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,*,Digital Image Processing,Chapter 5:,Image Restoration,23 June 2006,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Concept of Image Restoration,Image restoration is to restore a degraded image back to,the original image while,image enhancement is to,manipulate the image so that it is suitable for a specific,application.,Degradation model:,where,h,(,x,y),is a system that,causes image distortion and,h,(,x,y,),is noise.,Noise Models,Noise cannot be predicted,but can be approximately described in,statistical way using the probability density function (PDF),Gaussian noise:,Rayleigh noise,Erlang (Gamma) noise,Noise Models (cont.),Exponential noise,Uniform noise,Impulse (salt & pepper) noise,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,PDF: Statistical Way to Describe Noise,PDF tells how much,each z value occurs.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Image Degradation with Additive Noise,Original image,Histogram,Degraded images,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Original image,Histogram,Degraded images,Image Degradation with Additive Noise (cont.),(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Periodic Noise,Periodic noise,looks like,dots,In the frequency,domain,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Estimation of Noise,We cannot use the image,histogram to estimate,noise PDF.,It is better to use the,histogram of one area,of an image that has,constant intensity to,estimate noise PDF.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Periodic Noise Reduction by Freq. Domain Filtering,Band reject filter,Restored image,Degraded image,DFT,Periodic noise,can be reduced by,setting frequency,components,corresponding to,noise to zero.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Band Reject Filters,Use to eliminate frequency components in some bands,Periodic noise from the,previous slide that is,Filtered out.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Notch Reject Filters,A notch reject filter is used to eliminate some frequency components.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Notch Reject Filter:,Degraded image,DFT,Notch filter,(freq. Domain),Restored image,Noise,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Example: Image Degraded by Periodic Noise,Degraded image,DFT,(no shift),Restored image,Noise,DFT of noise,Mean Filters,Arithmetic mean filter or moving average filter,(from Chapter 3),Geometric mean filter,mn,= size of moving window,Degradation model:,To remove this part,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Geometric Mean Filter: Example,Original,image,Image,corrupted,by AWGN,Image,obtained,using a 3x3,geometric,mean filter,Image,obtained,using a 3x3,arithmetic,mean filter,AWGN: Additive White Gaussian Noise,Harmonic and Contraharmonic Filters,Harmonic mean filter,Contraharmonic mean filter,mn,= size of moving window,Works well for salt noise,but fails for pepper noise,Q,= the filter order,Positive,Q,is suitable for,eliminating pepper noise.,Negative,Q,is suitable for,eliminating salt noise.,For,Q,= 0, the filter reduces to an arithmetic mean filter.,For,Q,= -1, the filter reduces to a harmonic mean filter.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Contraharmonic Filters: Example,Image,corrupted,by pepper,noise with,prob. = 0.1,Image,corrupted,by salt,noise with,prob. = 0.1,Image,obtained,using a 3x3,contra-,harmonic,mean filter,With,Q,= 1.5,Image,obtained,using a 3x3,contra-,harmonic,mean filter,With,Q,=-1.5,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Contraharmonic Filters: Incorrect Use Example,Image,corrupted,by pepper,noise with,prob. = 0.1,Image,corrupted,by salt,noise with,prob. = 0.1,Image,obtained,using a 3x3,contra-,harmonic,mean filter,With,Q,=-1.5,Image,obtained,using a 3x3,contra-,harmonic,mean filter,With,Q,=1.5,Order-Statistic Filters: Revisit,subimage,Original image,Moving,window,Statistic parameters,Mean, Median, Mode,Min, Max, Etc.,Output image,Order-Statistics Filters,Median filter,Max filter,Min filter,Midpoint filter,Reduce “dark” noise,(pepper noise),Reduce “bright” noise,(salt noise),Median Filter : How it works,A median filter is good for removing impulse, isolated noise,Degraded image,Salt noise,Pepper noise,Moving,window,Sorted,array,Salt noise,Pepper noise,Median,Filter output,Normally, impulse noise has high magnitude,and is isolated. When we sort pixels in the,moving window, noise pixels are usually,at the ends of the array.,Therefore, its rare that the noise pixel will be a median value.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Median Filter : Example,Image,corrupted,by salt-and-pepper,noise with,p,a,=,p,b,= 0.1,Images obtained using a 3x3 median filter,1,4,2,3,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Max and Min Filters: Example,Image,corrupted,by pepper,noise with,prob. = 0.1,Image,corrupted,by salt,noise with,prob. = 0.1,Image,obtained,using,a 3x3,max filter,Image,obtained,using,a 3x3,min filter,Alpha-trimmed Mean Filter,where,g,r,(,s,t,) represent the remaining,mn,-,d,pixels after,removing,the,d,/2 highest,and,d,/2 lowest values,of,g,(,s,t,).,This filter is useful in situations involving multiple types,of noise such as a combination of salt-and-pepper and,Gaussian noise.,Formula:,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Alpha-trimmed Mean Filter: Example,Image,corrupted,by additive,uniform,noise,Image,obtained,using,a 5x5,arithmetic,mean filter,Image,additionally,corrupted,by additive,salt-and-,pepper,noise,1,2,2,Image,obtained,using,a 5x5,geometric,mean filter,2,Alpha-trimmed Mean Filter: Example (cont.),Image,corrupted,by additive,uniform,noise,Image,obtained,using,a 5x5,median filter,Image,additionally,corrupted,by additive,salt-and-,pepper,noise,1,2,2,Image,obtained,using,a 5x5,alpha-,trimmed,mean filter,with,d,= 5,2,Alpha-trimmed Mean Filter: Example (cont.),Image,obtained,using,a 5x5,arithmetic,mean filter,Image,obtained,using,a 5x5,geometric,mean filter,Image,obtained,using,a 5x5,median filter,Image,obtained,using,a 5x5,alpha-,trimmed,mean filter,with,d,= 5,Adaptive Filter,Filter behavior depends on,statistical characteristics of local areas,inside,m,x,n,moving window,More complex but,superior performance,compared with “fixed”,filters,Statistical characteristics:,General concept:,Local mean:,Local variance:,Noise variance:,Adaptive, Local Noise Reduction Filter,Purpose:,want to preserve edges,1. If,s,h,2,is zero,No noise,the filter should return,g,(,x,y,),because,g,(,x,y,),=,f,(,x,y,),2. If,s,L,2,is high relative to,s,h,2,Edges,(should be preserved),the filter should return the value close to,g,(,x,y,),3. If,s,L,2,=,s,h,2,Areas inside objects,the filter should return the arithmetic mean value,m,L,Formula:,Concept:,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Adaptive Noise Reduction Filter: Example,Image,corrupted,by,additive,Gaussian,noise with,zero mean,and,s,2,=1000,Image,obtained,using,a 7x7,arithmetic,mean filter,Image,obtained,using,a 7x7,geometric,mean filter,Image,obtained,using,a 7x7,adaptive,noise,reduction,filter,Algorithm:,Level,A,:,A1=,z,median, z,min,A2=,z,median, z,max,If A1 0 and A2 0, goto level B,Else increase window size,If window size 0 and B2 0, return,z,xy,Else return,z,median,Adaptive Median Filter,z,min,= minimum gray level value in,S,xy,z,max,= maximum gray level value in,S,xy,z,median,= median of gray levels in,S,xy,z,xy,= gray level value at pixel (,x,y,),S,max,= maximum allowed size of,S,xy,where,Purpose:,want to remove impulse noise while preserving edges,Level,A,:,A1=,z,median, z,min,A2=,z,median, z,max,Else, Window is not big enough,increase window size,If window size 0 and B2 0 and A2 0, goto level B,Level,B,:,Determine,whether,z,median,is an impulse or not,Determine,whether,z,xy,is an impulse or not,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Adaptive Median Filter: Example,Image corrupted,by salt-and-pepper,noise with,p,a,=,p,b,= 0.25,Image obtained,using,a 7x7,median filter,Image obtained,using,an adaptive,median filter with,S,max,= 7,More small details are preserved,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Estimation of Degradation Model,Degradation model:,Purpose:,to estimate,h,(,x,y),or,H,(,u,v,),Methods:,1. Estimation by Image Observation,2. Estimation by Experiment,3. Estimation by Modeling,or,Why?,If we know exactly,h,(,x,y,),regardless of noise, we can do,deconvolution to get,f,(,x,y,),back from,g,(,x,y,),.,Estimation by Image Observation,f,(,x,y,),f,(,x,y,)*,h,(,x,y,),g,(,x,y,),Subimage,Reconstructed,Subimage,DFT,DFT,Restoration,process by,estimation,Original image (unknown),Degraded image,Estimated Transfer,function,Observation,This case is used when we,know only,g,(,x,y,) and cannot,repeat the experiment!,Estimation by Experiment,Used when we have the same equipment set up and can repeat the,experiment.,Input impulse image,System,H,( ),Response image from,the system,DFT,DFT,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Estimation by Modeling,Used when we know physical mechanism underlying the image,formation process that can be expressed mathematically.,Atmospheric,Turbulence model,Example:,Original image,Severe turbulence,k,= 0.00025,k,= 0.001,k,= 0.0025,Low turbulence,Mild turbulence,Estimation by Modeling: Motion Blurring,Assume that camera velocity is,The blurred image is obtained by,where,T,= exposure time.,Estimation by Modeling: Motion Blurring (cont.),Then we get, the motion blurring transfer function:,For constant motion,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Motion Blurring Example,For constant motion,Original image,Motion blurred image,a,=,b,= 0.1,T,= 1,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Inverse Filter,after we obtain,H,(,u,v,), we can estimate,F,(,u,v,) by the inverse filter:,From degradation model:,Noise is enhanced,when,H,(,u,v,) is small.,To avoid the side effect of enhancing,noise, we can apply this formulation,to freq. component,(,u,v,),with in a,radius,D,0,from the center of,H,(,u,v,),.,In practical, the inverse filter is not,Popularly used.,Inverse Filter: Example,Original image,Blurred image,Due to Turbulence,Result of applying,the full filter,Result of applying,the filter with,D,0,=70,Result of applying,the filter with,D,0,=40,Result of applying,the filter with,D,0,=85,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Wiener Filter: Minimum Mean Square Error Filter,Objective: optimize mean square error:,Wiener Filter Formula:,where,H,(,u,v,) = Degradation function,S,h,(,u,v,) = Power spectrum of noise,S,f,(,u,v,) = Power spectrum of the undegraded image,Approximation of Wiener Filter,Wiener Filter Formula:,Approximated Formula:,Difficult to estimate,Practically,K,is chosen manually to obtained the best visual result!,Wiener Filter: Example,Original image,Blurred image,Due to Turbulence,Result of the,full inverse filter,Result of the inverse,filter with,D,0,=70,Result of the,full Wiener filter,Wiener Filter: Example (cont.),Original image,Result of the inverse,filter with,D,0,=70,Result of the,Wiener filter,Blurred image,Due to Turbulence,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Example: Wiener Filter and Motion Blurring,Image,degraded,by motion,blur +,AWGN,Result of the,inverse filter,Result of the,Wiener filter,s,h,2,=650,s,h,2,=325,s,h,2,=130,Note:,K,is,chosen,manually,Degradation model:,Written in a matrix form,Constrained Least Squares Filter,Objective: to find the minimum of a criterion function,Subject to the constraint,We get a constrained least square filter,where,P,(,u,v,) = Fourier transform of,p,(,x,y,) =,where,Constrained Least Squares Filter: Example,Constrained least square filter,g,is adaptively adjusted to achieve the best result.,Results from the previous slide obtained,from the constrained least square filter,Constrained Least Squares Filter: Example (cont.),Image,degraded,by motion,blur +,AWGN,Result of the,Constrained,Least square,filter,Result of the,Wiener filter,s,h,2,=650,s,h,2,=325,s,h,2,=130,Constrained Least Squares Filter:Adjusting,g,Define,It can be shown that,We want to adjust gamma so that,where a = accuracy factor,Specify an initial value of,g,Compute,Stop if is satisfied,Otherwise return step 2 after,increasing,g,if,or,decreasing,g,if,Use the new value of,g,to recompute,1,1,Constrained Least Squares Filter:Adjusting,g,(cont.),For computing,For computing,(Images from Rafael C.,Gonzalez and Richard E.,Wood, Digital Image,Processing, 2,nd,Edition.,Constrained Least Squares Filter: Example,Original image,Blurred image,Due to Turbulence,Results obtained from constrained least square filters,Use wrong noise,parameters,Correct parameters:,Initial,g,= 10,-5,Correction factor = 10,-6,a = 0.25,s,h,2,= 10,-5,Wrong noise parameter,s,h,2,= 10,-2,Use correct noise,parameters,Geometric Mean filter,This filter represents a family of filters combined into a,single expression,a,= 1, the inverse filter,a,= 0, the Parametric Wiener filter,a,= 0,b,= 1, the standard Wiener filter,b,= 1,a, 0.5, More like the Wiener filter,Another name: the spectrum equalization filter,Geometric Transformation,These transformations are often called,rubber-sheet transformations,:,Printing an image on a rubber sheet and then stretch this sheet according,to some predefine set of rules,.,A geometric transformation consists of 2 basic operations:,1.,A spatial transformation :,Define how pixels are to be rearranged in the spatially,transformed image.,2.,Gray level interpolation :,Assign gray level values to pixels in the spatially,transformed image.,Geometric Transformation : Algorithm,Distorted image,g,Select coordinate,(x,y),in,f,to be restored,Compute,3. Go to pixel,in a distorted image,g,Image,f,to be,restored,4. get pixel value at,By gray level interpolation,5. store that value in pixel,f,(,x,y,),1,3,5,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Spatial Transformation,To map between pixel coordinate,(,x,y,),of,f,and pixel coordinate,(,x,y,),of,g,For a bilinear transformation mapping between a pair of,Quadrilateral regions,To obtain,r,(,x,y,),and,s,(,x,y,),we need,to know 4 pairs of coordinates,and its corresponding,which are called,tiepoints,.,(Images from Rafael C. Gonzalez and Richard E.,Wood, Digital Image Processing, 2,nd,Edition.,Gray Level Interpolation: Nearest Neighbor,Since may not be at an integer coordinate, we need to,Interpolate the value of,Example interpolation methods that can be used:,1. Nearest neighbor selection,2. Bilinear interpolation,3. Bicubic interpolation,Geometric Distortion and Restoration Example,Original image and,tiepoints,Tiepoints of distorted,image,Distorted image,Restored image,Use nearest,neighbor,intepolation,(Images from Rafael C.,Gonzalez and Richard E.,Wood, Digital Image,Processing, 2,nd,Edition.,Geometric Distortion and Restoration Example,(cont.),Original image and,tiepoints,Tiepoints of distorted,image,Distorted image,Restored image,Use bilinear,intepolation,(Images from Rafael C.,Gonzalez and Richard E.,Wood, Digital Image,Processing, 2,nd,Edition.,Example: Geometric Restoration,(Images from Rafael C.,Gonzalez and Richard E.,Wood, Digital Image,Processing, 2,nd,Edition.,Original image,Geometrically distorted,image,Difference between,2 above images,Restored image,Use the same,Spatial Trans.,as in the previous,example,知识回顾,Knowledge Review,祝您成功！,
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