Results 1  10
of
90
Reconstruction and Representation of 3D Objects with Radial Basis Functions
 Computer Graphics (SIGGRAPH ’01 Conf. Proc.), pages 67–76. ACM SIGGRAPH
, 2001
"... We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs al ..."
Abstract

Cited by 500 (1 self)
 Add to MetaCart
We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBFpreviously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energyminimisation characterisation of polyharmonic splines result in a "smoothest" interpolant. This scaleindependent characterisation is wellsuited to reconstructing surfaces from nonuniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a noninterpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for realworld rangefinder data.
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
Abstract

Cited by 340 (27 self)
 Add to MetaCart
(Show Context)
This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Variational image reconstruction from arbitrarily spaced samples: A fast multiresolution spline solution
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2005
"... We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared errors at the specified poin ..."
Abstract

Cited by 33 (3 self)
 Add to MetaCart
(Show Context)
We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared errors at the specified points and 2) a quadratic functional that penalizes the lack of smoothness. We search for a solution that is a uniform spline and show how it can be determined by solving a large, sparse system of linear equations. We interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach. Using the twoscale relation for Bsplines, we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution. We use this relation to develop a fast multigrid algorithm. We demonstrate the effectiveness of our approach on some image reconstruction examples.
Realtime Visual Tracking under Arbitrary Illumination Changes
"... In this paper, we investigate how to improve the robustness of visual tracking methods with respect to generic lighting changes. We propose a new approach to the direct image alignment of either Lambertian or nonLambertian objects under shadows, interreflections, glints as well as ambient, diffuse ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
(Show Context)
In this paper, we investigate how to improve the robustness of visual tracking methods with respect to generic lighting changes. We propose a new approach to the direct image alignment of either Lambertian or nonLambertian objects under shadows, interreflections, glints as well as ambient, diffuse and specular reflections which may vary in power, type, number and space. The method is based on a proposed model of illumination changes together with an appropriate geometric model of image motion. The parameters related to these models are obtained through an efficient secondorder optimization technique which minimizes directly the intensity discrepancies. Comparison results with existing direct methods show significant improvements in the tracking performance. Extensive experiments confirm the robustness and reliability of our method. 1.
Inference of segmented color and texture description by tensor voting
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... Abstract—A robust synthesis method is proposed to automatically infer missing color and texture information from a damaged 2D image by ND tensor voting (N>3). The same approach is generalized to range and 3D data in the presence of occlusion, missing data and noise. Our method translates texture ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
Abstract—A robust synthesis method is proposed to automatically infer missing color and texture information from a damaged 2D image by ND tensor voting (N>3). The same approach is generalized to range and 3D data in the presence of occlusion, missing data and noise. Our method translates texture information into an adaptive ND tensor, followed by a voting process that infers noniteratively the optimal color values in the ND texture space. A twostep method is proposed. First, we perform segmentation based on insufficient geometry, color, and texture information in the input, and extrapolate partitioning boundaries by either 2D or 3D tensor voting to generate a complete segmentation for the input. Missing colors are synthesized using ND tensor voting in each segment. Different feature scales in the input are automatically adapted by our tensor scale analysis. Results on a variety of difficult inputs demonstrate the effectiveness of our tensor voting approach. Index Terms—Image restoration, segmentation, color, texture, tensor voting, applications. 1
Wavelets, Fractals, and Radial Basis Functions
 IEEE TRANS. SIGNAL PROCESSING
, 2002
"... Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that t ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
(Show Context)
Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that the two types of representation are closely linked together ... through fractals. First, we identify and characterize the whole class of selfsimilar radial basis functions that can be localized to yield conventional multiresolution wavelet bases. Conversely, we prove that for any compactly supported scaling function 9(0c), there exists a onesided central basis function p+ (x) that spans the same multireso lution subspaces. The central property is that the multiresolution bases are generated by simple translation of p+ without any dilation. We also present an explicit timedomain representation of a scaling function as a sum of harmonic splines. The leading term in the decomposition corresponds to the fractional splines: a recent, continuousorder generalization of the polynomial splines.
Temporally Coherent Completion of Dynamic Shapes
"... We present a novel shape completion technique for creating temporally coherent watertight surfaces from realtime captured dynamic performances. Because of occlusions and low surface albedo, scanned mesh sequences typically exhibit large holes that persist over extended periods of time. Most convent ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
We present a novel shape completion technique for creating temporally coherent watertight surfaces from realtime captured dynamic performances. Because of occlusions and low surface albedo, scanned mesh sequences typically exhibit large holes that persist over extended periods of time. Most conventional dynamic shape reconstruction techniques rely on template models or assume slow deformations in the input data. Our framework sidesteps these requirements and directly initializes shape completion with topology derived from the visual hull. To seal the holes with patches that are consistent with the subject’s motion, we first minimize surface bending energies in each frame to ensure smooth transitions across hole boundaries. Temporally coherent dynamics of surface patches are obtained by unwarping all frames within a time window using accurate interframe correspondences. Aggregated surface samples are then filtered with a temporal visibility kernel that maximizes the use of nonoccluded surfaces. A key benefit of our shape completion strategy is that it does not rely on longrange correspondences or a template model. Consequently, our method does not suffer from error accumulation typically introduced by noise, large deformations, and drastic topological changes. We illustrate the effectiveness of our method on several highresolution scans of human performances captured with a stateoftheart multiview 3D acquisition system.
Boundary knot method for 2D and 3D Helmholtz and convectiondiffusion problems under complicated geometry
 J. Numer. Methd. Engng
, 2001
"... The boundary knot method (BKM) of very recent origin is an inherently meshless, integrationfree, boundarytype, radial basis function collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of nons ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
The boundary knot method (BKM) of very recent origin is an inherently meshless, integrationfree, boundarytype, radial basis function collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of nonsingular general solution in the BKM avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain. The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convectiondiffusion problems under rather complicated irregular geometry. The method is also first applied to 3D problems. Numerical experiments validate that the BKM can produce highly accurate solutions using only relatively a small number of knots. For inhomogeneous cases, some inner knots are found necessary to guarantee accuracy and stability. The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed.
Implicit Meshes for Surface Reconstruction
 IEEE Trans. on Pattern Analysis and Machine Intelligence
, 2004
"... Deformable 3D models are used extensively in Computer Graphics and Computer Vision for Visualization, Animation and Modeling. They can be represented either as traditional explicit surfaces, such as triangulated meshes, or as implicit surfaces. Explicit surface representations are widely accept ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
(Show Context)
Deformable 3D models are used extensively in Computer Graphics and Computer Vision for Visualization, Animation and Modeling. They can be represented either as traditional explicit surfaces, such as triangulated meshes, or as implicit surfaces. Explicit surface representations are widely accepted because they are simple to deform and render. However, for fitting purposes, they suffer from the fact that using them typically involves minimizing a nondifferentiable distance function. By contrast, implicit surface representations allow fitting by minimizing a differentiable algebraic distance. However, they have not gained wide acceptance because they are harder to meaningfully deform and render.
Imageguided blended neighbor interpolation of scattered data
, 2009
"... Uniformly sampled images are often used to interpolate other data acquired more sparsely with an entirely different mode of measurement. For example, downhole tools enable geophysical properties to be measured with high precision near boreholes that are scattered spatially, and less precise seismic ..."
Abstract

Cited by 14 (10 self)
 Add to MetaCart
Uniformly sampled images are often used to interpolate other data acquired more sparsely with an entirely different mode of measurement. For example, downhole tools enable geophysical properties to be measured with high precision near boreholes that are scattered spatially, and less precise seismic images acquired at the earth’s surface are used to interpolate those properties at locations far away from the boreholes. Imageguided interpolation is designed specifically to enhance this process. Most existing methods for interpolation require distances from points where data will be interpolated to nearby points where data are known. Imageguided interpolation requires nonEuclidean distances in metric tensor fields that represent the coherence, orientations and shapes of features in images. This requirement leads to a new method for interpolating scattered data that I call blended neighbor interpolation. For simple Euclidean distances, blended neighbor interpolation resembles the classic natural neighbor interpolation.