Tangled Splines

Extracting shape information from object bound- aries is a well studied problem in vision, and has found tremen- dous use in applications like object recognition. Conversely, studying the space of shapes represented by curves satisfying certain constraints is also intriguing. In this paper, we model and analyze the space of shapes represented by a 3D curve (space curve) formed by connecting n pieces of quarter of a unit circle. Such a space curve is what we call a Tangle, the name coming from a toy built on the same principle. We provide two models for the shape space of n-link open and closed tangles, and we show that tangles are a subset of trigonometric splines of a certain order. We give algorithms for curve approximation using open/closed tangles, computing geodesics on these shape spaces, and to find the deformation that takes one given tangle to another given tangle, i.e., the Log map. The algorithms provided yield tangles upto a small and acceptable tolerance, as shown by the results given in the paper.

Active Contour Models for Manifold Valued Image Segmentation

Image segmentation is the process of partitioning a image into different regions or groups based on some characteristics like color, texture, motion or shape etc. Active contours is a popular variational method for object segmentation in images, in which the user initializes a contour which evolves in order to optimize an objective function designed such that the desired object boundary is the optimal solution. Recently, imaging modalities that produce Manifold valued images have come up, for example, DT-MRI images, vector fields. The traditional active contour model does not work on such images. In this paper, we generalize the active contour model to work on Manifold valued images. As expected, our algorithm detects regions with similar Manifold values in the image. Our algorithm also produces expected results on usual gray-scale images, since these are nothing but trivial examples of Manifold valued images. As another application of our general active contour model, we perform texture segmentation on gray-scale images by first creating an appropriate Manifold valued image. We demonstrate segmentation results for manifold valued images and texture images.

A Novel Active Contour Model for Texture Segmentation

Texture is intuitively defined as a repeated arrangement of a basic pattern or object in an image. There is no mathematical definition of a texture though. The human visual system is able to identify and segment different textures in a given image. Automating this task for a computer is far from trivial. There are three major components of any texture segmentation algorithm: (a) The features used to represent a texture, (b) the metric induced on this representation space and (c) the clustering algorithm that runs over these features in order to segment a given image into different textures. In this paper, we propose an active contour based novel unsupervised algorithm for texture segmentation. We use intensity covariance matrices of regions as the defining feature of textures and find regions that have the most inter-region dissimilar covariance matrices using active contours. Since covariance matrices are symmetric positive definite, we use geodesic distance defined on the manifold of symmetric positive definite matrices PD(n) as a measure of dissimlarity between such matrices. We demonstrate performance of our algorithm on both artificial and real texture images.