Survey of Gaussian-Based Edge-Detection Methods Mitra Basu

1. Survey of Gaussian-Based Edge-Detection MethodsMitraBasu Presented by: Ali Agha 23Feb2009

2. Motivation • What is the edge detection? And Why we need it? • Edge detection is the process which detects the presence and locations of intensity transitions. • drastically reduces the amount of data • important information about the shapes of objects • easy to integrate into a large number of object recognition algorithms

3. Problem of edge detection • The addition of noise to an image can cause the position of the detected edge to be shifted from its true location. • Any linear filtering or smoothing performed on these edges to suppress noise will also blur the significant transitions. • Solution?

5. Earlier methods: • Some of the earlier methods, such as the Sobel and Prewitt detectors, used local gradient operators which only detected edges having certain orientations and performed poorly when the edges were blurred and noisy. • Sobel operator:

6. Sobel Operator Figures adapted from: http://en.wikipedia.org/wiki/Sobel_operator

7. Problems of methods based on local gradient • Effects of noise Figures adapted from: http://en.wikipedia.org/wiki/Sobel_operator

8. Smoothing filter Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

9. Gaussian derivatives Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

10. Laplacian of Gaussian Laplacian of Gaussian operator Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

11. Scale-space representation For a given image f(x,y), its linear (Gaussian) scale-space representation is a family of derived signals L(x,y;t) defined by the convolution of f(x,y) with the Gaussian kernel Such that Figures adapted from: http://en.wikipedia.org/wiki/Scale-space

12. Multiscale edge detection • Procedure • Applying smoothing operators of different sizes • Extracting the edges at each scale • Combining the recovered edge information to create a single edge map. • Problems to be solved • how many filters should be used • how to determine the scales of the filters • how to combine the responses from each filter so as to create a single edge map.

14. SIGNIFICANCE OF THE GAUSSIAN FILTER • Babaudet al. proved that when one-dimensional (1-D) signals are smoothed with a Gaussian filter, the scale space representation of their second derivatives shows that new zero-crossings are never created. • Yuilleet al. extended this work to 2-D signals (proved that with Laplacian) • The best tradeoff between the conflicting goals of the localization in spatial and frequency domains • The only rotationally symmetric filter that is separable in Cartesian coordinates.

15. 2D edge detection filters Laplacian of Gaussian Gaussian derivative of Gaussian Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

17. Marr-Hildreth method • Consider the Gaussian operator in two dimensions given by • Applied Gaussian filters of different scales to an image. • They find the zero-crossings of their second derivatives using the LOG function • The Marr-Hildreth operator formally introduced Gaussian filter into the edge-detection process. This is a turning point in the low-level image processing research area.

18. Marr-Hildreth method’s problems • Zero-crossings are only reliable in locating edges if they are well separated and the SNR in the image is high. • The location shifts from the true edge location for the finite-width case. • Detection of false edges. Zero-crossings correspond to local maxima and minima. • Missing edges

19. Marr-Hildreth method’s problems • it is very difficult to combine LOG zero-crossings from different scales, because: • a physically significant edge does not match a zero-crossing for more than a few and very limited number of scales • zero-crossings in larger scales move very far away from the true edge position due to poor localization of the LOG operator • there are too many zero-crossings in the small scales of a LOG filtered image, most of which is due to noise.

21. Canny edge detector - Formulation Figures from:

22. Canny edge detector - Formulation • Canny developed an operator, based on optimizing three criteria • good detection • good localization • only one response to a single edge.

23. Canny’s method – Optimal filter • By variational methods, Canny showed that the optimal filter given these assumptions is a sum of four exponential terms. He also showed that this filter can be well approximated by first-order derivatives of Gaussians. For example for a 1-D step edge: Figures from:

24. Canny’s method – Optimal filter • an example of a 5x5 Gaussian filter

25. Canny’s method – image gradient • The edge detection operator (Roberts, Prewitt, Sobel for example) returns a value for the first derivative in the horizontal direction (Gy) and the vertical direction (Gx). • Magnitude and direction:

26. Canny’s method – image gradient original image (Lena) norm of the gradient Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

27. Canny’s method – Non-maxima suppression • Derivative directions are rounded to four angles • At each point, compute its edge gradient, compare with the gradients of its neighbors along the gradient direction. If smaller, turn 0; if largest, keep it. http://www.pages.drexel.edu/~weg22/can_tut.html Figures from: Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

28. Canny’s method – Non-maxima suppression thresholding thinning (non-maximum suppression) Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

29. Canny’s method – hysteresis thresholding • Therefore we begin by applying a high threshold. This marks out the edges we can be fairly sure are genuine. Starting from these, using the directional information derived earlier, edges can be traced through the image. While tracing an edge, we apply the lower threshold, allowing us to trace faint sections of edges as long as we find a starting point. Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

30. Effect of  (Gaussian kernel size) original Canny with Canny with • The choice of depends on desired behavior • large detects large scale edges • small detects fine features adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

31. Problems with Canny edge detector • The algorithm marks a point as an edge if its amplitude is larger than that of its neighbors without checking that the differences between this point and its neighbors are higher than what is expected for random noise. • The technique causes the algorithm to be slightly more sensitive to weak edges, but it also makes it more susceptible to spurious and unstable boundaries wherever there is an insignificant change in intensity (e.g., on smoothly shaded objects and on blurred boundaries).

32. Schunck method • The initial steps of Schunck’s algorithm are based on Canny’s method. • The gradient magnitudes over the chosen range of scales are multiplied to produce a composite magnitude image. • Ridges that appear at the smallest scale and correspond to major edges will be reinforced by the ridges at larger scales. Those that do not, will be attenuated by the absence of ridges at larger scales.

33. Schunck method - problems • he did not discuss how to determine the number of filters to use. • He chooses the width of the smallest Gaussian filter to be around 7. Choosing such a large size for the smallest filter, Schunck’s technique loses a lot of important details which may exist at smaller scales.

34. Witkin’s representation • Idea: • examine the smoothed signal at various scales • The zero-crossings of the second derivative are marked. • This scale-space representation of a signal contains the location of a zero-crossing at all scales starting from the smallest scale to the scale at which it disappears.

35. first derivative peaks Witkin’s representation larger Gaussian filtered signal Properties of scale space (w/ Gaussian smoothing) • edge position may shift with increasing scale () • two edges may merge with increasing scale • an edge may not split into two with increasing scale adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

36. Bergholm’s method • Bergholm proposed an algorithm which combines edge information moving from a coarse-to-fine scale. His method is called edge focusing. • The idea behind edge focusing is to reverse the effect of the blurring caused by the Gaussian operator. The most obvious way of undoing the blurring process is to start with edges detected at the coarse scale and gradually track or focus these edges back to their original locations in the fine scale.

37. Bergholm’s method - problems • how to determine the starting and ending scales of the Gaussian filter? This is a parameter which is critical in determining how well the algorithm performs • Since edge focusing is obtained at a finer resolution, some edges (i.e., the blurred ones, such as shadows) present a juggling effect at small scales. This is due to the splitting of a coarse edge into several finer edges, and tends to give rise to broken, discontinuous edges.

38. Lacroix’s method • Idea: avoids the problem of splitting edges by tracking edges from a fine-to-coarse resolution • Start with Canny method • then considers three scales • The smallest scale is the detection scale • The largest scale is the coarsest scale, at which the edgel still remains • An edgel is validated and then tracked if: 1) it is the local maximum of a Gaussian gradient and 2) the two regions it separates are significantly different from one another.

39. Lacroix’s method - problems • problem of localization error as it is the coarsest resolution that is used to determine the location of the edges. • No explanation as to how to decide which scales are to be used and under what conditions.

40. Williams-Shah method • Idea: Starts with Canny’s method and after thinning gradient maxima points, they linked based on four measures: • 1) noisiness; 2) curvature; 3) contour length; and 4) gradient magnitude. • The set of points having the highest average weight is chosen. • Then, repeatedly, the next smaller scale is used, and the regions around the end points of the contours are examined to determine if there are possible edge points at the smaller scale having similar directions to the end points of the contours.

41. Williams-Shah method - problems • They did not suggest the best way to choose the value of scales and under what conditions.

42. Goshtasby’s method • Idea: modified scale-space representation • Instead of the zero-crossings, the signs of pixels after filtering with LOG operator are recorded. Figures from:

43. Goshtasby’s method Algorithm B(1,2) • Assuming the sign images obtained at scales 1 and 2 are I1 and I2, respectively, then: • If a region in I1 falls on more than two regions of the same sign in I2 then make a convolution in scale  to determine the sign image at scale , where =(1+ 2)/2. Then make a recursive calls to B. Otherwise, exit B. Figures from:

44. Goshtasby’s method - problems • The major problem with Goshtasby’s edge focusing algorithm is the need for a considerable amount memory to store the three-dimensional (3-D) edge images.

45. Deng-Cahill method • Idea: adapting the variance of the Gaussian filter to the noise characteristics and the local variance of the image data • They proposed that the variance of a 1-D Gaussian filter at location is

46. Deng-Cahill method - problem • The major drawback of this algorithm is that it assumes the noise is Gaussian with known variance. In practical situations, however, the noise variance has to be estimated. • The algorithm is also very computationally intensive.

47. Bennamoun’s method • present a hybrid detector (GoG+LoG) that divides the tasks of edge localization and noise suppression between two subdetectors. Figures from:

48. Bennamoun’s method – Scale & threshold • The work is extended to automatically determine the optimal scale and threshold by: • 1) finding the probability of detecting an edge for a signal with noise P(A) • 2) finding the probability of detecting an edge in noise only P(B) • Maximizing below cost function

49. Bennamoun’s method - problem • As the authors’ results show, their technique is still susceptible to false edge-detection, especially in the presence of high noise levels.

50. Qian-Huang method • A new edge detection scheme that detects two-dimensional (2-D) edges by a curve-segment-based detection functional guided by the zero-crossing contours of the Laplacian-of-Gaussian (LOG) to approach the true edge locations. • Algorithm: • convolving an image with the LOG operator and finding the zero-crossing contours. • contours are then segmented at points with large curvatures. • 2-D edge detection functional. • Adaptive thresholding based on the global noise estimation • Edge segments are combined from different scales using a fine-to-coarse strategy.