![Derivative of Gaussian filter
Is this filter separable?
* [1 -1] =](https://image.slidesharecdn.com/lec06edge-210326054201/85/Lec06-edge-13-320.jpg)
Lec06 edge
- 1. Edge detection • Goal: Identify sudden changes (discontinuities) in an image • Intuitively, most semantic and shape information from the image can be encoded in the edges • More compact than pixels • Ideal: artist’s line drawing (but artist is also using object-level knowledge) Source: D. Lowe
- 2. Origin of Edges Edges are caused by a variety of factors depth discontinuity surface color discontinuity illumination discontinuity surface normal discontinuity Source: Steve Seitz
- 3. Characterizing edges • An edge is a place of rapid change in the image intensity function image intensity function (along horizontal scanline) first derivative edges correspond to extrema of derivative
- 4. The gradient points in the direction of most rapid increase in intensity Image gradient The gradient of an image: The gradient direction is given by • how does this relate to the direction of the edge? The edge strength is given by the gradient magnitude Source: Steve Seitz
- 5. Differentiation and convolution Recall, for 2D function, f(x,y): This is linear and shift invariant, so must be the result of a convolution. We could approximate this as (which is obviously a convolution) -1 1 Source: D. Forsyth, D. Lowe
- 6. Finite difference filters Other approximations of derivative filters exist: Source: K. Grauman
- 7. Finite differences: example Which one is the gradient in the x-direction (resp. y-direction)?
- 8. Effects of noise Consider a single row or column of the image • Plotting intensity as a function of position gives a signal Where is the edge? Source: S. Seitz
- 9. Effects of noise • Finite difference filters respond strongly to noise • Image noise results in pixels that look very different from their neighbors • Generally, the larger the noise the stronger the response • What is to be done? Source: D. Forsyth
- 10. Effects of noise • Finite difference filters respond strongly to noise • Image noise results in pixels that look very different from their neighbors • Generally, the larger the noise the stronger the response • What is to be done? • Smoothing the image should help, by forcing pixels different to their neighbors (=noise pixels?) to look more like neighbors Source: D. Forsyth
- 11. Solution: smooth first • To find edges, look for peaks in ) ( g f dx d ∗ f g f * g ) ( g f dx d ∗ Source: S. Seitz
- 12. • Differentiation is convolution, and convolution is associative: • This saves us one operation: g dx d f g f dx d ∗ = ∗ ) ( Derivative theorem of convolution g dx d f ∗ f g dx d Source: S. Seitz
- 13. Derivative of Gaussian filter Is this filter separable? * [1 -1] =
- 14. Derivative of Gaussian filter Which one finds horizontal/vertical edges? x-direction y-direction
- 15. Smoothed derivative removes noise, but blurs edge. Also finds edges at different “scales”. 1 pixel 3 pixels 7 pixels Tradeoff between smoothing and localization Source: D. Forsyth
- 16. • The gradient magnitude is large along a thick “trail” or “ridge,” so how do we identify the actual edge points? • How do we link the edge points to form curves? Implementation issues Source: D. Forsyth
- 17. Designing an edge detector • Criteria for an “optimal” edge detector: • Good detection: the optimal detector must minimize the probability of false positives (detecting spurious edges caused by noise), as well as that of false negatives (missing real edges) • Good localization: the edges detected must be as close as possible to the true edges • Single response: the detector must return one point only for each true edge point; that is, minimize the number of local maxima around the true edge Source: L. Fei-Fei
- 18. Canny edge detector • This is probably the most widely used edge detector in computer vision • Theoretical model: step-edges corrupted by additive Gaussian noise • Canny has shown that the first derivative of the Gaussian closely approximates the operator that optimizes the product of signal- to-noise ratio and localization J. Canny, A Computational Approach To Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8:679-714, 1986. Source: L. Fei-Fei
- 19. Canny edge detector 1. Filter image with derivative of Gaussian 2. Find magnitude and orientation of gradient 3. Non-maximum suppression: • Thin multi-pixel wide “ridges” down to single pixel width 4. Linking and thresholding (hysteresis): • Define two thresholds: low and high • Use the high threshold to start edge curves and the low threshold to continue them MATLAB: edge(image, ‘canny’) Source: D. Lowe, L. Fei-Fei
- 21. Example norm of the gradient
- 24. 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. Source: D. Forsyth
- 25. 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). Edge linking Source: D. Forsyth
- 26. Hysteresis thresholding Check that maximum value of gradient value is sufficiently large • drop-outs? use hysteresis – use a high threshold to start edge curves and a low threshold to continue them. Source: S. Seitz
- 27. Hysteresis thresholding original image high threshold (strong edges) low threshold (weak edges) hysteresis threshold Source: L. Fei-Fei
- 28. Effect of σ (Gaussian kernel spread/size) Canny with Canny with original The choice of σ depends on desired behavior • large σ detects large scale edges • small σ detects fine features Source: S. Seitz
- 29. Edge detection is just the beginning… Berkeley segmentation database: http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/ image human segmentation gradient magnitude
- 30. Next: Corner and blob detection