Restoration
With Image Restoration one tries to repair errors or distortions in an image, caused during the image creation process. In general our starting point is a degradation and noise model: g(x,y) = H ( f(x,y) ) + (x,y)
Determined by quality of equipment and image taking conditions: image restoration is computationally complex equipment as degradation free as possible seen technical and financial limitations medical: low radiation, little time in magnet-tube lowest image quality to achieve medical goals web-cams: cheap lens distortions corrected by CPU in cam
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Noise functions
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Only noise
See also enhancement, mean and median filters
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Mars, mariner 6
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Linear degradation
When the degradation process is linear: H( k1f1 + k2f2 ) = k1 H( f1) + k2 H( f2 )
we can write (we temporarily leave the noise out of consideration):
g(x,y) = H(f(x,y)) = H( f(,) (x- ,y- ) d d ) = f(,) H( (x- ,y- ) ) d d = f(,) h(x, ,y, ) d d h(x, ,y, ) is the "impulse response" or "point spread function", the degraded image of an ideal light point. The integral is called the "superposition or "Fredholm integral of the first kind.
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Position invariant, inverse filtering
When H is a spatial invariant: Hf(x- ,y- )=g(x- ,y- ) then: h(x, ,y, ) = h(x- ,y- ) and g(x,y) = f(,) h(x- ,y- ) d d a convolution integral, and taking into account the noise: G(u,v) = H(u,v)F(u,v) + N(u,v) Inverse filtering: G(u,v)/H(u,v) = F(u,v) + N(u,v)/H(u,v) Problems: if H(u,v) = 0, or small: noise is blown up pseudo-inverse filter: use only parts of H(u,v)
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Degradation function by experiment
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by modelling
H(u,v)= exp( -k(u2+v2)5/6 )
atmosferic turbulence model
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by calculation, linear motion
Suppose a movement of the image during shutter opening: g(x,y) = 0T f(x-x0(t),y-y0(t)) dt
G(u,v) = [0T f(x-x0(t),y-y0(t)) dt ] e -j2(ux+vy) dxdy = F(u,v) 0T e -j2[uxo(t)+vy0(t)] dt = F(u,v) H(u,v) With linear motion x0(t)=at/T and y0(t)=bt/T :
H(u,v) = {T/[ (ua+vb)] } sin[ (ua+vb)] e -j[ua+vb]
This has a lot of 0s : (ua+vb) = n (any integer) pseudo-inverse filter is useless
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Linear motion blur
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Pseudo-inverse filter
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Gaussian movement
A 1-D Gaussian kernel for distortions in the horizontal direction. The intensity of each pixel is spread out over the neighboring pixels according to this kernel.
Power spectrum
Inverse filter
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with noise
Uniform noise [0,1] added (rounding floating point to unsigned byte) Movement lines disappear due to noise
Inverse filter: nothing Pseudo-inverse filter, only when H(u,v) > T
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Wiener filtering
minimum mean square error: e2 = E{ (f-fc)2} Fc(u,v) =[1/H(u,v)] [ |H(u,v|2 / (|H(u,v|2 +S(u,v)/Sf(u,v))] G(u,v)
S(u,v) = |N(u,v)|2 power spectrum of noise
Approximations of S(u,v)/Sf(u,v): K (constant) |P(u,v)|2 (power spectrum of Laplacian) found by iterative method to minimize e2 (constrained least squares filtering)
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Example Wiener filter
Original
Noise added
Pseudo-inverse
Wiener filter
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Linear motion Wiener filter
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Geometric distorsion
Lenses often show a typical pincushion or barrel deviation. When the projection function x'=g(x) is known, for each measured pixel it can be determined from which parts of ideal pixels it is buit up. If the inverse function g-1 is known, then for each ideal pixel we can determine from which parts of the distorted pixels it is built up of.
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Corrections
More complex, slower: bilinear interpolation subsampling e.g. 5x5
Original
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Nearest neighbor
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Bilinear interpolation
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Calibration
Calibration, e.g. x = a +b x +c y and y = r +s x +t y : affine transformations
Also higher order terms like d x2 + e y2 + f xy
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Fish eye lens
x' = x + x*(K1*r2 + K2*r4 + K3*r6) + P1*(r2 + 2*x2) + 2*P2*x*y y' = y + y*(K1*r2 + K2*r4 + K3*r6) + P2*(r2 + 2*y2) + 2*P1*x*y
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