}\frac{\pi }{6}\\ 1& 1& 1& -1& -1& -1\\ 1& -\frac{1}{2}+i\text{cos}\frac{\pi }{6}& -\frac{1}{2}-i\text{cos}\frac{\pi }{6}& -1& \frac{1}{2}-i\text{cos}\frac{\pi }{6}& \frac{1}{2}+i\text{cos}\frac{\pi }{6}\end{array}\right].$$

Appendix C: available channels

A designer must choose channels by first specifying how many channels are needed prior to choosing the channel locations. The degrees of freedom available for a set of chosen channel locations, assuming maximal sampling, is given by the following algorithm:

• Count the channel locations which are entirely within the interior of the Nyquist square. We’ll label this count by N_int.
• Count the channel locations which lie on one and only one edge (not in a corner). We’ll label this count by N_edg.
• Count if any channel locations lies in a corner and only a corner. This condition is binary, either there are 4 corner locations, or there are 0 corner locations (due to symmetry). Set N_cor = 1 or N_cor = 0 respectively.

The total available degrees of freedom, i.e., the maximum number of filter types that you can have, is N_int + N_edg/2 + N_cor.

Funding

Asian Office of Aerospace Research and Development (FA2386-15-1-4098).

Acknowledgment

The authors thank Matthew A. Kupinski for asking questions about MPAs which led to this work.

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