Lecture 6
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The document discusses various nonlinear spatial domain filters for image processing, including median filters, threshold boolean filters, and rank-order filters. The median filter replaces pixel intensities with the median value within a local window. It is highly effective at removing impulse noise while preserving edges. Threshold boolean filtering processes separate binary slices of an image before recombining, allowing different processing of each intensity level. Rank-order filters analyze the variational series within a window to perform nonlinear averaging, with examples including the median and rank-order EV filters.
Lecture 6
- 1. SPATIAL DOMAIN FILTERING NONLINEAR FILTERS CS-467 Digital Image Processing 1
- 2. Nonlinear Filtering • As well as any linear filter, a nonlinear filter is defined by an operator where is a signal to be processed, T is a filtering operator and is a resulting signal • Linearity does not hold for a nonlinear filter 2 ( ),g x y ( )ˆ ,f x y ( ) ( )( )ˆ , ,f x y T g x y= ( )( ) ( )( ) ( ), ( , ) , ( , )T af x y b x y aT f x y bT x yη η+≠+
- 3. Nonlinear Spatial Domain Filtering • Nonlinear spatial filtering of an image of size M x N with a filter of size m x n is defined by the expression where is a local m x n window around the pixel g(x,y) in the image g to be processed, and T is the nonlinear filtering operator 3 ( ) ( )ˆ , xyf x y T S= xyS
- 4. Variational Series – Series of Order Statistics • An arrangement of the values of a random sample with distribution function F(x) in ascending sequence where • The series is used to construct the empirical distribution function , where is the number of terms of the series which are smaller than x. 4 1,..., nx x (1) (2) ( )... nx x x≤ ≤ ≤ (1) ( ) 1,..., 1,..., min ; maxi n i i n i n x x x x = = = = ( ) /n xF x m n= xm
- 5. Variational Series – Series of Order Statistics • A rank (order statistics) of an element in a variational series is its serial number in the series Rank 1 Rank 2 … Rank n 5 (1) (2) ( )... nx x x≤ ≤ ≤
- 6. Variational Series – Series of Order Statistics • Example1. Let us have the following sample 200, 101, 102, 125, 5, 10, 207, 180, 100 Its variational series is 5, 10, 100, 101, 102, 180, 200, 201, 207 • Example2. Let us have the following sample 200, 101, 102, 101, 207, 101, 100, 207, 5 Its variational series is 5, 100, 101, 101, 101, 102, 200, 207, 207 6
- 7. Median Filter – the Simplest Spatial Domain Nonlinear Filter • Median filter replaces an intensity value in each pixel by a local median taken over a local n x m processing window: where is a local m x n window around the pixel (x,y) in the image g (to be processed), and MED is the median value taken over the window 7 ( ) ( )Mˆ ED, xyf x y S= xyS xyS
- 8. Median Filter – Implementation • To implement the median filter with an n x m processing window, it is necessary to build a variational series from the elements of the processing window around the pixel (x,y) (to be processed) • The central element of the vatiational series is the median of the intensity values in the window 8 xyS xyS
- 9. Median Filter and Impulse Noise • Median filter is highly efficient for impulse noise filtering • Median filter with 3x3 window can almost completely remove impulse noise with the corruption rate up to 30% • Applied iteratively, it can remove even noise with a higher corruption rate 9
- 10. Median Filter and Impulse Noise • The ability of the median filter to remove impulse noise is based on the following: • Impulses have unusually high (or low) intensity values compared to their neighbors • This means that they are located in the left or right ends of the variational series built from the intensity values in a local window around the pixel (x,y) 10 xyS
- 11. Median Filter and Impulse Noise 11 25 36 36 47 48 104 11 201 0 120 25 36 36 47 104 110 148 20 201 25 36 36 47 48 104 1 48 10 120 3x3 window, 201 is the impulse Variational Series Median The processing result The impulse 201 was removed
- 12. Mean Filter and Impulse Noise • Unlike the median filter, the mean filter is unable to remove impulse noise. It only “washes” impulses 12 3x3 window, 201 is the impulse The processing result The impulse 201 was just “washed” 25 36 36 47 48 104 11 201 0 120 25 36 36 47 48 104 1 81 10 120 Mean = 81
- 13. Disadvantages of Median Filtering • Median filter removes impulse noise, but it also smoothes (“washes”) all edges and boundaries and may “erase” all details whose size is about n/2 x m/2 ,where n x m is a window size • As a result, an image becomes “fuzzy” • Median filter is not so efficient for additive Gaussian noise removal, it yields to linear filters 13
- 14. Detection of Impulse Noise • To reduce a image smoothing by the median filter, impulse detectors should be used • A detector analyses local statistical (and possibly other) characteristics in a local window around each pixel using some criteria, and marks those pixels that are corrupted by noise • Then only marked pixels shall be processed by the median filter 14
- 15. Differential Rank Impulse Detector • This detector is based on the following reasonable assumptions Impulses are located either in the ends of the variational series (salt-end-pepper and bipolar noise) or close to these ends (random noise) The difference between the intensity values of impulse and its neighbor located in the variational series between this impulse and the median should not exceed some reasonable threshold value 15
- 16. Differential Rank Impulse Detector • Let R(x) be the rank of an element x in the variational series and var(k) be the intensity value of the pixel whose rank is k 16 ( ) ( ) ( ) ( ) ( , ) var ( , ) 1 , if ( , ) ( , ) var ( , ) 1 , if ( , ) 0 , otherwise, xy xy S xy S g x y R g x y R g x y R MED d g x y R g x y R g x y R MED − − > = − + <
- 17. Differential Rank Impulse Detector • Let r be the length (in ranks) of the interval in the variational series where an impulse can be located • Let s be the maximal acceptable difference between the intensity value of the pixel of interest and its neighbor located in the variational series between this impulse and the median dxy= Var( )-Var( )≥ s dxy= Var( )-Var( ) ≥ s 17 r=2 r=2Impulses Differential Criteria of an impulse
- 18. Differential Rank Impulse Detector • Differential Rank Impulse Detector (DRID) works as follows • The pixel (x,y) is considered noisy if the following condition holds: the intensity value g(x,y) in the pixel of interest is located in the r-interval from one of the ends of the variational series, and the difference between this value and its neighbor located in the variational series closer to the median exceeds s 18 ( ) ( )( ) ( )( ( , ) ( ( , )) 1 xyR g x y R g x y mn dr r s≤ ∨ ≥ − + ∧ ≥ xyd
- 19. Filtering of Scratches • Any scratch is a collection of impulses – there are consecutive impulses extended in some direction along some virtual curve • To remove a scratch from an image, it is necessary to process it locally and to use the mask median filter 19
- 20. Filtering of Scratches • In the mask median filter, a median is taken not over all the pixels in a filtering window, but only over those marked in the mask • To remove a scratch, it is necessary to create a “counterscratch” mask including there pixels from the virtual perpendicular to the scratch 20
- 22. Threshold Boolean Filtering (TBF) • Threshold Boolean filtering (also often referred to as stack filtering) is reduced to: Splitting an image with M+1 gray levels into M binary planes (slices) by thresholding Separate processing of the binary planes (slices) using a processing Boolean function Merging the processed binary planes into a resulting image 22
- 23. Threshold Boolean Filtering (TBF) • Let g be a scale image with (M+1) gray levels • Let be a kth binary plane • Then a filtering operation is determined by where F is the processing Boolean function, is a kth binary plane of a local window around a pixel of interest 23 ( ) 1 , if ( , ) ( , ) ; 1,..., 0 , otherwise k f x y k g x y k M ≥ = = ( ) ( )( ) 1 ˆ , M k xy k Ff x j S = = ∑ ( )k ijS
- 24. Threshold Boolean Filter (TBF) Image M Binary slices Threshold Boolean Filter Result The idea behind the Threshold Boolean Filter is to process each binary plane of an image separately, and after filtering all the binary planes are combined together to form the resulting image. The following diagram illustrates this idea: ( ) ( )( ) 1, if , , 0, otherwise 1,2,..., the binary slice index k g x y k g x y k M ≥ = = − ( ) 1 ˆ , M k k f x y = ∑
- 25. Impulse Noise Filtering using TBF ),,,,,,,,( 987654321 xxxxxxxxxf ∧∨∧∨ ∨∧∧= ≠ == ≠ == )()( 5 1 5 5 1 5 11 k i k ik T j k i k ik T j xxxx tj j tj j (...)),...,( 25155 fxxf x = ∧∨∧∨ ∨∧∧∧ ≠ == ≠ == )()( 5 1 5 5 1 5 11 k i k ik P j k i k ik P j xxxx sj j sj j The filter is defined by the following functions. For 3x3 window: For 5x5 window where X5 is the central element of the window, i.e. the pixel under consideration, T is the number of all possible elementary conjunctions of t variables out of 8 (the number of pixels in a 3x3 window), P is the number of all possible elementary conjunctions of s variables out of 16 (5x5 window). 8 tT C= 16 pP C=
- 26. Impulse Noise Filtering using TBF • TBF is highly efficient for filtering of random impulse noise filtering with a low corruption rate (≤5%). It may preserve image details with a higher accuracy than other filters • Disadvantages: strictly dependent on the successful choice of the parameters and computationally costly (compared to median and rank-order ER filters, even used in conjunction with noise detector) 26
- 27. Rank-Order (Order Statistic) Filters • Rank-order filtering is based on the analysis of the variational series followed by some nonlinear averaging of a signal • The median filter is a representative of the rank-order filters family • We will consider rank-order filters using their classification done by Leonid Yaroslavsky 27
- 28. Rank-Order EV Filter • Let be a value, which determines the following interval (EV) around the intensity value of the pixel of interest • An EV interval contains those intensity values whose difference from the intensity value in the pixel of interest does not exceed • A suboptimal value of the parameter is - standard deviation measured over an image 28 vε ( ) ( ) ( ){ }, : ( , ) , ; ,v xyEV g i j g x y g i j g i j Sε= − ≤ ∈ vε σ= ( , )g x y vε
- 29. Rank-Order EV Filter • Rank-order EV filter is determined as follows • The first option leads to more careful and accurate reduction of additive noise (compared to linear filters) 29 ( ) ( )( ) ( ) ( )( ) ( ) MEAN , , , ˆ , or MED , , , g i j g i j EV f x y g i j g i j EV ∈ = ∈
- 30. Rank-Order EV Filter • Rank-order EV filter preserves small details and image boundaries with a higher accuracy than linear filters. With a careful choice of it is possible to avoid smoothing of n/2 x m/2 details • Disadvantage is the dependence on the choice of the parameter . • This filter may show better results if EV has been chosen adaptively for each window 30 vε xyS vε
- 31. Rank-Order ER Filter • Let is a value, which determines the following interval (ER) around the intensity value of the pixel of interest • An ER interval contains those intensity values whose ranks differ from the rank of the intensity value in the pixel of interest by not more than 31 rε ( ) ( ) ( )( ) ( ){ }, : ( , ) , ; ,r xyER g i j R g x y R g i j g i j Sε= − ≤ ∈ rε
- 32. Rank-Order ER Filter • Rank-order ER filter is determined as follows • The second option leads to more careful removal of random impulse noise with a low corruption rate (1-5%) (the filter should be applied iteratively). It can also be used to “close” impulse gaps (spots) – a window larger than a spot should be used • The second option is also better for speckle (multiplicative) noise 32 ( ) ( )( ) ( ) ( )( ) ( ) MEAN , , , ˆ , or MED , , , g i j g i j ER f x y g i j g i j ER ∈ = ∈
- 33. Rank-Order ER Filter • Advantage of the rank-order ER filter is that it preserves small details and image boundaries with a higher accuracy than the median filter. With a careful choice of , it is possible to remove salt-and-pepper impulse noise iteratively with more careful preservation of image details • Disadvantage is the dependence on the choice of the parameter ER. • This filter may show better results if ER has been chosen adaptively for each window 33 rε xyS
- 34. Rank-Order KNV Filter • Let k be a value, which determines the following interval (KNV) around the intensity value of the pixel of interest • A KNV interval contains k closest pixels from to the xyth pixel, in terms of the intensity values closest to the intensity 34 ( ) ( ) ( )( ) ( ) , : ( , ) , ; ; , s xy g g i j R g x y R g i j k nm KNV s g i j S = − ≤ = ≤ ∈ ( , )g x y xyS
- 35. Rank-Order KNV Filter • Rank-order KNV filter is determined as follows • The first option leads to more careful reduction of additive noise (compared to linear filters) 35 ( ) ( )( ) ( ) ( )( ) ( ) MEAN , , , ˆ , or MED , , , g i j g i j KNV f x y g i j g i j KNV ∈ = ∈
- 36. Rank-Order KNV Filter • Advantage of the rank-order KNV filter is that it preserves small details whose area is < k squared pixels • However, its ability to reduce noise is limited. It is better to use this filter only when it is absolutely necessary to preserve some small details • Another disadvantage is its dependence on the choice of the parameter k • This filter may show better results if k has been chosen adaptively for each window 36 xyS
- 37. Geometric Mean Filter • Can preserve details a little bit better than the arithmetic mean filter 37 ( ) 1 ( , ) ˆ , ( , ) xy mn s t S f x y S s t ∈ = ∏
- 38. Harmonic Mean Filter • Can be good for multiplicative (speckle) noise removal 38 ( ) ( , ) ˆ , 1 ( , )xys t S mn f x y g s t∈ = ∑
- 39. Contraharmonic Mean Filter where Q is the order of the filter • With positive Q eliminates pepper noise , while with negative Q Eliminates salt noise • With Q=0 reduces to the arithmetic mean filter, and with Q=1 – to the harmonic mean filter 39 ( ) 1 ( , ) ( , ) ( , ) ˆ , ( , ) xy xy Q s t S Q s t S g s t f x y g s t + ∈ ∈ = ∑ ∑