Image Restoration Image Degradation and Restoration

Transcription

1 Image Degradation and Restoration hxy Image Degradation Model: Spatial domain representation can be modeled by: g x y h x y f x y x y Freqency domain representation can be modeled by: G F N Prepared By: Dr. asan Demirel PhD

2 Common Noise Models: Most types of noise are modeled as known probability density fnctions Noise model is decided based on nderstanding of the physics of the sorces of noise. Gassian: poor illmination Rayleigh: range image Gamma/Exp: laser imaging Implse: falty switch dring imaging Uniform is least sed. Parameters can be estimated based on histogram on small flat area of an image Prepared By: Dr. asan Demirel PhD

3 Restoration methods: The following methods are sed in the presence of noise. Mean filters Arithmetic mean filter Geometric mean filter armonic mean filter Contra-harmonic mean filter Adaptie filters Adaptie local noise redction filter Adaptie median filter Order statistics filters Median filter Max and min filters Mid-point filter Prepared By: Dr. asan Demirel PhD

4 Restoration in the presence of Noise: Adaptie Local Noise Redction Filter: Mean and ariance are the simplest statistical measres of a random noise. Mean gies the measre of the aerage gray leel in a local region Variance gies the measre of the aerage contrast in the region. Consider a filter operating in a local region S xy where the response of the filter at any point xy depends on: a gxy the ale of the noisy image at xy b s the additie noise ariance c m L local mean of pixels in S xy. d s L the local ariance in S xy. Prepared By: Dr. asan Demirel PhD

5 Restoration in the presence of Noise: Adaptie Local Noise Redction Filter: Consider an adaptie filter where the following conditions are satisfied: 1. If s = the filter shold retrn gxy zero-noise case [ fxy=gxy ].. If s L >> s the filter shold retrn a ale close to gxy. igh local ariance is associated with edges and shold be presered. 3. If s L = s retrn arithmetic mean of S xy. This occrs when local noise has the same properties of the entire image. Aeraging simply redces the noise. According to the preceding assmptions the filter response can be modeled as: s s L f ˆ x y g x y g x y m L The only nknown parameter in the aboe adaptie filter model is s The other parameters can be calclated at the local neighborhood of S xy. Prepared By: Dr. asan Demirel PhD

6 Restoration in the presence of Noise: Adaptie Local Noise Redction Filter: Image corrpted by Gassian Noise with s =1 μ= 7x7 Arithmetic Mean Filter 7x7 Geometric Mean Filter 7x7 Adaptie Filter Prepared By: Dr. asan Demirel PhD

7 Restoration in the presence of Noise: Adaptie Median Filter: Adaptie median filter can filter implse noise with ery high probabilities. Additionally smoothes the nonimplse noise which is not the featre of a traditional median filter. The filter ses the following parameters in the neighborhood of S xy : z min = minimm gray leel ale in S xy. z max = maximm gray leel ale in S xy. z med = median of gray leels in S xy. z xy = gray leel at coordinates xy. S max = maximm allowed size of S xy. Note that nlike the other filters the size of S xy increases dring the filtering operation. Changing size of the filter mask does not change the fact that the otpt of the filter is still a single ale centering the mask. Prepared By: Dr. asan Demirel PhD

8 Restoration in the presence of Noise: Adaptie Median Filter: The Adaptie median filtering Algorithm: Two leels exist Leels A and B Leel A: Leel B: A1 = z med z min A = z med z max if A1> and A< goto Leel B else increase the window size if window size<s max repeat Leel A else otpt z xy. B1 = z xy z min B = z xy z max if B1> and B< otpt z xy. else otpt z med. Prepared By: Dr. asan Demirel PhD

9 Restoration in the presence of Noise: Adaptie Median Filter: Undesired discontinities Image corrpted by salt & pepper noise with P a =P b =.5 7x7 Median Filter Adaptie Median Filter with S max =7 Prepared By: Dr. asan Demirel PhD

10 Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: Bandreject bandpass and notch filters can be sed for periodic noise remoal. Bandreject filters remoe/attenate a band of freqencies abot the origin of the Forier transform. Prepared By: Dr. asan Demirel PhD

11 Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: An Ideal Bandreject filter is gien by: 1 1 if if if D D D W D D W D W D W W is the width of the band D is the radial center D distance from the origin. Btterworth Bandreject filter is gien by: n 1 D D W 1 D W is the width of the band D is the radial center D distance from the origin. n is the order of the filter Prepared By: Dr. asan Demirel PhD

12 Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: Gassian Bandreject filter is gien by: 1 e 1 D D D W W is the width of the band D is the radial center D distance from the origin. Image corrpted by sinsoidal noise Spectrm of corrpted image Btterworth Bandreject Filter n=4 Filtered Image Prepared By: Dr. asan Demirel PhD

13 Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: Bandpass filters perform the opposite fnction of the bandreject filters and the filter transfer fnction of a bandpass filter is gien by: 1 bp br Image corrpted by sinsoidal noise Spectrm of corrpted image Btterworth Bandpass filter Noise Image Prepared By: Dr. asan Demirel PhD

14 Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: Notch filters rejects/passes freqencies in a predefined neighborhoods abot the center freqency. Notch filters appear in symmetric pairs de to the symmetry of the Forier transform. The transfer fnction of ideal notch filter of radis of D with centers at and by symmetry at - - is gien by: D D 1 if D 1 otherwise D or D 1/ M / N / 1 M / N / 1/ D Prepared By: Dr. asan Demirel PhD

15 Prepared By: Dr. asan Demirel PhD Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: Notch filters: The transfer fnction of Btterworth notch filter of order n and of radis of D with centers at and by symmetry at - - is gien by: n D D D The transfer fnction of Gassian notch filter of radis of D with centers at and by symmetry at - - is gien by: D D D e

16 Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: Notch filters: Ideal Notch Filter Btterworth Notch Filter of order Gassian Notch Filter Prepared By: Dr. asan Demirel PhD

17 Restoration in the presence of Noise: Periodic Noise Remoal by Freqency Domain Filtering: Notch filters: Original Noisy image with ndesired horizontal scanning lines Spectrm of the image A simple ideal Notch filter along the ertical axis The filtered noise pattern Corresponding the horizontal artifacts filtered image free of horizontal scanning lines. Prepared By: Dr. asan Demirel PhD

18 Estimating the Degradation Fnction : There are 3 principal methods of estimating the degradation fnction for Image Restoration: 1 Obseration Experimentation 3 Mathematical Modeling. The degradation fnction can be estimated by isally looking into a small section of the image containing simple strctres with strong signal contents like part an object and the backgrond. Gien a small sbimage g s xy we can manally i.e. filtering remoe the degradation in that region with an estimated sbimage fˆ s x y and assming that the additie noise is negligible in sch an area with a strong signal content. s G Fˆ s s aing s estimated for sch a small sbimage the shape of this degradation fnction can be sed to get an estimation of for the entire image. Prepared By: Dr. asan Demirel PhD

19 Estimating the Degradation Fnction : There are 3 principal methods of estimating the degradation fnction for Image Restoration: 1 Obseration Experimentation 3 Mathematical Modelling. Estimation by Image Experimentation: If we hae the acqisition deice prodcing degradation on images we can se the same deice to obtain an accrate estimation of the degradation. This can be achieed by applying an implse bright dot as an inpt image. The Forier transform of an implse is constant therefore. G A Where A is a constant describing the strength of the implse. Note that the effect of noise on an implse is negligible. Prepared By: Dr. asan Demirel PhD

20 Estimating the Degradation Fnction : There are 3 principal methods of estimating the degradation fnction for Image Restoration: 1 Obseration Experimentation 3 Mathematical Modelling. Estimation by Image Experimentation: Implse Image Degraded Implse Image consider as hxy Simply take the Forier transform of the degraded image and after normalization by a constant A se it as the estimate of the degradation fnction. Prepared By: Dr. asan Demirel PhD

21 Estimating the Degradation Fnction : There are 3 principal methods of estimating the degradation fnction for Image Restoration: 1 Obseration Experimentation 3 Mathematical Modelling. Estimation by Mathematical Modeling: Sometimes the enironmental conditions that cases the degradation can be modeled by mathematical formlation. For example the atmospheric trblence can be modeled by: e k 5 / 6 k is a constant that depends on the natre of the Trblence This eqation is similar to Gassian LPF and wold prodce blrring in the image according to the ales of k. For example if k=.5 the model represents seere trblence if k=.1 the model represents mild trblence and if k=.5 the model represents low trblence. Once a reliable mathematical model is formed the effect of the degradation can be obtained easily. Prepared By: Dr. asan Demirel PhD

22 Estimating the Degradation Fnction : Estimation by Mathematical Modeling: Illstration of the atmospheric trblence model Negligible trblence Seere trblence k=.5 Mild trblence k=.1 Low trblence k=.5 Prepared By: Dr. asan Demirel PhD

23 Estimating the Degradation Fnction : Estimation by Mathematical Modeling: In some applications the mathematical model can be deried by treating that the image is blrred by niform linear motion between the image and the sensor dring image acqisition. The motion blr can be modeled as follows: g x Let fxy be sbject to motion in x- and y-direction by time arying motion components x t and y t. The total exposre is obtained by integrating the instantaneos exposre oer the time interal dring the shtter of the imaging deice is open. If T is the dration of the exposre than y G f x x t y y t dt T g x T f y e j xy dx dy x x t y y t dte gxy is the blrred image j xy dx dy Prepared By: Dr. asan Demirel PhD

24 Estimating the Degradation Fnction : Estimation by Mathematical Modeling: Reersing the order of integration yields: G j xy x x t y y t e dx dy dt T f Using the translation property of the Forier Transform the inner part can be simplified G T F e F T e j x j x t y t y t t dt dt Then T j x t y t e dt Prepared By: Dr. asan Demirel PhD

25 Estimating the Degradation Fnction : Estimation by Mathematical Modeling: By assming that the linear niform motion is in x-direction only at a rate of x t=at/t the image coers a distance when t=t. T e jx t dt T sin a e a ja If we allow the motion in y-direction with y t=bt/t the model becomes T e jat/ T dt T a b sin j ab a b e Prepared By: Dr. asan Demirel PhD

26 Estimating the Degradation Fnction : Estimation by Mathematical Modeling: The reslt of the modeled motion blr is demonstrated in the following example: Original Image Blrred image with a=b=.1 and T=1 T a b sin j ab a b e Prepared By: Dr. asan Demirel PhD

27 Inerse Filtering: Until now or focs was the calclation of degradation fnction. aing calclated/estimated the next step is the restoration of the degraded image. The simplest way of image restoration is by sing Inerse filtering: ˆ G F Fˆ is the Forier transform of the restored image Fˆ F N Unknown random fnction Mst not be ery small. Otherwise the noise dominates In Inerse filtering we simply take sch that the noise does not dominate the reslt. This is achieed by inclding only the low freqency components of arond the origin. Note that the origin M/N/ corresponds to the highest amplitde component. Prepared By: Dr. asan Demirel PhD

28 Inerse Filtering: Consider the degradation fnction of the atmospheric trblence for the origin of the freqency spectrm e k M / N / 5 / 6 If we consider a Btterworth Lowpass filter of arond the origin we will only pass the low freqencies high amplitdes of. As we increase the ctoff freqency of the LPF more smaller amplitdes will be inclded. Therefore instead of the degradation fnction the noise will be dominating. Fˆ F N Mst not be ery small. Otherwise the noise dominates Prepared By: Dr. asan Demirel PhD

29 Inerse Filtering: Consider the degradation fnction of the atmospheric trblence for the origin of the freqency spectrm Reslt of fll filter/degradation Ctoff otside of radis 4 Inpt image with Seere trblence k=.5 48x48 pixels Ctoff otside of radis 85 Ctoff otside of radis 7 Prepared By: Dr. asan Demirel PhD

30 Wiener Min Mean Sqare Error Filtering: Inerse filtering does not consider the additie noise for restoration. The Wiener Filter consider both the degradation fnction and the statistical characteristics of the noise in the restoration process. The method tries to minimize the mean sqare error MSE between the ncorrpted image and the estimate of the image by: e E f fˆ E{.} is the expected ale of the argment The noise and the image are assmed to be ncorrelated. The minimm of the error fnction gien aboe is achieed in the freqency domain by the following expression. Fˆ S f * S f G S Prepared By: Dr. asan Demirel PhD

31 Prepared By: Dr. asan Demirel PhD Wiener Min Mean Sqare Error Filtering: / 1 / * * ˆ G S S G S S G S S S F f f f f * * F S N S f Degradation fnction. Complex conjgate of Power spectrm of the noise. Power spectrm of the ndegraded image.

32 Prepared By: Dr. asan Demirel PhD Wiener Min Mean Sqare Error Filtering: / 1 ˆ G S S F f When the power spectrm of the ndegraded image and noise are not known the ratio of the power spectrms of the noise and image is assmed to be constant. When not known. Assmed to be constant 1 ˆ G K F Typically different ales of K are chosen and the image qality is measred by MSE. The ale of K is chosen in sch a way that the MSE is minimized.

33 Wiener Min Mean Sqare Error Filtering: Inpt image with Seere trblence blr k=.5 48x48 pixels Reslt of Fll inerse filtering Radially limited Inerse Filtering with a ctoff radis of 7 Reslt of Wiener filtering with and optimized K Prepared By: Dr. asan Demirel PhD

34 Wiener Min Mean Sqare Error Filtering: Image corrpted by motion blr and additie noise Variance of the noise is one order of magnitde less Variance of the noise is fie order of magnitde less Degraded Inpt Image Reslt of Inerse filtering Reslt of Wiener filtering Prepared By: Dr. asan Demirel PhD