Topology preserving linear filtering

Abstract : One of the central problems of medical imaging is the three-dimensional (3D) visualization of body parts. The 3D volume can be viewed in slices, but the extraction of a part requires a segmentation process. Inasmuch as body parts are distinguishable by their various densities, a widely accepted method for extracting an organ is to extract isodensity surfaces by a simple threshold. Unfortunately, the density of organs, arteries, etc.\ varies spatially due to morphology, and no unique threshold allows one to extract the organs boundaries. The snake or active contour methods have attempted to capture these boundaries as smooth and overall contrasted surfaces. The snake method suffers, however, from severe drawbacks. The contour has to be initialized near the boundary. In addition, many body parts have too complex a topology. In this paper we focus on another idea, which is to preprocess the image before thresholding. The preprocessing aims at the homogeneity of the different parts while preserving small features. Starting from a recent seminal work by Grady and Funka-Lea [in Computer Vision and Mathematical Methods in Medical and Biomedical Image Analysis: ECCV $2004$ Workshops CVAMIA and MMBIA, Prague, Czech Republic, May $2004$, Revised Selected Papers, Springer, Berlin, 2004, pp.~230--245], several linear heat equations on images will be compared. They stem from nonlinear partial differential equations or from their associated nonlinear filters. By linearizing these processes one obtains more accurate topology preserving methods. These linear filters will be tested comparatively to visualize challenging angiography images of arteries. A salient fact of the method will emerge. By a concentration phenomenon, peaks in the image histogram become much more concentrated under the linear heat equations, thus permitting us to fix the thresholds defining the surfaces without supervision. Automatic extraction can be performed in this way for angiography images taken at a one-year or longer delay.
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SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2008, 1 (1), pp.26-50
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Antoni Buades, Aichi Chien, Jean-Michel Morel, Stanley Osher. Topology preserving linear filtering. SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2008, 1 (1), pp.26-50. 〈hal-00271149〉

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