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EE 4780

EE 4780. Edge Detection. Detection of Discontinuities. Matched Filter Example >> a=[0 0 0 0 1 2 3 0 0 0 0 2 2 2 0 0 0 0 1 2 -2 -1 0 0 0 0]; >> figure; plot(a); >> h1 = [-1 -2 2 1]/10; >> b1 = conv(a,h1); figure; plot(b1);. Detection of Discontinuities. Point Detection Example:

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EE 4780

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  1. EE 4780 Edge Detection

  2. Detection of Discontinuities • Matched Filter Example >> a=[0 0 0 0 1 2 3 0 0 0 0 2 2 2 0 0 0 0 1 2 -2 -1 0 0 0 0]; >> figure; plot(a); >> h1 = [-1 -2 2 1]/10; >> b1 = conv(a,h1); figure; plot(b1);

  3. Detection of Discontinuities • Point Detection Example: • Apply a high-pass filter. • A point is detected if the response is larger than a positive threshold. • The idea is that the gray level of an isolated point will be quite different from the gray level of its neighbors. Threshold

  4. Detection of Discontinuities • Point Detection Detected point

  5. Detection of Discontinuities • Line Detection Example:

  6. Detection of Discontinuities • Line Detection Example:

  7. Detection of Discontinuities • Edge Detection: • An edge is the boundary between two regions with relatively distinct gray levels. • Edge detection is by far the most common approach for detecting meaningful discontinuities in gray level. The reason is that isolated points and thin lines are not frequent occurrences in most practical applications. • The idea underlying most edge detection techniques is the computation of a local derivative operator.

  8. Origin of Edges • Edges are caused by a variety of factors surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity

  9. Profiles of image intensity edges

  10. Image gradient • The gradient of an image: • The gradient points in the direction of most rapid change in intensity • The gradient direction is given by: • The edge strength is given by the gradient magnitude

  11. The discrete gradient • How can we differentiate a digital image f[x,y]? • Option 1: reconstruct a continuous image, then take gradient • Option 2: take discrete derivative (finite difference)

  12. Effects of noise • Consider a single row or column of the image • Plotting intensity as a function of position gives a signal

  13. Look for peaks in Solution: smooth first

  14. Derivative theorem of convolution • This saves us one operation:

  15. Laplacian of Gaussian • Consider Laplacian of Gaussian operator Zero-crossings of bottom graph

  16. 2D edge detection filters • is the Laplacian operator: Laplacian of Gaussian Gaussian derivative of Gaussian

  17. Edge Detection Possible filters to find gradients along vertical and horizontal directions: Averaging provides noise suppression This gives more importance to the center point.

  18. Edge Detection

  19. Edge Detection

  20. Edge Detection • The Laplacian of an image f(x,y) is a second-order derivative defined as Digital approximations:

  21. Edge Detection One simple method to find zero-crossings is black/white thresholding: 1. Set all positive values to white 2. Set all negative values to black 3. Determine the black/white transitions. • Compare (b) and (g): • Edges in the zero-crossings image is thinner than the gradient edges. • Edges determined by zero-crossings have formed many closed loops.

  22. Edge Detection • The Laplacian of a Gaussian filter A digital approximation:

  23. The Canny edge detector • original image (Lena)

  24. The Canny edge detector • norm of the gradient

  25. The Canny edge detector • thresholding

  26. The Canny edge detector • thinning • (non-maximum suppression)

  27. Edge detection by subtraction original

  28. Edge detection by subtraction smoothed (5x5 Gaussian)

  29. Edge detection by subtraction Why does this work? smoothed – original

  30. Gaussian - image filter Gaussian delta function Laplacian of Gaussian

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