Abstract
Deformable models are well known examples of artificially intelligent system (AIS). They have played an important role in the challenging problem of extracting useful information about regions and areas of interest (ROIs) imaged through different modalities. The challenge is also in extracting boundary elements belonging to the same ROI and integrate them into a coherent and consistent model of the structure. Traditional low-level image processing techniques that consider only local information can make incorrect assumptions during this integration process and generate unfeasible object boundaries. To solve this problem, deformable models were introduced (Ivins, 1994), (McInerney, 1996), (Wang, 2000). These AI models are currently important tools in many scientific disciplines and engineering applications (Duncan, 2000). Deformable models offer a powerful approach to accommodate the significant variability of structures within a ROI over time and across different individuals. Therefore, they are able to segment, match and track images of structures by exploiting (bottom-up) constraints derived from the image data together with (top-down) a priori knowledge about the location, size, and shape of these structures. 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 mechanisms for fitting the models to data. The physical interpretation views deformable models as elastic bodies which respond to applied force and constraints.
TopIntroduction
Deformable models are well known examples of artificially intelligent system (AIS). They have played an important role in the challenging problem of extracting useful information about regions and areas of interest (ROIs) imaged through different modalities. The challenge is also in extracting boundary elements belonging to the same ROI and integrate them into a coherent and consistent model of the structure. Traditional low-level image processing techniques that consider only local information can make incorrect assumptions during this integration process and generate unfeasible object boundaries. To solve this problem, deformable models were introduced (Ivins, 1994), (McInerney, 1996), (Wang, 2000). These AI models are currently important tools in many scientific disciplines and engineering applications (Duncan, 2000).
Deformable models offer a powerful approach to accommodate the significant variability of structures within a ROI over time and across different individuals. Therefore, they are able to segment, match and track images of structures by exploiting (bottom-up) constraints derived from the image data together with (top-down) a priori knowledge about the location, size, and shape of these structures.
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 mechanisms for fitting the models to data. The physical interpretation views deformable models as elastic bodies which respond to applied force and constraints.
TopBackground
The deformable model that has attracted the most attention to date is the active contour model (ACM), well-known as snakes, presented by Kass et al. (Kass, 1987), (Cootes & Taylor, 1992). The mathematical basis present in snake models is similar to all deformable models, which are based on energy minimizing techniques.
Recently, there has been an increasing interest in level set or geodesic segmentation methods, introduced in (Osher & Sethian, 1988), (Malladi, 1995) and (Caselles, 1997). Level set approach involves solving the ACM minimization problem by the computation of minimal distances curve. This method allows topological changes within the ROIs and extension to 3D. Therefore, for some applications it is an improvement on classical ACM.
Other approaches to deformable model are those based on dynamic models or physically based techniques, for example superquadrics (Terzopoulos, 1991) and the finite element model (FEM) (Pentland, 1991). The FEM accurately describes changes in position, orientation and shape. The FEM can be used to solve fitting, interpolation or correspondence problems. In the FEM, interpolation functions are developed that allow continuous material properties, such as mass and stiffness, to be integrated across the ROIs. This last property makes them different from the previous models and therefore more suitable for some artificial vision applications.
Key Terms in this Chapter
Modal Analysis: Study of the dynamic properties and response of structures and or fluids under vibrational excitation. Typical excitation signals can be classed as impulse, broadband, swept sine, chirp, and possibly others. The resulting response will show one or more resonances, whose characteristic mass, frequency and damping can be estimated from the measurements.
Conventional Mathematical Modeling: The applied science of creating computerized models. That is a theoretical construct that represents a system composed by set of region of interest, with a set of parameters, both variables together with logical and quantitative relationships between them, by means of mathematical language to describe the behavior of the system. Parameters are determined by finding a curve in 2D or a surface in 3D, each patch of which is defined by a net of curves in two parametric directions, which matches a series of data points and possibly other constraints.
Active Model: It is a numerical technique for tracking interfaces and shapes based on partial differential equations. The model is a curve or surface which iteratively deforms to fit to an object in an image.
Tracking: Tracking is the process of locating a moving object (or several ones) in time. An algorithm analyses the image sequence and outputs the location of moving targets within the image. There are two major components of a visual tracking system; Target Representation and Localization and Filtering and Data Association. The 1st one is mostly a bottom-up process which involve segmentation and matching. The 2nd one is mostly a top-down process, which involves incorporating prior information about the scene or object, dealing with object dynamics, and evaluation of different hypotheses.
Finite Element Method: Numerical technique used for finding approximate solution of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an equivalent ordinary differential equation, which is then solved using standard techniques such as finite differences.
Geodesic Curve: In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it. Geodesics describe the motion of point particles.
Karhunen-Loeve: Mathematical techniques equivalent to Principal Component Analysis transform aiming to reduce multidimensional data sets to lower dimensions for analysis of their variance.