a unified approach towards reconstruction of a planar point setraman/recons_smi15.pdf · 2017. 11....

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 Statement

Motivation

Related Work

Algorithm Overview

Theoretical Guarantee

Comparative Studies

Conclusion & Future Work


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)


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


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


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


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


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


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


Preliminaries

Voronoi diagram and Delaunay triangulation

Exterior triangle and exterior edges


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


Algorithm

Preliminaries

Circle Constraint: diametric circle, chord circle & midpointcircle

Regularity constraint: bridge, dangling edge & junction point


Algorithm

Illustration of Our Algorithm

Figure : (a) Input (b) Delaunay triangulation DT(Graph G)


Algorithm


Figure : (c) Non-empty diametric circle, (d) Two new edgesas EEs


Algorithm


Figure : (f) Empty diametric circle & non-empty midpointcircle (g) Two new edges as EEs


Algorithm


Figure : (i) Empty diametric circle & non-empty chord circle(j)Two new edges as EEs


Algorithm


Figure : (k) All circles empty (l) Final Graph


Algorithm


Figure : (l) Final Graph (m) ec-shape


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


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



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 .



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.


ec-shapes for Boundary Sample and Dot Pattern


Results

ec-shapes for Boundary Sample and Dot Pattern


Qualitative Comparison for Boundary Sample


Comparison

Qualitative Comparison for Boundary Sample


Comparison

Qualitative Comparison for Dot Pattern


Comparison

Parameter Turing for Other Methods

Figure : (a) & (b):α-shape, Figs (c) & (d): Simple shape,Figs (e) & (f): χ-shape


Comparison

Quantitative Comparison

Measure for quantitative comparison [8]∗

L2error norm =area((O − Re) ∪ (Re − O))

area(O)

∗ M.Duckham et al. 2008


Comparison



Comparison


Different distributions considered:I Non-Random (NR)I Semi Random Dense Boundary (SRDB)I Semi Random Sparse Boundary (SRSB)I Random (R)


Comparison

Outputs of Different Distributions


Comparison

Parameter Tuning for ec-shape


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


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


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.


Thank You



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



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



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



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



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


Algorithm


Long exterior edge - not part of resultant shape

Empty circle on EE

Non-empty circle on shorter adjacent side of EE


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


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.


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.


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.


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.


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.


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.


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.


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.

a unified approach towards reconstruction of a planar point setraman/recons_smi15.pdf · 2017. 11....

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