Let s and t denote scalar parameters that indicate the location along a given surface,

S. The surface, S, can be represented explicitly as x(s, t) = [x(s , t) , y(s , t) , z.s, t)]T : R2 ! R3. Assume that x(s , t), y(s,t), and z(s,t) are represented by linear combinations of basis functions, Ci(s, t), as follows:

where ui, vi, and wi are scalar coefficients of the basis functions. Then, the surface can be represented in an explicit manner as x = x(s, t, в), where в = [u0, v0, w0, u1..., vNB-1 , wNb_1]t. B-spline or Fourier functions can be employed for the basis functions. Here, the NURBS (nonuniform rational basis spline) functions are described because they are also widely employed for the surface representation. Letting a B-spline function be denoted by Bk(s), a NURBS function, Ь(s), is defined as

where Wj is a positive weighting coefficient, which determines the weight of the j-th control point. Bk(s) = bk(s) when Wj = 1 for all j. The NURBS functions in (2.137) can be used as basis functions for representing curves. NURBS basis functions for surfaces can be constructed thus:

Letting the coordinates of the (i, j)-th control point be denoted by в ij and letting в = [вl,0, ввNB_i,Nb_iF, a can be represented using the NURBS functions as follows [62-64]:

Analogous to Eq. (2.126), the distance between two surfaces, x(s, t, в “) and x(s, t, в^), can be defined as follows:

where S is a coefficient for the normalization with respect to the size of the domain of (s, t). Equation (2.140) can be written thus:


and the component of the metric tensor, B, is determined by bij,i,m.

When lines/surfaces are represented using the B-splines or NURBS, the coordinates of each point on the lines/surfaces are represented in a Cartesian coordinate system. Other coordinate systems can also be employed for representation. One of

(a) The spherical coordinates. (b) Spherical harmonics functions corresponding to l = 0,1,2,3, and 4

Fig. 2.14 (a) The spherical coordinates. (b) Spherical harmonics functions corresponding to l = 0,1,2,3, and 4

the most popular coordinate systems for non-Cartesian representation is a spherical coordinate system, in which the coordinates of a point in a three-dimensional space are represented with three parameters, the radial distance r, the polar angle в, and the azimuth angle p as shown in Fig. 2.14a. Spherical harmonics form a complete set of orthonormal functions defined on the unit sphere, on which the location of a point can be represented by the angular components, (в, p): Analogous to the Fourier series that are a complete set of orthonormal functions defined on the unit circle, the spherical harmonics can expand any square-integrable function as a linear combination of these functions. Letting a given square-integrable function be denoted by f (в, p) : R2 ! R and letting the spherical harmonics be denoted by У?(в,р), where l is a nonnegative integer and m is an integer satisfying —l < m < l enable representation of the given function as

where ff denotes the complex expansion coefficients. The spherical harmonics functions are given as follows:

where Pjm is an associated Legendre polynomial:

Given a function defined on the unit sphere, f (q), where q = (в, f)T, allows computation of the coefficients as follows:

where Yf is the complex conjugate of Yf. Some examples of the spherical harmonics are shown in Fig. 2.14b. Setting a two-dimensional coordinate system (в, ) on a given simple1 closed surface allows representation of the coordinates of each point on the surface in a Cartesian coordinate system by functions of the angular components, (в, iff), as x = x(e,f) = [x(e,f),y(e,f), z(e,f)]T. Expanding these functions, x, y, and z, to linear combinations of the spherical harmonics functions results in representation of the surface as

where c = {cfl e N, — l < m < lg are the coefficients, each of which is a three- vector, cf = (xf, yf, zm)T, where each of its components is computed from х(в, f ), у(в, f), and z(e,f ), respectively, as shown in Eq. (2.146). Changing the values of the coefficients, cf, allows deformation of the surface. It should be noted that the coefficients with larger values of l correspond to the higher frequencies of the surface and to the smaller details of its shape.

A PDM [8, 65-68] represents a surface using the coordinates of the points distributed on it: x(i) : N ! R3, where i denotes an index of each of the points on the surface. The distance between two surfaces represented by PDMs is defined as (2.132). Examples of liver surfaces represented by PDMs are shown in Fig. 2.15. Analogous to PDM-based curve representations shown in (2.134), surfaces can be represented using linear combinations of eigenvalues obtained by applying a PCA

'A simple surface does not have a self-crossing point.

Examples of liver surfaces represented by PDMs

Fig. 2.15 Examples of liver surfaces represented by PDMs

to a set of training surfaces for targets:

where x and u denote the mean of the training surfaces and the eigenvalues corresponding to the NB largest eigenvalues, respectively.

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