Prerna Sethi and Hilary W. Thompson
background to grow and when they converge, the points of contact define the boundary.
9.1.2.3. Edge detection
Edge detection is a feature-extraction technique that seeks to provide the information on the location of the regions in the image where the intensity changes sharply or discontinuities are detected. If a pixel lies on the boundary of the object, then the neighborhood pixels will show a region of transition. The two main characteristics of the edge detection operators are slope and direction. The points lying on an edge can be detected by:
(1) detecting local maxima or minima of the first derivative or (2) detecting the zero crossing of the second derivative. Edge detection methods are classified broadly in three categories: first-order derivative (gradient), second-order derivative, and optimal edge detection.
9.1.2.3.1. The first-order derivative (gradient) methods
The first-order derivative methods have kernel operators that calculate the strength of the slope in the vertical or horizontal direction. Consequently, the edge strength is calculated as an aggregate of the different components of the slope. The Prewitt,25 Roberts,26 and Sobel27 operators are classified as the first-order derivative methods.
The Prewitt edge operator measures the horizontal edge component using kernel Kx, and the vertical edge component using kernel, Ky. The gradient intensity of the current pixel is calculated by |Kx| + |Ky |. The Prewitt edge detector operator is easy to implement and is less computationally intensive than other methods. However, one of the major drawbacks of the Prewitt edge detector operator is its sensitivity to noise.24
The Roberts edge operator is a local differential operator used for detecting edges. The mathematical formulation is given as:
g(x, y) = I(r, c) − I(r + 1, c + 1 2 |
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+ I(r + 1, c) − I(r, c + 1) 2 1/2, |
(9.7) |
where, I(r, c) is the input image with the pixel coordinates (r, c). The Roberts edge operator marks the edge points in an image, providing no
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information about the orientation of the edge is available. The results have indicated that this method works best with binary images.28
The Sobel edge operator involves two convolution kernels denoted by g(x) and g(y), where
g(x) |
−2 |
0 |
2 |
and g(y) |
−0 |
−0 |
−0 |
. (9.8) |
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The convolution kernels smooth the image and, hence, are less prone to noise. However, the edge localization is poor since it produces thicker edges.27
9.1.2.3.2. The second-order derivative methods
The second-order derivative methods search for zero crossings to find edges in an image, which are computed from the second-order derivative expression.29,30 The Laplacian, Laplacian of Gaussian, and difference of Gaussian (DoG) methods are classified as second-order derivative methods.
The Laplacian of an input image I denoted by f(x, y) is defined as,
f(x, y) = |
∂2f(x, y) |
+ |
∂2f(x, y) |
(9.9) |
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∂x2 |
∂y2 |
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The Laplacian operator is usually susceptible to noise and requires filtering. The Laplacian of Gaussian method is also known as the Marr-Hildreth edge detector. It is defined as,
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= − |
1 |
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x2 y2 |
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+y2 |
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πσ4 |
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LoG(x, y) |
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e− 2σ2 . |
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The value of σ dictates the value of the Gaussian filter. The broader the Gaussian filter, the more smoothening is performed. However, the LoG is computationally intensive. The DoG can approximate the LoG. The LoG can be approximated by the difference of two Gaussians, DoG. The difference of the Gaussian also called the Mexican hat operator is defined as,
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x2+y2 |
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x2+y2 |
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2πσ2 |
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2πσ2 |
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DoG(x, y) = |
e |
1 |
− |
e |
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. |
(9.11) |
2πσ12 |
2πσ22 |
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Here, the width of the edge can be adjusted by changing the values of σ1 and σ2.
9.1.2.3.3. The optimal edge detector
The Canny edge detection31 is one of the most popular edge detection techniques. The algorithm utilizes an optimal edge detector based on a set of criteria to achieve the following optimization constraints:
•Achieve a good localization to mark the edges as closely as possible to the actual edges,
•Mark the edges only once when a single edge exists to minimize the number of responses to a single edge (to help detect the true negatives, that is the nonedges are not marked), and
•Maximize the signal-to-noise ratio in detecting the true positives.
According to Canny, the optimal filter that meets the above three criteria can be approximated as the first derivative of Gaussian function defined as,
G(x, y) = |
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1 x2+y2 |
(9.12) |
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e 2σ2 |
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2πσ2 |
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∂G(x, y) |
x2+y2 |
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∂G(x, y) |
x2+y2 |
(9.13) |
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αxe 2σ2 |
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αxe 2σ2 . |
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Hafez et al.32 have applied the Canny and Sobel edge detectors to find MAs in the fluorescein angiograms of the ocular fundus. They first detected the edges in the image using the Canny edge detector and then subtracted those segments that represented the vessel segments in the image. For the remaining objects, they calculated the edge threshold by computing the Sobel edge operator for each point in the object. The results obtained outperformed the Hough transform method in the computational time, as well as in detecting the number of false MAs.
We obtained a set of fluorescein angiographic images from the Louisiana State University Health Science Center, New Orleans and performed a series of preprocessing steps using a three-step preprocessing algorithm (Fig. 9.3.). First, we map low intensity pixels to high intensity values. The nonzero pixels of the resultant image correspond to the exudates or optic disc, and they are replaced by pixels with an average intensity value. This replacement aids in dismissing false alarms. Second, we smoothen the image for noise
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