Parametric Deformation Models

A parametric deformable template refers to the parametric shape model representing the a priori knowledge about the structural properties of a class of objects. By designing a global shape model, boundary gaps are easily bridged, and overall consistency is more likely to be achieved. By parameterizing the model, a compact description of the shape can be achieved. Parametric deformation models are commonly used when some prior information of the geometrical shape is available, which can be encoded using preferably, a small number of parameters. There are two general ways to parameterize the shape class and its variations:

• Analytical form-based parametric deformable models: One can represent the shape as a collection of parameterized curves, i.e., parameterize the geometric shape directly. The template is represented by a set of curves which is uniquely described by some parameters. The geometrical shape of the template can be changed by using different values of the parameters. Variations in the shape are determined by the distribution of the admissible parameter values. This representation requires that the geometrical shapes be well structured.
• Prototype-based parametric deformable models: A so called ``standard'' or ``prototype'' or ``generic'' template is specified to describe the ``most likely'' or ``average'' or ``characteristic'' shape of a class of objects which has a global conforming structure and possibly individual deviations. Each instance of the shape class is derived from the ``prototype'' via a parametric mapping. The use of different parameter values again gives rise to different shapes. Variations in the shape are also determined by the distribution of the admissible parameter values of the mapping.

In both the parametric models mentioned above, the deformable templates interact with the image features dynamically by adjusting the parameters according to the image forces. Similar to the active contour approach, an objective function which is a weighted sum of an internal energy term and an external energy term is used to quantify how well a deformed template matches the objects in the given image. Recall that in the active contour approach, the internal energy, in terms of the stretchness and the elasticity of the spline, actually imposes a rather general and weak a priori distribution on the contour model, i.e., the contour should be smooth and compact. In the parametric deformable template approaches, where the a priori shape preferences are explicitly encoded by the parameters, a similar internal energy term is defined based on the constraints and interactions on the geometrical structures. For example, it can be defined to penalize the deviation of the deformed template from the ``expected'' shape. The external energy term, which pertains to the fidelity of the deformed template to the input image, is introduced so that the template deforms according to the desired goal. It is perceived that the internal energy corresponds to a geometric measure of the fitness, and the external energy corresponds to an image fidelity measure of fitness. The two fitness measures are combined to give an overall measure of fitness, appropriately weighting both the prior knowledge and the image data. The set of parameters which optimizes the objective function gives a description of the detected or matched shape. The value of the objective function quantifies the plausibility of the detection.

• Analytical Form-based Parametric Deformable Models
• Prototype-based Parametric Deformable Models
  • A Prototype-based Model for Object Matching