# Workshop on Computational Manifolds

NOVEMBER 22-24, 2011

Location: Auditório 3, IMPA

**Overview**

The workshop on Computational Manifolds and Applications will consist

of keynote lectures, contributed talks and project presentations by the participants
of the program.

## Schedule

TUESDAY

13h30 - 14h15 :: Douglas Cedrim

14h15 - 15h00 :: Mattia Natali, Javier Lezama

15h00 - 15h30 :: Break

15:30 - 16h15 :: Francisco Ganacim

16:15 - 17h00 :: Andre Maximo

WEDNESDAY

13h30 - 14h15 :: Francisco Benavides

14h15 - 15h00 :: Leandro Cruz

15h00 - 15h30 :: Break

15:30 - 16h15 :: Maria Andrade, Leonardo Carvalho, Dalia Bonilla

16:15 - 17h00 :: Aldo Rene Zang

THURSDAY

13h30 - 14h30 :: Geovan Tavares

14h30 - 15h00 :: Break

15h00 - 16h00 :: Jean Gallier

16h00 - 17h00 :: Wolfgang Ziller

## Presentations

Douglas Cedrim (USP)

**Title**: Voronoi based clustering

**Abstract**:
Discrete Voronoi Diagrams are a common and powerful tool to handle
geometric properties of discrete surfaces. In some geometric processing
algorithms,
for sake of complexity, one has to work just with a subset of the original
data.
In this talk we discuss a particular construction of these subsets by using
a Voronoi clustering
algorithm regarding the adaptivity of the mesh.

Mattia Natali (University of Bergen), Javier Lezama (Universidad de Cordoba)

**Title**: Discussion of semi-regular remeshing

**Abstract**:
The described technique proposed by Igor Guskov aims to generate a new semi-regular
triangulation starting from an initial mesh and
passing through a manifold construction of the underlying surface. The
parameterization process is based on
the definition of a base mesh, on which a differential structure is
built. At the same time, it is possible
to control the anisotropy of the final regular mesh.

Francisco Benavides (IMPA)

**Title**: A Direct texture placement and editing surface

**Abstract**: The authors present an approach for texture placement and editing based on direct manipulation of textures on the surface. The theoretical foundations of the applied discrete operators can be found in the article "Discrete Differential-Geometry Operators
for Triangulated 2-Manifolds", which presents theoretical results on which the texture placement is based. These derivations are based on Voronoi cells and Finite Element-Volume methods.

Aldo Rene Zang (IMPA)

**Title**: BRDFs Evolution: from analytical models to measured data

**Abstract**: A fundamental problem in computer graphics rendering is modeling how lights
is reflected from surfaces. A class of functions called Bidirectional
Reflectance Distribution Functions (BRDFs) characterizes the process where
light transport occurs at an idealized surface point.
First we will talk a little bit about the physically inspired analytic
reflection models that provide the BRDFs used in computer graphics today.
Before talking about measured approaches for BRDFs we will talk about a
change of variables for get a efficient BRDF representation introduced by
Rusinkiewicz in 1998.
We will talk too about the new approaches for measuring the BRDFs from real
materials using the Rusinkiewicz parametrization, and techniques for
constructing new BRDFs from measured data sets.
Finally we will talk about a non-parametric Inverse Shade Tree (IST)
framework for representing and editing measured spatially- and
directionally-dependent surface appearance (SVBRDF). The IST framework
allow us to edit a captured SVBRDF editing simples 1D and 2D functions in
the leaves of the tree.

Leandro Cruz (IMPA)

**Title**: Fitting Subdivision Surfaces

**Abstract**: In this talk we will show how to fit a subdivision surface
using the Quasi-Interpolation method. This method calculates the
position of control points of a Catmul-Clark surface that better fit
the given surface. It obtains an approximation of original surface
with a desired accuracy, without need to solve a Linear System, like
the Least-Square Method. Thus, this is a fast, local, and scale well
algorithm. Its convergence rate is optimal for regular meshes and,
empirically, was observed that it behave well for irregular meshes.

Francisco Ganacim (IMPA)

**Title**: Parametrization of Triangular Meshes using Angle-Based Flattening

**Abstract**:
Computing a low distortion parameterization for a mesh is a key
step in many algorithms (e.g. texture mapping). The distortion occurs when
areas of high surface curvature are flattened. By cutting this areas, we
can reduce the parameterization's distortion at the expense of introducing
additional seams to the mesh. After eliminating the high distortion areas,
we compute a low-distortion parameterization using the angles in the mesh.
In this talk we are going to show how to compute the seams and the
parameterization using two techniques: Seamster and ABF.

Andre Maximo (IMPA)

**Title**: M4G: A Surface Representation for Adaptive CPU-GPU Coupled Computation

**Abstract**: Discrete surfaces are represented in different ways
depending on the application and the target computational domain. In
this work, we explore a new surface representation, called
Manifolds-for-GPUs (M4G),
aiming at describing discrete surfaces with adaptive control of the
resolution and combining the different computational granularities of
the CPU and the GPU.

Maria Andrade, Leonardo Carvalho, Dalia Bonilla (IMPA)

**Title**: Fluid simulation on surfaces

**Abstract**:
In this work, we show how to calculate the Navier-Stokes equations on a
smooth surface, where we want to simulate a fluid flow. We used a
parametrization
of a Catmull-Clark surface, from which we calculate a metric that depends
on its tangent vectors and use this metric to get the necessary differential
operators to solve the problem. The Navier-Stokes equations are solved on
a discretization of the surface, dealing with the transition of the
fluid between the patches that form the surface.

Geovan Tavares

**Title**: On the Combinatorics of Implicit Manifolds

**Abstract**:
Implicit manifolds have been at the forefront of geometric
modeling for over 20 years now. Its use in mathematics is more than a
century old and it is at the very definition of general manifolds. In
this talk we will describe a combinatorics for implicit manifolds and
its application to numerical methods in implicit differential
equations. Along the talk we will point several unsolved problems on
the mathematical and combinatorial aspects of implicit manifolds.

Jean Gallier

**Title**: The classification theorem for compact surfaces

**Abstract**:
The classification theorem for compact surfaces is one of
the great achievements
of early 20th century mathematics. The statement of this theorem is
quite intuitive but
it took about sixty years until a rigorous proof was finally given by
Brahana in 1921. Early versions of
the classification theorem were given by Mobius in 1861, and by Jordfan in 1866.
More definite proofs were given later by von Dyck in 1888 and Dehn and
Heegaard in 1907.
A complete and rigorous proof requires a significant amount of machinery.
In this talk, we will give a guided tour of the proof, pointing out
which tools from algebraic topology
are needed. We will also give an abbreviated history of the "proof", and
briefly explain how the theorem leads to "global parametrizations," using
fundamental domains.

Wolfgang Ziller

**Title**: Ricci Flow and the Uniformization Theorem

**Abstract**:
The Ricci flow is a natural flow that changes a metric on a manifold
to something that is "nicer". We will discuss some applications, in
particular how one can use it to prove the uniformization theorem for
surfaces.

**Submission of contributors**

Please send a one-page extended abstract of your presentation

in PDF format to cma2011@impa.br until October 28th, 2011.

Submissions will be selected based on suitability to the event

and time limitations.