Partial differential equations and variational methods applied to medical image analysis

Recently, variational and PDE-based methods have become very popular in the processing and analysis of medical images. In this presentation, we apply these methods to three key topics in medical image analysis namely (i) the correspondence problem, (ii) fuzzy segmentation, and (iii) registration.; The Correspondence Problem for curves has many applications to medical imaging, including motion-tracking, morphometrics and neural growth. The general problem is to find a continuous, non-rigid correspondence between two curves such that a shape-dissimilarity measure is minimized. Current methods solve this problem to its entirety only for parameterized curves. In this work, we look at the correspondence problem for implicitly represented curves. Given two signed distance functions, we search for a diffeomorphism between their zero-level sets that minimizes a shape dissimilarity measure. The diffeomorphisms are generated as flows of certain tangential vector fields, and curve-normals are chosen as the similarity criterion. We also show how this model can be extended to compare curves of different topologies. We have tested our model on synthetic and ultrasound cardiac data, and have obtained good results.; Segmentation of images is a widely researched field and is unarguably the first step to any problem in medical image analysis. In this work, we use fuzzy classification and active contours in a single variational framework, allowing the use of tools from both deformable geometry and clustering, and hence giving a useful technique for unsupervised segmentation. The model was tested on synthetic and MRI brain data, with promising results.; For techniques like functional MR imaging that work with sequences of images, proper registration or spatial alignment of the images is essential. A usual registration technique is to segment a key feature in each of the images and then align the images with respect to this feature. But due to low contrast and poor resolution of fMR images, usual techniques that use only image information for the segmentation step, give inaccurate registration results. In our work, we incorporate prior shape information into the segmentation step, within a variational framework and have obtained good registration results.