Image filtering in Digital image processing
- 1. DIGITAL IMAGE PROCESSING IMAGE FILTERING B.ABINAYA BHARATHI M.Sc (Cs&IT) , NADAR SARASWATHI COLLEGE OF ARTS AND SCIENCE , THENI. 1
- 3. FILTERING Filtering is a technique used for modifying or enhancing an image like highlight certain features or remove other features. Image filtering include smoothing, sharpening, and edge enhancement Term ‘convolution ‘ means applying filters to an image . It may be applied in either spatial domain frequency domain. 3
- 5. MECHANISM OF SPATIAL FILTERING This process shows moving filter mask point to point Filter at each point (x , y) are calculated by predefined relationship 5
- 6. LINEAR SPATIAL FILTERING For Linear spatial filtering, Result = sum of product of filter coefficient and the corresponding image pixels For above figure Response R is w(0,0) -indicates mask is centered at x , y R=result or response of linear filtering 1)y1,(1,1)f(x.y),(0,0)f(x.. y)1,-(-1,0)f(x1)-y1,-)f(x1,1(R ww ww 6
- 7. LINEAR SPATIAL FILTERING Linear filter of an image f of size M×N with a filter mask size of m × n is given by the expression a as b bt tysxftswyxg ),(),(),( 7
- 8. LINEAR SPATIAL FILTERING For a mask of size=m × n Assume that m=2a+1 n=2b+1 Where a and b are nonnegative integers Then m and n are odd. 8
- 9. CONVOLUTION The process of linear filtering is same as convolution. So linear spatial filtering is referred to as “convolving a mask with an image.” 9
- 10. When interest lies on the response, R, of an m x n mask at any point (x , y), and not on the mechanics of implementing mask convolution It is a simplified notation i mn i i mnmn zw zwzwzwR 1 2211 ... 10
- 11. w1 w2 W3 w4 w5 w6 w7 w8 w9 i i i zwzwzwzwR 9 1 992211 ... For above 3 x 3 general Mask ,response at any point (x, y) in the image is given by 11
- 12. SMOOTHING SPATIAL FILTERS Smoothing filters are used for blurring and for noise reduction Blurring is used in preprocessing steps, such as removal of small details from an image prior to (large) object extraction, and bridging of small gaps in lines or curves. Noise reduction can be accomplished by blurring with a linear filter and also by non-linear filtering. 12
- 13. Smoothing spatial filter is done in two ways Linear filters operation performed on a pixel Order statistics filter(non linear) based on ranking on pixel 13
- 14. SMOOTHING LINEAR FILTERS linear spatial filter is simply the average of the pixels contained in the neighborhood of the filter mask. These filters sometimes are called averaging filter By replacing the value of every pixel in an image by the average of the gray levels in the neighborhood defined by the filter mask This process results in an image with reduced “sharp” transitions in gray levels 14
- 15. Two mask averaging filter weighted average 15
- 16. AVERAGING FILTER A major use of averaging filters is in the reduction of “Irrelevant” detail in an image A spatial averaging filter in which all coefficients are equal is sometimes called a box filter. Also known as low pass filter. m x n mask would have a normalizing constant equal to 1/ m n. 16
- 17. AVERAGING FILTER 1 1 1 1 1 1 1 1 1 9 1 9 1 ,1 9 1 i zR 17
- 19. WEIGHTED AVERAGING FILTER Pixels are multiplied by different coefficients , the pixel at the center of the mask is multiplied by a higher value than any other, thus giving this pixel more importance in the calculation of the average. The other pixels are inversely weighted as a function of their distance from the center of the mask 19
- 20. WEIGHTED AVERAGE 1 2 1 2 4 2 1 2 1 16 1 20
- 21. The general implementation for filtering an M x N image with a weighted averaging filter of size m x n (m and n odd) is given by the expression a as b bt a as b bt tsw ysxftsw yxg ),( )1,(),( ),( 21
- 22. EXAMPLE OF WEIGHTED AVERAGING FILTER 22
- 23. ORDER STATISTICS FILTERS Order-statistics filters are nonlinear spatial filters whose response is based on ordering (ranking) the pixels Example for this filter is median filter 23
- 24. MEDIAN FILTER Median filters used for noise-reduction with less blurring than linear smoothing filters of similar size. Median filters are particularly effective in the presence of impulse noise also called salt-and-pepper noise because of its appearance as white and black dots superimposed on an image. 24
- 25. EXAMPLE FOR MEDIAN FILTER 25
- 27. SMOOTHING FREQUENCY DOMAIN FILTERS Smoothing is achieved by attenuating a specified range of high frequency component The concept of filter in frequency domain is same as the concept of a mask in convolution. G(u , v)=H(u , v)F(u , v) 27
- 28. SMOOTHING FREQUENCY DOMAIN FILTERS After converting an image to frequency domain, some filters are applied in filtering process to perform different kind of processing on an image. The processing include blurring an image, sharpening an image etc,. The three type of filters for these purposes are: Ideal low pass filter Butterworth low pass filter Gaussian low pass filter 28
- 29. IDEAL LOW PASS FILTER Low-pass filtering smooth a signal or image . ideal low pass filter (ILPF) is one whose transfer function satisfies the relation For cutoff frequency H(u , v)= 1 if D(u , v) < 0 if D(u , v) > 0D 0D 29
- 30. D is a specified nonnegative quantity, and D(u, v) is the distance from point (u, v) in the frequency domain and the center of frequency rectangle 2 1 22 )2()2(),( NvMuvuD 30
- 31. VISUALIZATION Perspective plot of an ILPL Transfer function Filter displayed as an image Filter radical cross section 31
- 32. The low pass filters are radially symmetric about the origin The complete filter transfer function can then be generated by rotating the cross section 360 about the origin For an ideal low pass filter cross section, the point of transition between H(u, v) = 1 and H(u, v) = 0 is often called the cutoff frequency The sharp cutoff frequencies of an ideal low pass filter cannot be realized with electronic components , although they can certainly be simulated in a computer. 32
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- 34. APPLYING ILPF TO A IMAGE IN DIFFERENT FREQUENCIES 34
- 35. BUTTERWORTH FILTER The transfer function of the Butterworth low pass (BLPF) of order n and with cutoff frequency locus at a distance Do, from the origin is defined by the relation. BLPF transfer function does not have a sharp discontinuity that establishes a clear cutoff between passed and filtered frequencies n DvuD vuH 2 0/),(1 1 ),( 35
- 36. BUTTERWORTH LOW PASS FILTER Filter displayed as an image Perspective plot of an BLPF Transfer function Filter radical cross section 36
- 37. APPLYING BLPF IN AN IMAGE IN DIFFERENT FREQUENCIES 37
- 38. GAUSSIAN IMAGE FILTERING The form of these filters in two dimensions is given by D(u, v) is the distance from the origin of the Fourier transform. σ is a measure of the spread of the Gaussian curve 22 2/),( ),( vuD evuH 38
- 39. GAUSSIAN LOW PASS FILTERS Perspective plot of an GLPF Transfer function Filter radical cross section Filter displayed as an image 39
- 40. APPLYING GLPF IN AN IMAGE IN DIFFERENT FREQUENCIES 40
- 41. SHARPENING FREQUENCY DOMAIN FILTERS Image sharpening is done by using high pass filters It attenuates the low frequency components without disturbing high frequency information The transformation of high pass function is represents high pass function Represents low pass function hpH ),(1),( vuHvuH lphp lpH 41
- 42. Sharpening frequency domain filters include Ideal low pass filter Butterworth low pass filter Gaussian low pass filter 42
- 43. Sharpening technique is reverse operation of low pass filters When the low pass filters attenuates frequencies , the high pass filter passes them When the high pass filters attenuates frequencies , the low pass filter passes them 43
- 44. IDEAL HIGH PASS FILTER It is defined as H(u , v)= 1 if D(u , v) < 0 if D(u , v) > Ideal high pass shows significant ringing artifacts 0D 0D 44
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- 46. BUTTERWORTH HIGH PASS FILTER Transformation function of BHPF BHPF shows sharp edges with minor ringing artifacts n DvuD vuH 2 0/),(1 1 1),( 46
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- 48. GAUSSIAN HIGH PASS FILTER Transformation function of GHPF Gaussian high pass filters shows high sharpness without ringing artifacts. 22 2/),( 1),( vuD evuH 48
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- 50. PERSPECTIVE PLOT , IMAGE REPRESENTATION AND CROSS SECTION OF IHPF,BHPF,GHPF 50
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