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Disclaimer: this page provides a short online tutorial on Topology Correction in Medical Imaging and some of the corresponding source codes. This page does not aim at providing an exhaustive list of all existing current techniques, neither does it try to rigorously define a mathematical theory of topology. Yet, this page introduces some key concepts of Topology in Medical Imaging and cites the correponding key references in the literatture. When available, source codes are provided (but no responsability will be taken - I am a terrible programmer :-) ). Most of the content of this page was extracted from the background section of my PhD dissertation.  

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This page presents background material of central importance in Topology in Medical Imaging. We first introduce some general notions of topology and present the strong connections of topology with differential geometry. The Euler-characteristic and the genus of a surface are defined. Next, we show how the continuous theory of topology can be applied to the segmentation of medical images under topological constraints. Particularly, we describe how topological notions can be adapted to the two most common data structures used in medical imaging: 3D voxel grids and surfaces. Also, we present methods for extracting topologically-consistent isocontours from digital images. Finally, we present the state-of-the-art segmentation of medical images under topological constraints. The most common segmentation algorithms are described and their limitations clearly reported. Some of the material presented in this section were taken from Mathworld~\cite{mathworld} and  the work of Bertrand~\cite{bertrand:94,bertrand:96}. We refer the reader to the following text books~\cite{munkres:75,gamelin-greene:99,hatcher:02} for a complete introduction to topology and algebraic topology.

I - General Notions of Topology

        I.A - A Continuous Theory
        I.B - Notions of Topological Equivalence
        I.C - Topology and Differential Geometry
        I.D - On Topological Defects

II - Topology and Discrete Imaging

        II.A - Digital Topology
        II.B - Surfaces in Discrete Imaging
                II.B.1 - Explicit Representations
                II.B.2 - Implicit Representations       
        II.C - From Images to Surfaces: Isocontour Extraction
                II.C.1 - The Original Marching Cubes Algorithm
                II.C.2 - Connectivity-Consistent Marching Cubes Algorithm

III - State of the Art Segmentation under Topological Constraints

        III.A - Topologically-Constrained Segmentation
                III.A.1 - Active Contours
                III.A.2 - Digital Homotopic Deformations     
                III.A.3 - Segmentation by Registration
                III.A.4 - Limitations of Topologically-Constrainted Segmentation 
        III.B -  Retropsective Topology Correction
                III.B.1 - Volume-based Approaches
                III.B.2 - Surface-based Approaches
                III.B.3 - Limitations of Retrospective Topology Correction

References

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Key words: topology, digital topology, isocontour extraction, level sets, Reeb graphs, segmentation, nombres topologiques, euler characteristic.

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