7. Image Segmentation
Society for Industrial and Applied Mathematics eBooks · 2005
Image segmentation is the bridge between low-level vision/image processing and high-level vision. Its goal is to partition a given image into a collection of “objects,” built upon which other high-level tasks such as object detection, recognition, and tracking can be further performed. In this chapter, we discuss several important and interconnected models pertinent to the segmentation task, including Active Contours, Geman and Geman's mixture model, and Mumford and Shah's free boundary model. From imaging and graphics points of view, segmentation is also an inverse problem, i.e., from images to the perception of objects, instead of from objects to the acquisition of images. The current chapter thus starts with a mathematical model or theory for the forward problem, as done in the previous chapters for denoising and deblurring. 7.1 Synthetic Images: Monoids of Occlusive Preimages In this first section, we develop a simplified mathematical theory or model for synthesizing images from individual objects. Without the complication of 3-D real imaging environments, the theory focuses on the topological and algebraic structures of image generation. Though only restricted to binary images, it provides a reasonable forward problem for the segmentation task. We have especially emphasized the important role of occlusion in vision. 7.1.1 Introduction and Motivation In human and computer vision, the occlusion phenomenon plays a key role in successful retrieval of 3-D structural information from 2-D images projected onto the retinas. Its importance has been repeatedly emphasized by many giants in vision sciences, including David Marr in computer vision and artificial intelligence [210], Gaetano Kanizsa in Gestalt and cognitive vision [163], and David Mumford in mathematical and statistical modeling of visual perception [191, 226, 234]. The left panel of Figure 7.1 shows a popular image in vision research, for which the lack of occlusion cues causes a perceptual illusion; i.e., two or more distinct 3-D scenes can be interpreted from the image. The right panel on the other hand shows the importance of occlusion cues in the visual perception of 3-D knots as in knot theory [253].