Bayesian Formulation for Deformable Model Matching

In many deformable matching problems, the objective function consists of two parts:

• The internal energy, prior, or geometrical information is related only to the geometric shape of the deformed template (contour) which is an intrinsic property of the template, independent of the input image and sensor data.
  1. In free-form deformation models, this term corresponds to the regularization constraints on the template, for example, the elasticity or stretchness. It is equivalent to a prior which reflects a preference for smooth and compact contours.
  2. In the analytic form based deformation models, where the template is parameterized, this term is specified in terms of the model parameters, which reflects either the choice of the parameter values, or the interactions between different parts of the template.
  3. In the prototype-based deformation models, this term is also a function of the deformation parameters. It biases the choice of the geometric shape. For example, it can specify the penalty for deviations from the expected shape.
• The likelihood term, in all the cases, pertains to the input image data. Via this term, the deformable model interacts with the image, being attracted to the desired salient image features. This term measures the fidelity, or goodness of fit, of the template to the input image.

The deformed template matching which optimizes an objective function leads to an interpretation of the image. When the deformable template modeling is cast in the Bayesian framework using Eq. (1), the prior model typically imposes the geometrical preferences of the shape model. It is related to the internal energy term, which is a measure of the geometrical structure on a deformed template or contour. The imaging model is a description of the noisy or stochastic process that relates the deformed template to the input image or sensor values . This likelihood captures the desired image cues. It is related to the external energy term, which describes the interaction between the template and the image. Bayes' rule combines these two probabilistic models to form a posterior probability which describes the best estimate of given the data and prior knowledge . Note that is a constant, given . Therefore, maximizing the posteriori density in Eq. (1) is equivalent to maximizing the product .