Image Enhancement in the Frequency Domain Periodicity and the need for Padding:

Transcription

1 Prepared B: Dr. Hasan Demirel PhD Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: Periodicit propert of the DFT: The discrete Forier Transform and the inverse Forier transforms are periodic. Hence: N v M F N v F v M F v F N M f N f M f f Conjgate Smmetr propert : The DFT is conjgate smmetric. The `*` indicates the conjgate operation on a comple nmber. v F v F v F v F

2 Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: Prepared B: Dr. Hasan Demirel PhD

3 Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: According to the convoltion theorem the mltiplication in the freqenc domain is the convoltion in the spatial domain. Consider the D convoltion of f and h f h M M m0 f m h m discrete fnctions with periodicit propert Prepared B: Dr. Hasan Demirel PhD

4 Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: The illstration of the D convoltion: Correct convoltion Wraparond error De to periodicit Prepared B: Dr. Hasan Demirel PhD

5 Prepared B: Dr. Hasan Demirel PhD Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: The soltion of the wraparond error: Given f and h consist of A and B points. Zeros are appended to both fnctions so that both of the fnctions have identical periods denoted b P. The D etended/padded fnctions are given b: P A A f f e 0 0 P B B h h e 0 0 B A P

6 Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: The soltion of the wraparond error: etended/padded fnctions. Etended/padded region in f Etended/padded region in h The padding provides enogh separation of the 2 signals. So there is no Wraparond error problem Prepared B: Dr. Hasan Demirel PhD

7 Prepared B: Dr. Hasan Demirel PhD Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: The soltion of the wraparond error: The 2D etended/padded fnctions f and h with sizes AB and CD respectivel can be defined b. Q B or P A B and A f f e Q D or P C D and C h h e 0 0 0

8 Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: The soltion of the wraparond error: The 2D etended/padded fnctions f and h with sizes AB and CD respectivel can be defined b. Wraparond Error No Wraparond Error Prepared B: Dr. Hasan Demirel PhD

9 Image Enhancement in the Freqenc Domain Periodicit and the need for Padding: The soltion of the wraparond error: Padded inpt image f Needs to be cropped for the final answer * Convoltion Padded spatial Filter h g= f * h Prepared B: Dr. Hasan Demirel PhD

10 Prepared B: Dr. Hasan Demirel PhD Image Enhancement in the Freqenc Domain Convoltion and Correlation: Convoltion: Discrete convoltion of two fnctions f and h with sizes MN is defined b. 0 0 M m N n n m h m n f MN h f v H v F h f v H v F h f Correlation: Discrete correlation of two fnctions f and h with sizes MN is defined b. 0 0 M m N n n m h m n f MN h f v H v F h f v H v F h f

11 Image Enhancement in the Freqenc Domain Convoltion and Correlation: Convoltion: The most important application of the convoltion is the filtering in the spatial and freqenc domains. Correlation: The principal application of the correlation is matching. In matching f is the image containing objects/regions and the h is the object/reqion that we are tring to locate. h is called the template. If there is a match the correlation of the two fnctions will be maimm at the location where the template h finds the highest similarit in fnction f. Note: Image padding is also reqired in Correlation as well as in convoltion. Prepared B: Dr. Hasan Demirel PhD

12 Prepared B: Dr. Hasan Demirel PhD Image Enhancement in the Freqenc Domain Convoltion and Correlation: Correlation: Cross correlation is the special term given to the correlation of two different images. In atocorrelation both images are identical where: 2 v F v F v F f f The spatial atocorrelation is identical to the power spectrm in the freqenc domain. 2 v F v F f

13 Image Enhancement in the Freqenc Domain Convoltion and Correlation: image f template h Correlation: The cross correlation of a template h with an inpt image f. This process is also known as the template matching. padded f padded h Note: The padded images are transformed into the Freqenc Domain with DFT. Comple conjgate of one of the images is mltiplied with the other. The reslting fnction is inverse transformed. Prepared B: Dr. Hasan Demirel PhD

14 Restoration is to attempt to reconstrct or recover an image that has been degraded b sing a a priori knowledge of the degradation. The restoration approach involves the modeling of the degradation and appling the reverse process to recover the original image. Image Degradation Model: Given some knowledge abot the degradation fnction h and some knowledge abot the additive noise the degraded otpt image g can be obtained b: g h f Freqenc domain representation can be modeled b: G v H v F v N v Prepared B: Dr. Hasan Demirel PhD

15 Image Degradation/Restoration Process h Noise Models: Image acqisition digitization and the transmission processes are the primar sorces of the noise. Assmption: In almost all considerations we will assme that the noise is ncorrelated with the image which means that there is no correlation between the piel vales of the image and the noise components. Prepared B: Dr. Hasan Demirel PhD

16 Noise models: Some important Noise PDFs The statistical behaviors of the noise components can be considered as the random variables characterized b the probabilit densit fnction PDF. The following noise PDFs are common for image processing applications: Gassian Noise: Gassian Noise models are freqentl sed in practice. Becase this tpe of noise model is easil tractable in both spatial and freqenc domains. The PDF of the Gassian random variable z is given b: p z e z / 2 Where z represents gra level is the average vale of z and the is the standard deviation and 2 is the variance. Approimatel 68% of the vales will be in the range of [ ] and abot 95% will be in the range of [2 2]. Prepared B: Dr. Hasan Demirel PhD

17 Noise models: Some important Noise PDFs Raleigh Noise: The PDF of Raleigh noise is given b: 2 pz b 0 The mean of the densit is: 2 za / b z a e for z a The variance of the densit is: a b/4 2 b4 4 for z Becase of its skewed distribtion it can be sefl for approimating the distribtion of the images characterized b skewed histograms. a Prepared B: Dr. Hasan Demirel PhD

18 Noise models: Some important Noise PDFs Erlang Gamma Noise: The PDF of Gamma noise is given b: b b az az e for z 0 pz b! 0 for z 0 The mean of the densit is: The variance of the densit is: 2 b a b 2 a Prepared B: Dr. Hasan Demirel PhD

19 Noise models: Some important Noise PDFs Eponential Noise: The PDF of Eponential noise is given b: pz az ae for z 0 0 for z 0 The mean of the densit is: The variance of the densit is: 2 a a 2 The PDF of the Eponential noise is the special case of the Gamma PDF where b=. Prepared B: Dr. Hasan Demirel PhD

20 Noise models: Some important Noise PDFs Uniform Noise: The PDF of Uniform noise is given b: pz b a 0 The mean of the densit is: The variance of the densit is: if a z b otherwise a b 2 2 b a 2 2 Prepared B: Dr. Hasan Demirel PhD

21 Noise models: Some important Noise PDFs Implse salt-and-peper Noise: The PDF of bipolar implse noise is given b: Pa for z a p z Pb for z b 0 otherwise If b>a then the gra-level b will appear as a light dot in the image. Otherwise the a will appear like a dark dot. If P a or P b is zero than implse noise is called nipolar. Tpicall implse noise can be positive or negative. The negative implse is qantized as zero which is black pepper and the positive implse is qantized as ma intensit vale i.e. 255 which is white salt. Prepared B: Dr. Hasan Demirel PhD

22 Noise models: Some important Noise PDFs Consider the following image which contains 3 gra levels. Histogram of the image If the onl degradation in the image is additive noise then the image degraded b the additive noise is given b. Degradation fnction h is ignored. g f Prepared B: Dr. Hasan Demirel PhD

23 Noise models: Some important Noise PDFs Nois Image Histogram of the Nois Image Gassian Raleigh Gamma Prepared B: Dr. Hasan Demirel PhD

24 Noise models: Some important Noise PDFs Nois Image Histogram of the Nois Image Eponential Uniform Salt & Pepper Prepared B: Dr. Hasan Demirel PhD

25 Estimation of Noise Parameters models: If the sensor device i.e. camera is available then the a solid gre area which is niforml illminated is acqired. Then the noise can be estimated b analzing the histogram of this area. If the sensor is not available and alread generated images are to be considered then PDF parameters can be obtained from small patches of reasonable constant gra level. This image strip can be sed to analze the histogram. Then the mean and variance approimations can be etracted from this image strip. Prepared B: Dr. Hasan Demirel PhD

26 Estimation of Noise Parameters models: The following histograms are obtained from the image strips etracted from 3 different images. The shapes of the histograms gives an idea abot the tpe of the noise distribtion. Gassian Raleigh Uniform Based on observation we can sa that the first image strip contains Gassian noise the second one is Raleigh and the third one is Uniform. Prepared B: Dr. Hasan Demirel PhD

27 Estimation of Noise Parameters models: Once we know the PDF tpe then we can estimate the mean and variance from the basic statistics b: z i p zi and 2 z S i z S i 2 z p z i where z i s are the gra-level vales of the piels in image strip S and pz i are the corresponding normalized histogram vales. The PDF parameters of other noise models can be estimated b sing the mean and variance approimations. The variables a and b can easil be calclated from the mean and variance. i In the case of salt&pepper the mean and variance is not needed. The heights of the peaks corresponding to the white and black piels are the estimates of the P a and P b respectivel. Prepared B: Dr. Hasan Demirel PhD

28 Restoration in the presence of Noise: If the onl degradation in an image is the noise then: and g f G v F v N v When the distribtion additive noise is estimated we cannot simpl sbtract the noise terms from the nois image. The method is to se spatial filtering for noise removal/redction. Note that there is no point to transform into the freqenc domain as there is no convoltion in the spatial domain. Onl if there is a periodic noise we can transform into the freqenc domain. Prepared B: Dr. Hasan Demirel PhD

29 Restoration in the presence of Noise: Onl-Spatial Filtering Mean Filters: There are for main tpes of noise redction mean filters that can be sed for image restoration/enhancement. Arithmetic Mean Filter: Let S be the coordinates in a sbimage window of size m n centered at point. The vale of the restored image at an point is given b: fˆ mn s t S g s t This operation can be implemented b a convoltion mask in which all its components have a vale /mn. Local variations in the image is smoothed and noise is redced as a reslt of blrring. Prepared B: Dr. Hasan Demirel PhD

30 Restoration in the presence of Noise: Onl-Spatial Filtering Mean Filters: Geometric Mean Filter: Let S be the coordinates in a sbimage window of size m n centered at point. The vale of the restored image at an point is given b: fˆ g s t s t S In this filter each piel is given b the prodct of the piels in the sbimage windows raised to the power of /mn. Achieves more smoothing bt looses less image details. mn Prepared B: Dr. Hasan Demirel PhD

31 Restoration in the presence of Noise: Onl-Spatial Filtering Mean Filters: Harmonic Mean Filter: Let S be the coordinates in a sbimage window of size m n centered at point. The vale of the restored image at an point is given b: fˆ s t g s t S mn Harmonic mean filter works well with the salt noise bt fails for the pepper noise. It works well with other tpes of noise as well sch as Gassian noise. Prepared B: Dr. Hasan Demirel PhD

32 Restoration in the presence of Noise: Onl-Spatial Filtering Mean Filters: Contraharmonic Mean Filter: Let S be the coordinates in a sbimage window of size m n centered at point. The vale of the restored image at an point is given b: fˆ s ts s ts g s t g s t Q is the order of the filter. This filter is well sited for redcing the effects of salt-and-pepper noise. For negative vales of Q it eliminates the salt noise and for positive vales of Q it eliminates the pepper noise. The filter becomes the arithmetic mean filter for Q=0 and harmonic mean filter for Q= -. Q Q Prepared B: Dr. Hasan Demirel PhD

33 Restoration in the presence of Noise: Onl-Spatial Filtering Mean Filters: Original Additive Gassian noise with =0 2 = Arithmetic Mean Filter f ˆ mn s t S g s t 33 Geometric Mean Filter Sharper details fˆ s ts Prepared B: Dr. Hasan Demirel PhD g s t mn

34 Restoration in the presence of Noise: Onl-Spatial Filtering Mean Filters: Pepper Noise with P p =0. Salt Noise with P s =0. fˆ s ts s ts g s t g s t Q Q 33 Contraharmonic With Q=.5 good for Pepper 33 Contraharmonic With Q=-.5 good for Salt Prepared B: Dr. Hasan Demirel PhD

35 Restoration in the presence of Noise: Onl-Spatial Filtering Order Statistics Filters: The response of these spatial filters is based on the ordering/ranking the piels in the filter mask. Median Filter: Let S be the coordinates in a sbimage window of size m n centered at point. The vale of the restored image at an point is given b: fˆ median s t S g s t The vale of a piel in is replaced b the median of the gra level in the neighborhood characterized b S sbimage. Provides less blrring than the other linear smoothing filters. Median filters are ver effective in bipolar or nipolar implse Salt&Pepper noise. Prepared B: Dr. Hasan Demirel PhD

36 Restoration in the presence of Noise: Onl-Spatial Filtering Order Statistics Filters: Ma and Min Filters : Let S be the coordinates in a sbimage window of size m n centered at point. The vale of the restored image at an point is given b: fˆ ma s t S g s t Usefl for finding the brightest points in the image. Also redces the pepper noise. fˆ min s t S g s t Usefl for finding the darkest points in the image. Also redces the salt noise. Prepared B: Dr. Hasan Demirel PhD

37 Restoration in the presence of Noise: Onl-Spatial Filtering Order Statistics Filters: Midpoint filter : Let S be the coordinates in a sbimage window of size m n centered at point. The vale of the restored image at an point is given b: fˆ 2 ma s g s t ming s t t S s t S This filter combines the order statistics and averaging. Works best for Gassian and niform noise. Prepared B: Dr. Hasan Demirel PhD

38 Restoration in the presence of Noise: Onl-Spatial Filtering Order Statistics Filters: Some traces of pepper noise Salt & Pepper Noise with P a =P b =0. 33 Median Filter first pass Completel removed 33 Median Filter second pass 33 Median Filter third pass Prepared B: Dr. Hasan Demirel PhD

39 Restoration in the presence of Noise: Onl-Spatial Filtering Order Statistics Filters: Some dark border piels are removed Pepper Noise with P b =0. 33 Ma Filter Good for pepper Prepared B: Dr. Hasan Demirel PhD

40 Restoration in the presence of Noise: Onl-Spatial Filtering Order Statistics Filters: Some white dots are darker now. Salt Noise with P a =0. 33 Min Filter Good for Salt Prepared B: Dr. Hasan Demirel PhD