Mathematical Foundations of Deformable Models

The mathematical foundations of deformable models represent the confluence of geometry, physics, and approximation theory. Geometry serves to represent object shape, physics imposes constraints on how the shape may vary over space and time, and optimal approximation theory provides the formal underpinnings of mechanisms for fitting the models to measured data.

Deformable model geometry usually permits broad shape coverage by employing geometric representations that involve many degrees of freedom, such as splines. The model remains manageable, however, because the degrees of freedom are generally not permitted to evolve independently, but are governed by physical principles that bestow intuitively meaningful behavior upon the geometric substrate. The name "deformable models" stems primarily from the use of elasticity theory at the physical level, generally within a Lagrangian dynamics setting. The physical interpretation views deformable models as elastic bodies that respond naturally to applied forces and constraints. Typically, deformation energy functions defined in terms of the geometric degrees of freedom are associated with the deformable model. The energy grows monotonically as the model deforms away from a specified natural or "rest shape" and often includes terms that constrain the smoothness or symmetry of the model. In the Lagrangian setting, the deformation energy gives rise to elastic forces internal to the model. Taking a physics-based view of classical optimal approximation, external potential energy functions are defined in terms of the data of interest to which the model is to be fitted. These potential energies give rise to external forces that deform the model such that it fits the data.

Deformable curve, surface, and solid models gained popularity after they were proposed for use in computer vision [138] and computer graphics [135] in the mid-1980s. Terzopoulos introduced the theory of continuous (multidimensional)

deformable models in a Lagrangian dynamics setting [139] based on deformation energies in the form of (controlled-continuity) generalized splines [140]. Ancestors of the deform-able models now in common use include Fischler and Elshlager's templates [49] and Widrow's rubber mask technique [154].

The deformable model that has attracted the most attention to date is popularly known as "snakes" [73]. Snakes or "deformable contour models'' represent a special case of the general multidimensional deformable model theory [139]. We will review their simple formulation in the remainder of this section in order to illustrate with a concrete example the basic mathematical machinery that is present in many deformable models.

Snakes are planar deformable contours that are useful in several image analysis tasks. They are often used to approximate the locations and shapes of object boundaries in images based on the reasonable assumption that boundaries are piecewise continuous or smooth (Fig. 1). In its basic form, the mathematical formulation of snakes draws from the theory of optimal approximation involving functionals.

2.1 Energy-Minimizing Deformable Models

Geometrically, a snake is a parametric contour embedded in the image plane (x, y) e R2. The contour is represented as v(s) = (x(s),y(s))T, where x and y are the coordinate functions and 5 e [0,1] is the parametric domain. The shape of the contour subject to an image I(x, y) is dictated by the functional

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