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The first mask is used to compute Gx while the second is used to compute Gy. The gradient magnitude image is generated by combining Gx and Gy using Eq. (5). Figure 7B shows an edge magnitude image obtained with the 3x 3 Sobel operator applied to the magnetic resonance angiography (MRA) image of Fig. 7A.

The results of edge detection depend on the gradient mask. Some of the other edge operators are Roberts, Prewitt, Robinson, Kirsch, and Frei-Chen [36,41,53,96,97].

Many edge detection methods use a gradient operator, followed by a threshold operation on the gradient, in order to decide whether an edge has been found [12,16,25,36, 41,72,96,97,107,113]. As a result, the output is a binary image indicating where the edges are. Figures 7C and 7D show the

FIGURE 7 Edge detection using Sobel operator. (A) Original angiography image showing blood vessels, (B) edge magnitude image obtained with a 3 x 3 Sobel mask, (C) edge image thresholded with a low threshold (300), (D) edge image thresholded with a high threshold (600).

5 Edge-Based Segmentation Techniques

An edge or boundary on an image is defined by the local pixel intensity gradient. A gradient is an approximation of the firstorder derivative of the image function. For a given image f(x, y), we can calculate the magnitude of the gradient as

Gx2 + Gy and the direction of the gradient as D ^ tan-(g where Gx and Gy are gradients in directions x and y, respectively. Since the discrete nature of digital image does not allow the direct application of continuous differentiation, calculation of the gradient is done by differencing [36].

Both magnitude and direction of the gradient can be displayed as images. The magnitude image will have gray levels

FIGURE 7 Edge detection using Sobel operator. (A) Original angiography image showing blood vessels, (B) edge magnitude image obtained with a 3 x 3 Sobel mask, (C) edge image thresholded with a low threshold (300), (D) edge image thresholded with a high threshold (600).

results of thresholding at two different levels. Please note that the selection of the appropriate threshold is a difficult task. Edges displayed in Fig. 7C include some background pixels around the major blood vessels, while edges in Fig. 7D do not enclose blood vessels completely.

The edge-based techniques are computationally fast and do not require a priori information about image content. The common problem of edge-based segmentation is that often the edges do not enclose the object completely. To form closed boundaries surrounding regions, a postprocessing step of linking or grouping edges that correspond to a single boundary is required. The simplest approach to edge linking involves examining pixels in a small neighborhood of the edge pixel (3x3, 5x5, etc.) and linking pixels with similar edge magnitude and/or edge direction. In general, edge linking is computationally expensive and not very reliable. One solution is to make the edge linking semiautomatic and allow a user to draw the edge when the automatic tracing becomes ambiguous. For example, Wang et al. developed a hybrid algorithm (for MR cardiac cineangiography) in which a human operator interacts with the edge tracing operation by using anatomic knowledge to correct errors [121]. A technique of graph searching for border detection has been used in many medical applications [6,14,64,81,105,106,112]. In this technique each image pixel corresponds to a graph node and each path in a graph corresponds to a possible edge in an image. Each node has a cost associated with it, which is usually calculated using the local edge magnitude, edge direction, and a priori knowledge about the boundary shape or location. The cost of a path through the graph is the sum of costs of all nodes that are included in the path. By finding the optimal low-cost path in the graph, the optimal border can be defined. The graph searching technique is very powerful, but it strongly depends on an application-specific cost function. A review of graph searching algorithms and cost function selection can be found inRef. [107].

Since the peaks in the first-order derivative correspond to zeros in the second-order derivative, the Laplacian operator (which approximates second-order derivative) can also be used to detect edges [16,36,96].

The Laplace operator V2 of a function f (x, y) is defined as

FIGURE 8 Results of Laplacian and Laplacian of Gaussian (LoG) applied to the original image shown in Fig. 7A. (A) 3 x 3 Laplacian image, (B) result of a 7 x 7 Gaussian smoothing followed by a 7 x 7 Laplacian, (C) zero-crossings of the Laplacian image A, (D) zero-crossings of the LoG image B.

The Laplacian is approximated in digital images by an N by N convolution mask [96,107]. Here are three examples of 3 x 3 Laplacian masks that represent different approximations of the Laplacian operator:

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