\documentclass[a4paper]{article} 


%%%%%%%% DO NOT MODIFY THIS PART %%%%%%%%%%
\usepackage{url}
\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
\newcommand{\mytitle}[1]{\title{\textsf{\bf{#1}}}}
\date{} %
\pagestyle{empty} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%% INCLUDE MACRO HERE IF NEEDED %%%%%%%%%%
\def \matV {\mathbf{V}}
\def \matB {\mathbf{B}}
\def \matP {\mathbf{P}}
\def \matZ {\mathbf{Z}}
\def \vecP {\mathbf{p}}
\def \vecV {\mathbf{v}}
\def \vecb {\mathbf{b}}
\def \vecz {\mathbf{z}}
\def \vecE {\mathbf{e}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


%%% TITLE %%%
\mytitle{Sparse $G^1$ Spline Manifold and Salient Feature Map Approximations with Topological Accuracy}

%%%% SPEAKER (OR PRESENTATOR OF POSTER) %%%
\author{Ahmed Blidia\\
Univ. C\^ote d'Azur, Inria, \textsc{aromath}
}


\begin{document}
\maketitle
\thispagestyle{empty}


%%%% MAIN BODY -- MODIFY THIS %%%%%
We approximate a surface manifold, (or a point set sampling of a
surface manifold), and recover salient features of the manifold by a
sparse set of $G^1$ B-spline functions. The final solution possesses
the same topology as that of the original manifold as well as the
recovered salient features while providing a sparse representation
using smooth spline functions supported on quad meshes. We utilize an
alternating optimization method, between progressively improving a
quad mesh with well distributed  sampling of vertices using
$\epsilon$-nets (or $Delone$ sets) \cite{Agar05,Clark06,LL00} and
approximating the surface manifold (or point set) based on the use of
compactly supported $G^1$ spline function basis.  For the first
alternating step we optimize the coarse complex approximation and
infer the topology of the manifold using a sparsified Rips complex,
along with the homology  recovery of salient shape features
\cite{Carl2003,Sheehy2013,GIC2015}. The second alternating
optimization step produces a sparse parametric encoding of the surface
manifold and of recovered salient features. We use distribution
sampling strategies for producing low discrepancy sampling of
vertices and also allow for  a choice of different atomic $G^1$ spline
function basis \cite{bajaj2002hierarchical,BLIDIA2017106,TOSUN2011,Ying2004,Wang:2016:CMV}.

Joint work with
Chandrajit Bajaj, Qixing Huang, Yi Wang (ICES, University of Texas at Austin, Texas, USA),
Bernard Mourrain (Univ. C\^ote d'Azur, Inria, \textsc{aromath}).


%%%% BIBLIOGRAPHY -- MODIFY THIS %%%%%
% \bibliographystyle{alpha}
% \bibliography{CS2018}
\newcommand{\etalchar}[1]{$^{#1}$}
\begin{thebibliography}{WLY{\etalchar{+}}16}

\bibitem[AHPV05]{Agar05}
Pankaj Agarwal, Shmuel Har-Peled, and Kasturi Varadarajan.
\newblock Geometric approximation via coresets.
\newblock In {\em Combinatorial and Computational Geometry, MSRI Pubs},
  volume~52, page~x, 2005.

\bibitem[BMV17]{BLIDIA2017106}
Ahmed Blidia, Bernard Mourrain, and Nelly Villamizar.
\newblock G$^{1}$-smooth splines on quad meshes with 4-split macro-patch elements.
\newblock {\em Computer Aided Geometric Design}, 52:106 -- 125, 2017.
\newblock Geometric Modeling and Processing 2017.

\bibitem[BXHN02]{bajaj2002hierarchical}
Chandrajit~L Bajaj, Guoliang Xu, Robert~J Holt, and Arun~N Netravali.
\newblock Hierarchical multiresolution reconstruction of shell surfaces.
\newblock {\em Computer Aided Geometric Design}, 19(2):89--112, 2002.

\bibitem[CdS03]{Carl2003}
G.~Carlsson and V~de~Silva.
\newblock Topological approximation by small simplicial complexes.
\newblock In {\em Preprint online}, 2003.

\bibitem[Cla06]{Clark06}
Kenneth Clarkson.
\newblock Building triangulations using $\epsilon$-nets.
\newblock In {\em Proceedings of the thirty-eighth annual ACM symposium on
  Theory of computing, STOC}, pages 326--335, 2006.

\bibitem[DFY15]{GIC2015}
Tamal Dey, Fengtao Fan, and Wang Yusu.
\newblock Graph induced complex on point data.
\newblock {\em Computational Geometry:Theory and Applications}, 48(8):575--588,
  2015.

\bibitem[LL00]{LL00}
G.~Leibon and D.~Letscher.
\newblock Delaunay triangulations and voronoi diagrams for riemannian
  manifolds.
\newblock In {\em Proceedings of the Sixteenth Annual Symposium on
  Computational geometry, SOCG00}, pages 341--349, 2000.

\bibitem[She]{Sheehy2013}
D.~Sheey.
\newblock Linear-size approximations to the vietoris–rips filtration.
\newblock {\em Discrete Comput. Geom.}

\bibitem[TZ11]{TOSUN2011}
Elif Tosun and Denis Zorin.
\newblock Manifold-based surfaces with boundaries.
\newblock {\em Computer Aided Geometric Design}, 28(1):1 -- 22, 2011.

\bibitem[WLY{\etalchar{+}}16]{Wang:2016:CMV}
Ruimin Wang, Ligang Liu, Zhouwang Yang, Kang Wang, Wen Shan, Jiansong Deng, and
  Falai Chen.
\newblock Construction of manifolds via compatible sparse representations.
\newblock {\em ACM Trans. Graph.}, 35(2):14:1--14:10, February 2016.

\bibitem[YZ04]{Ying2004}
Lexing Ying and Denis Zorin.
\newblock A simple manifold-based construction of surfaces of arbitrary
  smoothness.
\newblock {\em ACM Trans. Graph.}, 23(3):271--275, August 2004.

\end{thebibliography}


\end{document}
