DEL OBJETIVO ESPECÍFICO N° 3

The shape o f the human head is not strictly symmetric and there should be a unique transformation which could be expected to match exactly two images o f the head. However, a number o f transformations exist which produce incorrect matches having local minima in the distance function. This is due to the partially regularity and symmetricity o f the head shape. The problem implies the need to check all the possible promising transformations which rnight be a candidate for obtaining the global true match. On the other hand, evaluating the function at an insufficient number o f search locations (transformation parameters) might fail to find correct match location, that is, to give proper transformation parameters for registration. As discussed in section 3.2.5, a mean distance error (MDE) is required by the surface fitting process introduced so far, due to the nature o f the minimization process (because the MDE value which is essential for setting the next search direction, is obtained after evaluating all the sample points). Accordingly, all the n extracted features (corresponding points) have to be evaluated at each transformation, prior to any decision on the correctness o f the fit. Considering the two above requirements would result in a very slow convergence algorithm due to evaluating all points at all search locations. On the other hand, not to do this is the cause o f the major disadvantage o f the algorithm, that is, the detection o f local minima (wrong minimization).

However, the accuracy which can be obtained by evaluating all the sample points is required only for those relatively few locations (transformations) near the match location. Hence, at a vast majority o f locations, it would be a considerable waste o f time to perform high- accuracy calculation using all sample points. This is the basic idea behind the introduction o f a new technique for the surface fitting algorithm. In the minimization process applied in this new technique, all the possible transformation parameters are searched in order to find the best parameters. This process is known as a global search and is based on the grid search explained in section 5.2. Assuming that the scaling parameters o f the images are known by the system

searched during the minimization. The way that the error is computed and subsequently the decision is made about integrity o f any transformation, is a sequential process. Accordingly, this new technique is known as the sequential method,

A class o f sequential similarity detection algorithms was introduced by Bamea and Silverman 1972. In their method, a translational registration was implemented by a template based matching process. The method was extended and a more complicated sequential matching algorithm was introduced by Ramapriyan 1976, in which a template was partitioned and translational shift was applied in different size-level. The method was then assessed and used by other workers (Rosenfeld and Vanderbrug 1977, Vanderbrug and Rosenfeld 1977, Wong and Hall 1978, Wong 1978, Wong and Hall 1979). However, all the modifications made so far to the algorithm, only concern the applicability o f the process to 2-D images (specifically optical and radar images) undergoing translational registration.

The concept of the sequential method and its algorithm are presented in this chapter (see also IPMI’91; Oghabian & Todd-Pokropek 1991). In section 6.2, the theoretical discussion and properties of the method are explained. The algorithm is also outlined in this section. The properties (e.g. size) of search space (transformation parameters) used in the sequential process are outlined in section 6.2.1. The sequentiality o f the cumulative distance error and its properties are described in section 6.2.2. Different sources of error and their contribution to the expected distance error are outlined in section 6.2.3. Section 6.2.4 and 6.2.5 will describe two types of threshold methods (i.e. constant and variable) used in this technique. The expression for the threshold sequence which depends on the number o f sample points involved in the registration process, is presented in section 6.2.5. Some properties o f these sample points are presented in section 6.2.6. Computational aspect of the process and cost expectation are presented in section 6.3. Section 6.3.1 provides a quantitative analysis o f the search locations and sample points based on which the cost expectation was made. Finally, a brief summary is given in section 6.4.

6.1.1- Terminology

Matching process is used synonymously with ^registration process" to find the best transformation parameters by which the two registering surfaces are matched. The term "location"

rotations in x, y, and z direction) at which the matching is evaluated. Search window is used to denote the whole proposed search locations over which the best transformation is searched. Two analogous terms, match location and registration pointy refer to the best transformation parameters yielding the minimum distance error between two surfaces. Mismatch location in this concept refers to any location not giving a match. The basic variables and expressions used in this chapter are listed below. A type o f subscription may be used in each case which specifies that expression with a particular type o f aspect (feature) as defined in the relevant text.

W search window

P sample points; that is the sampled points o f the transforming surface

V tracing voxels; the voxels traced in 3-D space for detecting an intersection o f P with the objective surface

e individual distance error

ê Mean distance error (MDE)

a standard deviation (S.D) o f the individual distance errors E cumulative distance error

r MDE error at true match location

R cumulative distance error at true match location g number o f standard deviations (S.D.) from mean (r) T threshold used in the sequential methods

C computational cost

n number o f entities (denotes number o f sample points if it is used without any subscript).