## Wavelet Edge Detection and Segmentation

Edge detection plays an important role in image segmentation. In many cases, boundary delineation is the ultimate goal for an image segmentation and a good

Figure 6.23: Sample results using multiscale texture segmentation. (a) Synthetic texture image. (b) Segmentation result for image (a) with a 2-class labeling. (c) MRI T1 image of a human brain. (d) Segmentation result for image (c) with a 4-class labeling.

Figure 6.23: Sample results using multiscale texture segmentation. (a) Synthetic texture image. (b) Segmentation result for image (a) with a 2-class labeling. (c) MRI T1 image of a human brain. (d) Segmentation result for image (c) with a 4-class labeling.

edge detector itself can then fulfill the requirement of segmentation. On the other hand, many segmentation techniques require an estimation of object edges for their initialization. For example, with standard gradient-based deformable models, an edge map is used to determine where the deforming interface must stop. In this case, the final result of the segmentation method depends heavily on the accuracy and completeness of the initial edge map. Although many research works have made some efforts to eliminate this type of interdependency by introducing nonedge constraints [86, 87], it is necessary and equally important to improve the edge estimation process itself.

As pointed out by the pioneering work of Mallat et al. [16], first- or second-derivative-based wavelet functions can be used for multiscale edge detection. Most multiscale edge detectors smooth the input signal at various scales and detect sharp variation locations (edges) from their first or second derivatives. Edge locations are related to the extrema of the first derivative of the signal and the zero crossings of the second derivative of the signal. In [16], it was also pointed out that first-derivative wavelet functions are more appropriate for edge detection since the magnitude of wavelet modulus represents the relative "strength" of the edges, and therefore enable to differentiate meaningful edges from small fluctuations caused by noise.

Using the first derivative of a smooth function 0 (x, y) as the mother wavelet of a multiscale expansion results in a representation where the two components of wavelet coefficients at a certain scale s are related to the gradient vector of the input image f (x, y) smoothed by a dilated version of 0 (x, y) at scale s:

The direction of the gradient vector at a point (x, y) indicates the direction in the image plane along which the directional derivative of f(x, y) has the largest absolute value. Edge points (local maxima) can be detected as points (xo, yo) such that the modulus of the gradient vector is maximum in the direction toward which the gradient vector points in the image plane. Such computation is closely related to a Canny edge detector [88]. Extension to higher dimension is quite straightforward.

Figure 6.24 provides an example of a multiscale edge detection method based on a first derivative wavelet function.

To further improve the robustness of such a multiscale edge detector, Mallat and Zhong [16] also investigated the relations between singularity (Lipschitz regularity) and the propagation of multiscale edges across wavelet scales. In [89], the dyadic expansion was extended to an M-band expansion to increase directional selectivity. Also, continuous scale representation was used for better adaptation to object sizes [90]. Continuity constraints were applied to fully recover a reliable boundary delineation from 2D and 3D cardiac ultrasound in [91]

Figure 6.24: Example of a multiscale edge detection method finding local maxima of wavelet modulus, with a first-derivative wavelet function. (a) Input image and (b)-(e) multiscale edge map at expansion scale 1 to 4.

Figure 6.24: Example of a multiscale edge detection method finding local maxima of wavelet modulus, with a first-derivative wavelet function. (a) Input image and (b)-(e) multiscale edge map at expansion scale 1 to 4.

and [92]. In [93], both cross-scale edge correlations and spatial continuity were investigated to improve the edge detection in the presence of noise. Wilson et al. in [94] also suggested that a multiresolution Markov model can be used to track boundary curves of objects from a multiscale expansion using a generalized wavelet transform.

Given their robustness and natural representation as boundary information within a multiresolution representation, multiscale edges have been used in deformable model methods to provide a more reliable constraint on the model deformation {Yoshida, 1997 #3686; de Rivaz, 2000 #3687; Wu, 2000 #3688; Sun, 2003 #3689}, as an alternative to traditional gradient-based edge map. In [99], it was used as a presegmentation step in order to find the markers that are used by watershed transform.

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