The 2015 Symposium on Geometry Processing is accompanied by a graduate school on the weekend before the actual conference which is specifially targeted towards graduate students.

Saturday July 4, 2015

Sunday July 5, 2015


Course 1: Optimization Techniques for Geometry Processing

Slides by J. Solomon: [pdf], [ppt] Slides by D. Bommes: [keynote], [pdf], [ppt],

Countless techniques in geometry processing can be described variationally, that is, as the minimization or maximization of an objective function measuring shape properties.  Algorithms for parameterization, mapping, quad meshing, alignment, smoothing, and other tasks can be expressed and solved in this powerful language.  With this motivation in mind, this tutorial will summarize typical use cases for optimization in geometry processing.  In particular, we will show how procedures for generating, editing, and comparing shapes can be expressed variationally and will summarize computational tools for solving the relevant optimization problems in practice.  Along the way, we will highlight tools for prototypical problems in this class, including algorithms for unconstrained/constrained/convex optimization, lightweight schemes recently popularized in the machine learning literature, and procedures for relaxing discrete problems.   [go to top]

Justin Solomon (Stanford) and David Bommes (RWTH Aachen).

Course 2: Variational time integrators

[keynote] [pdf] [ppt]

This course is a gentle introduction to the idea behind and construction of a class of time integration schemes, often used for physical simulation, known as variational time integrators. We will initially focus on Hamilton’s principle of extremal (least) action, where we will show, in particular, that the smooth equations of motion of a physical system arise as critical paths of an action integral, the time integral of a system’s kinetic minus potential energy. Throughout the derivation we touch on ideas from the calculus of variations and reduce the infinite dimensional search for critical paths to much more familiar territory: single variable calculus. This variational approach to equations of motions can be readily discretized yielding a variational time integrator: direct discretization of the action integral using your favorite finite difference and quadrature scheme (in both time and space) followed by an analogous application of Hamilton’s principle leads to discrete equations of motion. It turns out that this construction guarantees conservation of linear and angular momentum. While conservation of energy is lost in the discretization, variational time integrators are also symplectic, a somewhat mysterious property that guarantees bounded oscillation around the true energy level, even after exponentially many time steps. Comparisons with other integration schemes is provided where careful attention is given to teasing apart the notions of higher order discretization for short term accuracy versus long term conservation. After this course students should be able to: (i) build their own variational time integrator for physical simulations and (ii) have a solid foundation from which to explore in detail the example applications of variational time integrators (and their generalizations) to various aspects of computer graphics.   [go to top]

Andy Sageman-Furnas (Univ. Göttingen)

Course 3: Mappings

Slides Part A: [pdf] [ppt], Slides Part B: [pdf] [ppt]

Map-computation is one of the most fundamental research topics in Geometry Processing. Many problems in this field either use maps extensively, or are actually map-computation problems themselves, prime examples being deformations, shape-registration\shape-correspondence and mesh parameterization. The goals of this tutorial are: (1) to get acquainted with the various concepts of this field, such as piecewise linear maps on simplicial meshes, deformation energies, distortion and optimization techniques; and (2) survey algorithms and applications as well as the state of the art.   [go to top]

Noam Aigerman, Shahar Kovalsky, and Roi Poranne (Weizmann Institute).

Course 4:Spectral Processing

Slides: [pdf] [ppt] [Demo]

In this course, we will look at the problem of signal-processing on meshes. Although meshes do not have the symmetry that enables the definition of a Fast Fourier Transform, we will see how an analogous frequency decomposition can be derived by considering the eigenvectors/eigenvalues of the Laplace-Beltrami operator. We will begin by formulating the generalized eigenvalue problem that defines the spectrum and then explore the utility of such a decomposition in a number of applications, including the solution of PDEs, mesh segmentation, and signal/geometry filtering. Time permitting, we will also look at how the spectral perspective facilitates the analysis of methods like gradient domain processing and discrete time integration.   [go to top]

Misha Kazhdan (Johns Hopkins University).

Course 5: Skinning


Real-time deformation brings 3D characters to life. Methods developed in recent years are found in abundance throughout computer games, film production, medical simulations and augmented reality systems. By ensuring fast performance, these methods also find applications outside of character deformation, such as 2D graphic design, interactive variational modeling and computer vision. Vibrant research from different subdisciplines of computer graphics has sculpted a diverse landscape of techniques, but also an intimidating field of (often inconsistent) terminology and related work. This course cuts a consistent narrative through the core techniques and recent advances in the realm of real-time deformation.

Computer graphics is first and foremost concerned with the outward visual appearance of shapes, hence character deformation techniques that parameterize the way a character's "skin" moves are referred to under the umbrella term of "skinning." Parameterized skinning has been in use nearly as long as polygonal meshes have been representing 3D shapes. In a traditional skinning setup or rig, an animator articulates a given 3D character by propagating transformations of an internal skeleton made of rigid bones to the vertices of the character mesh via scalar blending or weighting functions. The character's geometry may be arbitrary complex, but so long as the number of bones is small skinning a character is fast. Deformation computation is embarrassingly parallel and fits neatly into the SIMD optimized standard graphics pipeline: deformations may be computed just in time during vertex shading. Many linear and non-linear techniques were introduced to blend the deformation contributions of individual bones, using either the single rest-pose of a given shape or a database of example poses of the shape. Blending weights have been traditionally painted manually by highly trained rigging artists, but several automatic techniques are now available, reducing the barrier of entry and opening new doors to automated character deformation scenarios and applications. This course navigates recent developments and offers outlook toward future research.   [go to top]

Alec Jacobson, Columbia University

Course 6: Machine Learning Techniques for Geometric Modeling

[Slides as pdf]

Creating compelling shapes and scenes is a fundamental goal of computer graphics. Despite decades of advances in computer graphics and geometry processing research, it remains a tough challenge to provide users with modeling tools that are simple to use, easy to learn, enable rapid prototyping, and encourage interactive exploration of design alternatives. The course will cover recent developments in algorithms and techniques that learn how geometric representations of shapes can be edited and generated from data. In contrast to traditional modeling approaches that rely on laborious user interaction, low-level selection and editing commands, or explicitly programmed instructions, machine learning techniques can help users to easily perform complex geometric modeling operations through intuitive input such as linguistic terms, intelligent deformation handles, sketches, images and so on. The course will cover both theoretical and practical aspects of these techniques, discuss various application cases, and present possible future research directions related to latest advances in deep learning.   [go to top]

Evangelos Kalogerakis (Univ. Massachusets, Amherst).

Course 7: Registration

[Slides as pdf] [Movie]

Image and geometry registration algorithms are an essential component of many computer graphics and computer vision systems. With recent technological advances in RGB-D sensors, such as the Microsoft Kinect or Intel RealSense  robust algorithms that combine 2D image and 3D geometry registration have become an active area of research. The goal of this course is to introduce the basics of 2D/3D registration algorithms and to provide theoretical explanations and practical tools to design computer vision and computer graphics systems based on RGB-D devices.   [go to top]

Sofien Bouaziz and Andrea Tagliasacchi  (EPFL, Lausanne)