Image Restoration
Examples
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Image and Multidimensional Signal Processing
Image Restoration Metrics
• RMS error is easy to compute (ie, square root of the
mean squared error).
– However, it (and similar metrics) are biased towards oversmoothed (i.e., blurry) results. Namely, an algorithm that
removes not only the noise but also part of the structure (e.g.,
edges) will have a good score.
• The structural similarity index (SSIM) takes into account
the similarity of the edges (high frequency content)
between the denoised image and the ideal one.
– To have a good SSIM measure, an algorithm needs to remove
the noise while also preserving the edges of the objects.
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Other Image Restoration Metrics
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Example 1 – Gaussian filter for denoising
clear all
close all
% Get an image.
Iinput = imread('liftingbody.png');
Iinput = double(Iinput);
imshow(Iinput, []); % Input image
% Add noise to the image. Let's use Gaussian noise, with sigma = sn.
sn = 20.0;
I = Iinput + sn*randn(size(Iinput));
figure, imshow(I, []); % Noisy image
sigmaGaussian = 3.0;
Iout = imfilter(I, fspecial('gaussian', 6*sigmaGaussian, sigmaGaussian));
figure, imshow(Iout, []), title('gaussian filter');
Idiff = Iout - Iinput;
rms = sqrt(mean2(Idiff.^2));
figure, imshow(Idiff, []);
title(sprintf('Errors (gaussian filter), rms = %f', rms));
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Example 2 - Bilateral Filter
• The “bilateral filter” is similar to the “adaptive mean
filter” discussed in the textbook.
• It is useful to reduce image noise, but doesn’t blur
across boundaries.
• A good tutorial is available on this website:
– http://people.csail.mit.edu/sparis/bf_course/
– Matlab code available there too.
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Example 2 (continued)
clear all
close all
% Get an image.
Iinput = imread('liftingbody.png');
Iinput = double(Iinput);
imshow(Iinput, []); % Input image
% Add noise to the image. Let's use Gaussian noise, with sigma = sn.
sn = 20.0;
I = Iinput + sn*randn(size(Iinput));
figure, imshow(I, []); % Noisy image
maxval = max(I(:));
minval = min(I(:));
sigmaSpatial = sigmaGaussian;
sigmaRange = (maxval-minval)/10;
% Default is (maxval-minval)/10
Iout = bilateralFilter(I, ...
[], ...
% optional edge image
minval, maxval, ...
% range of values
sigmaSpatial, sigmaRange); % sigmaSpatial, sigmaRange
figure, imshow(Iout, []), title('bilateral filter');
Idiff = Iout - Iinput;
rms = sqrt(mean2(Idiff.^2));
figure, imshow(Idiff, []);
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title(sprintf('Errors
(bilateral
filter),Signal
rms
= %f', rms));
6
Example 3 - Blur Degradation
• A blurring degradation can be modeled by a Gaussian
h ( x, y ) =
1
2πσ 2
−
e
x2 + y2
2σ 2
• If you take an image of a dot (impulse), the output
blurred image is just the Gaussian blurring function
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Example 3 - Blurred image of a line
• Assume that the input image is a bright vertical line located at
x=a in the image, and zero everywhere else.
• Namely, f(x,y) = δ(x-a). What is the degraded output image
g(x,y)?
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Example 3 (solution)
• Solution:
h ( x, y ) ∗ δ ( x − a ) =
∞ ∞
=
∫ ∫ h(x' , y') δ ((x − a ) − x') dx' dy'
−∞ −∞
∞
= ∫ h( x − a, y ' ) dy '
−∞
∞
= ∫ e − (( x − a )
2
+ y '2
) dy '
−∞
=e
− ( x − a )2
∞
− y'
e
∫ dy'
2
−∞
2
−𝑦𝑦
𝑒𝑒
over
• The integral of
all y is equal to the square root of pi.
The output image g(x,y) is a vertical line that is blurred in the x
direction. Namely, .
2
g ( x, y ) = π e − (( x − a ) )
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Example 4 – Compare filters
• Compare the performance of these filters, on reducing added
Gaussian noise:
–
–
–
–
Gaussian low pass filter
Bilateral filter
Adaptive mean filter
Anisotropic diffusion filter
• Try on these images:
– ckt-board.tif
– Fig0222(a)(face).tif
– cameraman.tif
% Add noise to the image. Let's use Gaussian noise, with sigma = sn.
sn = 20.0;
I = Iinput + sn*randn(size(Iinput));
figure, imshow(I, []); % Noisy image
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Example 4 (continued)
• For each image and each filter, display the error
image between the original (noise-free) image and
the filtered reconstructed image. Notice where
errors occur.
• For each image and each filter, give the RMS error
between the restored image and the original. See if
a low RMS error gives the “best” reconstruction in
terms of preserving edges as well as reducing noise.
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Example 4 (continued)
clear all
close all
% Get an image.
%Iinput = imread('ckt-board.tif');
%Iinput = imread('Fig0222(a)(face).tif');
Iinput = imread('cameraman.tif');
Iinput = double(Iinput);
imshow(Iinput, []); % Input image
% Add noise to the image. Let's use Gaussian noise, with sigma = sn.
sn = 20.0;
I = Iinput + sn*randn(size(Iinput));
figure, imshow(I, []); % Noisy image
sigmaGaussian = 3.0;
Iout = imfilter(I, fspecial('gaussian', 6*sigmaGaussian, sigmaGaussian));
figure, imshow(Iout, []), title('gaussian filter');
Idiff = Iout - Iinput;
rms = sqrt(mean2(Idiff.^2))
figure, imshow(Idiff, []);
title(sprintf('Errors (gaussian filter), rms = %f', rms));
maxval = max(I(:));
minval = min(I(:));
sigmaSpatial = sigmaGaussian;
sigmaRange = (maxval-minval)/10;
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% Default is (maxval-minval)/10
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Iout = bilateralFilter(I, ...
[], ...
% optional edge image
minval, maxval, ...
% range of values
sigmaSpatial, sigmaRange); % sigmaSpatial, sigmaRange
figure, imshow(Iout, []), title('bilateral filter');
Idiff = Iout - Iinput;
rms = sqrt(mean2(Idiff.^2))
figure, imshow(Idiff, []);
title(sprintf('Errors (bilateral filter), rms = %f', rms));
Iout = wiener2(I, [3*sigmaGaussian 3*sigmaGaussian]);
figure, imshow(Iout, []), title('adaptive mean filter');
Idiff = Iout - Iinput;
rms = sqrt(mean2(Idiff.^2))
figure, imshow(Idiff, []);
title(sprintf('Errors (adaptive mean filter), rms = %f', rms));
% Parameters:
niter - number of iterations.
%
kappa - conduction coefficient 20-100 ?
%
lambda - max value of .25 for stability
%
option - 1 Perona Malik diffusion equation No 1
%
2 Perona Malik diffusion equation No 2%
Iout = anisodiff(I, ...
10, ...
% niter - number of iterations
30, ...
% conduction coefficient 20-100
0.25, ...
% max value of .25 for stability
1);
% option: Perona Malik diffusion equation No 1 or No 2
figure, imshow(Iout, []);
Idiff = Iout - Iinput;
rms = sqrt(mean2(Idiff.^2))
figure, imshow(Idiff, []);
title(sprintf('Errors (anisotropic diffusion filter), rms = %f', rms));
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Example 4 (solutions)
Image
ckt-board.tif
Fig0222(a)(face).
tif
cameraman.tif
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Gaussian low
pass
29.9
13.8
Bilateral
Adaptive mean
10.4
9.4
15.9
8.7
Anisotropic
diffusion
9.7
7.3
24.9
10.3
13.2
10.2
Image and Multidimensional Signal Processing
14