Edge detection

1. Edge detection Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

2. Image edges • Points of sharp change in an image are interesting: • changes in reflectance • changes in object • changes in illumination • noise • Sometimes called edge points or edge pixels • We want to find the edges generated by scene elements and not by noise Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

3. Edge detection • Convert a 2D image into a set of curves • Extracts salient features of the scene • More compact than pixels Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro 3

4. Origin of edges surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity • Edges are caused by a variety of factors Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro 4

5. Edge detection How can you tell whether a pixel is on an edge? Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro 5

6. Edge detection • Basic idea: look for a neighborhood with lots of change • Questions: • What is the best neighborhood size? • How should change be detected? 81 82 26 24 82 33 25 25 81 82 26 24 Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

7. Finding edges • General strategy: • Determine image gradients after smoothing (gradients are directional derivatives computed using finite differences) • Mark points where the gradient magnitude is large with respect to neighboring points • Ideally this yields curves of edge points. Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

8. Image gradients • We use the image gradient to determine whether a pixel is an edge. • Two components: [gx, gy] • Both components use finite differencing to approximate derivatives • Gradients have magnitude and orientation • Vertical edges respond strongly to the x component • Horizontal edges respond strongly to the y component • Diagonal edges will respond less strongly, but to both components • Overall magnitude should be the same Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

9. Sobel operator • The Sobel operator is a simple example that is common. -1 0 1 1 2 1 Sx = -2 0 2 Sy = 0 0 0 -1 0 1 -1 -2 -1 • On a pixel of the image I • let gx be the response to Sx • let gy be the response to Sy Then the gradient is I = [gx gy] T g = (gx + gy ) is the gradient magnitude.  = atan2(gy,gx) is the gradient direction. 2 2 1/2 Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

10. Smoothing and differentiation • Issue: noise • Need to smooth image before determining image gradients • Should we perform two convolutions (smooth, then differentiate)? • Not necessarily: we can use a derivative of Gaussian filter • Differentiation is convolution and convolution is associative • D * (G * I) = (D * G) * I – What are D, G, and I? Gaussian Gaussian derivative in x Plot of Gaussian derivative Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

11. Smoothing and differentiation • Shape of Gaussian derivative: • Light on one side (positive values) • Dark on other side (negative values) • Values fall off from horizontal center line • After initial peaks, values fall off from vertical center line Gaussian derivative in x Plot of Gaussian derivative Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

12. Smoothing and differentiation • Important implementation trick – we don’t need to convolve by a 2D kernel • A 2D Gaussian function is “separable” • Gσ(x, y) = Gσ(x) * Gσ(y) • This means we can convolve the image with two 1D functions (rather than one 2D function) • This results in considerable savings for an n x n image and k x k kernel: • 1 2D kernel: approximately n2k2 multiplications and additions • 2 1D kernels: approximately 2n2k • The gradient operator is convolved with the appropriate 1D kernel or applied in succession Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

13. Gradient magnitudes after smoothing Original Sigma = 1 Sigma = 5 As the scale (sigma) increases, finer features are lost, but diffuse edges are gained. Note that the gradient magnitude encompasses horizontal, vertical, and diagonal edges. Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

14. Gradient magnitudes after smoothing Original Sigma = 1 Sigma = 5 There are three major issues: 1) The gradient magnitude at different scales is different; which should we choose? 2) The gradient magnitude is large along thick trail; how do we identify the significant points? 3) How do we link the points up into curves? Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

15. We wish to mark points along the curve where the gradient magnitude is largest. We can do this by looking for a maximum along a slice along the gradient direction. These points should form a curve. There are two algorithmic issues: at which point is the maximum, and where is the next one along the curve? Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

16. Non-maximum suppression At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values. Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

17. Non-maxima suppression • At q, the gradient Gq is a vector perpendicular to the edge direction • The locations p and r are one pixel in the direction of the gradient and the opposite direction. • One pixel in the gradient direction is g = [Gx/Gmag, Gy/Gmag] • Recall that Gmag is the length of the gradient vector [Gx, Gy] • r = q + g, p = q - g Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

18. Non-maxima suppression • At p and r, the gradient magnitude should be interpolated from the surrounding four pixels. • If the gradient magnitude at q is larger than the interpolated value at p and r, then q is marked as an edge Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

19. Predicting the next edge point Assume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s). Only necessary if following edges. Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

20. Remaining issues • Must check that the gradient magnitude is sufficiently large. • A common problem is that at some points along the curve the gradient magnitude will drop below the threshold, but not at others. • Use hysteresis: a high threshold to start edge curves and a lower threshold to continue them. • Performance at corners is poor. Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

21. Canny edge detector • The Canny edge detector (1986) is still used most often in practice. It is essentially what we have discussed: • Smooth and differentiate the image using derivative of Gaussian filters in x and y • Detect initial candidates by thresholding the gradient magnitude • Apply non-maxima suppression at the candidates • Aggregate edge pixels into contours by following edges perpendicular to the gradient • When aggregating, allow contour gradient magnitude to fall below initial threshold, but must remain above lower threshold • Note that this detector (and others) is sensitive to the parameters used (sigma, thresholds) Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

22. Zero-crossing detectors Edge detection using the zero-crossing of the 2nd derivative is historically important. Performance at corners is poor, but zero-crossings always form closed contours. step edge smoothed 1st derivative zero crossing 2nd derivative Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

23. original image Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

24. fine scale (sigma=1), medium threshold, no hysteresis Much detail (and noise) that disappears at coarser scales Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

25. coarse scale (sigma=4), high threshold, no hysteresis Curves are often broken, not closed contours Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro

26. coarse scale (sigma=4), low threshold, no hysteresis Additional edges found are questionable. Computer Vision Set: Edge detection Slides by D.A. Forsyth, C.F. Olson, S.M. Seitz, L.G. Shapiro