Figure 1119 Three edges, a texture

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Figure 1120 Three corners, a texture

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In other words, the assumption that fg = fh allows us to link the geometric and probabilistic interpretations of the spectrum Points on a smooth contour have an such that: fh ( ) = 1 because a smooth contour lls the space as a line fg ( ) = 1 because a smooth contour has a given probability to appear In fact, we may de ne the point type (ie edge, corner, smooth region, etc) through its associated f ( ) value; a point x with f ( (x)) = 1 is called an edge point, a point x with f ( (x)) = 2 is called a smooth point, and, more generally, for t [0, 2], x is called a point of type t if f ( (x)) = t A bene t of the multifractal approach is thus that it allows us to de ne not only edge points, but a continuum of various types of points An important issue lies in the choice of a relevant sequence of capacities for describing the scene The problem of nding an optimal c in a general setting is unsolved In practice, we often use the following Assume the image is de ned n on [0, 1] [0, 1] Let P := ((Ik )0 k< n )1 n N be a sequence of partitions of n n n n [0, 1] [0, 1] and (xk , yk ) be any point in Ik Each Ik is made of an integer number n n of pixels Let L(Ik ) denote the sum of the gray levels in Ik Let (x, y) denote a generic pixel in the image and L(x, y) denote the gray level at (x, y) Let (pn )1 n N be a xed sequence of positive integers and be a region in the image sum measure: cs ( ) =

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L(x, y)

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max capacities: cm ( ) = n

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n+p n+p Ik n /(xn+pn ,yk n ) k

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n+p L(Ik n )

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cM ( ) = max L(x, y)

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iso capacities:

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n+p n+p ci ( ) = max # {k / L(Ik n ) = l, (xn+pn , yk n ) } n k l

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c ( ) = max # {(x, y) / L(x, y) = l, (x, y) }

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It is easy to see that: cs ( ) depends on both the gray level values and their distribution in ,

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cm ( ) and cM ( ) only depend on the gray level values, n ci ( ) and cI ( ) only depend on the gray level distribution n Thus, (cm , cM ) and (ci , cI ) give in some loose sense orthogonal information n n about the image Furthermore, it can be shown that they are more robust to noise than cs 11811 Edge detection The simplest procedure for extracting edges using multifractal analysis is as follows: Choose a sequence c of capacities Calculate the H lder exponent of c at each point of the image Calculate the multifractal spectrum of c Declare as smooth edge points those belonging to the set(s) T whose dimension is 1 Declare as irregular edge points those belonging to the set(s) T whose dimension is between two xed values, typically 11 and 15 A result of segmentation using this approach is shown in Figure 1121

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Figure 1121 (a) Original image; (b) H lder exponents with max capacity; (c) smooth edges; (d) irregular edges

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The sum measure, max and iso capacities are the basic tools used for analyzing images There are cases where speci c capacities, and/or more elaborate schemes than the one just described, must be designed in order to get robust results See [LEV 98] for more details 119 Change detection in image sequences using multifractal analysis In [CAN 96], the authors propose a multifractal approach to the problem of change detection in image sequences Multifractal analysis proves to be useful for detecting changes without any a priori knowledge of the object to be extracted If a change occurs in an incoming image, it is re ected in the global description provided by the multifractal spectrum graph The abscissa of the spectrum part ( , fh ( )) that has changed makes it possible to extract binary images corresponding to the changes As can be seen in Figure 1122c and Figure 1123b, the extracted changes using multifractal analysis are much more relevant than the absolute difference between the two images

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Figure 1122 (a) First image, (b) second image and (c) absolute difference (pixel to pixel) between the two images