Appendix A

A	Formulation of B-Spline and Hermite Finite Element Models

The formulation of the finite element active contour models described in Chapters 2 and 3 relies upon the specification of shape functions defining the piecewise continuous curve. The cubic B-Spline shape functions and the cubic Hermite shape functions are defined in A.1. The resulting stiffness matrix for each set of shape functions is derived in A.2.

A.1 Specification of Finite Element Shape Functions

Cubic Hermite Shape Functions

The set of four cubic hermite blending functions are defined as (Bartels et al., 1987):

(1)

Define a vector of the above shape functions:

(2)

The vector of control vertices defining one element of a piecewise parametric curve required for use with these equations is given by:

(3)

where the end-derivatives, D_n, are specified as the default slopes:

(4)

Thus any point in one element of the curve is defined by:

(5)

It is helpful at this stage to rewrite (5) making use of the definition for D_n and D_n+1 given by (4). Thus the matrix equation for a point on the curve may be written explicitly in terms of the control vertices. This provides a modified set of cubic Hermite shape functions:

(6)

To be clear, the corresponding vector of control vertices is given by:

(7)

Thus the curve defined in (5) is now equivalently specified by the inner product of (6) and (7). This results in a more manageable form that will aid in the development of the stiffness matrix arising from the definition of the internal energy terms of the active contour model.

Cubic B-Spline Shape Functions

The cubic B-Spline shape functions are defined by Bartels et al., 1987:

(8)

Thus any point on the curve v(s) contained within an element e is given by:

(9)

Note that the cubic curve defined by set of control vertices using B-Spline elements will differ from the cubic curve defined by the same set of control vertices when the elements are hermites. Hermite interpolation allows the curve to pass through the control vertices which are located at the end points of each element. B-Splines do not possess this ability to interpolate the control vertices so that in general the control vertices and element end points do not coincide.

A.2 Development of Finite Element Matrix Equations

Application of the Rayleigh-Ritz finite element method with cyclic boundary conditions and constant internal parameters:

(1) With reference to the energy functional (2.1) in Chapter 2: for each element e derive the following

(10.1)

(10.2)

where X_e and Y_e are the first and second columns of the elemental control-vertex vector, V_e, respectively and N_s(s) and N_ss(s) respectively are matrices of the first and second derivatives of the elemental shape functions (regardless of specific form).

(2) Performing the differentiation with respect to the control vertices in (10.1) and (10.2) yields:

(11.1)

(11.2)

(11.1) and (11.2) may be combined into a single matrix equation:

(12)

Without regard to the specific form of shape equations to be used, (12) will reduce to the form of the following equation:

(13)

Clearly, to proceed any further with (12) it is necessary to specify the form of the shape functions. This will allow direct evaluation of the first integral term on the RHS of (12). Due to the complex nature of the external potential function, P, the integration of its partial spatial derivatives along each element must be performed numerically. Thus no further reduction of f_e is possible.

A.2.1 Hermite Cubic Elemental Stiffness Matrix

Substituting the cubic hermite shape functions from (6) for N(s) in (12) and performing the necessary calculus operations yields a matrix equation in the form of (13) where K₁ and K₂ have the following specific definitions:

(14)

(15)

Using the definition K_e=aK₁+bK₂ for each element e in the domain of the solution, one may deduce that the final system assemblage stiffness matrix will be a cyclic sparse matrix (hepta-diagonal), the rows of which will be formed by shifted versions of the following row vector:

(16)

A.2.2 Cubic B-Spline Shape Functions

In an identical manner to the above, substituting the cubic B-Spline shape functions (8) into (12) yields:

(17)

(18)

As in the previous section, the B-Spline equivalent of (16) becomes:

(19)

Þ Appendix B

Title Page	4 Manual Delineation	Appendix A
Contents	5 Parameter Setting	Appendix B
1 Introduction	6 Results	Appendix C
2 Literature Survey	7 Discussion	Colour Plates
3 Algorithm Development	8 Conclusion	References