Hierarchical and variational geometric modeling with wavelets

Gortler, Steven J.; Cohen, Michael F.

doi:10.1145/199404.199410

Cited by 79 publications

(63 citation statements)

References 28 publications

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“…They offer both theoretical characterization of smoothness, insights into the structure of functions and operators, and practical numerical tools which lead to faster computational algorithms. Examples of their use in computer graphics include surface and volume illumination computations [16,29], curve and surface modeling [17], and animation [18] among others. Given the high computational demands and the quest for speed in computer graphics, the increasing exploitation of wavelets comes as no surprise.…”

Section: Waveletsmentioning

confidence: 99%

Spherical wavelets: Efficiently representing functions on a sphere

Schröder¹,

Sweldens²

Wavelets in the Geosciences

262

409

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Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

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Section: Waveletsmentioning

confidence: 99%

Spherical wavelets: Efficiently representing functions on a sphere

Schröder¹,

Sweldens²

Wavelets in the Geosciences

262

409

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show abstract

“…We draw our wavelet motivation from the variational modelling technique developed by Gortler and Cohen [15] in the field of computer graphics. However, we note a fundamental difference in the problems being solved.…”

Section: Related Workmentioning

confidence: 99%

A hierarchical wavelet decomposition for continuous-time SLAM

Anderson

Dellaert

Barfoot

2014

2014 IEEE International Conference on Robotics and Automation (ICRA)

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Abstract-This paper proposes using hierarchical wavelets as a basis in parametric continuous-time batch estimation. The need for a continuous-time robot pose in the simultaneous localization and mapping (SLAM) problem has arisen as stateof-the-art batch SLAM algorithms attempt to handle more challenging hardware; specifically, the continuous-time framework is particularly beneficial when using high-rate sensors, multiple unsynchronized sensors, or scanning sensors, such as lidar and rolling-shutter cameras, during motion. Although the traditional discrete-time SLAM formulation can be adapted by using temporal pose interpolation, approaches using the continuous-time framework are able to generate smooth robot trajectories with less state variables. In this paper, we focus on the parametric approach using temporal basis functions to develop a finite-element representation of the continuous-time robot trajectory. While the majority of current implementations have utilized a uniformly spaced B-spline basis, we note that trajectory richness is often quite variable; in this paper, we show how a hierarchical system of wavelet basis functions can be used to increase the resolution of the solution only in the temporally local regions of the trajectory that require additional detail. We validate our approach by contrasting uniform Bsplines and wavelets in a six-dimensional pose-graph SLAM experiment, using both simulated and real data.

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“…These constraints offer the advantage of efficient processing, allowing for interactive manipulation of the free form geometry. Non-linear constraints that are commonly considered are the area [Elb01,HSB05], the volume [RSB95], second order differential constraints such as convexity [KS95,PE98], curve constraints [CW92, GL96, PGL + 02], and first and second order fairing functionals such as the bending energy of a thin plate [CG91,GC95,FRSW87,HS92,BH94,Hah98]. Satisfying non-linear constraints, however, requires intense computational effort, so that their use for interactive shape manipulation is generally very limited.…”

Section: Introductionmentioning

confidence: 99%

Detail preserving deformation of B-spline surfaces with volume constraint

Sauvage

Hahmann

Bonneau

et al. 2008

Computer Aided Geometric Design

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Geometric constraints have proved to be helpful for shape modeling. Moreover, they are efficient aids in controlling deformations and enhancing animation realism. The present paper addresses the deformation of B-spline surfaces while constraining the volume enclosed by the surface. Both uniform and non-uniform frameworks are considered. The use of level-of-detail (LoD) editing allows the preservation of fine details during coarse deformations of the shape. The key contribution of this paper is the computation of the volume with respect to the appropriate basis for LoD editing: the volume is expressed through all levels of resolution as a trilinear form and recursive formulas are developed to make the computation efficient. The volume constrained is maintained through a minimization process for which we develop closed solutions. Real-time deformations are reached thanks to sparse data structures and efficient algorithms.

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Hierarchical and variational geometric modeling with wavelets

Cited by 79 publications

References 28 publications

Spherical wavelets: Efficiently representing functions on a sphere

Spherical wavelets: Efficiently representing functions on a sphere

A hierarchical wavelet decomposition for continuous-time SLAM

Detail preserving deformation of B-spline surfaces with volume constraint

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