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Image segmentation
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== Partial differential equation-based methods == Using a [[partial differential equation]] (PDE)-based method and solving the PDE equation by a numerical scheme, one can segment the image.<ref>{{cite journal | last1 = Caselles | first1 = V. | last2 = Kimmel | first2 = R. | last3 = Sapiro | first3 = G. | year = 1997 | title = Geodesic active contours | url = https://www.cs.technion.ac.il/~ron/PAPERS/CasKimSap_IJCV1997.pdf | journal = International Journal of Computer Vision | volume = 22 | issue = 1| pages = 61β79 | doi = 10.1023/A:1007979827043 | s2cid = 406088 }}</ref> Curve propagation is a popular technique in this category, with numerous applications to object extraction, object tracking, stereo reconstruction, etc. The central idea is to evolve an initial curve towards the lowest potential of a cost function, where its definition reflects the task to be addressed. As for most [[inverse problems]], the minimization of the cost functional is non-trivial and imposes certain smoothness constraints on the solution, which in the present case can be expressed as geometrical constraints on the evolving curve. === Parametric methods === [[Lagrangian relaxation|Lagrangian]] techniques are based on parameterizing the contour according to some sampling strategy and then evolving each element according to image and internal terms. Such techniques are fast and efficient, however the original "purely parametric" formulation (due to Kass, [[Andrew Witkin|Witkin]] and [[Demetri Terzopoulos|Terzopoulos]] in 1987 and known as "[[Snake (computer vision)|snakes]]"), is generally criticized for its limitations regarding the choice of sampling strategy, the internal geometric properties of the curve, topology changes (curve splitting and merging), addressing problems in higher dimensions, etc.. Nowadays, efficient "discretized" formulations have been developed to address these limitations while maintaining high efficiency. In both cases, energy minimization is generally conducted using a steepest-gradient descent, whereby derivatives are computed using, e.g., finite differences. === Level-set methods === The [[level-set method]] was initially proposed to track moving interfaces by Dervieux and Thomasset<ref>Dervieux, A. and Thomasset, F. 1979. A finite element method for the simulation of Raleigh-Taylor instability. Springer Lect. Notes in Math., 771:145β158.</ref><ref>Dervieux, A. and Thomasset, F. 1981. [https://www.researchgate.net/profile/Alain_Dervieux/publication/226529379_Multifluid_Incompressible_Flows_by_a_Finite_Element_Method/links/57176a3e08ae2679a8c766ac.pdf Multifluid incompressible flows by a finite element method]. Lecture Notes in Physics, 11:158β163.</ref> in 1979 and 1981 and was later reinvented by Osher and Sethian in 1988.<ref name="OsherSethian1988">{{cite journal|last1=Osher|first1=Stanley|last2=Sethian|first2=James A|title=Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations|journal=Journal of Computational Physics|volume=79|issue=1|year=1988|pages=12β49|issn=0021-9991|doi=10.1016/0021-9991(88)90002-2|bibcode=1988JCoPh..79...12O|citeseerx=10.1.1.46.1266}}</ref> This has spread across various imaging domains in the late 1990s. It can be used to efficiently address the problem of curve/surface/etc. propagation in an implicit manner. The central idea is to represent the evolving contour using a signed function whose zero corresponds to the actual contour. Then, according to the motion equation of the contour, one can easily derive a similar flow for the implicit surface that when applied to the zero level will reflect the propagation of the contour. The level-set method affords numerous advantages: it is implicit, is parameter-free, provides a direct way to estimate the geometric properties of the evolving structure, allows for change of topology, and is intrinsic. It can be used to define an optimization framework, as proposed by Zhao, Merriman and Osher in 1996. One can conclude that it is a very convenient framework for addressing numerous applications of computer vision and medical image analysis.<ref>S. Osher and N. Paragios. [http://www.mas.ecp.fr/vision/Personnel/nikos/osher-paragios/ Geometric Level Set Methods in Imaging Vision and Graphics], Springer Verlag, {{ISBN|0-387-95488-0}}, 2003.</ref> Research into various [[level-set data structures]] has led to very efficient implementations of this method. === Fast marching methods === The [[fast marching method]] has been used in image segmentation,<ref>{{cite web|url=http://math.berkeley.edu/~sethian/2006/Applications/Medical_Imaging/artery.html|title=Segmentation in Medical Imaging|author=James A. Sethian|access-date=15 January 2012}}</ref> and this model has been improved (permitting both positive and negative propagation speeds) in an approach called the generalized fast marching method.<ref>{{Citation| journal=Numerical Algorithms| date=July 2008|volume=48|issue=1β3|pages=189β211| title=Generalized fast marching method: applications to image segmentation| first1=Nicolas|last1=Forcadel | first2=Carole | last2=Le Guyader | first3= Christian | last3= Gout| doi=10.1007/s11075-008-9183-x| s2cid=7467344}}</ref>
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