1. EDGE DETECTION IN COMPUTER VISION SYSTEMS PRESENTATION BY : ATUL CHOPRA JUNE 28 2004 EE-6358 COMPUTER VISION UNIVERSITY OF TEXAS AT ARLINGTON

2. CONTENTS: Introduction One-dimensional Formulation Finding Optimal detectors by numerical Optimization Detectors for step edges. An efficient approximation Noise estimation and Threshold

3. Cont. Two or more dimensions Need for multiple widths Conclusion References

4. INTRODUCTION Edge Detection: Simplify the analysis on the image by drastically reducing the amount of data to be processed, while at the same time it preserves the useful structural information about edge boundaries. Criterions for edge detection :  Low error rate  Well localized edge points Need to add third criterion to circumvent the possibility of multiple response to single edge Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

5. One dimensional Formulation We will assume 2-D edges have a constant cross- section in some direction. Performance criterion are as follows  Good Detection 1) Low probability of failing to mark real edge 2) Low probability of falsely marking non edge point.  Good Localization: The points marked as edge points by the operator should be as close as possible to the centre of true edge.  Only one response to single edge Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

6. MATHEMATICAL FORMULATION OF PERFORMANCE CRITERIONS Detection and Localization criterions Let, impulse response of filter : f(x) Let, edge be : G(x) Assume edge is centered at x=0  The response of the filter to the edge at its centre is given by a convolution Reference: A computational approach to Edge detection John Canny, 1986 IEEE

7. a.) First criterion SNR Assume the filter has finite response bounded by [-W,W] The root mean squared noise will be given by Now the first criterion, the output signal-to- noise ratio (SNR) is given by Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

8. b.)Second criterion Localization For localization we need some measure which increases as localization increases. We will use the reciprocal of root-mean squared distance of the marked edge from the centre of true edge. Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

9. Contd. Root mean squared distance is Localization is defined as reciprocal of Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

10. Maximizing the product Reference: A computational approach to Edge detection, John Canny, 1986 IEEE The designing problem is reduced to maximize the product of localization and SNR.

11. c.) Third criterion Multiple responses. When maxima are close together it is difficult to separate the step from noise We need to obtain an expression for the distance between adjacent noise peaks. Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

12. Contd. The mean distance between adjacent maxima in the output is twice the distance between adjacent zero- crossings in the derivative of the output operator. Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

13. Contd. Expected number of noise maxima in region of width 2W is given by Fixing k fixes the number of noise maxima that could lead to false response

14. Finding optimal detectors by Numerical Optimization It is impossible to find an optimal filter f which maximizes the SNR localization product in presence of multiple response constraint. If the function f is discrete the computational problem is reduced to calculation of four inner terms. Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

15. Contd. Penalty function: non- zero values when one of the constraints is violated We then find f which maximizes

16. Detector for Step Edges. Formulation of the SNR and Localization criterion Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

17. Contd. A spatially scaled filter is formed as show below

18. Contd. The uncertainty principle Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

19. Gaussian Edge Detector[2] In real images step edges are not perfectly sharp Images are severely corrupted by noise The type of linear operator that provides the best compromise between noise immunity and localization while retaining the advantages of Gaussian filter is FIRST DERIVATIVE OF GAUSSIAN

20. First derivative of Gaussian[2]  This operator corresponds to smoothing an image with Gaussian function and then computing the gradient.  The operator is symmetric along the edge and antisymmetric perpendicular to edge (i.e. along the direction of gradient.)  Sensitive to edge in direction of steepest change, but insensitive to the edge.  Acts as smoothing operator in the direction of edge

21. Canny edge detector -an Efficient Approximation The optimal operator is similar to the first derivative of Gaussian The reason for doing this is that there are very efficient ways to compute the 2-D extension of filter if it can be represented as some derivative of Gaussian. Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

22. Noise Estimation Weiner Filtering is used to estimate the noise component in the image  Requires knowledge of autocorrelation function of two components i.e. noise and response due to step edges.  Requires knowledge of autocorrelation function of combined signal. Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

23. Contd. Global histogram estimation is used to estimate the noise strength after noise component has been optimally separated.  Noise response should be Gaussian  While step edge response should be composed of large values occurring very infrequently.  If we take a histogram of the filter values, we should find that the position of low percentiles (less than 80%) will be determined mainly the noise energy. [1]

24. Thresholding Broken edge Caused by the operator output fluctuating above and below the threshold along the length of contour. Streaking Single threshold scheme and limitation Possible solution, used by Pentland with Marr-Hildreth zero-crossing

25. Two or more dimensions[1] Definition of edge direction.  Edge direction i.e. the direction of the tangent to the contour that the edge defines in 2-D. In 2-D an edge has one position coordinate and an orientation.

26. Contd. Detection of edges in 2-D[1]  2-D mask for orientation is created by convolving a linear edge detection function aligned normal to the edge direction with a projection function parallel to the edge direction. Projection function is Gaussian with same deviation as that of detection function.[1]

27. The need for multiple widths Choosing the width so as to give best detection/localization tradeoff in particular direction. SNR will be different for each edge in the image [1] Feature Synthesis approach

28. Contd. Reference: A computational approach to Edge detection, John Canny, 1986 IEEE Feature Synthesis  Mark edges from smallest operator.  Synthesis large operator outputs from these edges (convolve with Gaussian normal to the edge direction)  Compare the actual operator outputs with these synthesized outputs.  Additional edges are marked only if the large operator has significantly larger response.

29. Synopsis Edge detection criterions and mathematical formulation Numerical optimization technique to find optimal operators. Detection and localization tradeoff Impulse response of the optimal operator-first derivative of Gaussian.

30. Contd. Adaptive thresholding according to noise estimation Feature synthesis

31. Reference [1] A computational approach to Edge detection, John Canny, 1986 IEEE [2] Machine vision, Ramesh Jain