Fourier domain design of microgrid imaging polarimeters with improved spatial resolution

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1 Fourier domain design of microgrid imaging polarimeters with improved spatial resolution Keigo Hirakawa a and Daniel A. LeMaster b a University of Dayton, 300 College Park, Dayton, OH 45469, USA; b Air Force Research Laboratory, 2241 Avionics Circle, Wright-Patterson AFB, OH, USA ABSTRACT Microgrid polarimetric imagers sacrifice spatial resolution for sensitivity to states of linear polarization. We have recently shown that a 2 4 microgrid analyzer pattern sacrifices less spatial resolution than the conventional 2 2 case without compromising polarization sensitivity. In this paper, we discuss the design strategy that uncovered the spatial resolution benefits of the 2 4 array. Keywords: microgrid, polarimetric imaging, polarimetry 1. INTRODUCTION In a recently published paper, 1 we showed that a Microgrid Polarimetric Array (MPA) design with a 2 4 repeating pattern provides better spatial resolution than the conventional 2 2 MPA repeating pattern without affecting Stokes parameter estimation performance in the presence of noise. This insight was built upon the Fourier analysis of MPAs by Tyo et al 2 and the color filter array design of Hirakawa et al. 3 The purpose of this paper is to provide additional details on the design strategy that was used to develop the 2 4 array. The design arguments in this paper are hinged upon maximization of spatial frequency bandwidth in the Discrete Space Fourier Transform (DSFT) domain. To review, the bandwidth of the j th Stokes parameter image, S j, is the smallest ν j satisfying Ŝ j (ω) = 0 ω > ν j, (1) where Ŝj is the DSFT of S j and ω = (ω 1, ω 2 ) T {R/2π} 2 are spatial frequency corrdinates. The connection between bandwidth and spatial resolution in the image domain can be expressed in terms of the Ground Resolved Distance 4 (GRD): GRD j = R, (2) fν j where R is the range to the target and f is the focal length of the imaging system. Consequently, maximizing bandwidth is tantamount to minimizing GRD. In Equation 2, the units of bandwidth are cycles per unit length. Since sample spacing is irrelevant to the result, we use the equivalent and more convenient units of radians per sample throughout the rest of this paper. Our Fourier domain MPA design strategy is laid out in Section 2. An analysis of the conventional 2 2 MPA repeating pattern is provided in Section 3. In Section 4 we design the 2 4 case for improved spatial resolution (reduced GRD) over the 2 2 MPA repeating pattern. Further author information: K.H.: khirakawa1@udayton.edu, Telephone: D.A.L: daniel.lemaster@us.af.mil, Telephone: Polarization: Measurement, Analysis, and Remote Sensing XI, edited by David B. Chenault, Dennis H. Goldstein, Proc. of SPIE Vol. 9099, SPIE CCC code: X/14/$18 doi: / Proc. of SPIE Vol

2 2. FREQUENCY DOMAIN DESIGN OF MPA The MPA takes power measurement X(n) at detector location n = (n 1, n 2 ) T Z 2 as follows: X(n) = S 0 (n) + a(n)s 1 (n) + b(n)s 2 (n), (3) where S j (n) is the j th Stokes parameter at location n. Here, the modulation terms a(n) and b(n) stem from the MPA pattern, and are periodic functions related to each other through the analyzer orientation angles θ(n), i.e., a(n) = cos (2θ(n)), b(n) = sin (2θ(n)). (4) The Discrete Space Fourier Transform (DSFT) of X is ˆX(ω) = Ŝ0(ω) + Â(ω) Ŝ1(ω) + ˆB(ω) Ŝ2(ω), (5) where is the convolution operator. Because they are periodic, both  and ˆB will have the form (  (ω) = A[m 1, m 2 ]δ ω 1 2π m ) ( 1 δ ω 2 2π m 2 N 1 N 2 m 1= m 2= ), (6) where A(B) is the Discrete Fourier Transform (DFT) of the unit cell of a(b) with periods N 1 and N 2. More explicitly, A(B) is periodic in both m 1 and m 2, i.e. A[m 1, m 2 ] = A[m 1 + kn 1, m 2 + ln 2 ] for all k and l Z, etc. The unit cell is the smallest non-repeating sequence of values within each modulation function. Designing MPA for maximum bandwidth proceeds from Equations 5 and 6: 1. N 1 and N 2 determine the possible locations of the modulated S 1 and S 2 spectra. 2. Spectral content that is most susceptible to (bandwidth limiting) aliasing may be suppressed by selectively zeroing out DFT coefficients in A and B. 3. Any valid configuration must obey the analyzer angle constraint in Equation 4. and two assumptions: 1. The Stokes spectra are assumed to be bandlimited and circularly symmetric. 2. The bandwidth of Ŝ0 is greater than the bandwidths of Ŝ1 and Ŝ2. Assumption 2 requires that the bandwidth of Ŝ1 and Ŝ2 must be less than π 2 radians per sample or, equivalently, π 2 < ν 0 π. The MPA analyzer orientation angles, θ, designed using these observations are calculated from the inverse DFT of A and B and Equation ANALYSIS OF 2 2 MPA It is useful to introduce this design philosophy through the example of the conventional 2 2 MPA. In this case, all possible locations for S 1 and S 2 spectral content are shown in Figure 1. The position indices of the DFT unit cell are labeled in the image following the convention established for A and B. Colors in the figure are used to identify spectral components that are replicas of each other (e.g. all green sites are replicas). It is clear from Equation 5 that A[0, 0] = B[0, 0] = 0, (7) is required for any MPA to avoid interference with the unmodulated S 0 spectrum. The goal of maximizing bandwidth by minimizing the risk of aliasing drives selection of the remaining DFT coefficients. In this scenario, this risk of aliasing is minimized when the center locations of the modulated spectra are spread out as much as possible. Two variations on the 2 2 MPA are used to demonstrate this concept. For Case 1: ( ) ( ) ( ) θ = = A = B =, (8) Proc. of SPIE Vol

3 Figure 1: Modulation points in the DSFT of the 2 2 MPA. Grey coordinates represent the DSFT domain, ω, with only one period shown. The black coordinates indicate the component of A or B that will weigh the modulated Stokes spectra at this location..0. I % i I /1 I i I -. (a) Case 1 (b) Case2 Figure 2: Spectral component mapping for two variations of the 2 2. MPA and in Case 2: θ = ( ) ( ) 0 0 = A = 1 1 B = ( ) 0 0. (9) 1 1 The spectra that result from these two MPA options are shown in Figure 2. For reference, the location of the S 0 spectrum is labeled with an x. The figure also shows the available bandwidth about S 0 and for each modulation point as dashed curves. Both cases would allow Ŝ1 and Ŝ2 to have a bandwidth up to π radians per sample so, from a spatial resolution point of view, both are equally valid options. Proc. of SPIE Vol

4 4. EXTENSION TO THE 2 4 MPA All possible locations for S 1 and S 2 spectral content in the 2 4 case are shown in Figure 3. W2 0 [3m 0 0 [01] _. [2,1]0 (0,n) (71/2,70 (7t,7t) (-71/2,7) [3,0]s [0,0] [1,0] [2,0] 001 (-7E/2,0) (0,0) (R/2,0) (7r,0) Figure 3: Modulation points in the DSFT of the 2 4 MPA. Grey coordinates represent the DSFT domain, ω, with only one period shown. The black coordinates indicate the component of A or B that will weigh the modulated Stokes spectra at this location. The goal will now be to devise a larger array pattern that results is better separation of spectral components over the 2 2 case. As before, the [0, 0] component must be eliminated. Similarly, A[1, 0] = B[1, 0] = 0 (10) A[3, 0] = B[3, 0] = 0, (11) because each of these components would bring the modulated Ŝ1 and Ŝ2 content closer to Ŝ0 than in the 2 2 case. If the [3, 1], [0, 1], and [2, 0] components are also eliminated, i.e. A[3, 1] = B[3, 1] = 0 (12) A[0, 1] = B[0, 1] = 0 (13) A[2, 0] = B[2, 0] = 0, (14) then the bandwidth available to Ŝ1 and Ŝ2 is maintained from the 2 2 case but the available bandwidth for Ŝ0 is increased by almost 24%. The chosen 2 4 distribution of available bandwidth is shown in Figure 4. An MPA pattern that meets these requirements is θ = ( ), (15) which results in ( ) A = (16) 0 i 0 i ( ) B =. (17) Proc. of SPIE Vol

5 o 1 W2., 601 o -i / o 1 Figure 4: Frequency domain modulation points for the 2 4 MPA. 5. IMAGERY COMPARISON Figure 5 illustrates how the additional bandwidth of the 2 4 MPA improves resolution over the conventional 2 2 case. Degree of Linear Polarization (DOLP) images are constructed from synthesized microgrid data. The underlying Stokes images were collected using a rotating analyzer polarimeter. The Stokes imagery is demosaicked 1 from the synthesized microgrid data using identical reconstruction filters across both 2 2 and 2 4 cases. The inset 2 2 spectrum shows significant overlap between the modulated and unmodulated spectra leading to inevitable aliasing artifacts in the DOLP image. The spectral components in the 2 4 case are spread out more and aliasing is minimized throughout the scene. As a result, resolution and image quality are demonstrably improved. Figure 5: Comparison of DOLP images reconstructed with 2 2 and 2 4 MPA. Proc. of SPIE Vol

6 6. CONCLUSIONS In this paper, we shown a method for optimizing the design of microgrid polarimetric imaging arrays for spatial resolution using frequency domain analysis. The result is that a 2 4 MPA pattern provides a 24% wider bandwidth for the S 0 image over the conventional 2 2 MPA while mantaining the available bandwidth for S 1 and S 2. This paper is a companion piece to our previous work 1 where we also show that the 2 4 MPA does not adversely affect the conditioning of the polarimetric data reduction matrix; explain how the raw data are demosaiced into Stokes images; and provide example results through simulation. REFERENCES [1] LeMaster, D. A. and Hirakawa, K., Improved microgrid arrangement for integrated imaging polarimeters, Opt. Lett. 39, 1 4 (April 2014). [2] Tyo, J. S., LaCasse, C. F., and Ratliff, B. M., Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters, Opt. Lett. 34, (Oct 2009). [3] Hirakawa, K. and Wolfe, P. J., Spatio-spectral color filter array design for optimal image recovery, Image Processing, IEEE Transactions on 17(10), (2008). [4] Eismann, M. T., [Hyperspectral Remote Sensing], SPIE Press, Bellingham, WA (2012). Proc. of SPIE Vol