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Computer Vision

Computer Vision. Spring 2012 15-385,-685 Instructor: S. Narasimhan WH 5409 T-R 10:30 – 11:50am. Edge Detection Lecture #6. Edge Detection. Convert a 2D image into a set of curves Extracts salient features of the scene More compact than pixels. Origin of Edges.

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Computer Vision

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  1. Computer Vision Spring 2012 15-385,-685 Instructor: S. Narasimhan WH 5409 T-R 10:30 – 11:50am

  2. Edge Detection Lecture #6

  3. Edge Detection • Convert a 2D image into a set of curves • Extracts salient features of the scene • More compact than pixels

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

  5. Edge Types Step Edges Line Edges Roof Edge

  6. Real Edges • Edge Magnitude • Edge Orientation • High Detection Rate and Good Localization Noisy and Discrete! We want an Edge Operator that produces:

  7. Gradient • Gradient equation: • Represents direction of most rapid change in intensity • Gradient direction: • The edge strength is given by the gradient magnitude

  8. Ideal edge Unit step function: Image intensity (brightness): Theory of Edge Detection

  9. Partial derivatives (gradients): • Squared gradient: Edge Magnitude: (normal of the edge) Edge Orientation: Rotationally symmetric, non-linear operator Theory of Edge Detection • Image intensity (brightness):

  10. Partial derivatives (gradients): • Laplacian: Rotationally symmetric, linear operator zero-crossing Theory of Edge Detection • Image intensity (brightness):

  11. Discrete Edge Operators • How can we differentiate a discrete image? Finite difference approximations: Convolution masks :

  12. Second order partial derivatives: • Laplacian : Convolution masks : or Discrete Edge Operators (more accurate)

  13. The Sobel Operators • Better approximations of the gradients exist • The Sobel operators below are commonly used

  14. Comparing Edge Operators Good Localization Noise Sensitive Poor Detection Gradient: Roberts (2 x 2): Sobel (3 x 3): Sobel (5 x 5): Poor Localization Less Noise Sensitive Good Detection

  15. Where is the edge?? Effects of Noise • Consider a single row or column of the image • Plotting intensity as a function of position gives a signal

  16. Look for peaks in Solution: Smooth First Where is the edge?

  17. Derivative Theorem of Convolution …saves us one operation.

  18. Laplacian of Gaussian (LoG) Laplacian of Gaussian Laplacian of Gaussian operator Where is the edge? Zero-crossings of bottom graph !

  19. 2D Gaussian Edge Operators Gaussian Derivative of Gaussian (DoG) Laplacian of Gaussian Mexican Hat (Sombrero) • is the Laplacian operator:

  20. Canny Edge Operator • Smooth image I with 2D Gaussian: • Find local edge normal directions for each pixel • Compute edge magnitudes • Locate edges by finding zero-crossings along the edge normal directions (non-maximum suppression)

  21. Non-maximum Suppression • Check if pixel is local maximum along gradient direction • requires checking interpolated pixels p and r

  22. The Canny Edge Detector original image (Lena)

  23. The Canny Edge Detector magnitude of the gradient

  24. The Canny Edge Detector After non-maximum suppression

  25. Canny Edge Operator original Canny with Canny with • The choice of depends on desired behavior • large detects large scale edges • small detects fine features

  26. Difference of Gaussians (DoG) • Laplacian of Gaussian can be approximated by the difference between two different Gaussians

  27. (a) (b) DoG Edge Detection (b)-(a)

  28. Gaussian – Image filter Fourier Transform Gaussian delta function

  29. Unsharp Masking – = + a =

  30. MATLAB demo g = fspecial('gaussian',15,2); imagesc(g) surfl(g) gclown = conv2(clown,g,'same'); imagesc(conv2(clown,[-1 1],'same')); imagesc(conv2(gclown,[-1 1],'same')); dx = conv2(g,[-1 1],'same'); imagesc(conv2(clown,dx,'same'); lg = fspecial('log',15,2); lclown = conv2(clown,lg,'same'); imagesc(lclown) imagesc(clown + .2*lclown)

  31. Edge Thresholding • Standard Thresholding: • Can only select “strong” edges. • Does not guarantee “continuity”. • Hysteresis based Thresholding (use two thresholds) Example: For “maybe” edges, decide on the edge if neighboring pixel is a strong edge.

  32. Edge Relaxation • Parallel – Iterative method to adjust edge • values on the basis of neighboring edges. a f No Edge e g b Edge to be updated Edge c h (1) • Vertex Types: (0)

  33. Edge Relaxation • Vertex Types (continued): (2) (3)

  34. Edge Relaxation Algorithm Leave as is Increment • Action Table: Decrement 0 - 1 0 - 0 1 - 1 2 - 2 0 - 2 1 - 2 Edge Type 2 - 3 0 - 3 1 - 3 3 - 3 • Algorithm: Step 0: Compute Initial Confidence of each edge e: Step 1: Initialize Step 2: Compute Edge Type of each edge e Step 3: Modify confidence based on and Edge Type Step 4: Test to see if all have CONVERGED to either 1 or 0. Else go to Step 2.

  35. Edge Relaxation

  36. Next Class • Boundary Detection and Hough Transform

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