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Representation Using Parametric Functions

Parametric functions, whose values denote the coordinates in an image space, are also widely employed for representing figures in images.


Let s denote a scholar parameter that indicates the location along a given curve, C. Then, the curve, C, can be explicitly represented as x(s) = (x(s) , y(s) , z(s))T : R ! R3. Different from the representations described in the previous subsection, the functions x(s), y(s), and z(s) are functions of a parameter, and the values of the functions denote the coordinates in the image. Here, assume that x(s), y(s), and z(s) are represented by linear combinations of basis functions, C(s) (i = 1,2,... ,Nb), as follows:

where ai, fo, and yi are scalar coefficients of the basis functions. Then, a curve with explicit functions, x = x(si в)canberepresentedthus : R3Nb ! R3, where в = [ai,Pi,yi,a2, ••• , yNB]T.

One of the most popular basis functions employed for representing curves is B-spline functions (“B” comes from basis), which are piecewise polynomials. Let the i-th B-spline basis function of degree k in the variable, s, be denoted by B(s). The B-spline functions, Bk(s), have the following properties: first, the functions are not negative, Bk(s) > 0, have a compact support [ui, ui+k+1], and are k — 1 times continuously differentiable. In addition, by convention, the B-spline functions are constructed in a way such that they sum to 1 at all points:

A variety of functions can have the properties described above, and a simple example of the B-spline bases for representing closed curves is described below. More details of the spline representations can be found in [60].

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Using the de Boor’s algorithm, the k-th order B-spline function can be constructed from the k — 1-th order functions (k > 1) as follows:


For example, the second-order (k = 2) function, B^(s) has the form as follows when Ui = i:

The graphs of the second-order functions are demonstrated in Fig. 2.13. A closed curve can be represented by a periodic function, which satisfies x(s) = x(s + qL) for any integer q where L denotes the period, and L = uNb when the closed curve is represented by NB B-spline basis functions. The basis functions should be appropriately wrapped as shown in Fig. 2.12 for representing closed curves. Using these NB wrapped basis functions enables the representation of closed curves as

An example of the second-order B-spline functions (This figure comes from the website of active contours) follows

Fig. 2.13 An example of the second-order B-spline functions (This figure comes from the website of active contours) follows:

where в, (i = 0,1,... ,NB) are three-vector, each denoting the coordinates in an image space. The NB points whose coordinates are represented by в i are called control points, and their coordinates, в = [в[, вT,..., вNB-1]T 2 R3Nb, are the parameters for the representation of the curves. Changing the locations of the control points allows changing of locations, sizes, and shapes of the curves. It should be noted that, by changing the location of a control point, the local portion of the curve can be deformed because each of the B-spline functions has a compact support. Let two curves be denoted by x(s; в“) and x(s; в ^). The distance between the two curves, d(x(s; в“), x(s; в?)), can be defined as follows:

where [0, L] denotes the domain of s. Substituting (2.125) with (2.126) results in Let

Then, Eq. (2.127) can be written as

The distance between the two curves is defined as the distance between the two corresponding parameter vectors with a metric defined by by. It should be noted that the metric, by, can be determined only by the basis functions and can be computed before figures are given. It should also be noted that the distance is defined based on the distance between two points whose locations are indicated by the same value of s on both curves. This means that the locations, sizes, and orientations (poses) of curves should be normalized before the distances among them are measured and that each of the curves should be appropriately parameterized so that corresponding points on the curves have the same values of s.

A Fourier series can also represent a closed curve in an explicit way. Again, letting a positive scalar, L, denote the period of x(s), wherex(s) = [x(s) , y(s), z(s)]T = [x(s + L), y(s + L), z(s + L)]T for any s, allows the representation of x(s), y(s), and z(s) by linear combinations of the Fourier basis functions:

Letting the Fourier coefficients be denoted by в = [a0x),al(y),...,^(z_^1]T, an explicit representation of a closed curve can be obtained such that x = x(s; в). Changing the value of each of the Fourier coefficients can change the amount of the component of the corresponding frequency and can deform the whole shape of the target curve. For example, setting the values of the coefficients corresponding to higher frequencies to zero can smooth the target curve.

The same definition (Eq. (2.126)) of the distances between figures can be employed, and the distances can be computed in the same way (Eq. (2.129)) even when using different basis functions for the representation. A difference appears in the metric tensor, B. Because of the unity and the orthogonality of the sinusoidal basis functions, an identity matrix for the metric tensor is obtained, and hence the following equation holds:

A PDM of a curve [61], C, is another example of explicit representations of curves, where it represents a curve with a series of the coordinates of L points. Letting i (i = 0,2,..., L— 1) denote an index number for each of the points along the curve, and letting x(i) denote the coordinates of the i-th point, x = x(i) is an explicit representation of a curve. Letting a 3L-vector, uC = [xT(0) , xT(1) ,... ,xT(L — 1)]T denote the coordinates of the L points and represent the curve, the distance between two curves represented by a PDM can be defined as

The distances between two corresponding points with identical numbers are used to define the distance between the two curves.

A PDM can also be represented by a linear combination of basis functions. Letting a 3L-vector, x, denote a representation of some curves and letting 3L- vectors, ui (i = 0,1,..., — 1), denote basis functions, where NB < 3L, allows

representation of a curve with a linear combination of the basis functions as follows:

where 9i is a scalar and в = [0O , в1, 9Nb ]r. Changing the values of the

parameters, в causes deformation of the shape of the curve. In many applications, the basis functions are constructed from a set of training data, D = {x( j)j =

1,2 , ..., Mg, where M denotes the number of the data: A PCA is applied to the data set, D, and the mean of D and the eigenvectors corresponding to the largest NB eigenvalues are selected for the basis functions. Then the variety of a target curve can be represented by an NB-dimensional linear space as follows:

where X is the mean vector and ui denotes the eigenvectors. As described in Sect. 2.2.3, the Mahalanobis distance between the mean shape, X, and a described curve, х), is given as

where Xi are the eigenvalues corresponding to ui.

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