Introduction to linear and nonlinear programming

We present a method for solving the following problem: Given a set of data points scattered in three dimensions and an initial triangular mesh M0, produce a mesh M, of the same topological type as M0, that fits the data well and has a small number of vertices. Our approach is to minimize an energy function that explicitly models the competing desires of conciseness of representation and fidelity to the data. We show that mesh optimization can be effectively used in at least two applications: surface reconstruction from unorganized points, and mesh simplification (the reduction of the number of vertices in an initially dense mesh of triangles).

Mesh optimization

We present an image-based modeling and editing system that takes a single photo as input. We represent a scene as a layered collection of depth images, where each pixel encodes both color and depth. Starting from an input image, we employ a suite of user-assisted techniques, based on a painting metaphor, to assign depths and extract layers. We introduce two specific editing operations. The first, a “clone brushing tool,” permits the distortion-free copying of parts of a picture, by using a parameterization optimization technique. The second, a “texture-illuminance decoupling filter,” discounts the effect of illumination on uniformly textured areas, by decoupling large- and small-scale features via bilateral filtering. Our system enables editing from different viewpoints, extracting and grouping of image-based objects, and modifying the shape, color, and illumination of these objects.

/pdf/image-based-modeling-and-photo-editing-2ye7pvltxg.pdf

Image-based modeling and photo editing

Geological modelling from field data and geological knowledge. Part I. Modelling method coupling 3D potential-field interpolation and geological rules

We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices. In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.

/pdf/spectral-geometry-processing-with-manifold-harmonics-c7aizixidn.pdf

Spectral Geometry Processing with Manifold Harmonics

In such fields as geology and biology, a common problem is that of modelling complex surfaces that are defined by data of various types. Classical modelling techniques based on Bezier and spline interpolations can account for only some of these types of data. The paper proposes a different approach that is based on the discrete smooth interpolation method. In this approach, surfaces are modelled as 2D graphs whose node locations are determined for a wide variety of heterogeneous data.

Discrete smooth interpolation in geometric modelling

Interpolation of a function ƒ (.) known at some data points of RP is a common problem. Many computer applications (e.g., automatic contouring) need to perform interpolation only at the nodes of a given grid. Whereas most classical methods solve the problem by finding a function defined everywhere, the proposed method avoids explicitly computing such a function and instead produces values only at the grid points. For two-dimensional regular grids, a special case of this method is identical to the Briggs method (see “Machine Contouring Using Minimum Curvature,” Geophysics 17, 1 (1974)), while another special case is equivalent to a discrete version of thin plate splines (see J. Duchon, Fonctions Splines du type Plaque Mince en Dimention 2, Seminaire d'analyse numerique, n 231, U.S.M.G., Grenoble, 1975; and J. Enriquez, J. Thomann, and M. Goupillot, Application of bidimensional spline functions to geophysics, Geophysics 48, 9 (1983)).

Discrete smooth interpolation

Interpolating physical properties in the subsurface is a recurrent problem in geology. In sedimentary geology, the geometry of the layers is generally known with a precision much superior to that which one can reasonably expect for the properties. The geometry of the layers is affected by folding and faulting since the time of deposition, whereas the distribution of properties is, to a certain extent, determined at the time of deposition. As a consequence, it may be wise to model first the geometry of the layers and then, “simplify the geologic equation” by removing the influence of that geometry. Inspired from the work of H. E. Wheeler on “Time-Stratigraphy,” we define, mathematically, a new space where all horizons are horizontal planes and where faults, if any, have disappeared. We surmise that this new space, however approximative, is better to model physical properties of the subsurface whatever the subsequent interpolation method used. The proposed mathematical framework also provides solutions to complex problems such as determination of strains resulting from tectonic events and up-scaling of permeabilities on structured and unstructured 3D grids.

Space–Time Mathematical Framework for Sedimentary Geology

3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data

This paper presents a discrete technique specially designed for modeling the geometry and the properties of natural objects as those encountered in biology and geology. Contrary to classical Computer-Aided Design methods based on continuous (polynomial) functions, the proposed approach is based on a discretization of the objects close to the finite-element techniques used for solving differential equations. Each object is modeled as a set of interconnected nodes holding the geometry and the physical properties of the objects and the Discrete Smooth Interpolation method is used for fitting the geometry and the properties to complex data. Data are turned into linear constraints and some constraints related to typical information encountered in geology are presented.

Jean-Laurent Mallet

Papers

Discrete smooth interpolation in geometric modelling

Discrete smooth interpolation

Space–Time Mathematical Framework for Sedimentary Geology

3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data

Discrete modeling for natural objects