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CAP 5415 Computer Vision Fall 2004. Dr. Alper Yilmaz Univ. of Central Florida www.cs.ucf.edu/courses/cap5415/fall2004 Office: CSB 250. Recap (Edge Detection). Prewitt and Sobel edge detectors Compute derivatives In x and y directions Find gradient magnitude

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CAP 5415 Computer Vision Fall 2004


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    1. CAP 5415 Computer VisionFall 2004 Dr. Alper Yilmaz Univ. of Central Florida www.cs.ucf.edu/courses/cap5415/fall2004 Office: CSB 250 Alper Yilmaz, Fall 2004 UCF

    2. Recap (Edge Detection) • Prewitt and Sobel edge detectors • Compute derivatives • In x and y directions • Find gradient magnitude • Threshold gradient magnitude • Difference between Prewitt and Sobel is the derivative filters Alper Yilmaz, Fall 2004 UCF

    3. Prewitt Edge Detector Prewitt’s edges in x direction Prewitt’s edges in y direction Alper Yilmaz, Fall 2004 UCF

    4. Sobel Edge Detector Sobel’s edges in x direction Sobel’s edges in y direction Alper Yilmaz, Fall 2004 UCF

    5. Edge Detection Continued Marr Hildreth Edge Detector Canny Edge Detector Alper Yilmaz, Fall 2004 UCF

    6. Marr Hildreth Edge Detector • Smooth image by Gaussian filter  S • Apply Laplacian to S • Used in mechanics, electromagnetics, wave theory, quantum mechanics and Laplace equation • Find zero crossings • Scan along each row, record an edge point at the location of zero-crossing. • Repeat above step along each column Alper Yilmaz, Fall 2004 UCF

    7. Marr Hildreth Edge Detector • Gaussian smoothing • Find Laplacian • is used for gradient (derivative) •  is used for Laplacian Alper Yilmaz, Fall 2004 UCF

    8. Marr Hildreth Edge Detector • Deriving the Laplacian of Gaussian (LoG) Homework Alper Yilmaz, Fall 2004 UCF

    9. LoG Filter Y X Alper Yilmaz, Fall 2004 UCF

    10. Finding Zero Crossings • Four cases of zero-crossings : • {+,-} • {+,0,-} • {-,+} • {-,0,+} • Slope of zero-crossing {a, -b} is |a+b|. • To mark an edge • compute slope of zero-crossing • Apply a threshold to slope Alper Yilmaz, Fall 2004 UCF

    11. On the Separability of LoG • Similar to separability of Gaussian filter • Two-dimensional Gaussian can be separated into 2 one-dimensional Gaussians n2 multiplications 2n multiplications Alper Yilmaz, Fall 2004 UCF

    12. On the Separability of LoG Requires n2 multiplications Requires 4n multiplications Alper Yilmaz, Fall 2004 UCF

    13. Seperability Gaussian Filtering Image + g(x) g(y) Laplacian of Gaussian Filtering gxx(x) g(x) Image + gyy(y) g(y) Alper Yilmaz, Fall 2004 UCF

    14. Example Alper Yilmaz, Fall 2004 UCF

    15. Example Alper Yilmaz, Fall 2004 UCF

    16. Algorithm • Compute LoG • Use 2D filter • Use 4 1D filters • Find zero-crossings from each row • Find slope of zero-crossings • Apply threshold to slope and mark edges Alper Yilmaz, Fall 2004 UCF

    17. Quality of an Edge • Robust to noise • Localization • Too many or too less responses Alper Yilmaz, Fall 2004 UCF

    18. Quality of an Edge True edge Poor robustness to noise Poor localization Too many responses Alper Yilmaz, Fall 2004 UCF

    19. Canny Edge Detector • Criterion 1: Good Detection: The optimal detector must minimize the probability of false positives as well as false negatives. • Criterion 2: Good Localization: The edges detected must be as close as possible to the true edges. • Single Response Constraint: The detector must return one point only for each edge point. Alper Yilmaz, Fall 2004 UCF

    20. Canny Edge Detector Steps • Smooth image with Gaussian filter • Compute derivative of filtered image • Find magnitude and orientation of gradient • Apply “Non-maximum Suppression” • Apply “Hysteresis Threshold” Alper Yilmaz, Fall 2004 UCF

    21. Canny Edge DetectorFirst Two Steps • Smoothing • Derivative Homework Alper Yilmaz, Fall 2004 UCF

    22. Canny Edge DetectorDerivative of Gaussian Alper Yilmaz, Fall 2004 UCF

    23. Canny Edge DetectorFirst Two Steps Alper Yilmaz, Fall 2004 UCF

    24. Canny Edge DetectorThird Step • Gradient magnitude and gradient direction image gradient magnitude Alper Yilmaz, Fall 2004 UCF

    25. Canny Edge DetectorFourth Step • Non maximum suppression We wish to mark points along the curve where the magnitude is biggest. We can do this by looking for a maximum along a slice normal to the curve (non-maximum suppression). These points should form a curve. There are then two algorithmic issues: at which point is the maximum, and where is the next one? Alper Yilmaz, Fall 2004 UCF

    26. Canny Edge DetectorNon-Maximum Suppression • Suppress the pixels in |S|which are not local maximum x’ and x’’ are the neighbors of x along normal direction to an edge Alper Yilmaz, Fall 2004 UCF

    27. Canny Edge DetectorQuantization of Normal Directions 2 3 1 0 Alper Yilmaz, Fall 2004 UCF

    28. Canny Edge DetectorNon-Maximum Suppression Alper Yilmaz, Fall 2004 UCF

    29. Canny Edge DetectorHysteresis Thresholding • If the gradient at a pixel is • above “High”, declare it an ‘edge pixel’ • below “Low”, declare it a “non-edge-pixel” • between “low” and “high” • Consider its neighbors iteratively then declare it an “edge pixel” if it is connected to an ‘edge pixel’ directly or via pixels between “low” and “high”. Alper Yilmaz, Fall 2004 UCF

    30. Canny Edge DetectorHysteresis Thresholding • Connectedness x x x 8 connected 6 connected 4 connected Alper Yilmaz, Fall 2004 UCF

    31. Canny Edge DetectorHysteresis Thresholding High Gradient magnitude low Alper Yilmaz, Fall 2004 UCF

    32. Canny Edge DetectorHysteresis Thresholding • Scan the image from left to right, top-bottom. • The gradient magnitude at a pixel is above a high threshold declare that as an edge point • Then recursively consider the neighbors of this pixel. • If the gradient magnitude is above the low threshold declare that as an edge pixel. Alper Yilmaz, Fall 2004 UCF

    33. Canny Edge DetectorHysteresis Thresholding Alper Yilmaz, Fall 2004 UCF

    34. Homework • Derive Laplacian of Gaussian • Drive gradient of 2D Gaussian • Show that gradient magnitude is rotation invariant. • Due date 27 September 2004 (Monday) Alper Yilmaz, Fall 2004 UCF

    35. Programming Assignment 1 • Implement Marr Hildreth Edge Detector • Implement Canny Edge Detector • Deliverables • Short report, problems, results, program code, etc. • Step by step outputs of images • Program code • Program should ask for a PGM image from user • Ask for the threshold value, sigma of Gaussian • Write out or display the image • Due Date 18 October 2004 Alper Yilmaz, Fall 2004 UCF

    36. Suggested Reading • Chapter 4, Emanuele Trucco, Alessandro Verri, "Introductory Techniques for 3-D Computer Vision" • Chapter 2, Mubarak Shah, “Fundamentals of Computer Vision” Alper Yilmaz, Fall 2004 UCF