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{f (x z) + g(z)} .
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Its complementary operation, the erosion is given by (f g)(x) =
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The multiscale dilation erosion of the image f (x) by g (x) is de ned by [21] (f g )(x) (f g )(x) = f (x) (f g| | )(x) if > 0, if = 0, if < 0.
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where denotes the scale parameter of the structuring function. In reference 19, it was shown that the choice of structuring function does not lead to statistically signi cant changes in the face veri cation performance. However, it affects the computational complexity of feature calculation. Such morphological operations can highlight and capture important information for key facial features, such as eyebrows, eyes, nose tip, nostrils, lips, and face contour, but can be affected by different illumination conditions and noise [19]. To compensate for these conditions, the normalized multiscale dilation erosion has been also proposed for facial image analysis [27, 30]. It is well known that the different illumination conditions affect the facial region in a nonuniform manner. However, it can safely be assumed that the illumination changes are locally uniform inside the area of the structuring element used for multiscale analysis. The proposed morphological features are calculated by subtracting the mean value of the intensity of the image f inside the area of structuring element from the corresponding maximum (dilation) or minimum (erosion) of the area. Formally, the normalized multiscale morphological analysis is given by (f g )(x) m (f, x, G ) if > 0, if = 0, (f g )(x) = f (x) (f g| | )(x) m+ (f, x, G ) if < 0,
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6.3 Facial Region Modeling
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Figure 6.4. Output of normalized multiscale dilation erosion for seven scales.
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where m (f, x, G ) and m+ (f, x, G ) are the mean values of the image f (x z), x z D and f (x + z), x + z D inside the support area of the structuring element G = {z G : ||z|| < }, respectively. Another implementation for the operators m+ (f, x, G ) and m (f, x, G ) would be the median of the values of the image inside the support area of the structuring element. The output of these morphological operations forms the jet j(xl ), at the graph node l that is located in the image coordinates xl : j(xl ) = ((f g )(xl ), . . . , (f g 1 )(xl ), f (xl ), (f g 1 )(xl ), . . . , (f g )(xl )), (6.14) where is the number of different scales used. Figure 6.4 depicts the output of normalized dilation erosion for various scales used. The lower right image is the original image extracted from the XM2VTS database. The rst seven images, starting from the upper left corner, are the normalized eroded images and the remaining nine are the normalized dilated images.
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Another option for analyzing a gray-scale facial image region f (x) is by employing the morphological signal decomposition (MSD) [22]. Given f (x) : D Z2 Z and a structuring function g(x) : G Z2 Z, the gray-scale dilation of the image f (x) by the structuring function g(x) is noted by (f g)(x) and its dual operation, the gray-scale erosion, is de ned as (f g)(x), [29, 31]. Let f (x) be approximated by
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where fi (x) denotes the ith component and K is the total number of components. MSD provides a simple method to determine the components and can then be implemented recursively as follows. Step 1. Initialization: f 0 (x) = 0. Step 2. Find the ith level of decomposition. Starting with ni = 1, increment ni until (f f i 1 ) (ni + 1)g (x) 0. (x). (6.17) (6.16)
Step 3. Calculate the ith component by (f f i 1 ) fi (x) =
li (x)