2007Theo Schouten1 Restoration With Image Restoration one tries to repair errors or distortions in an image, caused during the image creation process.

2007 Theo Schouten 1

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

2007 Theo Schouten 2

Noise functions

2007 Theo Schouten 3

Only noise

See also enhancement, mean and median filters

2007 Theo Schouten 4

Mars, mariner 6

2007 Theo Schouten 5

Linear degradationWhen 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.

2007 Theo Schouten 6

Position invariant, inverse filteringWhen 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 uppseudo-inverse filter: use only parts of H(u,v)

2007 Theo Schouten 7

Degradation function by experiment

2007 Theo Schouten 8

by modelling

H(u,v)= exp(-k(u2+v2)5/6 )

atmosfericturbulencemodel

2007 Theo Schouten 9

by calculation, linear motionSuppose a movement of the image during shutter opening:g(x,y) = 0

T 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 0’s : (ua+vb) = n (any integer)pseudo-inverse filter is useless

2007 Theo Schouten 10

Linear motion blur

2007 Theo Schouten 11

Pseudo-inverse filter

2007 Theo Schouten 12

Gaussian movementA 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

2007 Theo Schouten 13

with noiseUniform 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

2007 Theo Schouten 14

Wiener filteringminimum 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)

2007 Theo Schouten 15

Example Wiener filter

Original

Noise added

Pseudo-inverse

Wiener filter

2007 Theo Schouten 16

Linear motion Wiener filter

2007 Theo Schouten 17

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.

2007 Theo Schouten 18

Corrections

Original Nearest neighbor Bilinear interpolation

More complex, slower:•bilinear interpolation•subsampling e.g. 5x5

2007 Theo Schouten 19

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

2007 Theo Schouten 20

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