The Analytic Structure of Image Flows: Deformation and Segmentation

Abstract Time-varying imagery is often described in terms of image flow fields (i.e., image motion), which correspond to the perceptive projection of feature motions in three dimensions (3D). In the case of multiple moving objects with smooth surfaces, the image flow possesses an analytic structure that reflects these 3D properties. This paper describes the analytic structure of image flow fields in the image space-time domain, and its use for segmentation and 3D motion computation. First we discuss thelocal flow structure as embodied in the concept ofneighborhood deformation. The local image deformation is effectively represented by a set of 12 basis deformations, each of which is responsible for an independent deformation. This local representation provides us with sufficient information for the recovery of 3D object structure and motion, in the case of relative rigid body motions. We next discuss theglobal flow structure embodied in the partitioning of the entire image plane intoanalytic regions separated byboundaries of analyticity, such that each small neighborhood within the analytic region is described in terms of deformation bases. This analysis reveals an effective mechanism for detecting the analytic boundaries of flow fields, thereby segmenting the image into meaningful regions. The notion ofconsistency which is often used in the image segmentation is made explicit by the mathematical notion ofanalyticity derived from the projection relation of 3D object motion. The concept of flow analyticity is then extended to the temporal domain, suggesting a more robust algorithm for recovering image flow from multiple frames. Finally, we argue that the process of flow segmentation can be understood in the framework of grouping process. The general concept ofcoherence orgrouping through local support (such as the second-order flows in our case) is discussed.