The iterative closest point algorithm cite{Mckay_1992} is based on finding the correspondences based on the minimum distance criterion.

Different shape registration methods based on this technique are provided in the literature (e.g. cite{Zhang_1994}). It is used in registeringeither $2D$ or $3D$ objects.Different shape registration approaches were proposed in the literature cite{Cohen_1998,Fitzgibbon_2001,Kozinska_1997}. These approaches suffer from various problems, including scale variations and dependency on initialization. Also local deformations can not be covered.

Image boundaries are used for registration in cite{Farin_2004}. Images are registered using the iterative closest point algorithm to match these boundaries(shapes). A hierarchical B-spline technique is used to match these surfaces. The algorithm depends mainly on the number of points that represent the source and target images in addition to the number of control points which create a huge matrix equation system. The solution of such equations is sometimes impossible.In cite{Malik_2006}, internal and external contours of an object are extracted from the image to be matched with a database to mark joints of the human body. Edge detectors are important to extract the features for matching in such cases.A shape registration framework was proposed in cite{Paragios_eccv_2002} using the signed distance map representation.

A global transformation which includes homogeneous scale, rotation, and translation, is used to align planar shapes. The local deformations are handled, by minimizing sum of squaredifferences of the shapes representation. The form of local deformations does not give the desired local correspondences.In cite{Rouss_2004}, a 3D shape-segmentation approach was proposed which included a shape registration process. A shape model was built from a set of training shapes using signed distance functions (the conventional representation).

A level set function evolved, thus minimizing the shape alignment and the intensity gray level energies. Using a simple global transformation with homogenous scales creates a problem when gathering training shapes fromdifferent patients scans. The registration process fails when the target shape requires inhomogeneous scales.An interesting shape registration approach was proposed recently in cite{Paragios_PAMI_2006} using maximization of mutual information. Images of signed distance maps of the registered shapes are involved.

An affine transformation is used to handle the global registration problem through a gradient descent optimization technique. The local deformations are represented by the incremental free form deformations method. The control points minimize a sum ofsquared differences energy. Using such a method makes the problem complicated since a large number of control points may be required to cover detailed local deformations. The gradient descent optimization may have a problem with these large number of variables associated with different levels of the IFFD.Vector level set functions are used to represent and register shapes in cite{Hossam_ICCV_2005}. The vector components represent the vector projections from any point in space to the nearest point on the shape boundary.

A positive sign is used to mark points inside the shape while outside points are given negative values. A simple dissimilarity measure is used to handle the problem of inhomogeneous scaling in the shape registration framework used to solve the shape-based segmentation problem.In this chapter, we present a global and elastic shape registration technique using the iterative closest point algorithm. The global motion is described by an affine transformation while local deformations are handled by the incremental free form deformations. In both cases, a closed form solution is illustrated to estimate the correspondence motion directly. The elastic motion uses control points which are computed by a closed form solution for energy minimization as a point-based registration problem.

Sum of least squares and smoothing constrains are demonstrated to formulate an energy function which is quadratic in terms of control points deformations. We will show that the multi level resolution control lattice approach gives exact and accurate correspondences compared to the single high resolution level case. The minimization of the energy function results in a linear system of size dependenceonly on the number of control points, which makes the solution possible and reasonable. Different synthetic and real shapes registration will be demonstrated to illustrate the efficiency of the approach through the following application.

Many medical image application require building a detailed 3D model of the organ under investigation. These details can be achieved using different imaging radiology like CT scans and MRI’s. CT scans which are expensive and the radiation dose is considered to be high and therefore not accepted as a routine practice. To achieve this goal, we need to build a system that gives detailed volumetric information about the human jaw.

This system will startfrom that surface model which is divided into individual crowns (the teeth upper parts). Each crown will be matched with a $ extbf{database}$ of different individual teeth types. So, a database of anatomical structures is of great interest for such a problem.The generation of robust, accurate and accessible 3-D anatomical models is now a ridder mark for future research and understanding in the medical world.

Diverse 3-D anatomical libraries provide a passive means by which the motivations of education, simulation and experimentation may all greatly benefit. Thus, an efficient technique for autonomously creating a diverse, 3-D anatomical library is a viable medical resource with a great emphasis onautonomous. A reliable and autonomous procedure for obtaining 3-D anatomical models provides an efficient means for enrolling large amounts of subjects into a 3-D library that can be made available to students and professionals in the medical field.The focus of this application is to outline the framework for autonomously building a 3-D library of human teeth. Currently research in this field includes a tooth library generated by Nagasawa and Yoshida~ extit{et.al.}~cite{sakaeDent10}, using X-ray images of 55 human teeth that were obtained by 3-D micro-CT.Real extracted human teeth of different types are fixed over wax rods.

Then, we use Cone-beam CT to scan the wax and teeth. Total 280 teeth have been processed in this fashion. A point-based 3-D shape registration method has been further applied to align all subjects enrolled into the library, providing a more uniform orientation for ease of use. Thus, an autonomous algorithm is proposed for reconstructing human teeth, enrolling each into a comprehensivelibrary and further carrying out global, 3-D registration. This framework will be outlined in detail and some preliminary results will be given.

The rest of this book chapter is organized as follows: Global shape registration is demonstrated in Sec.~ef{sec:GlobalRegistration}. Elastic deformations are estimated in Sec.~ef{sec:Point-based-Elastic}. Experimental results and validation are demonstrated in Sec.~ef{sec:result}, and conclusion inSection~ef{sec:conclusion}.