Globally minimal contours and surfaces for image segmentation
Ben Appleton · The University of Queensland · 2005
This thesis develops a new geometric optimisation framework that can provide significantnimprovements over both traditional active contour and graph-based approaches to imagensegmentation. It combines active contour and graph-based methods to produce powerfulnimage segmentation methods sharing their strengths: continuity and reliability.n n We begin by reviewing the relevant literature in image segmentation and in geometricnoptimisation. This includes the role of image derivatives in segmentation, active contournmethods such as snakes and level sets, and graph-based methods such as shortest pathsnand minimum cuts. We also discuss hybrid methods which combine the best propertiesnof active contours and graph-based methods. Finally we introduce a previously unsolvednproblem that is central to this thesis: the computation of globally minimal surfaces inncontinuous spaces. n We then investigate the recursive computation of derivatives in finite and discretenimages. Previous approaches to recursive filtering are reviewed including convolution andnFourier methods. A matrix viewpoint is put forward which deals naturally with imagenboundaries. We give an efficient decomposition and solution for this matrix formulation.nIt is proven to exist for stable filters and shown both in theory and practice to have goodnaccuracy in finite precision implementations. We demonstrate its application to the fastncomputation of image derivatives at arbitrary scales.n n Next we consider in detail the segmentation of planar images by optimal curves. Firstnwe give a simple technique for object segmentation by a minimal convex polygon. We thennextend this technique to continuous curves, convoluted boundaries, and oriented metrics.nExperiments demonstrate that the optimality of these techniques produces significantlynmore accurate and robust segmentations than standard active contour methods. An application is given to the segmentation of the left ventricle in the heart from multiple-slicenmagnetic resonance image sequences.n n Finally, we propose a general method for segmentation by globally minimal surfacesnin higher dimensions. This method is based on the computation of a continuous maximalnflow whose bottleneck forms the globally minimal surface. We investigate its behaviournin planar images and show that it is equivalent to our method for obtaining optimalncontinuous curves. We also develop a general scheme for removing the bias of minimalnsurfaces toward small objects. Unlike existing graph-based or active contour methods thennew minimal surface method is simultaneously optimal and grid-invariant whilst beingnmore efficient than either method. Results in 2D and 3D image segmentation and in stereonreconstruction demonstrate the practical benefits which are predicted by the theoreticalnproperties of globally minimal surfaces. An application is given to the segmentation ofnphysiological structures in the brain from volumetric magnetic resonance images.n