GNR401M Remote Sensing and Image Processing

Transcription

1 GNR401M Remote Sensing and Image Processing Guest Instructor: Prof. B. Krishna Mohan CSRE, IIT Bombay Slot 5 Guest Lectures 3 4 Neighborhood Operations Sept. 5, AM AM IIT Bombay Slide 1 Sept. 5, GL 3 4 Neighborhood Opns Contents of the Lectures Neighborhood Operations Concept of Neighborhood Operations Utility of neighborhood in smoothing and edge enhancement Image smoothing algorithms Gradient operations Edge enhancement using gradient operators 1

2 IIT Bombay Slide NEIGHBORHOOD OPERATIONS IIT Bombay Slide 3 Pixel and Neighborhood A B C D X E F G H Pixel under consideration X Neighbors of X are A, C, F,H, B,D,E,G Size of neighborhood = 3x3 Neighborhoods of size mxn m and n are odd; Unique pixel at the centre of the neighborhood

3 IIT Bombay Slide 4 4-neighborhoods A B C D X E F G H B,D,E and G are the 4-neighborhood of X 4-neighbors are physically closest to X, at one-unit distance IIT Bombay Slide 5 8-neighborhood A B C D X F G H A,C,F and H are ALSO included with B,D,E,G as neighbors; 8-pixel set is the 8-neighborhood of X A,C,F and H are the diagonal neighbors, sqrt() times farther from X E 3

4 IIT Bombay Slide 6 o o o o o o o o o o o o X o o o o o o o o o o o o Larger Neighborhoods 5 x 5 neighborhood Larger neighborhoods used based on need; computational load varies exponentially with size of neighborhood 3x3 9 neighbors; 5x5 5 neighbors IIT Bombay Slide 7 Point Operations v/s Neighborhood Operations Point operations do not alter the sharpness or resolution of the image Gray level associated with a pixel is manipulated independent of the gray levels associated with neighbors Pixel operations cannot deal with noise in the image, nor highlight local features like object boundaries 4

5 IIT Bombay Slide 8 Neighborhood Effect Normal Noise? Abnormalities can be located by comparing a pixel with neighboring pixels IIT Bombay Slide 9 Neighborhood Effect Normal region Boundary Sharp transitions from one region to another are marked by large differences in pixel values at neighboring positions 5

6 IIT Bombay Slide 10 Neighborhood Operations Results of operations performed on the neighborhood are posted at the location of the central pixel The values in the input image are not overwritten, instead the results are stored in an output array or file Cannot be computed in real time since the configurations of gray levels in the neighborhood are very large IIT Bombay Slide 11 Neighborhood Operations Simple averaging A B C D X E F G H g(x) = (1/9)[f(A) + f(b) + f(c) + f(d) + f(x) + f(e) + f(f) + f(g) + f(h)] The output gray level is the average of the gray levels of all the pixels in the 3x3 neighborhood 6

7 IIT Bombay Slide 1 Example Case 1 Case In case 1, after averaging, the central element 17 is replaced by the local average 16 negligible change In case, after averaging, the central element 37 is replaced by 18 significant change Averaging is a powerful tool to deal with random noise IIT Bombay Slide 13 Neighborhood Operations - Procedure Apply the computational step at every pixel, considering its value and the values at the neighboring pixels Shift the neighborhood by one pixel to the right Centre pixel of the new neighborhood is in focus This process continues from left to right, top to bottom 7

8 Processing step IIT Bombay Slide 14 Image IIT Bombay Slide 15 Mathematical form for averaging In general, we can write g(x) = K i= 1 f ( A ) N( X ) i where K is the number of neighbors A i. A 5 refers to X, the central pixel for a 3x3 neighborhood. It is obvious that all neighbors are given equal weightage during the averaging process 8

9 IIT Bombay Slide 16 General form for averaging To assign different weights for different neighbors, g(x) = w f ( A ) For simple averaging over a 3x3 neighborhood, w i = (1/9), i=1,,,9 Weights can be altered for 4-neighbors and 8-neighbors. In such a case, w i is not a constant for all values of i. K i i= 1 K i= 1 w i i IIT Bombay Slide 17 Averaging as Space Invariant Linear Filtering Simple averaging can be represented as a linear space invariant operation: i, j k, l i k, j l k = w l= j w k, l k = w l= j+ w g = h f h 1 = (w + 1)(w + 1) k,l= -w,, 0,, w For a 3x3 window, w=1; For 5x5 window, w=, -d discrete convolution of h with f; g = f*h 9

10 IIT Bombay Slide 18 Concept of Convolution Convolution is a weighted summation of inputs to produce an output; weights do not change anytime during the processing of the entire data If the input shifts in time or position, the output also shifts in time or position; character of the processing operation will not change The weights with which the pixels in the image are modified are represented by the term filter IIT Bombay Slide 19 Filter Mask The filter can be compactly represented using the weights or multiplying coefficients: e.g., 3x3 averaging filter or (1/9) This implies that the pixels in the image are multiplied with corresponding filter coefficients and the products are added 10

11 IIT Bombay Slide 0 Reduced neighborhood influence Central pixel is given 0% weight, 4- neighbors 15% weight. Diagonal neighbors given 5% weight. Note that the weights are all positive, and sum to unity IIT Bombay Slide 1 Discrete Convolution In general, all the filter coefficients need not be equal or symmetric In that case, the weighted averaging operation has to be performed using a discrete convolution procedure This is a general operation, assuming that the process is space invariant. 11

12 IIT Bombay Slide g i,j = Discrete Convolution w w k = w l= w h f k, l i k, j l The filter coefficients are mirror-reflected around the central element, and then the filter is slid on the input image The filter moves from top left to bottom right, moving one position at a time For each position of the filter, an output value is computed IIT Bombay Slide 3 Filter Mask Image 1

13 IIT Bombay Slide 4 Border Effect The computation of the filtering operation is applicable at those positions of the image where the filter completely fits inside. At the boundary positions, only part of the filter fits inside the image. At such positions, the computation is arbitrarily defined IIT Bombay Slide 5 Smoothing Weighted averaging also referred to as image smoothing By smoothing, local differences between pixels are reduced Images are often filtered using the same operator throughout, implying shift-invariance Most image display adaptors, have hardware convolvers built in to perform 3x3 convolutions in real-time. Shift-variant filtering is chosen when local information is to be preserved. 13

14 IIT Bombay Slide 6 Original Image IIT Bombay Slide 7 3x3 averaging 14

15 IIT Bombay Slide 8 Gaussian smoothing Gaussian filter: linear smoothing weight matrix w 1 r + c ( σ ( r, c) = ke ) for all ( r, c) W, k = e ( r, c) W 1 1 r + c ( ) σ W: one or two σ from center IIT Bombay Gaussian smoothing Slide 8a Specify neighborhood size, and σ, get W(r,c) by varying r,c in the range [ W/ +W/] Alternatively, find the size of the neighbourhood from 3 σ limits About 99% of the Gaussian distribution is covered within the range mean±3 3 σ -3 σ r,c 3 σ If σ = 1, -3 r,c 3, size of neighbourhood is 7x7 15

16 IIT Bombay p(x) Slide 8b Gaussian curve x µ σ µ σ µ µ+σ µ+σ IIT Bombay Slide 9 Shift-Variant Filtering To adapt to local intensity variations filter coefficients should vary according to the position in the image. Shift-variant filters can preserve the object boundaries better, while smoothing the image One example is the sigma filter 16

17 IIT Bombay Slide 30 Sigma filter The underlying principle here is to take the subset of pixels in the neighborhood whose gray levels lie within c.σ of the central pixel gray level k = i+ w l= j+ w g h f = i, j i, j, k, l i k, j l k = i w l= j w h i,j,k,l = 0 if f i-j,k=l f ij > c.σ ij ; h i,j,k,l = 1 otherwise σ ij is the local standard deviation of the gray levels within the neighborhood centred at pixel (i,j) To save time, one can also use global std. dev. c = 1 or depending on the size of neighborhood IIT Bombay Slide 31 Sigma Filter Algorithm Consider neighborhood size, and value of c Find the mean and standard deviation of the pixels within the neighborhood Find the neighbors of the central pixel whose gray levels are within c.σ of the central pixel s gray level Compute the average of the pixels meeting the above criterion Replace the central pixel s value by the average This cannot be replaced by a convolution since the filter response varies for each position in the image 17

18 IIT Bombay Comments on Sigma Filter Slide 31a Degradation of a smoothed image is due to blurring of object boundaries Here boundaries are better preserved by limiting the smoothing only to a homogeneous subset of pixels in the neighborhood The selected subset comprises those pixels that have similar intensities Pixels with very different intensities are excluded by making corresponding weights equal to 0 IIT Bombay Slide 3 Simple Lee filter Lee filter g ij = f mean + k.(f ij f mean ) k varies between 0 and 1 for smoothing k = 0, g ij = f mean simple averaging k = 1, g ij = f ij no smoothing at all k =, g ij = f ij + (f ij f mean ) 18

19 IIT Bombay Slide 33 Lee filter a. Original image b. Wallis filter c. K= d. K=3 e. K=0.5 f. K=0 IIT Bombay Slide 34 General form of Lee filter The general form of Lee filter is given by gij = fmean + kij ( fij fmean ) k ij is given by k ij = f ij meanσ v + σ ij Greater noise, smaller k ij, hence more smoothing σ 19

20 IIT Bombay Comments on General form of Lee filter Noise variance has to be estimated from homogeneous areas Unless noise variance is very low, this filter smoothes the image like average filter kij = fmeanσ v + σ ij Greater noise, smaller k ij, hence more smoothing σ ij Slide 34a IIT Bombay Slide 35 Gradient Inverse Filter The gradient inverse filter applies weights to the neighbors in an inverse proportion to their difference to the central pixel (i,j) s gray level Let u(i,j,k,l)= 1, if f ( i + k, j + l) f ( i, j) f ( i + k, j + l) f ( i, j) Else, u(i,j,k,l) =.0 0

21 IIT Bombay Slide 36 Gradient Inverse Filter The gradient inverse filter is defined by k =+ w l=+ w g h f = i, j i, j, k, l i k, j l k= w l= w h i,j,0,0 = p (=weight for centre pixel), p 1 h i,j,k,l = (1-p)[u i,j,k,l / ui,j,k,l k,l ] for other pixels IIT Bombay Slide 37 K-Nearest Neighbor algorithm Find k-nearest neighbors k neighbors whose gray levels are closest to the central pixel in the neighborhood Sort the neighbors on the basis of similarity of gray level to the central pixel Compute the average of centre pixel and the k nearest neighbors 1

22 IIT Bombay Slide 38 Example Consider the neighborhood K = 4 Closest 4 gray levels to 46 are 41, 39, 37, 33 Including the central pixel, the average is (1/5)( ) = 39.0 ~ 39 IIT Bombay Slide 39 Non-linear filtering Nonlinear filters have certain advantages over linear filters when dealing with noise Common examples are the rank order filters A typical rank order filter is of the form g ij = H[f i,j,k,l ], where H represents a userspecified rank criterion

23 IIT Bombay Slide 40 Modal filter Rank filtering Central pixel is assigned the gray level that occurs most frequently in the neighborhood g ij = mode {f i-k,j-l k,l=-w,, o,, w} e.g., f n = Modal filter output = 15 IIT Bombay Slide 41 Median Filter Most common non-linear filter for image smoothing Images corrupted by random salt-andpepper noise, are effectively smoothed, without degrading the input image g ij = median {f i-k,j-l k,l=-w,, o,, w} 3

24 IIT Bombay Slide 4 Example Mean v/s Median filter Case 1 Case Mean=16 Mean=3 Median=16 Median=17 In arithmetic averaging, noise is distributed over the neighbours; in median filtering, the extreme values are pushed to the extremes of the sequence IIT Bombay Slide 43 Algorithm Consider the size of the window around the pixel Collect all the pixels in the window and sort them in ascending / descending order Select the gray level after sorting, according to the rank criterion It can easily be verified that median and mode filters are nonlinear, according to the definition of linearity 4

25 IIT Bombay Slide 44 Median filtering Example here is over 7x7 neighborhood Example IIT Bombay Slide 45 Trimmed Mean Filter Trimmed-Mean Operator: trimmed-mean: first k and last k gray levels not used trimmed-mean: equal weighted average of central N-k elements z trimmed mean = 1 N k N k n= 1+ k x ( n) 5

26 IIT Bombay Slide 46 Some Comments Shift variant filters can adapt to the image conditions better More computations are involved in shift variant filtering Gaussian smoothing has some optimal properties for which it is popular Degree of smoothing can be controlled by varying the width σ of the Gaussian filter IIT Bombay Slide 47 Comments contd Simple averaging type filters fare poorly in case of signal dependent noise Particularly with SAR images noise suppression is challenging Noise filtering is performed in case of SAR prior to image formation or after image formation Shift variant and nonlinear filters more successful with SAR images 6

27 IIT Bombay Slide 48 Comments contd An important requirement of image smoothing: the sharpness in the image should be least affected Many comparative studies to evaluate methods Estimating noise statistics key to improving quality of data like SAR images Additional techniques based on mathematical morphology Edge Enhancement Methods 7

28 IIT Bombay Slide 49 Edge: Edge boundary where brightness values significantly differ among neighbors edge: brightness value appears to abruptly jump up (or down) IIT Bombay Slide 50 Original image (left), Sharpened Image (right) 8

29 IIT Bombay Slide 51 Edge Detection Essential to mark the boundaries of objects Area, shape, size, perimeter, etc. can be computed from clearly identified object boundaries Intensity / color / texture / surface orientation gradient employed to detect edges Gradient magnitude denotes the strength of edge Gradient direction relates to direction of change of intensity / color IIT Bombay Slide 5 How is an edge perceived? An edge is a set of connected pixels that lie on the boundary between two regions The pixels on an edge are called edge points Gray level / color / texture discontinuity across an edge causes edge perception Position & orientation of edge are key properties 9

30 IIT Bombay Slide 53 Different Edges A Different colors Different Intensities GNR607 GL 3-4 Different B. Krishna brightness Mohan IIT Bombay Slide 54 Different Edges Different textures Different surfaces 30

31 IIT Bombay Slide 55 Step edge: Types Of Edges Gray level profile derivatives 1st Ramp edge: nd Peak edge: 1st IIT Bombay Slide 56 Locating an Edge Locating an edge is important, since the shape of an object, its area, perimeter and other such measurements are possible only when the boundary is accurately determined Edge is a local feature, marked by sharp discontinuity in the image property on either side of it 31

32 IIT Bombay Slide 57 Edge: Edge boundary where brightness values significantly differ among neighbors edge: brightness value appears to abruptly jump up (or down) IIT Bombay Slide 58 Principle of Gradient Operator The interpretation of this operator is that the intensity gradient is computed in two perpendicular directions, followed by the resultant whose magnitude and orientation are computed by treating the values from the two masks as two projections of the edge vector 3

33 IIT Bombay Slide 59 Gradient Edge Detection Given an image f(x,y), compute f f f =, x y Squared gradient magnitude f f f = + x y f f Gradient direction = arctan y x IIT Bombay Slide 60 Gradient Directions Vertical gradient Horizontal gradient Diagonal gradient 33

34 IIT Bombay Slide 61 Gradient Edge Detectors As seen, two mutually perpendicular gradient detectors are required to detect edges in an image, since edges may occur in any orientation. Using two mutually perpendicular orientations, an edge in any direction can be resolved in terms of these two orthogonal components IIT Bombay Slide 6 Roberts Operator Roberts operator: two X masks to calculate gradient; Operates on x size neighborhood r + 1 r gradient magnitude: r 1 = f(a) f(d); r = f(b) f(c) r 1, r gradient outputs from the masks; direction = arctan(r /r 1 ) A C B D 34

35 IIT Bombay Slide 63 Gradient Edge Detectors Prewitt Operator gradient magnitude: Prewitt 1 Prewitt g = + p 1 p θ = arctan( p 1 p) gradient direction: clockwise w.r.t. column axis p 1, p are gradient outputs from the masks IIT Bombay Slide 64 Gradient Edge Detectors Prewitt Edge Detector (one part of it) x-1 x x f f ' ( ' ( x) = x 1) = f ( x + 1) f ( x) = f (x+1) f (x -1) f ( x) f ( x 1) More stable than Roberts, robust to noise in the image, and produces better edges. More time consuming, 35

36 IIT Bombay Slide 65 Input image IIT Bombay Slide 66 Prewitt Operator Output 36

37 IIT Bombay Slide 67 Gradient Edge Detectors Sobel edge detector Compare with Prewitt! gradient magnitude: gradient direction: θ = Sobel 1 Sobel arctan( s s ) 1 IIT Bombay Slide 68 Gradient Edge Detectors Frei and Chen Operators 1 sqrt() sqrt() 0 sqrt() -1 - sqrt() The above are two of nine masks, four of which are formed by rotation in steps of 90 o, four are line detectors, and one is a simple 3x3 smoothing operator 37

38 IIT Bombay Slide 69 Compass Gradient Operators Frei and Chen edge detector: nine orthogonal masks (3X3) IIT Bombay Slide 70 Kirsch Compass Gradient Operator K 1 K K 3 K K 5 K 6 K 7 K 8 38

39 IIT Bombay Slide 71 Kirsch Gradient Edge Detectors Kirsch: set of eight compass template edge masks gradient magnitude: gradient direction: g = max e k, k = 1,,..., 9 k θ = 45 arg max e k IIT Bombay Slide 7 Robinson Gradient Detector Robinson: compass template mask set with only 0, 1, R 1 R R 3 R 4 Other four masks are mirror reflections of the first four Gradient magnitude and direction same as Kirsch 39

40 IIT Bombay Slide 73 Actual Edges The edge enhanced images are thresholded in order to suppress the interior portions of the image and retain only the edges This helps in identifying the outlines of the objects of interest IIT Bombay Slide 74 Prewitt operator output thresholded at 40 40

41 IIT Bombay Slide 75 Laplacian Operator The Laplacian operator is based on the Laplace equation given by f x f y + = 0 Laplacian operator is discretized version of the above equation and is based on second derivatives along x and y directions IIT Bombay Slide 76 Laplacian Operator Filter coefficients The discrete version of the second derivative operator: [1-1] and [1-1] T in the horizontal and vertical directions Superimposing the two, we get the discrete Laplace operator

42 IIT Bombay Slide 77 Properties of Laplace Operator Isotropic operator cannot give orientation information Any noise in image gets amplified Faster since only one filter mask involved Smoothing the image first prior to Laplace operator is often needed for reliable edges IIT Bombay Slide 78 Zero-Crossing Edge Detectors First derivative maximum: exactly where second derivative zero crossing In order to detect edges, we look at pixels where the intensity gradient is high, or the first derivative magnitude is maximum First derivative maximum implies a zero when the second derivative is computed Edges are located at those positions where there is a positive value on one side and a negative value on the other side, in other words a zero-crossing 4

43 IIT Bombay Slide 79 A step edge, whose first derivative is an impulse, and whose second derivative shows a transition from a positive to a negative Edge location corresponds to the point where a sign change occurs from positive to negative (or vice versa) IIT Bombay Slide 80 Zero-Crossing Edge Detectors Laplacian of a function I(r,c) I = ( r + c ) I = I r I + c Two commonly used masks for Laplacian operator 43

44 IIT Bombay Slide 81 Zero Crossing Edge Detector Direct operation on the image using the Laplacian operator results in a very noisy result Derivative operator amplifies the high frequency noise Preprocess the input image by a smoothing operator prior to application of the Laplacian IIT Bombay Slide 8 Zero Crossing Edge Detector The Gaussian shaped smoothing operator is found to be ideal as a preprocessing operator Therefore the Laplacian operator is applied on Gaussian smoothed input image ZC(image) = Laplacian [gaussian(image)] 44

45 IIT Bombay Slide 83 LOG operator Both Laplacian operator and Gaussian operator are linear, and hence can be combined into one Laplacian of Gaussian (LoG) operator Laplacian[Gaussian(image)] = [Laplacian(Gaussian)](image) IIT Bombay Slide 84 LOG operator Laplacian[Gaussian(image)] = [Laplacian(Gaussian)](image) Verify! LOG r c 1 r + c ( ) 1 r = + 4 πσ σ σ (, ) e (1 ) 1 = πσ 1 πσ 1 r + c ( ) σ [ e (1 )] 4 1 r + c r + c ( ) ( ) 4 e σ σ c σ 45

46 IIT Bombay Slide 85 LOG operator LoG operator is a sampled version of the function 1 r + c ( ) 1 r + c σ LOG( r, c) = ( ) e 4 πσ σ For a given value of σ, the size of the Gaussian filter is -3σ to +3σ Computationally more expensive due to convolution with large filter masks IIT Bombay Slide 86 Zero-Crossing Edge Detectors Properties Edges depend on the value of σ For small value of σ all edges are detected For large value of σ only major edges are detected Any minor difference in intensity between neighbors can be captured using LoG filter Significant zero crossings can be identified using suitable threshold 46

47 IIT Bombay Slide 87 Zero-Crossing Edge Detectors A pixel at (m,n) is declared to have a zero crossing if OR f (m,n) > T and f (m+δm, n+δn) < -T f (m,n) < -T and f (m+δm, n+δn) > T IIT Bombay Slide 88 Edge Detection in Multispectral Images Simple approaches: Compute gradient by taking Euclidean distance between multispectral vectors of data at adjacent pixels instead of differences in gray levels Find independent gradients for different bands, edges and combine edges Find independent gradients, combine gradients, and find edge from multiband gradient 47

48 IIT Bombay Slide 89 Edge Detection in Multispectral Images IIT Bombay Slide 90 Edge Detection in Multispectral Images 48

49 IIT Bombay Slide 9 For example, Image Sharpening Sharpened image = Original image + k. gradient magnitude Scale factor k can determine whether gradient magnitude is added as it is or a fraction of it. The sum may be rescaled to 0-55 to display like an image IIT Bombay Slide 93 Original image (left), Sharpened Image (right) 49

50 IIT Bombay Slide 94 Unsharp Masking Sample convolution mask (1/9) G = F + (F Fmean) NR607 GL 3-4 IIT Bombay NR607 B. Krishna Mohan Slide 95 GL 3-4 B. Krishna Mohan 50

51 IIT Bombay Slide 96 Line Enhancement Difference between a line and an edge Line is a physical entity Edge is a perceptual entity IIT Bombay Slide 97 Lines NR607 Lecture 8 B. Krishna Mohan 51

52 IIT Bombay Slide 98 Line Enhancement Detection of a physical line involves High to low transition Low to high transition OR Low to high transition High to low transition IIT Bombay Slide 99 Line Enhancement Masks These masks look for positive to negative and negative to positive transitions in vertical/horizontal/diagonal directions 5

53 IIT Bombay Slide 100 Summary of Gradient Operators Edges or boundaries convey very important information for image understanding Gradient operators emphasize the local intensity or other property differences thereby making visible object boundaries Gradient operations in normal course are only the first step in reliable edge extraction IIT Bombay Slide 101 Summary of Neighborhood Operators Image processing operations involving neighborhoods of pixels are important in many tasks Smoothing filters are composed of nonnegative coefficients which add up to 1 Gradient filters are composed of both positive and negative coefficients which must add up to 0 so that in images where there is no edge, the output is zero. 53

54 IIT Bombay Slide 10 Shape Fitting by Hough Transform IIT Bombay Slide 103 Hough Transform A method for finding global relationships between pixels. Example: We want to find straight lines in an image Apply Hough transform to the edge enhanced thresholded image Any curve that can be represented by a parametric equation can be extracted by Hough transform 54

55 IIT Bombay Slide 104 Line Fitting Edges Lines fit to the edges IIT Bombay Slide 105 Hough transform Procedure Consider a set of points x i, y i on a line y = a.x + b; a and b are parameters (slope and intercept) All the above points satisfy the equation y i = a.x i + b Let x i and y i be the parameters; then b = -a.x i + y i Vary a and find corresponding b. In the a-b space, it is a line For each x i, y i there is a line in the a-b space 55

56 IIT Bombay Slide 106 Hough transform Procedure Consider an array (called accumulator array) with a varying along the columns and b along the rows Initialize the array with count 0. Vary a for a given point (x i,y i ) and compute corresponding b. Increment count in cell (a,b) by 1 When points (x i,y i ), i=1,,,n lie on the same line, the lines in the a-b space pass through a common cell corresponding to the slope and intercept of the line in the x-y space In other words, the count in the accumulator array will be high for one cell corresponding to the line IIT Bombay Slide 107 y b x a xy-space ab- or parameter space 56

57 IIT Bombay Slide 108 Problem with the line model y=ax+b In reality we have a problem with y=ax+b because the slope a reaches infinity for vertical lines. In 197, Duda & Hart proposed a Standard HT (SHT). They used the polar coordinate equation of a straight line: x.cosθ + y.sinθ = ρ For vertical lines, θ=π/; No problem projecting any line into (θ, ρ) space IIT Bombay Slide 109 Standard Hough Transform Y ρ = x cos( θ ) + y sin( θ ) y = ax + b θ ρ X θ 57

58 IIT Bombay Slide 110 Accumulator Array Creation Select the point (x,y) on a line Create an array in which θ varies along columns, from θ = 0 to θ = π in small increments (e.g., 15 o ) For each θ, find the value of ρ. Increment the count in accumulator array for cell (θ,ρ θ,ρ) by 1 For each point (x,y) there is a sinusoid in the (θ,ρ θ,ρ) space. All points (x i, y i ) on a given line will have some (θ,ρ θ,ρ) common IIT Bombay Slide 111 Original artwork from the book Digital Image Processing by R.C. Gonzalez and R.E. Woods R.C. Gonzalez and R.E. Woods, reproduced with permission granted to instructors by authors on the website ace.com Hough Space 58

59 IIT Bombay Slide 11 Example of Line and Accumulator Theta = 45º = rad ρ = (1 ) / = Theta: 0 to 3.14 (rad) ρ: 0 to 1.55 Brightest point gets 0 votes Original artwork from the book Digital Image Processing by R.C. Gonzalez and R.E. Woods R.C. Gonzalez and R.E. Woods, reproduced with permission granted to instructors by authors on the website IIT Bombay Slide 113 Mechanics of the Hough transform Difficulties how big should the cells be? (too big, and we cannot distinguish between quite different lines; too small, and noise causes lines to be missed) How many lines? count the peaks in the Hough array Who belongs to which line? tag the votes Hardly ever satisfactory in practice, because problems with noise GNR607 GL 3-4 B. Krishna and cell Mohan size defeat it 59

60 IIT Bombay Slide 114 Noisy Line Brightest point = 6 votes Original artwork from the book Digital Image Processing by R.C. Gonzalez and R.E. Woods R.C. Gonzalez and R.E. Woods, reproduced with permission granted to instructors by authors on the website IIT Bombay Slide 115 Totally Chaotic! Original artwork from the book Digital Image Processing by R.C. Gonzalez and R.E. Woods R.C. Gonzalez and R.E. Woods, reproduced with permission granted to instructors by authors on the website 60

61 IIT Bombay Slide 116 Improvements to Simple Hough Transform Noise tolerance: Most edge detectors give edge direction. Consider only those directions in accumulator array corresponding to edge direction at pixels Speed up: A two-stage process can be considered. First, generate coarse (ρ,θ) array. Find approximate lines. Next, find precise values of (ρ,θ) by searching around the coarse values IIT Bombay Slide 117 High Order Parametric Curves Circle: (x-a) + (y-b) = r Parameter space is 3-dimensional Highly computation intensive Searching for maxima in 3-D arrays is computationally expensive Efficient data structures are important Ref: A.M. Cross, Detection of circular geological features using the Hough transform, International Journal of Remote Sensing, vol. 9, no. 9, pp ,

62 IIT Bombay Slide 118 Circle Detection a b Contd 6