# Linear Filtering – Part I Selim Aksoy Department of Computer Engineering Bilkent University

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Linear Filtering – Part I Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr

CS 484, Fall 2012©2012, Selim Aksoy2 Importance of neighborhood Both zebras and dalmatians have black and white pixels in similar numbers. The difference between the two is the characteristic appearance of small group of pixels rather than individual pixel values. Adapted from Pinar Duygulu, Bilkent University

CS 484, Fall 2012©2012, Selim Aksoy3 Outline We will discuss neighborhood operations that work with the values of the image pixels in the neighborhood. Spatial domain filtering Frequency domain filtering Image enhancement Finding patterns

CS 484, Fall 2012©2012, Selim Aksoy4 Spatial domain filtering What is the value of the center pixel? What assumptions are you making to infer the center value? 333 3?3 333 343 2?3 342 3 3

CS 484, Fall 2012©2012, Selim Aksoy5 Spatial domain filtering Some neighborhood operations work with the values of the image pixels in the neighborhood, and the corresponding values of a subimage that has the same dimensions as the neighborhood. The subimage is called a filter (or mask, kernel, template, window). The values in a filter subimage are referred to as coefficients, rather than pixels.

CS 484, Fall 2012©2012, Selim Aksoy6 Spatial domain filtering Operation: modify the pixels in an image based on some function of the pixels in their neighborhood. Simplest: linear filtering (replace each pixel by a linear combination of its neighbors). Linear spatial filtering is often referred to as convolving an image with a filter.

Linear filtering CS 484, Fall 2012©2012, Selim Aksoy7 g [m] h [m] 0 1 2 2 Linear system: Input: Output? h [-k] f [m=0]=-2 h [1-k] 0 1 2 3 2222 1 1 1 0 0 0 1 2 2 0 1 2 2 f [m=1]=-4 f [m=2]=0

Linear filtering CS 484, Fall 2012©2012, Selim Aksoy8 g [m,n] f [m,n] For a linear spatially invariant system 2 2 2 g[m,n] h[m,n] f[m,n] = 111115113111112111112111 135138137139145146149147 163168188196206202206207 180184206219202200195193 189193214216104798377 191201217220103596068 195205216222113686983 199203223228108687177 m=0 1 2 … ???????? ?-59-921-1210? ?-2918244-75? ?-5040142-88-3410? ?-4141264-175-710? ?-2437349-224-120-10? ?-2333360-217-134-23? ????????

CS 484, Fall 2012©2012, Selim Aksoy9 Linear filtering Filtering process: Masks operate on a neighborhood of pixels. The filter mask is centered on a pixel. The mask coefficients are multiplied by the pixel values in its neighborhood and the products are summed. The result goes into the corresponding pixel position in the output image. This process is repeated by moving the filter mask from pixel to pixel in the image.

CS 484, Fall 2012©2012, Selim Aksoy10 Linear filtering This is called the cross-correlation operation and is denoted by Input image F[r,c] Mask overlaid with image at [r,c] Output image G[r,c] H[-1,-1]H[-1,0]H[-1,1] H[0,-1]H[0,0]H[0,1] H[1,-1]H[1,0]H[1,1] Filter

CS 484, Fall 2012©2012, Selim Aksoy12 Smoothing spatial filters Often, an image is composed of some underlying ideal structure, which we want to detect and describe, together with some random noise or artifact, which we would like to remove. Smoothing filters are used for blurring and for noise reduction. Linear smoothing filters are also called averaging filters.

CS 484, Fall 2012©2012, Selim Aksoy13 Smoothing spatial filters Averaging (mean) filterWeighted average

CS 484, Fall 2012©2012, Selim Aksoy14 Smoothing spatial filters 1/9.(10x1 + 11x1 + 10x1 + 9x1 + 10x1 + 11x1 + 10x1 + 9x1 + 10x1) = 1/9.( 90) = 10 1/9.( 90) = 10 10 1110 9 10 11 10910 1 10 10 2 9 0 9 0 9 9 9 9 0 1 99 10 1011 10 1 11 11 11 11 1010 I 1 1 1 1 1 1 1 1 1 F X XX X 10 X X X X X X X X X X X X X X XX 1/9 O Adapted from Octavia Camps, Penn State

CS 484, Fall 2012©2012, Selim Aksoy15 Smoothing spatial filters 1/9.(10x1 + 9x1 + 11x1 + 9x1 + 99x1 + 11x1 + 11x1 + 10x1 + 10x1) = 1/9.( 180) = 20 1/9.( 180) = 20 I 1 1 1 1 1 1 1 1 1 F X XX X 20 X X X X X X X X X X X X X X XX 1/9 O 10 1110 9 10 11 10910 1 10 10 2 9 0 9 0 9 9 9 9 0 1 99 10 1011 10 1 11 11 11 11 1010 Adapted from Octavia Camps, Penn State

CS 484, Fall 2012©2012, Selim Aksoy16 Smoothing spatial filters Common types of noise: Salt-and-pepper noise: contains random occurrences of black and white pixels. Impulse noise: contains random occurrences of white pixels. Gaussian noise: variations in intensity drawn from a Gaussian normal distribution. Adapted from Linda Shapiro, U of Washington

CS 484, Fall 2012©2012, Selim Aksoy17 Adapted from Linda Shapiro, U of Washington

CS 484, Fall 2012©2012, Selim Aksoy18 Smoothing spatial filters Adapted from Gonzales and Woods

CS 484, Fall 2012©2012, Selim Aksoy19 Smoothing spatial filters Adapted from Gonzales and Woods

CS 484, Fall 2012©2012, Selim Aksoy20 Smoothing spatial filters Adapted from Darrell and Freeman, MIT

CS 484, Fall 2012©2012, Selim Aksoy21 Smoothing spatial filters A weighted average that weighs pixels at its center much more strongly than its boundaries. 2D Gaussian filter Adapted from Martial Hebert, CMU

CS 484, Fall 2012©2012, Selim Aksoy22 Smoothing spatial filters If σ is small: smoothing will have little effect. If σ is larger: neighboring pixels will have larger weights resulting in consensus of the neighbors. If σ is very large: details will disappear along with the noise. Adapted from Martial Hebert, CMU

CS 484, Fall 2012©2012, Selim Aksoy23 Smoothing spatial filters Result of blurring using a uniform local model. Produces a set of narrow horizontal and vertical bars – ringing effect. Result of blurring using a Gaussian filter. Adapted from David Forsyth, UC Berkeley

CS 484, Fall 2012©2012, Selim Aksoy24 Smoothing spatial filters Adapted from Martial Hebert, CMU

CS 484, Fall 2012©2012, Selim Aksoy25 Smoothing spatial filters Adapted from Martial Hebert, CMU

CS 484, Fall 2012©2012, Selim Aksoy26 Order-statistic filters Order-statistic filters are nonlinear spatial filters whose response is based on ordering (ranking) the pixels contained in the image area encompassed by the filter, and then replacing the value of the center pixel with the value determined by the ranking result. The best-known example is the median filter. It is particularly effective in the presence of impulse or salt-and-pepper noise, with considerably less blurring than linear smoothing filters.

CS 484, Fall 2012©2012, Selim Aksoy27 Order-statistic filters 10 1110 9 10 11 10910 1 10 10 2 9 0 9 0 9 9 9 9 0 1 99 10 1011 10 1 11 11 11 11 1010 I X XX X 10 X X X X X X X X X X X X X X XX O 10,11,10,9,10,11,10,9,109,9,10,10,10,10,10,11,11 sort median Adapted from Octavia Camps, Penn State

CS 484, Fall 2012©2012, Selim Aksoy28 Order-statistic filters 10 1110 9 10 11 10910 1 10 10 2 9 0 9 0 9 9 9 9 0 1 99 10 1011 10 1 11 11 11 11 1010 I X XX X 10 X X X X X X X X X X X X X X XX O 10,9,11,9,99,11,11,10,109,9,10,10,10,11,11,11,99 sort median Adapted from Octavia Camps, Penn State

CS 484, Fall 2012©2012, Selim Aksoy29 Salt-and-pepper noise Adapted from Linda Shapiro, U of Washington

CS 484, Fall 2012©2012, Selim Aksoy30 Gaussian noise Adapted from Linda Shapiro, U of Washington

CS 484, Fall 2012©2012, Selim Aksoy31 Order-statistic filters Adapted from Gonzales and Woods

CS 484, Fall 2012©2012, Selim Aksoy32 Order-statistic filters Adapted from Martial Hebert, CMU

CS 484, Fall 2012©2012, Selim Aksoy33 Sharpening spatial filters Objective of sharpening is to highlight or enhance fine detail in an image. Since smoothing (averaging) is analogous to integration, sharpening can be accomplished by spatial differentiation. First-order derivative of 1D function f(x) f(x+1) – f(x). Second-order derivative of 1D function f(x) f(x+1) – 2f(x) + f(x-1).

CS 484, Fall 2012©2012, Selim Aksoy34 Sharpening spatial filters Adapted from Gonzales and Woods

CS 484, Fall 2012©2012, Selim Aksoy35 Sharpening spatial filters Adapted from Gonzales and Woods

CS 484, Fall 2012©2012, Selim Aksoy36 Sharpening spatial filters Observations: First-order derivatives generally produce thicker edges in an image. Second-order derivatives have a stronger response to fine detail (such as thin lines or isolated points). First-order derivatives generally have a stronger response to a gray level step. Second-order derivatives produce a double response at step changes in gray level.

CS 484, Fall 2012©2012, Selim Aksoy38 Sharpening spatial filters

CS 484, Fall 2012©2012, Selim Aksoy39 Sharpening spatial filters Adapted from Gonzales and Woods

CS 484, Fall 2012©2012, Selim Aksoy40 Sharpening spatial filters High-boost filtering Adapted from Darrell and Freeman, MIT

CS 484, Fall 2012©2012, Selim Aksoy41 Sharpening spatial filters Adapted from Darrell and Freeman, MIT

CS 484, Fall 2012©2012, Selim Aksoy42 Combining spatial enhancement methods

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