Optimized design of N optical filters for color and polarization imaging

Xingzhou Tu; Stanley Pau

doi:10.1364/OE.24.003011

1. Introduction

Without optical filters, conventional focal plane arrays (FPA), such as charge-coupled device (CCD) and complementary metal-oxide semiconductor (CMOS) device, have broad color response and are both color and polarization blind. For example, silicon sensor is sensitive from ultraviolet (~300nm) to near infrared (~1100nm) wavelength ranges. To create a color image, a color filter array (CFA) such as the Bayer filter mosaic, with red-green-blue (RGB) color filters, is often used in digital image sensors to measure colors [1]. To create a polarization image, a micro-polarizer mosaic, with linear, circular, or elliptical polarizer, is integrated with the digital image sensors to measure the Stokes vector [2–5]. Most existing designs are optimized to measure three colors and four polarization states. In this paper, optimized designs for N colors and polarization states are presented. The designs have applications in multispectral imaging where many wavelength bands are sampled simultaneously [6] and in precision polarization imaging where multiple polarization states must be measured repeatedly to increase signal to noise ratio (SNR) [7]. The detection of color and polarization information in a single-sensor solution can be achieved by utilizing both CFA and micro-polarizer array on a FPA. Such sensor design can lead to a compact, high speed and vibration-insensitive instrument that can simultaneously measure the complete Stokes vector at multiple wavelength bands. Applications of such an instrument include biomedical imaging, material characterization and classification and remote sensing. In Section 2, the optimized tiling configuration for N filters are determined for N = 2 to N = 9 by minimizing the error of the bicubic spline interpolation. A generalized procedure for arbitrary N filters is also presented. In Section 3, the optimized color-filter designs for N = 2 to N = 7 are presented. In Section 4, the optimized micro-polarizer designs for N = 4 to N = 30 are analyzed, and the solutions are compared with the solutions of the Thomson problem. In Section 5, the results are applied to the design of a three colors full Stokes camera. This design combines one layer of color filter array and one layer of polarizer array, both with an optimized tiling, to achieve a single-sensor solution. The conclusion is presented in Section 6.

2. Tiling of N filters on a square grid

In this section, we examine the problem of tiling N different type of filters. The filters can be color filter, polarization filter or both. A square grid is considered as the pixel in many existing FPA is designed to have a square arrangement. The N filters are tiled periodically to cover the entire FPA. A fraction of 1/N of the total number of pixel, representing each type of filter, is sampled on the square grid by putting one set of filters on the selected pixel (Fig. 1). The pixel pattern of each type of filter forms a two dimensional (2D) lattice. The sampling pattern is repeated for adjacent pixels and another set of filters is put in, until the entire grid is filled up with N different types of filters. For the purpose of our discussion, the edge effect is neglected. For each set of filters, data is only measured on the sampled pixels, and interpolation technique is required to reconstruct data on other pixels. The optimal tiling configuration is determined by minimizing the error of the interpolation. In this paper, the standard technique of bicubic spline interpolation is used. For each type of filter, the sampled pixels can be connected by parallel lines, and cubic-spline interpolation is performed along these lines. A second cubic-spline interpolation is done in the orthogonal direction to recover the entire plane, based on the result of the first interpolation. Figure 2 shows two ways to do interpolation with the same sampling pattern, based on different connection lines. Hereby the interval of first cubic-spline interpolation is denoted by a, the second is denoted by b, and N = ab.

Fig. 1 Schematics of two different ways for interpolation with the same sampling method. The connection lines are represented by dashed lines.

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Fig. 2 The optimized tiling configuration for N = 2 to N = 9. The direction of first interpolation is represented by a black arrow, while the second is represented by the white one. The unit cell is encircled by bold line. Note that the unit cell is not unique. There can be more than one unit cell for each N.

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The error bound for bicubic-spline interpolation has been analyzed in a previous paper [8]. The maximum deviation between original function f and interpolated function s_f is related to the maximum value of original function’s derivatives and the interpolation intervals on x-axis and y-axis, which are hereby denoted by h_x and h_y. This relation is given by

\begin{matrix} {‖ {(f - s_{f})}^{(k, 1)} ‖}_{\infty} \leq ε_{4 - 1, k} {‖ f^{(k - 1, 1)} ‖}_{\infty} h_{x}^{(4 - k)} \\ + ε_{2 k} ε_{21} {‖ f^{(2, 2)} ‖}_{\infty} h_{x}^{2 - k} h_{y}^{2 - 1} \\ + ε_{4 - 1, k} {‖ f^{(k, 4 - k)} ‖}_{\infty} h_{y}^{4 - 1} . \end{matrix}

Here k and l are the order of derivative. Equation (1) can be re-written in the following form:

{‖ I_{true} {-I}_{interpolated} ‖}_{\infty} \leq ε_{4,0} ({‖ I_{true}^{(4,0)} ‖}_{\infty} a^{4} + {‖ I_{true}^{(0,4)} ‖}_{\infty} b^{4}) + ε_{2,0}^{2} {‖ I_{true}^{(2,2)} ‖}_{\infty} N^{2}

The optimal tiling is found by minimizing the error of interpolation with a given N. For a general image, it is reasonable to assume that the local distribution of the image is approximately isotropic, which means the maximum derivative is identical in different directions, so the first term of the right side of Eq. (2) is proportional to a⁴ + b⁴. Given that N is a constant, the second term of Eq. (2) is a constant. Therefore, the optimal tiling configuration should have a minimum value of a⁴ + b⁴.

For arbitrary N, the optimized solution can be obtained by calculating the a⁴ + b⁴ of all possible sampling patterns and then finding the minimum one. Figure 2 gives the optimized tiling configuration for N = 2 to N = 9. These configurations along with other non-optimized configurations are applied to the interpolation of several images to verify that the optimal tiling has a minimum value of a⁴ + b⁴. Note that since N = ab, this is also equivalent to a configuration with minimum value of a + b. A commonly utilized measure of evaluating interpolation algorithms is the mean square error (MSE)

MSE = \frac{1}{MN} \sum_{1 \leq i \leq M} \sum_{1 \leq j \leq N} {{(I}_{true} (i,j) - I_{interpolated} (i,j))}^{2},

where I_true(i,j) represents the value of the reconstruction point in original image, and I_interpolated(i,j) represents the interpolated value of the reconstruction point after performing the interpolation.

Figure 3 shows the MSE curves of bicubic-spline interpolation for three different images with different tiling patterns from N = 2 to N = 9. As expected, MSE increases as the separation between adjacent sampling pixels is increased and is found to vary roughly linearly with N. The MSE of the optimized tiling configuration, represented in Fig. 2, is found to be the smallest in general, as shown by the red curve in Fig. 3. Although calculation is performed up to N = 9 in this paper, our conclusion is expected to hold true for larger value of N.

Fig. 3 Images used for interpolation: (A) vegetables. (B) fruits and (C) chess pieces. The MSE of different tiling methods for the images: (D) vegetables, (E) fruits, and (F) chess pieces.

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3. Optimized sampling of color spectrum using N measurements

Traditional camera utilizes Bayer filter to obtain a color image. However, this kind of camera can only measure three wavelength bands, which represent red, green and blue lights. To capture a multispectral image, additional filters for different bands are required. In this section, the configuration of N color filters for optimized sampling of the color spectrum is presented. The optimized set of filters can be found by minimizing different noise components or by maximizing transmission. In general, the measurement noise has two components: a signal dependent part such as the shot noise of light and a signal independent part such as dark current noise and read-out noise [9]. The exact magnitude of the two noise components depends on the type of sensors and operating conditions. For the purpose of illustrating the different designs, the cases where only one component of the noise is dominant are considered. Our analysis can be easily extended to the case where the exact magnitudes of the noise components are known.

For the analysis of measurement of N spectral bands, the absorption spectrum of N filters can be expressed by an N-by-N transmission matrix [10]. Each filter corresponds to one row in the matrix. A filter can absorb or transmit one specific band, which is respectively represented by 0 or 1 in a row component. The overlap of individual transmission band is assumed to be small. For more elaborate analysis, this number can be represented by a number between 0 and 1. Figure 4 shows an example of three-band filters and the corresponding transmission matrix.

Fig. 4 The transmission spectrum of N = 3 color filters and the corresponding transmission matrix.

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By denoting the intensity passing through the n^th filter by i_n, the intensity of the incoming n^th band by b_n, and the transmission matrix by T, the following equation relates the incoming and transmitted light at different bands.

\begin{matrix} {[\begin{matrix} i_{1} & .. & .. & i_{N} \end{matrix}]}^{T} = T {[\begin{matrix} b_{1} & .. & .. & b_{N} \end{matrix}]}^{T} \\ I = T B \end{matrix}

B can be derived from I by multiplication with the inverse of transmission matrix,

\begin{matrix} {[\begin{matrix} b_{1} & .. & .. & b_{N} \end{matrix}]}^{T} = T^{- 1} {[\begin{matrix} i_{1} & .. & .. & i_{N} \end{matrix}]}^{T} \\ b_{n} = \sum_{m = 1}^{N} {{(T}^{- 1})}_{nm} i_{m} \end{matrix}

When a signal independent noise, such as detector noise, is dominant, it is reasonable to assume that all measurements have the same noise variance, given by σ. It follows from Eq. (5) that the variance of b_n is

v_{n} = σ \times \sqrt{\sum_{m = 1}^{N} {{(T}^{- 1})}_{nm}^{2}}

Including all the components of B, the summation of all variances is

v = σ \times \sqrt{\sum_{m = 1}^{N} \sum_{n = 1}^{N} {{(T}^{- 1})}_{mn}^{2}}

A detector noise factor (NF_detector) can be defined to be

N F_{detector} = v / Nσ = \sqrt{\sum_{m = 1}^{N} \sum_{n = 1}^{N} {{(T}^{- 1})}_{mn}^{2}} / N

Here the factor is related to the sum of the squares of all elements in the transmission matrix’s inverse and includes only the signal independent noise. An optimized transmission matrix can be found by minimizing the noise factor in Eq. (8). For a given N, there are $2^{N^{2}} {/(N!)}^{2}$ combinations of transmission matrices. MATLAB is used to find all unique transmission matrices for a given N, representing all possible absorption spectrums for N bands. The matrix with the minimum noise factor is found for the cases of N = 2 to N = 7, and the results are listed in Table 1. The optimized matrix for N = 2 is a submatrix of the optimized matrix for N = 3. This trend continues until N = 6 and ends at N = 7.

Table 1. Transmission Matrices for N = 2 to N = 7 Optimized for Detector Noise Factor

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Alternatively, the optimized transmission matrix can be found for the case where the noise is dominated by photon shot noise. In this case, the noise variances of all measurements are no longer constant. Assuming that the incident light is equal for all bands, the result of the m^th measurement will be proportional to the summation of all elements in the m^th row of transmission matrix. Since the shot noise is proportional to the square root of intensity, Eq. (6) can be rewritten as

v_{n} \propto \sqrt{\sum_{m = 1}^{N} [{{(T}^{- 1})}_{nm}^{2} \sum_{k = 1}^{N} T_{mk}]}

Similarly, by summing v_n² for n = 1 to N and then taking the average and square root, a noise factor for photon shot noise (NF_sn) can be defined as

{NF}_{sn} = \sqrt{\sum_{m = 1}^{N} \sum_{n = 1}^{N} \sum_{k = 1}^{N} {{(T}^{- 1})}_{nm}^{2} T_{mk}} / N

Again MATLAB is used to calculate all combinations of transmission matrices for N = 2 to N = 7. The matrix with the minimum photon shot noise is found to be the identity matrix. In this configuration, each filter transmits only light for one specific band. The identity matrix represents a configuration with minimum amount of light transmission. Both the detector noise factor and shot noise factor for the identity matrix are $1 / \sqrt{N}$ .

The optimized transmission matrix can also be defined to be the configuration where the total amount of transmitted light is the highest. In this case, the figure of merit is given by the summation of all the elements in the transmission matrix. Thus, every row of the matrix should have as many ‘1’s as possible. To avoid a singular transmission matrix, only one row can have all its components to be 1, and other rows will have N-1 ‘1’s. The optimized transmission matrices in this case therefore have a general form, in which all off-diagonal elements and the first element are filled by 1 and others are filled by 0,

T_{o p t i m i z e d} = | \begin{matrix} 1 & 1 & 1 & 1 & . & . & 1 \\ 1 & 0 & 1 & 1 & . & . & 1 \\ 1 & 1 & 0 & 1 & . & . & 1 \\ 1 & 1 & 1 & 0 & . & . & 1 \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ 1 & 1 & 1 & 1 & . & . & 0 \end{matrix} |

The detector noise factor and shot noise factor in this case are equal to $\sqrt{1 - 1 / N + 1 / N^{2}}$ and $\sqrt{N - 1 - 1 / N + 2 / N^{2}}$ respectively. Figure 5 shows both the detector noise and shot noise factor curves for the configurations optimized for minimum noise factor, both Eq. (8) and Eq. (10), and optimized for maximum transmission. The curve for maximum transmission is always above the curve for optimized noise factor.

Fig. 5 Detector and shot noise factors are plotted as a function of N, the number of spectral band, for the case of minimum detector noise, minimum shot noise, and maximum transmission.

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4. Optimized sampling of Stokes vector using N measurements

In addition to color, another important property of light is the polarization state, which can be described by the four component Stokes vector. The Stokes vector can be estimated by a minimum of four intensity measurements of light passing through different polarizers. Higher accuracy can be achieved by additional measurements. Figure 6 shows two common types of imaging polarimeter [7]. In the division of time polarimeter, images are taken sequentially as a retarder is being rotated in front of a fixed linear polarizer. The Stokes vector at each pixel is calculated by measurement of a minimum of four images taken at different retarder angle orientations at different time. In the division of focal plane polarimeter, a pixelated retarder and a linear polarizer are placed in front of the sensor array. A minimum of four types of micro-retarder is used, oriented at four different angles, to measure the Stokes vector locally [4]. The Stokes vector at each pixel is calculated by using interpolated values of intensity measurement taken at adjacent pixels [11–14]. The fast-axis orientation angles of the retarder in the optimized design for N measurements are the same for both the division of time and division of focal plane polarimeters. In this section, the orientations of single and two-layer retarder for N measurements are optimized for minimum measurement error. The results are compared to each other and to the solutions of the Thomson problem.

Fig. 6 Two common designs of imaging polarimeter: (A) division of time. (B) division of focal plane.

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The relationship between the Stokes vector and the measurement result can be expressed as $I = W S$ , where S represents the Stokes vector of incoming light, and I is an N-element vector representing the intensity that passes through N polarizers. The measurement matrix, denoted by W, is an N-by-4 matrix, with each row representing one polarizer. To estimate the Stokes vector from the measurement, a pseudo-inverse of W is adopted to achieve minimum least squares error:

S = W^{+} I

Three figures of merit have been used in the literature [15,16], which are RAD (reciprocal absolute determinant), CN (condition number), and EWV (equally weighed variance). For all figures of merit, smaller value denotes a better design. The expressions of each are expressed in Eq. (13). Here, µ represents the singular value of the measurement matrix.

\begin{array}{l} R A D = Π_{j = 0}^{R - 1} 1 / μ_{j} \\ C N = μ_{m a x} / μ_{m i n} \\ E W V = \sum_{j = 0}^{R - 1} 1 / μ_{j}^{2} \end{array}

For the purpose of our analysis, all polarizers have a common retardance but different fast-axis angles as shown in Fig. 6 (one-layer configuration). The configuration of the one-layer retarder is considered first. The k^th row of the measurement matrix can be expressed by a function of the common retardance δ and the fast-axis angle of k^th filter θ_k. This expression has been found to be proportional to [17]

{[1, \cos^{2} (2 θ_{k}) + \cos (δ) \sin^{2} (2 θ_{k}), \sin^{2} (δ / 2) \sin (4 θ_{k}), - \sin (δ) \sin (2 θ_{k})]}^{}

Based on Eq. (14), MATLAB is used to find the optimized N-filter design for N = 4 to N = 30 by minimizing the three figures of merit respectively. The optimization variables are the common retardance and the fast-axis angles of the N filters. In other words, N filters are chosen from one constant retardance curve on Poincaré sphere as shown in Fig. 7. The MATLAB function fminuc is used for the optimization calculations. All parameters are floating points. The optimization is repeated 60 times with random starting points, and the solution which gives the minimal CN/EWV/RAD is adopted to be the global minimum. In our calculation, almost all 60 optimizations give the same minimal CN/EWV/RAD. We believe that the calculated optimization result is the global minimum, since our calculation also provides the correct Thomson solutions.

Fig. 7 Constant retardance curves on the Poincaré sphere.

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The optimization result shows that all three figures of merit produce the same result, except for the case of N = 5. Table 2 summarizes the optimized retardance and fast-axis angles for N = 4 to N = 7, including the results for the three figures of merit respectively when N = 5. When $N \geq 8$ , the optimized designs are not unique anymore. Each optimization calculation gives a different result, and the results have the same retardance of 131.81 degree and a different fast-axis angle, although all of the results have the same value of figures of merit. Figure 8 shows the optimized designs for N = 4 to N = 30. Each point on the Poincaré sphere represents a measurement using a retarder of different angle. The optimized value for N = 4 has been calculated by Sabatke et al. [17], which represents a regular tetrahedron on the Poincaré sphere. For $N \geq 8$ , only one of the optimized designs is presented.

Table 2. Retardance and Fast Axis Angles of the Optimized N Measurement Designs

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Fig. 8 Optimized micro-polarizer designs (one-layer configuration) for N = 4 to N = 30. The polarizer is represented by blue spots on the Poincaré Sphere. The red curve represents constant retardance.

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The results for the one-layer configuration can be compared with another configuration where the polarization states are measured by two layers of retarders. This two-layer configuration is shown in Fig. 9. The measurement matrix is obtained by combining the Mueller matrices of two retarders and one linear polarizer for N different filters and then by taking the first row of the Mueller matrices of the N different filters. Since a second layer of retarder is added, more degree of freedom is given to the system, which may lead to smaller noise.

Fig. 9 The schematic of two-layer configuration.

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Optimization of the two-layer configuration (Fig. 9) is considered next. The optimization process is the same as one-layer configuration, except that the optimization variables now include two different retardances for the two layers respectively and 2N fast-axis angles. The optimization results for N = 4 to N = 30 are listed in Table 3 and compared with the results of the one-layer configuration. Comparison shows that the optimized design of one-layer configuration has the same CN/EWV/RAD as the two-layer configuration. For practical implementation, the simpler one-layer configuration is sufficient to realize an optimized polarimeter.

Table 3. CN/EWV/RAD of the Optimized Designs of One-layer and Two-layer Configuration

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The optimized design can also be compared to the solution of the Thomson problem, which is a model of N electrons on a sphere. The Thomson problem is to find the position for each electron so that the electric potential energy is minimal [18]. The positions of all electrons form a symmetric polyhedron of large volume, and one conjectures that this configuration may also provide the optimized polarimeter design of the lowest noise, since it is true for the case of N = 4. Figure 10 shows the solution of the Thomson problem for N = 4 to N = 30 calculated using MATLAB. The three figures of merit, CN, EWV and RAD, are calculated for both the optimized design (Fig. 8) and the Thomson solution (Fig. 10). A comparison between them is made in Fig. 11. The optimized design is same or better than the Thomson solution for all three figures of merit. To realize the Thomson solutions for N measurement of polarization state, more than one layer of retarder will generally be needed [15]; thus, the optimized solution not only has better figures of merit but also a simpler retarder configuration, i.e. only a single retarder with constant phase at different orientations.

Fig. 10 Thomson solution for N = 4 to N = 30.

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Fig. 11 Comparison of two solutions for three figures of merit. (A) CN, (B) EWV, (C) RAD, (D) CN difference between optimized design and Thomson solution, (E) EWV difference between optimized design and Thomson solution, (F) RAD difference between optimized design and Thomson solution.

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5. Design of an optimized three color full Stokes camera

The solutions for optimized sampling of color and full Stokes vector are presented in the previous sections. Here, an optimized design that combines the measurement of both color and Stokes vector is presented. A three colors full Stokes camera design is determined by minimizing the noise factor and the error of interpolation. While the three colors can represent any band in the electromagnetic spectrum, the design is ideal for imaging application in the visible spectrum. The design can measure the full Stokes vector of RGB light in a single frame.

For each of the three colors, four polarizers are needed to measure the full Stokes vector. Hence, there are a total of 12 filters to tile in a two dimensional filter array. An optimized tiling for N = 12 can be found based on the technique presented in Section 2. Each pixel includes a color filter and a polarizer. For the color filter, traditional cameras utilize RGB filters to measure a color image. The optimized configuration for minimum photon shot noise is the RGB configuration, whereas the optimized configuration for minimum detector noise is green & blue, red & blue, and red & green, which is the cyan-magenta-yellow (CMY) configuration (Table 1). For the polarizer array, the optimized configuration of the four polarizers has a common retardance at 131.81 degree and different fast-axis angles at ± 74.88, ± 38.31 degree, representing four elliptical polarizers (Table 2). The elliptical polarizers are achromatic and are designed to have a flat diattenuation across the visible spectrum [19]. Figure 12 shows the basic design of the camera. The tiling pattern for 12 filters is found by minimizing the interpolation error. The structure of the camera is shown with a layer of RGB color filters and a layer of achromatic elliptical polarizers. The incoming light first goes through the color filter array and then the polarizer array to reach the pixel of the FPA. Bicubic spline interpolation can be used to calculate the full Stokes vector for each color at each pixel.

Fig. 12 Design of an optimized three color full Stokes camera. Left side is the optimized tiling pattern of the filter array for N = 12.The direction of first interpolation is represented by a black arrow, while the second is represented by the white one. The unit cell is encircled by bold line. Right side is the structure of the camera made of RGB color filters and achromatic elliptical polarizers.

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6. Discussion and conclusion

In this paper, the tiling and design of optical filters in a division of focal plane configuration are presented for the measurement of color and polarization. Advantages of the division of focal plane design include low form factor, single frame acquisition, and insensitivity to vibration. Disadvantages include loss of spatial resolution, complex filter design and fabrication, and additional overhead and error associated with image interpolation. The optimal tiling of optical filters is found by minimizing the interpolation error and the optimal filter transmission is found by minimizing the noise factors. For N = 3 operating in the visible spectrum, the filter transmission for minimum photon shot noise is found to be the RGB configuration and agrees with results from existing literature [9]. It is interesting to find that the filter transmission for minimum detector noise in this case is the CMY configuration. Both the RGB and CMY implementations are realized in existing digital cameras and a combination of RGB and CMY implementation may have to be utilized for situation where both detector and photon shot noise are important. Measurement of the full Stokes vector requires at least four different polarizers. The optimal designs of the polarizers are found by minimizing the CN, EWV and RAD of the measurement matrix. The results can be implemented by using a single layer of retarder, and the performance is found to be better than that of the Thomson solutions. By combining the results for color and polarization filters, a design of a three colors full Stokes imager is presented.

Acknowledgments

This work is funded by the Arizona Technology Research Infrastructure Fund (TRIF), and National Science Foundation (NSF) Award 1455630. The authors thank Prof. Russell Chipman and Prof. Amit Ashok for helpful discussion of the manuscript.

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N	2	3	4
Optimized Transmission Matrix	$\| \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \|$	$\| \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \|$	$\| \begin{matrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{matrix} \|$
N	5	6	7
Optimized Transmission Matrix	$\| \begin{matrix} 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 \end{matrix} \|$	$\| \begin{matrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 \end{matrix} \|$	$\| \begin{matrix} 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 \end{matrix} \|$

N	Retardance(degree)	Fast axis angles(degree)
4	131.81	± 74.88, ± 38.31
5(CN)	136.16	0, ± 33, ± 57
5(EWV)	132.60	0, ± 58.02, ± 31.98
5(RAD)	131.81	0, ± 64.62, ± 35.78
6	131.81	± 10.08, ± 36.78, ± 59.62
7	131.81	0, ± 18.17, ± 42.39, ± 59.77
8 or more	131.81	vary*

N	CN		EWV		RAD
N	One-layer	Two-layer	One-layer	Two-layer	One-layer	Two-layer
4	1.732	1.732	10.000	10.000	5.196	5.196
5	1.794	1.794	8.119	8.119	3.373	3.373
6	1.732	1.732	6.667	6.667	2.309	2.309
7	1.732	1.732	5.714	5.714	1.697	1.697
8	1.732	1.732	5.000	5.000	1.299	1.299
9	1.732	1.732	4.444	4.444	1.026	1.026
10	1.732	1.732	4.000	4.000	0.831	0.831
11	1.732	1.732	3.636	3.636	0.687	0.687
12	1.732	1.732	3.333	3.333	0.577	0.577
13	1.732	1.732	3.077	3.077	0.492	0.492
14	1.732	1.732	2.857	2.857	0.424	0.424
15	1.732	1.732	2.667	2.667	0.370	0.370
16	1.732	1.732	2.500	2.500	0.325	0.325
17	1.732	1.732	2.353	2.353	0.288	0.288
18	1.732	1.732	2.222	2.222	0.257	0.257
19	1.732	1.732	2.105	2.105	0.230	0.230
20	1.732	1.732	2.000	2.000	0.208	0.208
21	1.732	1.732	1.905	1.905	0.189	0.189
22	1.732	1.732	1.818	1.818	0.172	0.172
23	1.732	1.732	1.739	1.739	0.157	0.157
24	1.732	1.732	1.667	1.667	0.144	0.144
25	1.732	1.732	1.600	1.600	0.133	0.133
26	1.732	1.732	1.538	1.538	0.123	0.123
27	1.732	1.732	1.481	1.481	0.114	0.114
28	1.732	1.732	1.429	1.429	0.106	0.106
29	1.732	1.732	1.379	1.379	0.099	0.099
30	1.732	1.732	1.333	1.333	0.092	0.092

N	2	3	4
Optimized Transmission Matrix	$\| \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \|$	$\| \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \|$	$\| \begin{matrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{matrix} \|$
N	5	6	7
Optimized Transmission Matrix	$\| \begin{matrix} 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 \end{matrix} \|$	$\| \begin{matrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 \end{matrix} \|$	$\| \begin{matrix} 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 \end{matrix} \|$

N	Retardance(degree)	Fast axis angles(degree)
4	131.81	± 74.88, ± 38.31
5(CN)	136.16	0, ± 33, ± 57
5(EWV)	132.60	0, ± 58.02, ± 31.98
5(RAD)	131.81	0, ± 64.62, ± 35.78
6	131.81	± 10.08, ± 36.78, ± 59.62
7	131.81	0, ± 18.17, ± 42.39, ± 59.77
8 or more	131.81	vary*

Optimized design of N optical filters for color and polarization imaging

Abstract

1. Introduction

2. Tiling of N filters on a square grid

3. Optimized sampling of color spectrum using N measurements

4. Optimized sampling of Stokes vector using N measurements

5. Design of an optimized three color full Stokes camera

6. Discussion and conclusion

Acknowledgments

References and links

Cited By

Figures (12)

Tables (3)

Equations (14)

Optics Express