Artificial Neural Networks for 3-D Nonrigid Motion Analysis (1992)

A novel approach to 3-D nonrigid motion analysis using artificial neural networks is presented in this paper. A set of neural networks is proposed to tackle the problem of nonrigidity in 3-D motion estimation. This is the first attempt in applying neural network techniques to 3-D nonrigid motion analysis. The proposed neural networks differ from other existing neural network models in their unique structure and dynamics. Experiments on both synthetic and real data are conducted to corroborate the proposed techniques. 1 Introduction Most proposed algorithms and approaches to 3-D motion analysis are based on the rigidity assumption (SPSS). Not until recent years have researchers begun to tackle the issue of nonrigidity in 3-D motion analy- sis (CHASl) (GLHSO) (PHSl) (SP86). The analysis of the motion of nonrigid objects are in general far more complicated than that of rigid objects. A real-world example is the heart motion. As a heart pumps, its left ventricle undergoes a rather complex motion. It shifts, rotates, twists, expands or contracts. The traditional transformations in 3-D rigid motion analysis are obviously insufficient to represent these motions. More research is needed to find precise mathematical descriptions of various 3-D nonrigid motions. Much of the early research in nonrigid motion analysis can be characterized as exploiting the alternatives to relax the nonrigidity condition in special cases. Chen et. al. (SP86) considered nonrigid elastic bodies under the perspective projection. In order to obtain an essentially unique solution in a closed form, the authors limited their research solely on a class of isometric generalized-motion. This class includes rigid motions and bendings as well as locally rigid motions and bendings, but not shearing. Webb etal. (WA83) demonstrated that the rigidity assumption in 3-D motion interpretation can be relaxed for textured objects. Using assumptions on the lighting conditions of a scene, the authors concluded that the shape of an object can be recovered through the use of only local rigidity. They therefore applied the rigidity assumption locally under the condition that the motion of the object studied was nearly rigid in any small area. Goldgof etal. in their early work of motion analysis of nonrigid surfaces (GLHSO) described a curvature-based approach to nonrigid motion analysis. Depending on the changes in the mean and Gaussian curvatures during motion, they classified the motion of a surface at each point as rigid, isometric, homothetic, conformal and general. Their discussion was limited to homothetic motion and the motion of piecewise rigid objects, i.e., objects consisting of finite numbers of rigid parts with nonrigid connections. A 3-D nonrigid motion can be decomposed into a global rigid motion and a set of local nonrigid deformations, where the local deformations are coupled with the global motion at every moment (CHASl) (GLHSO). The global rigid motion can be described using standard rigid motion parameters including a translation vector and a rota- tion matrix. Local deformations, by contrast, are more difficult to estimate given their complex characteristics in terms of transformations in motion. In heart motion, for example, when a heart pumps, local deformations are coupled with the global motion throughout the whole cardiac cycle. The local deformation here refers to a combination of all the local motions, including translation, rotation, shearing, and expansion or contraction. A novel approach to 3-D nonrigid motion analysis using artificial neural networks is proposed in this paper. It differs from the previous works in the way it tackles the problem of local deformation estimation in nonrigid motion analysis. A set of neural networks that are similar in structure and dynamics but different in physical size is proposed. These neural networks are parallel in operation and connected to each other through feedbacks. Each neural network consists of two layers, the input layer and the output layer. The activation function of the output layer is selected in such a way that a feedback is involved in the output updating. Constraints are specified to ensure a stable and global consistent estimation of local deformations. The assignments of weights between two layers, the initial values of the outputs, and the connections between each network reflect the con- straints defined. The objective of the proposed neural networks is to find the optimal deformation matrices that satisfy the constraints for all the points on the surface of the nonrigid object. This approach represents the first