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EE 4780

EE 4780. Bilateral Filter. Bilateral Filter. K is the normalization constant. Intensity (range) proximity. Spatial (domain) proximity. N is a fixed value used to define the spatial neighborhood of the filter. Bilateral Filter – Matlab implementation.

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EE 4780

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  1. EE 4780 Bilateral Filter

  2. Bilateral Filter K is the normalization constant Intensity (range) proximity Spatial (domain) proximity N is a fixed value used to define the spatial neighborhood of the filter

  3. Bilateral Filter – Matlab implementation n=1:1:500; % Generate a vector from 1 to 500; the increment is 1. I0=zeros(size(n)); % Generate a vector of zeros; the size of the vector is equal to the size of n. I0(1:250)=15; I0(251:end)=10; % Set the first 250 values to 15, and the rest to 10. I = I0 + 0.5*randn(size(I0)); % 0.5 is the standard deviation of the noise figure; subplot(2,1,1); plot(n,I0); axis ([190 310 6 18]); title('Original signal'); subplot(2,1,2); plot(n,I); axis ([190 310 6 18]); title('Noisy signal');

  4. Bilateral Filter – Matlab implementation sigma_d=10; N=round(4*sigma_d); % N determines the spatial neighborhood sigma_r=1.3; d = -N:1:N; weights_d = exp(-d.*d/(2*sigma_d*sigma_d)); The weights depend on the spatial distance (to the center pixel x) only; therefore, it is calculated once and saved.

  5. Bilateral Filter – Matlab implementation sigma_d=10; N=round(4*sigma_d); % N determines the spatial neighborhood sigma_r=1.3; d = -N:1:N; weights_d = exp(-d.*d/(2*sigma_d*sigma_d)); x=260; % An example pixels = I(x-N:x+N); % Put the pixels within the neighborhood of the center pixel into a vector. weights = weights_d .* exp(-(pixels-I(x)).*(pixels-I(x))/(2*sigma_r*sigma_r)) + 0.0001; weights = weights./sum(weights); Add a small number in case weights=0;

  6. Bilateral Filter – Matlab implementation sigma_d=10; N=round(4*sigma_d); % N determines the spatial neighborhood sigma_r=1.3; d = -N:1:N; weights_d = exp(-d.*d/(2*sigma_d*sigma_d)); x=260; pixels = I(x-N:x+N); % Put the pixels within the neighborhood of the center pixel into a vector. weights = weights_d .* exp(-(pixels-I(x)).*(pixels-I(x))/(2*sigma_r*sigma_r)) + 0.0001; weights = weights./sum(weights); % Normalize the weights so that its sum is equal to 1. I_output(x) = sum(weights.*pixels);

  7. Bilateral Filter figure; plot([x-N:x+N],weights)

  8. Bilateral Filter – Matlab implementation d = -N:1:N; weights_d = exp(-d.*d/(2*sigma_d*sigma_d)); % Repeat for all pixels I_output = I; for i=1+N:length(I)-N, % Be careful with the borders; do not exceed the dimensions. pixels = I(i-N:i+N); weights = weights_d .* exp(-(pixels-I(i)).*(pixels-I(i))/(2*sigma_r*sigma_r)) + 0.0001; weights = weights./sum(weights); I_output(i) = sum(weights.*pixels); end figure; plot(n,I_output);

  9. Bilateral Filter Input Gaussian Bilateral

  10. Bilateral Filter vs. Gaussian LPF Gaussian MSE=49.8 MSE=100.0 sigma_d=10 MSE=99.57 MSE=30.3 MSE=42.5

  11. Wiener Filter Noisy image Original image Noise Wiener Filter Signal variance Noise variance When sigma_x << sigma_w, (noise is very large), X goes to 0. When sigma_x >> sigma_w, (noise is very small), X goes to Y.

  12. Wiener Filter is estimated by Estimate manually by looking at the variance in a smooth region. Since variance is nonnegative, it is modified as Estimate signal variance locally: N N

  13. Wiener Filter Denoised (3x3neighborhood) Mean Squared Error is 56 Noisy, =10 wiener2 in Matlab

  14. Image Enhancement This is an high-pass filter. It removes low-frequency components.

  15. Image Enhancement • High-boost or high-frequency-emphasis filter • Sharpens the image but does not remove the low-frequency components unlike high-pass filtering

  16. Image Enhancement • High-boost or high-frequency-emphasis filter • High pass = Original – low pass • High boost = A*(Original) + High pass  Part of the low-frequency components are added back to the high frequency components

  17. Image Enhancement A high-pass filter A high-boost filter

  18. Image Enhancement • High-boost or high-frequency-emphasis filter

  19. Spatial Filtering

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