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Introduction to Computer Vision Lecture 8

Introduction to Computer Vision Lecture 8

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Introduction to Computer Vision Lecture 8

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  1. Introduction to Computer VisionLecture 8 Roger S. Gaborski

  2. Straight Line Detection • Man made object often contain straight lines • Straight lines can be detected by first filtering the image with line kernels and then linking together edges that are on a straight line • The Hough Transform is another method for detecting straight lines- works with EDGE DATA • Hough transforms can also detect other shapes, such as, circles • Chapter 11 in textbook Roger S. Gaborski

  3. image26.jpg Roger S. Gaborski

  4. >> im = imread('image26.jpg'); >> figure, imshow(im) >> im = imcrop(im); >> figure, imshow(im) Roger S. Gaborski

  5. >> imGray = rgb2gray(im); >> imGray = im2double(imGray); >> whos Name Size Bytes Class Attributes ans 1x1 1 uint8 im 355x410x3 436650 uint8 imGray 355x410 1164400 double >> max(imGray(:)) ans = 1 >> min(imGray(:)) ans = 0.0745 Roger S. Gaborski

  6. >> imSobelH = edge(imGray, 'sobel'); >> figure, imshow(imSobelH), title('Sobel'); Roger S. Gaborski

  7. >> imEdge = edge(imGray, 'canny', [ ], 3); >> figure, imshow(imEdge) Roger S. Gaborski

  8. Sobel – Canny Comparison Roger S. Gaborski

  9. Consider a set of points • We would like to find a subsets of the points that lie on straight lines • We would like to know the end points, orientation, location, join segments, etc. Roger S. Gaborski

  10. X-Y Coordinate System • Consider x-y coordinate system • Assume we have a straight line in the x-y coordinate system • The slope and intercept are constant parameters for a particular line y y = mix+bi or y = aix+bi mi , ai : slope of line bi : y intercept x Roger S. Gaborski

  11. The Line is a Point in Parameter Space bi (intercept) y b ai x a (slope) Spatial Domain Parameter Domain Roger S. Gaborski

  12. A Single Point in the Spatial Domain is a Line in Parameter Space y1 y b b = -ax1 + y1 Infinite number of Lines can pass through x1 y1 each with its own a,b x1 x a Spatial Domain Parameter Domain (xi , yi ) is constant, a and b are the variables Roger S. Gaborski

  13. GOAL: Given Two Points in the spatial domain use the Hough transform to find the equation of the line passing through the points y1 y2 y x2 x1 x Spatial Domain Roger S. Gaborski

  14. A Two Points in the Spatial Domain are Two Lines in Parameter Space y1 y2 b’ y b x2 x1 b = -ax2+ y2 b = -ax1 + y1 a’ x a Spatial Domain Parameter Domain a’ and b’ are the parameters of the straight line that passes through the two points in the spatial domain (xi , yi ) is constant, a and b are the variables Roger S. Gaborski

  15. Mapping from Spatial Domain to a Parameter Space • Consider a particular point (xi,yi) in the x,y plane • An infinite number of lines can pass through this point • All satisfy slope-intercept equation: yi= a xi + b for some a,b • Solve for b: b = -a xi + yi • Parameter space a-b plane yields single line for (xi,yi) Roger S. Gaborski

  16. Parameter Space • Consider a second point (xj,yj) in the x,y plane • Also has a line associated with it in parameter space • This line intersects the line associated with (xi,yi) at ( a’,b’ ) • a’ is the slope and b’ is the intercept of the line containing both (xi,yi) and (xj,yj) in the x-y space • All points on this line have lines in the parameter space that intersect at ( a’, b’ ) Roger S. Gaborski

  17. Chapter 10 Image Segmentation Roger S. Gaborski

  18. Parameter Space • (a, b) space usually referred to as (, ) space • Each line plots to a point in (, ) parameter space, but must satisfy: •  = x1cos + y1 sin  where x1 y1 are constants (, ) ( rho, theta) Roger S. Gaborski

  19. Parameter Space • Locus of such lines in x,y space is a sinusoid in parameter space • Any point in x,y plane corresponds to a sinusoid curve in ,  space Roger S. Gaborski

  20. Chapter 10 Image Segmentation Roger S. Gaborski

  21. ( ,  ) Space • Define  and  • For each point from from edge detector • Plug in values for x and y:  = x cos + ysin • For each value of  in parameter space, solve for  • For each   pair (from previous step) record in quantized space (this is a hit) • After evaluation of all edge points, number of hits in each block corresponds to the number of pixels on the line as defined by the values   Roger S. Gaborski

  22. Hough Transform – General Idea (one point) min max min  Sample of two lines  (x1y1)  max Two of an infinite number of lines through point (x1,y1)   Space Roger S. Gaborski

  23. Hough Transform – General Idea (three points) min max min   Line through three Points (count=3)  (x1y1)   max Two of an infinite number of lines through point (x1,y1)   Space Roger S. Gaborski

  24. YouTube Videos • EGGN 512 - Lecture 13-1 Hough • http://www.youtube.com/watch?v=uDB2qGqnQ1g • EGGN 512 - Lecture 13-2 Hough • http://www.youtube.com/watch?v=o-n7NoLArcs&feature=relmfu • EGGN 512 - Lecture 13-3 Hough • http://www.youtube.com/watch?v=GZ61BRH9KFE&feature=relmfu Roger S. Gaborski

  25. Simulated Vertical Edge %'Edge' image for Hough Transform imH = zeros(100); imH(:, 80) = 1; figure, imshow(imH, [ ],'InitialMagnification', 'fit' ),title('Edge Map') [H,theta,rho] = hough(imH); figure, imshow(H,[],'XData',theta,'YData',rho,'InitialMagnification','fit'); %figure, imshow(theta,rho,H, [ ], 'notruesize'),title('Vertical Line') axis on, axis normal; xlabel('\theta'), ylabel('\rho'), colormap(gray), colorbar Roger S. Gaborski

  26. Edge image and Hough results rho = 80 units Roger S. Gaborski

  27. Simulated Horizontal Edge >> %Horizontal line imH = imH'; figure, imshow(imH, [ ],'InitialMagnification', 'fit' ),title('Edge Map') [H,theta,rho] = hough(imH); figure, imshow(H,[],'XData',theta,'YData',rho,'InitialMagnification','fit'); axis on, axis normal; xlabel('\theta'), ylabel('\rho'), colormap(gray), colorbar Roger S. Gaborski

  28. Roger S. Gaborski

  29. >> %Line at -45 Degrees imH45 = zeros(100); for a=1:100 imH45(a,a) = 1; end figure, imshow(imH45, [ ],'InitialMagnification', 'fit' ),title('Edge Map, -45 degrees') [H,theta,rho] = hough(imH45); figure, imshow(H,[],'XData',theta,'YData',rho,'InitialMagnification','fit'); axis on, axis normal; xlabel('\theta'), ylabel('\rho'), colormap(gray), colorbar Roger S. Gaborski

  30. >> %Line at 45 Degrees imH45_2 = zeros(100); for a=1:100 imH45_2(a,101-a) = 1; end figure, imshow(imH45_2, [ ],'InitialMagnification', 'fit' ),title('Edge Map, 45 degrees') [H,theta,rho] = hough(imH45_2); figure, imshow(H,[],'XData',theta,'YData',rho,'InitialMagnification','fit'); axis on, axis normal; xlabel('\theta'), ylabel('\rho'), colormap(gray), colorbar Roger S. Gaborski

  31. Roger S. Gaborski

  32. Simple square %Simple square im = zeros(100); im(20:80,20)=1; im(20:80,80)=1; im(20, 20:80)=1; im(80, 20:80)=1; figure, imshow(im, [ ],'InitialMagnification', 'fit' ),title('Box') [H,T,R] = hough(im); imshow(H,[],'XData',T,'YData',R,'InitialMagnification','fit'); xlabel('\theta'), ylabel('\rho'); axis on, axis normal Roger S. Gaborski

  33. Roger S. Gaborski

  34. Peak Detection • Need to find peaks in Hough Transform • Due to quantization, pixels not in perfectly straight line, peaks lie in more than one cell • Approach: • Find Hough cell containing maximum number of entries and record location • Set immediate neighbors to zero • Look for next maximum peak, • etc Roger S. Gaborski

  35. Find Peaks %Simple square im = zeros(100); im(20:80,20)=1; im(20:80,80)=1; im(20, 20:80)=1; im(80, 20:80)=1; figure, imshow(im, [ ],'InitialMagnification', 'fit' ),title('Box') [H,T,R] = hough(im); P = houghpeaks(H,4); imshow(H,[],'XData',T,'YData',R,'InitialMagnification','fit'); xlabel('\theta'), ylabel('\rho'); axis on, axis normal, hold on; plot(T(P(:,2)),R(P(:,1)),'s','color','white'); Roger S. Gaborski

  36. Peak Roger S. Gaborski

  37. % Find lines and plot them lines = houghlines(im,T,R,P,'FillGap',5,'MinLength',7); figure, imshow(im), hold on max_len = 0; for k = 1:length(lines) xy = [lines(k).point1; lines(k).point2]; plot(xy(:,1),xy(:,2),'LineWidth',2,'Color','green'); % plot beginnings and ends of lines plot(xy(1,1),xy(1,2),'x','LineWidth',2,'Color','yellow'); plot(xy(2,1),xy(2,2),'x','LineWidth',2,'Color','red'); % determine the endpoints of the longest line segment len = norm(lines(k).point1 - lines(k).point2); if ( len > max_len) max_len = len; xy_long = xy; end end % highlight the longest line segment plot(xy_long(:,1),xy_long(:,2),'LineWidth',2,'Color','cyan'); Roger S. Gaborski

  38. Roger S. Gaborski

  39. >> [H,T,R] = hough(imEdge); P = houghpeaks(H,4); imshow(H,[],'XData',T,'YData',R,'InitialMagnification','fit'); xlabel('\theta'), ylabel('\rho'); axis on, axis normal, hold on; plot(T(P(:,2)),R(P(:,1)),'s','color','white'); % Find lines and plot them lines = houghlines(imEdge,T,R,P,'FillGap',5,'MinLength',7); figure, imshow(imEdge), hold on max_len = 0; for k = 1:length(lines) xy = [lines(k).point1; lines(k).point2]; plot(xy(:,1),xy(:,2),'LineWidth',2,'Color','green'); % plot beginnings and ends of lines plot(xy(1,1),xy(1,2),'x','LineWidth',2,'Color','yellow'); plot(xy(2,1),xy(2,2),'x','LineWidth',2,'Color','red'); % determine the endpoints of the longest line segment len = norm(lines(k).point1 - lines(k).point2); if ( len > max_len) max_len = len; xy_long = xy; end end % highlight the longest line segment plot(xy_long(:,1),xy_long(:,2),'LineWidth',2,'Color','cyan'); Roger S. Gaborski

  40. Canny Edge Image Roger S. Gaborski

  41. P=8, Gap Fill = 5 P=12 Gap Fill = 5 P = 8, Gap Fill = 25 P = 12, Gap Fill = 25 Roger S. Gaborski

  42. Chapter 10 Image Segmentation Roger S. Gaborski

  43. Pineapple and Orange Examples Roger S. Gaborski

  44. Roger S. Gaborski

  45. Histograms • A histogram is just a method for summarizing data, for example, gray level pixel values • Can also summarize edge data Roger S. Gaborski

  46. GRAY PIXEL VALUE HISTOGRAM >> imP1=double(imread('Pineapple1.jpg’)); >> figure, imshow(imP1) >> %Convert to Gray scale >> imP1gray = rgb2gray(imP1); >> figure, imshow(imP1gray) >>%Minimum and Maximum code values >> MAXcode = max(imP1gray(:)) MAXcode = 214 >> MINcode = min(imP1gray(:)) MINcode = 1 >>%Histogram of Gray Level Pixel Values Roger S. Gaborski

  47. Edges >> imP1=double(imread('Pineapple1.jpg')); >> figure, imshow(imP1) >> %Convert to Gray scale >> imP1gray = rgb2gray(imP1); >> figure, imshow(imP1gray) >> figure, hist(imP1gray(:)) >> >> w=[1 1 1;0 0 0; -1 -1 -1] w = 1 1 1 0 0 0 -1 -1 -1 >> imP1edge = imfilter(imP1gray,w); >> figure, imshow(imP1edge, [ ]) Roger S. Gaborski

  48. >> figure, imshow(imP1edge,[ ]) >>figure, imshow(abs(imP1edge),[ ]) Roger S. Gaborski

  49. Edge Histogram Edge histogram Absolute Edge Histogram Roger S. Gaborski

  50. Edge Image Thresholds >> figure, imshow(abs(imP1edge)>1.0),title('Threshold 1.0') Roger S. Gaborski