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Hermite Radial Basis Functions Implicits

The Hermite radial basis functions (HRBF) implicits reconstruct an implicit function which interpolates or approximates scattered multivariate Hermite data (i.e. unstructured points and their corresponding normals). Experiments suggest that HRBF implicits allow the reconstruction of surfaces rich in details and behave better than previous related methods under coarse and/or non‐uniform samplings, even in the presence of close sheets. HRBF implicits theory unifies a recently introduced class of surface reconstruction methods based on radial basis functions (RBF), which incorporate normals directly in their problem formulation. Such class has the advantage of not depending on manufactured offset‐points to ensure existence of a non‐trivial implicit surface RBF interpolant. In fact, we show that HRBF implicits constitute a particular case of Hermite–Birkhoff interpolation with radial basis functions, whose main results we present here. This framework not only allows us to show connections between the present method and others but also enable us to enhance the flexibility of our method by ensuring well‐posedness of an interesting combined interpolation/regularization approach.

Sketching Variational Hermite-RBF implicits

We present techniques for modeling Variational Hermite Radial Basis Function (VHRBF) Implicits using a set of sketch-based interface and modeling (SBIM) operators. VHRBF Implicits is a simple and compact representation well suited for SBIM. It provides quality reconstructions, preserving the intended shape from a coarse and non-uniform number of point-normal samples extracted directly from the input strokes. In addition, it has a number of desirable properties such as parameter-free modeling, invariance under geometric similarities on the input strokes, suitable estimation of differential quantities, good behavior near close sheets, and both linear fitting and reproduction. Our approach uses these properties of VHRBF Implicits to quickly and robustly generate the overall shape of 3D models. We present examples of implicit models obtained from a set of SBIM language operators for contouring, cross-editing, kneading, oversketching and merging.

Expression Transfer between Photographs through Multilinear AAM's

Expression transfer is a method for mapping a photographed expression performed by a given subject onto the photograph of another persons face. Building on well succeeded previous works by the vision researchers (facial expression decomposition, active appearance models and multilinear analysis, we propose a novel approach for expression transfer based on color images. We attack this problem with methods developed by the computer vision community for facial expression analysis and recognition. Combining active appearance models and multilinear analysis, its possible to suitably represent and analyze expressive facial images, while separating both style (subjects identity) and content (expressive flavor) from the captured performance

Hermite Interpolation of Implicit Surfaces with Radial Basis Functions

We present the Hermite radial basis function (HRBF) implicits method to compute a global implicit function which interpolates scattered multivariate Hermite data (unstructured points and their corresponding normals). Differently from previous radial basis functions (RBF) approaches, HRBF implicits do not depend on offset points to ensure existence and uniqueness of its interpolant. Intrinsic properties of this method allow the computation of implicit surfaces rich in details, with irregularly spaced points even in the presence of close sheets. Comparisons to previous works show the effectiveness of our approach. Further, the theoretical background of HRBF implicits relies on results from generalized interpolation theory with RBFs, making possible powerful new variants of this method and establishing connections with previous efforts based on statistical learning theory.

Gauge Optimization and Duality

Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by [R. M. Freund, Math. Programming, 38 (1987), pp. 47--67], seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be used to model a remarkable array of useful problems, including a special case of conic optimization, and related problems that arise in machine learning and signal processing. The gauge structure of these problems allows for a special kind of duality framework. This paper explores the duality framework proposed by Freund, and proposes a particular form of the problem that exposes some useful properties of the gauge optimization framework (such as the variational properties of its value function), and yet maintains most of the generality of the abstract form of gauge optimization.

Extended papers from NPAR 2010: Shape and tone depiction for implicit surfaces

We present techniques for rendering implicit surfaces in different pen-and-ink styles. The implicit models are rendered using point-based primitives to depict shape and tone using silhouettes with hidden-line attenuation, drawing directions, and stippling. We present sample renderings obtained for a variety of models. Furthermore, we describe simple and novel methods to control point placement and rendering style. Our approach is implemented using HRBF Implicits, a simple and compact representation, that has three fundamental qualities: a small number of point-normal samples as input for surface reconstruction, good projection of points near the surface, and smoothness of the gradient field. These qualities of HRBF Implicits are used to generate a robust distribution of points to position the drawing primitives.

Low-Rank Spectral Optimization via Gauge Duality

Various applications in signal processing and machine learning give rise to highly structured spectral optimization problems characterized by low-rank solutions. Two important examples that motivate this work are optimization problems from phase retrieval and from blind deconvolution, which are designed to yield rank-1 solutions. An algorithm is described that is based on solving a certain constrained eigenvalue optimization problem that corresponds to the gauge dual which, unlike the more typical Lagrange dual, has an especially simple constraint. The dominant cost at each iteration is the computation of rightmost eigenpairs of a Hermitian operator. A range of numerical examples illustrate the scalability of the approach.

Sketch-based warping of RGBN images

While current image deformation methods are careful in making the new geometry seem right, little attention has been given to the photometric aspects. We introduce a deformation method that results in coherently illuminated objects. For this task, we use RGBN images to support a relighting step integrated in a sketch-based deformation method. We warp not only colors but also normals. Normal warping requires smooth warping fields. We use sketches to specify sparse warping samples and impose additional constraints for region of interest control. To satisfy these new constraints, we present a novel image warping method based on Hermite-Birkhoff interpolation with radial basis functions that results in a smooth warping field. We also use sketches to help the system identify both lighting conditions and material from single images. We present results with RGBN images from different sources, including photometric stereo, synthetic images, and photographs.

PCA-Based 3D Face Photography

This paper presents a 3D face photography system based on a small set of training facial range images. The training set is composed by 2D texture and 3D range images (i.e. geometry) of a single subject with different facial expressions. The basic idea behind the method is to create texture and geometry spaces based on the training set and transformations to go from one space to the other. The main goal of the proposed approach is to obtain a geometry representation of a given face provided as a texture image, which undergoes a series of transformations through the texture and geometry spaces. Facial feature points are obtained by an active shape model (ASM) extracted from the 2D gray-level images. PCA then is used to represent the face dataset, thus defining an orthonormal basis of texture and range data. An input face is given by a gray-level face image to which the ASM is matched. The extracted ASM is fed to the PCA basis representation and a 3D version of the 2D input image is built. The experimental results on static images and video sequences using seven samples as training dataset show rapid reconstructed 3D faces which maintain spatial coherence similar to the human perception, thus corroborating the efficiency of our approach.

3D face computational photography using PCA spaces

In this paper, we present a 3D face photography system based on a facial expression training dataset, composed of both facial range images (3D geometry) and facial texture (2D photography). The proposed system allows one to obtain a 3D geometry representation of a given face provided as a 2D photography, which undergoes a series of transformations through the texture and geometry spaces estimated. In the training phase of the system, the facial landmarks are obtained by an active shape model (ASM) extracted from the 2D gray-level photography. Principal components analysis (PCA) is then used to represent the face dataset, thus defining an orthonormal basis of texture and another of geometry. In the reconstruction phase, an input is given by a face image to which the ASM is matched. The extracted facial landmarks and the face image are fed to the PCA basis transform, and a 3D version of the 2D input image is built. Experimental tests using a new dataset of 70 facial expressions belonging to ten subjects as training set show rapid reconstructed 3D faces which maintain spatial coherence similar to the human perception, thus corroborating the efficiency and the applicability of the proposed system.