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EE3190 Optical Sensing and Imaging

EE3190 Optical Sensing and Imaging. Computing PSFs with a digital computer. Sampling the Pupil Function. For square arrays (equal number of samples in both dimensions), select the sample spacing so that the pupil function fills up half of the array:.

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EE3190 Optical Sensing and Imaging

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  1. EE3190 Optical Sensing and Imaging Computing PSFs with a digital computer

  2. Sampling the Pupil Function For square arrays (equal number of samples in both dimensions), select the sample spacing so that the pupil function fills up half of the array: When using the FFT it is good practice to select N to be even. Better yet, is to selectN to be a power of 2. D is the pupil diameter in meters N is the number of samples across the array

  3. Sampling the Pupil Function • Create a sampling grid

  4. Sampling the Pupil Function • Create a sampling grid d2 = 2*D/N; u = [-N/2:1:N/2-1]*d2; [u1,u2] = meshgrid(u); example: d2 = 1/2; N = 6 -1.5000 -1.0000 -0.5000 0 0.5000 1.0000 u -1.5000 -1.0000 -0.5000 0 0.5000 1.0000 -1.5000 -1.0000 -0.5000 0 0.5000 1.0000 -1.5000 -1.0000 -0.5000 0 0.5000 1.0000 -1.5000 -1.0000 -0.5000 0 0.5000 1.0000 -1.5000 -1.0000 -0.5000 0 0.5000 1.0000 -1.5000 -1.0000 -0.5000 0 0.5000 1.0000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 0 0 0 0 0 0 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 u1 u2

  5. Sampling the Pupil Function • Example: Square Pupil (DxD meters) • Example: Circular Pupil (diameter: D meters) p = (abs(u1)<=D/2)&(abs(u2)<=D/2); r = sqrt(u1.^2 + u2.^2); p = r<=D/2;

  6. 1 1 2 2 3 3 4 4 Compute the PSF • Using the FFT algorithm P = fftshift(fft2(fftshift(p)); d1 = lambda*d/(N*d2); u = [-N/2:1:N/2-1]*d1; lambda is wavelength d is distance to object or image fftshift

  7. Example: Circular pupil D = 1; lambda = 1e-6; do = 10e3; N = 256; d2 = 2*D/N; u = [-N/2:1:N/2-1]*d2; [u1,u2] = meshgrid(u); r = sqrt(u1.^2+u2.^2); p = abs(r)<=D/2; figure(1); colormap(gray); subplot(2,1,1); imagesc(u,u,p); axis image; P = fftshift(fft2(fftshift(p))); d1 = lambda*do/(N*d2); x = [-N/2:1:N/2-1]*d1; subplot(2,1,2); imagesc(x,x,abs(P)); axis image;

  8. Example: Circular pupil D = 1; lambda = 1e-6; do = 10e3; N = 256; d2 = 2*D/N; u = [-N/2:1:N/2-1]*d2; [u1,u2] = meshgrid(u); r = sqrt(u1.^2+u2.^2); p = abs(r)<=D/2; figure(1); colormap(gray); subplot(2,1,1); imagesc(u,u,p); axis image; P = fftshift(fft2(fftshift(p))); d1 = lambda*do/(N*d2); x = [-N/2:1:N/2-1]*d1; subplot(2,1,2); imagesc(x,x,abs(P)); axis image; axis([-1 1 -1 1]*10*lambda*do/D);

  9. Example: Circular pupil (defocus) D = 1; lambda = 1e-6; do = 10e3; di = 10e-2; f = 1/(1/do + 1/di); ferror = 1/(1/(do*1.06) + 1/di); N = 256; d2 = 2*D/N; u = [-N/2:1:N/2-1]*d2; [u1,u2] = meshgrid(u); r = sqrt(u1.^2+u2.^2); p = (abs(r)<=D/2).*... exp(-j*(pi/(lambda*f))*r.^2).*... exp(j*(pi/(lambda*ferror))*r.^2); figure(1); colormap(gray); subplot(2,1,1); imagesc(u,u,angle(p)); axis image; P = fftshift(fft2(fftshift(p))); d1 = lambda*do/(N*d2); x = [-N/2:1:N/2-1]*d1; subplot(2,1,2); imagesc(x,x,abs(P)); axis image; axis([-1 1 -1 1]*8*lambda*do/D);

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