Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
A Unified Approach towards Reconstruction of aPlanar Point Set
Subhasree Methirumangalath, Amal Dev Parakkat &Ramanathan Muthuganapathy
Department of Engineering DesignIndian Institute of Technology Madras
June 25, 2015
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Introduction
Problem Statement
Motivation
Related Work
Algorithm Overview
Theoretical Guarantee
Comparative Studies
Conclusion & Future Work
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Problem Statement
Reconstruction: computation of a good approximation of theunknown shape induced by a given point setTwo types of inputs: Boundary Sample and Dot Pattern
Figure : (a)BS, (b)Reconstruction of BoundarySample(RBS), (c)DP, (d)Reconstruction of DotPattern(RDP)
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Problem
Relevance of a unified method
RBS and RDP are different problems
3D equivalent problem for RBS - Surface reconstruction [2]@,[7]$
No 3D equivalent problem for RDP
Current approaches (except simple shape [12]∗) are tuned foreither RBS or RDP
I Algorithms tuned for RBS - eg: Crust [2]@ and NN Crust [7]$
I Algorithm tuned for RDP - eg: RGG from boundary directedsample [15]#
Simple shape [12]∗ has different termination conditions forRBS and RDP
Our unified method works irrespective of the input type∗A.Gheibi et al. 2010, @ N. Amenta 1998, $ T.K.Dey et al. 1999, # Jiju Peethambaran et al. 2014
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Motivation
Reconstruction is an ill posed problem [9]∗
I Different outputs for the same input [11]#
∗H. Edelsbrunner et al. 1998, #A. Galton et al. 2006
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Motivation
Other Challenges of Reconstruction Problem
Quantifying how output approximates the input is a difficulttask [9]∗
Output differs with human cognition and perception [9]∗
Output is dependent on heterogeneity in density anddistribution
Application specific nature of the output
Varied applications∗ H. Edelsbrunner et al. 1998
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Motivation
Applications of 2D Reconstruction
Product design - eg:Initial design of an aircraft [18]∗
Geographical information systems - eg: Map generalization[11]#
Computer graphics - eg: Geometric modeling [17] @
Figure : (a) Points on surface with constraints, (b) Points inparametric 2D space, (c) Reconstruction, (d) Trimmed patch
Point set matching [19]$, Geometric computing [20]%,Bio medical image analysis[16]&.
∗ R.C.Veltkamp 1994, # A.Galton et al. 2006, @ B.R.Sundar et al. 2014, $ R.C.Veltkamp et al. 2001, % C. Villani
2003, & Sachdeva et al. 2012
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Related Work
Delaunay-based methodsI α-shape [10]∗
I Geometric Structures for 3D shape [5]#
I Crust [2]@ and NN Crust [7]$
I Automatic Surface Reconstruction [3]%
I χ-shape [8]&
I Optimal transport driven approach [6]!
I RGG from boundary directed sample [15]+
Non-Delaunay methodsI Ball Pivoting algorithm [4]?
I Simple Shape [12]∗∗
I Methods using implicit functions [13]##, [14]$$
∗ H.Edlesbrunner et al. 1983, # J.D.Boissonnat, 1984, @ N. Amenta 1998, $ T.K.Dey et al. 1999, % M. Attene et
al. 2000, & M.Duckham et al. 2008, ! F. de Goes et al. 2011, + Jiju Peethambaran et al. 2014, ? F.Bernardini,
1999, ∗∗ A.Gheibi et al. 2010, ## M.Kazhdan et al. 2006, $$ M.Kazhdan et al. 2013
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Our Contributions
Proposed a unified approach for RBS as well as RDP
Developed an empty circle method using DelaunayTriangulation(DT)
Provided theoretical guarantee as well as extensiveexperiments to evaluate our approach
Demonstrated that our method works well where otheralgorithms have restrictions
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Preliminaries
Voronoi diagram and Delaunay triangulation
Exterior triangle and exterior edges
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Intution of our algorithm
Long exterior edge - not part of resultant shape
Non-empty circle on EE
The points far apart on EE are not adjacent on the resultantshape
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Preliminaries
Circle Constraint: diametric circle, chord circle & midpointcircle
Regularity constraint: bridge, dangling edge & junction point
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Illustration of Our Algorithm
Figure : (a) Input (b) Delaunay triangulation DT(Graph G)
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Illustration of Our Algorithm
Figure : (c) Non-empty diametric circle, (d) Two new edgesas EEs
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Illustration of Our Algorithm
Figure : (f) Empty diametric circle & non-empty midpointcircle (g) Two new edges as EEs
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Illustration of Our Algorithm
Figure : (i) Empty diametric circle & non-empty chord circle(j)Two new edges as EEs
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Illustration of Our Algorithm
Figure : (k) All circles empty (l) Final Graph
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Illustration of Our Algorithm
Figure : (l) Final Graph (m) ec-shape
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
ec-shape Algorithm
The algorithm to create ec-shape from point set S:
Algorithm 1 Shape Reconstruction Algorithm
1: procedure Construct ec-shape(Point set S)2: Construct a graph G = Delaunay Triangulation, DT (S).3: Construct a Priority Queue (PQ) of EE s of G in the de-
scending order of edge lengths.4: repeat5: Delete the EE of ET from the head of PQ and remove it
from G, if it satisfies the circle constraint and G−EE is regular.6: If EE is removed from G, add the adjacent sides of the
ET to PQ maintaining the descending order of the edge lengths.7: until No more EE in G can be removed8: return ec-shape, the exterior edges of the graph G.9: end procedure
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Complexity Analysis
Time complexityI Construction of Delaunay Triangulation - O(n log n )I Construction of Priority queue - O(n log n )I One edge removal - O(1)
Circle constraint check - O(1)Regularity check - O(1)
Space complexityI No extra space - O(n)
Implementation: CGAL 4.3 [1]∗
∗ Computational Geometry Algorithms Library, http://www.cgal.org
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Theoretical Guarantee
To prove that ec-shape is homeomorphic to simple closedcurve
Sufficient to prove non-boundary edges are removed andboundary edges are retained from the original shape
Sampling ModelI An input point set S is sampled from a polygonal object O
under r -sampling if it satisfies the following constraints:I In RBS,S is sampled from O under r -sampling:
Each pair of adjacent boundary samples lies at a distance ofat most 2r .Any pair of non-adjacent boundary samples lies at a minimumdistance of 2r .
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Theoretical Guarantee
Proof of Theoretical Guarantee
Lemma
In RBS, assuming the input point set S is sampled from apolygonal object O using r-sampling, ec-shape algorithm removesthe exterior edges that are not boundary edges.
Lemma
In RBS, assuming the input point set S is sampled from apolygonal object O using r-sampling, ec-shape algorithm does notremove any of the boundary edges.
Corollary
Following the two Lemmas, ec-shape is homeomorphic to a simpleclosed curve.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
ec-shapes for Boundary Sample and Dot Pattern
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Results
ec-shapes for Boundary Sample and Dot Pattern
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Results
ec-shapes for Boundary Sample and Dot Pattern
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Qualitative Comparison for Boundary Sample
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Qualitative Comparison for Boundary Sample
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Qualitative Comparison for Boundary Sample
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Qualitative Comparison for Dot Pattern
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Qualitative Comparison for Dot Pattern
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Qualitative Comparison for Dot Pattern
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Parameter Turing for Other Methods
Figure : (a) & (b):α-shape, Figs (c) & (d): Simple shape,Figs (e) & (f): χ-shape
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Quantitative Comparison
Measure for quantitative comparison [8]∗
L2error norm =area((O − Re) ∪ (Re − O))
area(O)
∗ M.Duckham et al. 2008
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Quantitative Comparison
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Quantitative Comparison
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Quantitative Comparison
Different distributions considered:I Non-Random (NR)I Semi Random Dense Boundary (SRDB)I Semi Random Sparse Boundary (SRSB)I Random (R)
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Outputs of Different Distributions
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Comparison
Parameter Tuning for ec-shape
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Discussion
Comparison with other methods
Method Sculpting/Non-DT strategy
Guarantee Unified Non-Param
Noise
1stsculpt
Number of faces,edges, points
No NA Yes No
ASR EGH & EMST No NA No No
OTA Greedy sim-plification ofDT
No No Yes Yes
RGG Circum radius &circumcenter
Yes,onlyDBS
No No No
BPA Non DT-multiplepasses
No NA No Possible
SSA Non DT,2 condns No Yes No No
ec Three circles Yes Yes Nonsparse
No
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Conclusion & Future work
A Delaunay based, unified method developedI For reconstruction with theoretical guaranteeI For good approximation of wide variety of shapes with
different featuresI Experiments show that ec-shape is equal or better than
outputs of other methods
LimitationsI Parameter tuning is required for sparse point setI Not able to detect open curvesI Noisy inputs can not be handled
Current & Future workI Island detectionI 3D extensionI Sparse point set without parameter tuning
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References[1] Cgal, Computational Geometry Algorithms Library.
http://www.cgal.org.
[2] Nina Amenta, Marshall Bern, and Manolis Kamvysselis.A new voronoi-based surface reconstruction algorithm.In Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’98, pages 415–421, New York, NY, USA, 1998.ACM.
[3] Marco Attene and Michela Spagnuolo.Automatic surface reconstruction from point sets in space.Comput. Graph. Forum, 19(3):457–465, 2000.
[4] Fausto Bernardini, Joshua Mittleman, Holly Rushmeier, Claudio Silva, and Gabriel Taubin.The ball-pivoting algorithm for surface reconstruction.IEEE Transactions on Visualization and Computer Graphics, 5:349–359, 1999.
[5] Jean-Daniel Boissonnat.Geometric structures for three-dimensional shape representation.ACM Trans. Graph., 3(4):266–286, October 1984.
[6] F. de Goes, D. Cohen-Steiner, P. Alliez, and M Desbrun.An optimal transport approach to robust reconstruction and simplification of 2d shapes.Comput. Graph. Forum, 30:1593–1602, 2011.
[7] Tamal K. Dey and Piyush Kumar.A simple provable algorithm for curve reconstruction.In Robert Endre Tarjan and Tandy Warnow, editors, SODA, pages 893–894. ACM/SIAM, 1999.
[8] Matt Duckham, Lars Kulik, Michael F. Worboys, and Antony Galton.Efficient generation of simple polygons for characterizing the shape of a set of points in the plane.Pattern Recognition, 41(10):3224–3236, 2008.
[9] Herbert Edelsbrunner.Shape reconstruction with delaunay complex.In Claudio L. Lucchesi and Arnaldo V. Moura, editors, LATIN, volume 1380 of Lecture Notes in Computer Science, pages 119–132. Springer-Verlag BerlinHeidelberg, 1998.
[10] Herbert Edelsbrunner, David G. Kirkpatrick, and Raimund Seidel.On the shape of a set of points in the plane.IEEE Transactions on Information Theory, 29(4):551–558, 1983.
[11] Antony Galton and Matt Duckham.What is the region occupied by a set of points?In Martin Raubal, Harvey J. Miller, Andrew U. Frank, and Michael F. Goodchild, editors, GIScience, volume 4197 of Lecture Notes in Computer Science, pages81–98. Springer-Verlag Berlin Heidelberg, 2006.
[12] Amin Gheibi, Mansoor Davoodi, Ahmad Javad, Fatemeh Panahi, Mohammad M Aghdam, Mohammad Asgaripour, and Ali Mohades.Polygonal shape reconstruction in the plane.IET computer vision, 5(2):97–106, 2011.
[13] Michael Kazhdan, Matthew Bolitho, and Hugues Hoppe.Poisson surface reconstruction.In Proceedings of the Fourth Eurographics Symposium on Geometry Processing, SGP ’06, pages 61–70, Aire-la-Ville, Switzerland, Switzerland, 2006. EurographicsAssociation.
[14] Michael M. Kazhdan and Hugues Hoppe.Screened poisson surface reconstruction.ACM Trans. Graph., 32(3):29, 2013.
[15] Jiju Peethambaran and Ramanathan Muthuganapathy.A non-parametric approach to shape reconstruction from planar point sets through delaunay filtering.Computer-Aided Design, 62(0):164 – 175, 2015.
[16] Jainy Sachdeva, Vinod Kumar, Indra Gupta, Niranjan Khandelwal, and Chirag Kamal Ahuja.A novel content-based active contour model for brain tumor segmentation.Magnetic resonance imaging, 30(5):694–715, 2012.
[17] Bharath Ram Sundar, Abhijith Chunduru, Rajat Tiwari, Ashish Gupta, and Ramanathan Muthuganapathy.Footpoint distance as a measure of distance computation between curves and surfaces.Computers & Graphics, 38(0):300 – 309, 2014.
[18] Remco C. Veltkamp.Closed Object Boundaries from Scattered Points, volume 885 of Lecture Notes in Computer Science.Springer-Verlag, 1994.
[19] Remco C. Veltkamp and Michiel Hagedoorn.Principles of visual information retrieval.chapter State of the Art in Shape Matching, pages 87–119. Springer-Verlag, London, UK, UK, 2001.
[20] Cedric Villani.Topics in Optimal Transportation.Graduate studies in mathematics. American Mathematical Society, 2003.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Thank You
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Thank You
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Theoretical Guarantee
To prove that ec-shape is homeomorphic to simple closedcurve
Sufficient to prove non-boundary edges are removed andboundary edges are retained from the original shape
Sampling ModelI An input point set S is sampled from a polygonal object O
under r -sampling if it satisfies the following constraints:I In RBS,S is sampled from O under r -sampling:
Each pair of adjacent boundary samples lies at a distance ofat most 2r .Any pair of non-adjacent boundary samples lies at a minimumdistance of 2r .
In RDP,S is sampled from O under r -sampling with anadditional constraint:
I A boundary sample is at a minimum distance of 2r withrespect to any non-boundary sample
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Proof of Theoretical Guarantee
Lemma
In RBS, assuming the input point set S is sampled from apolygonal object O using r-sampling, ec-shape algorithm removesthe exterior edges that are not boundary edges.
Consider an ET 4ijk ∈ DT
Assume exterior edge (pi ,pj) with length dij is a non-boundaryedge
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Proof of Theoretical Guarantee
Proof.
To prove that there exists at least one non-empty circle sothat non-boundary edge is removed
Case-1:dij > 2r , dik < 2r and djk < 2r : non-empty diametriccircle
Case-2: dij > 2r , dik < 2r and djk > 2r : either non-emptydiametric circle or either of the chord or midpoint circle isnon-empty, else invalid DT
Case 3- dij > 2r , dik > 2r and djk > 2r : either of the threecircles is non-empty, else it reduces to case-2 of proof
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Proof of Theoretical Guarantee
Lemma
In RBS, assuming the input point set S is sampled from apolygonal object O using r-sampling, ec-shape algorithm does notremove any of the boundary edges.
Consider an ET 4ijk ∈ DT
Assume exterior edge (pi ,pj) is a boundary edge
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Proof of Theoretical Guarantee
Proof.
To prove that there does not exist a non-empty circle
Non-empty diametric circle violates r -sampling
Non-empty chord circle results in non-simple polygon
Non-empty midpoint circle implies an invalid DT
Corollary
Following the two Lemmas, ec-shape is homeomorphic to a simpleclosed curve.
Proof.
Non-boundary edge- removed, boundary edge- retained, ec-shape -linear approximation of original boundary and homeomorphic tosimple closed curve
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Intution of our algorithm
Long exterior edge - not part of resultant shape
Empty circle on EE
Non-empty circle on shorter adjacent side of EE
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
Algorithm
Intution of our algorithm
Short EE - not part of resultant shapeEmpty circle on EENon-empty circle on longer adjacent side of EE
IntutionI Non empty circle - shorter edges than EEI Longer EE in the local neighbourhood - not part of resultant
shape
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References I
[1] Cgal, Computational Geometry Algorithms Library.http://www.cgal.org.
[2] Nina Amenta, Marshall Bern, and Manolis Kamvysselis.A new voronoi-based surface reconstruction algorithm.In Proceedings of the 25th Annual Conference on ComputerGraphics and Interactive Techniques, SIGGRAPH ’98, pages415–421, New York, NY, USA, 1998. ACM.
[3] Marco Attene and Michela Spagnuolo.Automatic surface reconstruction from point sets in space.Comput. Graph. Forum, 19(3):457–465, 2000.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References II
[4] Fausto Bernardini, Joshua Mittleman, Holly Rushmeier,Claudio Silva, and Gabriel Taubin.The ball-pivoting algorithm for surface reconstruction.IEEE Transactions on Visualization and Computer Graphics,5:349–359, 1999.
[5] Jean-Daniel Boissonnat.Geometric structures for three-dimensional shaperepresentation.ACM Trans. Graph., 3(4):266–286, October 1984.
[6] F. de Goes, D. Cohen-Steiner, P. Alliez, and M Desbrun.An optimal transport approach to robust reconstruction andsimplification of 2d shapes.Comput. Graph. Forum, 30:1593–1602, 2011.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References III
[7] Tamal K. Dey and Piyush Kumar.A simple provable algorithm for curve reconstruction.In Robert Endre Tarjan and Tandy Warnow, editors, SODA,pages 893–894. ACM/SIAM, 1999.
[8] Matt Duckham, Lars Kulik, Michael F. Worboys, and AntonyGalton.Efficient generation of simple polygons for characterizing theshape of a set of points in the plane.Pattern Recognition, 41(10):3224–3236, 2008.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References IV
[9] Herbert Edelsbrunner.Shape reconstruction with delaunay complex.In Claudio L. Lucchesi and Arnaldo V. Moura, editors, LATIN,volume 1380 of Lecture Notes in Computer Science, pages119–132. Springer-Verlag Berlin Heidelberg, 1998.
[10] Herbert Edelsbrunner, David G. Kirkpatrick, and RaimundSeidel.On the shape of a set of points in the plane.IEEE Transactions on Information Theory, 29(4):551–558,1983.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References V
[11] Antony Galton and Matt Duckham.What is the region occupied by a set of points?In Martin Raubal, Harvey J. Miller, Andrew U. Frank, andMichael F. Goodchild, editors, GIScience, volume 4197 ofLecture Notes in Computer Science, pages 81–98.Springer-Verlag Berlin Heidelberg, 2006.
[12] Amin Gheibi, Mansoor Davoodi, Ahmad Javad, FatemehPanahi, Mohammad M Aghdam, Mohammad Asgaripour, andAli Mohades.Polygonal shape reconstruction in the plane.IET computer vision, 5(2):97–106, 2011.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References VI
[13] Michael Kazhdan, Matthew Bolitho, and Hugues Hoppe.Poisson surface reconstruction.In Proceedings of the Fourth Eurographics Symposium onGeometry Processing, SGP ’06, pages 61–70, Aire-la-Ville,Switzerland, Switzerland, 2006. Eurographics Association.
[14] Michael M. Kazhdan and Hugues Hoppe.Screened poisson surface reconstruction.ACM Trans. Graph., 32(3):29, 2013.
[15] Jiju Peethambaran and Ramanathan Muthuganapathy.A non-parametric approach to shape reconstruction fromplanar point sets through delaunay filtering.Computer-Aided Design, 62(0):164 – 175, 2015.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References VII
[16] Jainy Sachdeva, Vinod Kumar, Indra Gupta, NiranjanKhandelwal, and Chirag Kamal Ahuja.A novel content-based active contour model for brain tumorsegmentation.Magnetic resonance imaging, 30(5):694–715, 2012.
[17] Bharath Ram Sundar, Abhijith Chunduru, Rajat Tiwari,Ashish Gupta, and Ramanathan Muthuganapathy.Footpoint distance as a measure of distance computationbetween curves and surfaces.Computers & Graphics, 38(0):300 – 309, 2014.
Introduction Problem Motivation Related Work Algorithm Theoretical Guarantee Results Comparison Conclusion
References VIII
[18] Remco C. Veltkamp.Closed Object Boundaries from Scattered Points, volume 885of Lecture Notes in Computer Science.Springer-Verlag, 1994.
[19] Remco C. Veltkamp and Michiel Hagedoorn.Principles of visual information retrieval.chapter State of the Art in Shape Matching, pages 87–119.Springer-Verlag, London, UK, UK, 2001.
[20] Cedric Villani.Topics in Optimal Transportation.Graduate studies in mathematics. American MathematicalSociety, 2003.
Top Related