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Representation Using Functions of Voxels

Target regions and boundaries in images can be represented by using functions defined on an image space. As will be described below, one of the advantages of this category of representations is that they can straightforwardly depict topological changes in the figures, and one of the disadvantages is that the representations are largely redundant.

Letting x denote the coordinates of a voxel in a three-dimensional image space, where x = [x, y, z]T denotes the coordinates of a voxel in a given image, the figures in the images can be explicitly represented by using discrete functions, /label(x) : R3 ! N. A value of/label (x) at x describes a label corresponding to each figure in an image. Letting k = 1,2,..., K denote the label that identifies the figures, where K is the number of target figures to be represented, a labeling function explicitly represents the figures as

where k(x) denotes the label of the figure to which the voxel, x, belongs. Letting a binary function, fk(x), denote if a voxel, [x], belongs to the k-th figure or not, the figures in a given image can be represented using following vector labeling function, fvec(x) : R3 ! {0,1}K such that

where к = [/1(x),f2(x),... ,fK(x)]r. When each voxel belongs to a single figure, then Xk/k (x) = 1 should be satisfied.

Continuous functions, fcont(x) : R3 ! R, defined on an image space can represent figures implicitly. For example, regions, R, in a given figure can be represented as follows:

where T e R is a threshold. Then, the closed boundary surfaces of the regions, R, can be represented as the voxels where the value of fcont(x) passes through T. For example, zero-crossings of the Laplacian of a (smoothed) image are widely employed for the detection of edge points [55-57].

Level-set representation [58] is also widely employed for implicitly representing surfaces in images. Let a level-set function be denoted by flevel(x) : R3 ! R. Surfaces in an image can be implicitly represented by the zero-crossings offlevel(x) where the sign of the function changes. Different from the representations using the discrete functions, you can differentiate the level-set functions and can compute some geometric properties of the surfaces from the differential coefficients. Let a closed surface represented byflevel(x) be denoted by S and assume thatflevel(x) < 0 is satisfied inside S. Let the differential coefficients of flevel(x) be denoted as фх = 3flevel/Эх or as фх = 92flevel/9z9x. Then, for example, the outward unit normal vector at x on S, n(x), can be obtained as follows:

The mean curvature, km, and the Gaussian curvature, kg, can be computed as follows:

An example of level-set representations corresponding to topological changes

Fig. 2.12 An example of level-set representations corresponding to topological changes. Changing a level-set function continuously enables representation of topological changes of the target figures (These figures are appeared in [58])

This property of level-set representation enabling direct computation of the geometric properties with the differential coefficients of the level set function is used in the level methods for propagating surfaces [58, 59]. One of the strongest advantages of level-set representations is the ability to represent the topological changes of the surfaces (e.g., a single closed surface split into two surfaces) straightforwardly as shown in Fig. 2.12.

It is often necessary to measure quantitatively the distance between figures. This measurement is performed by comparing the values of the functions that represent the figures at each voxel. For example, assume two regions, Ra and Rp, are represented by the binary functions, fa(x) and fp(x), where a and p are the labels of the regions, respectively. Then, the Jaccard index (JI), dJ(Ra, Rp), defined below can be employed for measuring the distance:

where |R| denotes the number of voxels belonging to the region, R. |Ra П Rp denotes the number of voxels at which fa(x) = 1 and fp(x) = 1 are satisfied and |Ra U Rp| denotes the number of voxels where fa(x) = 1 or fp(x) = 1 are satisfied. When the regions are represented by the level-set functions,f^(x) and/^(x), then the distance dlevel(Ra, Rp) can be expressed as

where Q denotes the region of interest in which the two regions are included, under the assumption that a unique level-set function that represents a given region is obtained.

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