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Image RestorationWhat is Image RestorationImage Degradation ModelNoise ModelsNoise Removal Restoration MethodMean FiltersContra-Harmonic FiltersMedian FilterLSI Degradation ModelsTurbulence Blur ExamplesMotion BlurInverse FilterWiener FilteringDerivation of Wiener FiltersConstrained Least Square (CLS) FilterIntuitive Interpretation of CLSSolution and Iterative AlgorithmCLS Demonstration© 2002-2003 by Yu Hen Hu1ECE533 Digital Image ProcessingImage Restoration© 2002-2003 by Yu Hen Hu2ECE533 Digital Image ProcessingWhat is Image RestorationThe purpose of image restoration is to restore a degraded/distorted image to its original content and quality.Distinctions to Image Enhancement»Image restoration assumes a degradation model that is known or can be estimated.»Original content and quality ≠ Good looking© 2002-2003 by Yu Hen Hu3ECE533 Digital Image ProcessingImage Degradation ModelSpatial variant degradation modelSpatial-invariant degradation model»Frequency domain representation( , ) ( , , , ) ( , ) ( , )g x y h x y m n f m n x yh= +��( , ) ( , ) ( , ) ( , )g x y h x m y n f m n x yh= - - +��( , ) ( , ) ( , ) ( , )G u v H u v F u v N u v= +© 2002-2003 by Yu Hen Hu4ECE533 Digital Image ProcessingNoise ModelsMost types of noise are modeled as known probability density functionsNoise model is decided based on understanding of the physics of the sources of noise. »Gaussian: poor illumination»Rayleigh: range image»Gamma, exp: laser imaging»Impulse: faulty switch during imaging, »Uniform is least used.Parameters can be estimated based on histogram on small flat area of an image© 2002-2003 by Yu Hen Hu5ECE533 Digital Image ProcessingNoise Removal Restoration MethodMean filters»Arithmetic mean filter»Geometric mean filter»Harmonic mean filter»Contra-harmonic mean filterOrder statistics filters»Median filter»Max and min filters»Mid-point filter»alpha-trimmed filtersAdaptive filters»Adaptive local noise reduction filter»Adaptive median filter© 2002-2003 by Yu Hen Hu6ECE533 Digital Image ProcessingMean Filters,( , )1ˆ( , ) ( , )x ys t Sf x y g s tmn�=�,1( , )ˆ( , ) ( , )x ymns t Sf x y g s t�� �=� �� �� ��© 2002-2003 by Yu Hen Hu7ECE533 Digital Image ProcessingContra-Harmonic Filters[ ][ ],,1( , )( , )( , )ˆ( , )( , )x yx yQs t SQs t Sg s tf x yg s t+��=��© 2002-2003 by Yu Hen Hu8ECE533 Digital Image ProcessingMedian Filter{ },( , )ˆ( , ) ( , )x ys t Sf x y median g s t�=Effective for removing salt-and-paper (impulsive) noise.© 2002-2003 by Yu Hen Hu9ECE533 Digital Image ProcessingLSI Degradation ModelsMotion Blur»Due to camera panning or fast motionAtmospheric turbulence blur»Due to long exposure time through atmosphere»Hufnagel and StanleyUniform out-of-focus blur:Uniform 2D Blurmin max1 0,( , )0 .ai bj i i ih i jotherwise+ = � ��=��2 22( , ) exp2i jh i j Ks� �+= � -� �� �2 2 221( , )0 .i j Rh i jRotherwisep�+ ��=���21/ 2 , / 2( , )0 .L i j Lh i jLotherwise�- � ��=���( )()5/ 62 2( , ) exph i j k i j= - � +© 2002-2003 by Yu Hen Hu10ECE533 Digital Image ProcessingTurbulence Blur Examples( )()5/ 62 2( , ) exph i j k i j= - � +© 2002-2003 by Yu Hen Hu11ECE533 Digital Image ProcessingMotion BlurOften due to camera panning or fast object motion. Linear along a specific direction.blurring filter20 40 60204060blurring filter mask2 4 6 82468original image20 40 60102030405060blurred image20 40 60102030405060Blurdemo.m© 2002-2003 by Yu Hen Hu12ECE533 Digital Image ProcessingInverse FilterRecall the degradation model:Given H(u,v), one may directly estimate the original image byAt (u,v) where H(u,v)  0, the noise N(u,v) term will be amplified!( , ) ( , ) ( , ) ( , )G u v H u v F u v N u v= +ˆ( , ) ( , ) / ( , )( , )( , )( , )F u v G u v H u vN u vF u vH u v== +original, f20 40 60204060degraded: g20 40 60204060inverse filter20 40 60204060Invfildemo.m© 2002-2003 by Yu Hen Hu13ECE533 Digital Image ProcessingWiener FilteringMinimum mean-square error filter»Assume f and  are both 2D random sequences, uncorrelated to each other.»Goal: to minimize »Solution: Frequency selective scaling of inverse filter solution!»White noise, unknown Sf(u,v):{ }2ˆE f f-22( , )( , )ˆ( , )( , )( , ) ( , ) / ( , )fH u vG u vF u vH u vH u v S u v S u vh= �+22( , )( , )ˆ( , )( , )( , )H u vG u vF u vH u vH u v K= �+original, f20 40 60204060degraded: g20 40 60204060Wiener filter, K=0.220 40 60204060inverse filter20 40 60204060© 2002-2003 by Yu Hen Hu14ECE533 Digital Image ProcessingDerivation of Wiener FiltersGiven the degraded image g, the Wiener filter is an optimal filter hwin such that E{|| f – hwing||2} is minimized. Assume that f and  are uncorrelated zero mean stationary 2D random sequences with known power spectrum Sf and Sn. Thus,{ } { }{ }{ }{ }{ }( )2 222 22 2( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) (win winHwinH Hwin winf win f nH Hwin f winC E f h g E F u v H u v G u vE F u v H u v E F u v G u vH u v E F u v G u v H u v E G u vS u v H u v H u v S u v S u vH u v H u v S u v H u v H u= - = -= - �- � + �= + � � +- � � - � , ) ( , )fv S u v�{ }{ }{ }{ }22( , ) ( , )( , ) ( , )( , ) ( , )( , ) ( , ) 0fnHHE F u v S u vE N u v S u vE F u v N u vE F u v N u v=== =*2( , ) 0( , ) ( , )( , )( , ) ( , ) ( , )Set C/winfwinf nH u vH u v S u vH u vH u v S u v S u v� � = �=+© 2002-2003 by Yu Hen Hu15ECE533 Digital Image ProcessingConstrained Least Square (CLS) FilterFor each pixel, assume the noise  has a Gaussian distribution. This leads to a likelihood function:A constraint representing prior distribution of f will be imposed:the exponential form of pdf of f is known as the Gibbs’ distribution.Since L(f)  p(g|f), use Bayes rule, since g is given, to maximize the posterior probability, one should minimizeq is an operator based on prior knowledge about f. For example, it may be the Laplacian operator!221( ) exp **2L f g h fs� �� - -� ��{ }2( ) exp **p f q fa� -( | ) ( | ) ( ) / ( )p f g p g f p f p g=2 2** **g h f q fg- +© 2002-2003 by Yu Hen Hu16ECE533 Digital Image ProcessingIntuitive …


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UW-Madison ECE 533 - Image Restoration

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