The Persistent Morse Complex Segmentation of a 3-Manifold

We describe an algorithm for segmenting three-dimensional medical imaging data modeled as a continuous function on a 3-manifold. It is related to watershed algorithms developed in image processing but is closer to its mathematical roots, which are Morse theory and homological algebra. It allows for the implicit treatment of an underlying mesh, thus combining the structural integrity of its mathematical foundations with the computational efficiency of image processing.

[1]  Jerry L Prince,et al.  Current methods in medical image segmentation. , 2000, Annual review of biomedical engineering.

[2]  Kannan,et al.  ON IMAGE SEGMENTATION TECHNIQUES , 2022 .

[3]  Theodosios Pavlidis,et al.  Picture Segmentation by a Tree Traversal Algorithm , 1976, JACM.

[4]  Valerio Pascucci,et al.  Morse-smale complexes for piecewise linear 3-manifolds , 2003, SCG '03.

[5]  Kecheng Liu,et al.  Shape recovery algorithms using level sets in 2-D/3-D medical imagery: a state-of-the-art review , 2002, IEEE Transactions on Information Technology in Biomedicine.

[6]  S. Beucher,et al.  Watersheds of functions and picture segmentation , 1982, ICASSP.

[7]  R P Velthuizen,et al.  MRI segmentation: methods and applications. , 1995, Magnetic resonance imaging.

[8]  Serge Beucher,et al.  THE WATERSHED TRANSFORMATION APPLIED TO IMAGE SEGMENTATION , 2009 .

[9]  J Sijbers,et al.  Watershed-based segmentation of 3D MR data for volume quantization. , 1997, Magnetic resonance imaging.

[10]  Luc Vincent,et al.  Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Demetri Terzopoulos,et al.  Deformable models in medical image analysis: a survey , 1996, Medical Image Anal..

[12]  P.K Sahoo,et al.  A survey of thresholding techniques , 1988, Comput. Vis. Graph. Image Process..

[13]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[14]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[15]  R. Ho.  Algebraic Topology , 2022 .

[16]  Guang-Zhong Yang,et al.  Structure adaptive anisotropic image filtering , 1996, Image Vis. Comput..

[17]  David Cohen-Steiner,et al.  Extraction and simplification of iso-surfaces in tandem , 2005, SGP '05.

[18]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[19]  Yukio Matsumoto.  An Introduction to Morse Theory , 2001 .

[20]  S. Hyakin,et al.  Neural Networks: A Comprehensive Foundation , 1994 .

[21]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[22]  Jos B. T. M. Roerdink,et al.  The Watershed Transform: Definitions, Algorithms and Parallelization Strategies , 2000, Fundam. Informaticae.

[23]  Herbert Edelsbrunner.  Surface tiling with differential topology , 2005, SGP '05.

[24]  G. G. Cameron,et al.  Split-and-merge segmentation of magnetic resonance medical images: performance evaluation and extension to three dimensions. , 1998, Computers and biomedical research, an international journal.