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2007Theo Schouten1 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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2007Theo Schouten2 Noise functions

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2007Theo Schouten3 Only noise See also enhancement, mean and median filters

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2007Theo Schouten4 Mars, mariner 6

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2007Theo Schouten5 Linear degradation When the degradation process is linear: H( k 1 f 1 + k 2 f 2 ) = k 1 H( f 1 ) + k 2 H( f 2 ) 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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2007Theo Schouten6 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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2007Theo Schouten7 Degradation function by experiment

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2007Theo Schouten8 by modelling H(u,v)= exp( -k(u 2 +v 2 ) 5/6 ) atmosferic turbulence model

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2007Theo Schouten9 by calculation, linear motion Suppose a movement of the image during shutter opening: g(x,y) = 0 T f(x-x 0 (t),y-y 0 (t)) dt G(u,v)= [ 0 T f(x-x 0 (t),y-y 0 (t)) dt ] e -j2 (ux+vy) dxdy = F(u,v) 0 T e -j2 [uxo(t)+vy0(t)] dt = F(u,v) H(u,v) With linear motion x 0 (t)=at/T and y 0 (t)=bt/T : H(u,v)= {T/[ (ua+vb)] } sin[ (ua+vb)] e -j [ua+vb] This has a lot of 0’s : (ua+vb) = n (any integer) pseudo-inverse filter is useless

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2007Theo Schouten10 Linear motion blur

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2007Theo Schouten11 Pseudo-inverse filter

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2007Theo Schouten12 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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2007Theo Schouten13 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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2007Theo Schouten14 Wiener filtering minimum mean square error: e 2 = E{ (f-f c ) 2 } F c (u,v) =[1/H(u,v)] [ |H(u,v| 2 / (|H(u,v| 2 +S (u,v)/S f (u,v))] G(u,v) S (u,v) = |N(u,v)| 2 power spectrum of noise Approximations of S (u,v)/S f (u,v): K (constant) |P(u,v)| 2 (power spectrum of Laplacian) found by iterative method to minimize e 2 (constrained least squares filtering)

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2007Theo Schouten15 Example Wiener filter Original Noise added Pseudo-inverse Wiener filter

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2007Theo Schouten16 Linear motion Wiener filter

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2007Theo Schouten17 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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2007Theo Schouten18 Corrections Original Nearest neighbor Bilinear interpolation More complex, slower: bilinear interpolation subsampling e.g. 5x5

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2007Theo Schouten19 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 x 2 + e y 2 + f xy

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2007Theo Schouten20 Fish eye lens x' = x + x*(K 1 *r 2 + K 2 *r 4 + K 3 *r 6 ) + P 1 *(r 2 + 2*x 2 ) + 2*P 2 *x*y y' = y + y*(K 1 *r 2 + K 2 *r 4 + K 3 *r 6 ) + P 2 *(r 2 + 2*y 2 ) + 2*P 1 *x*y

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CS654: Digital Image Analysis

CS654: Digital Image Analysis

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