. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...

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.

This paper presents a two pass global illumination method based on the concept of photon maps. It represents a significant improvement of a previously described approach both with respect to speed, accuracy and versatility. In the first pass two photon maps are created by emitting packets of energy (photons) from the light sources and storing these as they hit surfaces within the scene. We use one high resolution caustics photon map to render caustics that are visualized directly and one low resolution photon map that is used during the rendering step. The scene is rendered using a distribution ray tracing algorithm optimized by using the information in the photon maps. Shadow photons are used to render shadows more efficiently and the directional information in the photon map is used to generate optimized sampling directions and to limit the recursion in the distribution ray tracer by providing an estimate of the radiance on all surfaces with the exception of specular...

We are presenting a new method for adaptive surface meshing and triangulation which controls the local level-of-detail of the surface approximation by local spectral estimates. These estimates are figured out by a wavelet representation of the surface data. The basic idea is to decompose the initial data set by means of an orthogonal or semi-orthogonal tensor product wavelet transform (WT) and to analyze the resulting coefficients. In surface regions, where the partial energy of the resulting coefficients is low, the polygonial approximation of the surface can be performed with larger triangles without loosing too much fine grain details. However, since the localization of the WT is bound by the Heisenberg principle the meshing method has to be controlled by the detail signals rather than directly by the coefficients. The dyadic scaling of the WT stimulated us to build an hierachical meshing algorithm which transforms the initially regular data grid into a quadtree representation by...

We present a new clustering algorithm for global illumination in complex environments. The new algorithm extends previous work on clustering for radiosity to allow for nondiffuse (glossy) reflectors. We represent clusters as points with directional distributions of outgoing and incoming radiance and importance, and we derive an error bound for transfers between these clusters. The algorithm groups input surfaces into a hierarchy of clusters, and then permits clusters to interact only if the error bound is below an acceptable tolerance. We show that the algorithm is asymptotically more efficient than previous clustering algorithms even when restricted to ideally diffuse environments. Finally, we demonstrate the performance of our method on two complex glossy environments.

In this paper we identify sources of error in global illumination algorithms and derive bounds for each distinct category. Errors arise from three sources: inaccuracies in the boundary data, discretization, and computation. Boundary data consist of surface geometry, reflectance functions, and emission functions, all of which may be perturbed by errors in measurement or simulation, or by simplifications made for computational efficiency. Discretization error is introduced by replacing the continuous radiative transfer equation with a finite-dimensional linear system, usually by means of boundaryelements and a corresponding projection method. Finally, computational errors perturb the finite-dimensional linear system through imprecise form factors, inner products, visibility, etc., as well as by halting iterative solvers after a finite number of steps. Using the error taxonomy introduced in the paper we examine existing global illumination algorithms and suggest new avenues of research. ...

This paper discusses how wavelet techniques may be applied to a variety of geometric modeling tools. In particular, wavelet decompositions are shown to be useful for hierarchical control point or least squares editing. In addition, direct curve and surface manipulation methods using an underlying geometric variational principle can be solved more efficiently by using a wavelet basis. Because the wavelet basis is hierarchical, iterative solution methods converge rapidly. Also, since the wavelet coefficients indicate the degree of detail in the solution, the number of basis functions needed to express the variational minimum can be reduced, avoiding unnecessary computation. An implementation of a curve and surface modeler based on these ideas is discussed and experimental results are reported.

We present a novel framework for dynamic simulation of elastically deformable solids. Our approach combines classical finite element methodology with subdivision wavelets to meet the needs of computer graphics applications. We represent deformations using a wavelet basis constructed from volumetric Catmull-Clark subdivision. Catmull-Clark subdivision solids allow the domain of deformation to be tailored to objects of arbitrary topology. The domain of deformation can correspond to the interior of a subdivision surface or can enclose an arbitrary surface mesh. Within the wavelet framework we develop the equations of motion for elastic deformations in the presence of external forces and constraints. We solve the resulting differential equations using an implicit method, which lends stability. Our framework allows trade-off between speed and accuracy. For interactive applications, we accelerate the simulation by adaptively refining the wavelet basis while avoiding visual "popping" artifacts. Off-line simulations can employ a fine basis for higher accuracy at the cost of more computation time. By exploiting the properties of smooth subdivision we can compute less expensive solutions using a trilinear basis yet produce a smooth result that meets the constraints.

We present a signal-processing framework for light transport. We study the frequency content of radiance and how it is altered by phenomena such as shading, occlusion, and transport. This extends previous work that considered either spatial or angular dimensions, and it offers a comprehensive treatment of both space and angle. We show that occlusion, a multiplication in the primal, amounts in the Fourier domain to a convolution by the spectrum of the blocker. Propagation corresponds to a shear in the space-angle frequency domain, while reflection on curved objects performs a different shear along the angular frequency axis. As shown by previous work, reflection is a convolution in the primal and therefore a multiplication in the Fourier domain. Our work shows how the spatial components of lighting are affected by this angular convolution. Our framework predicts the characteristics of interactions such as caustics and the disappearance of the shadows of small features. Predictions on the frequency content can then be used to control sampling rates for rendering. Other potential applications include precomputed radiance transfer and inverse rendering.

We describe a system that computes radiosity solutions for polygonal environments much larger than can be stored in main memory. The solution is stored in and retrieved from a database as the computation proceeds. Our system is based on two ideas: the use of visibility oracles to find source and blocker surfaces potentially visible to a receiving surface; and the use of hierarchical techniques to represent interactions between large surfaces efficiently, and to represent the computed radiosity solution compactly. Visibility information allows the environment to be partitioned into subsets, each containing all the information necessary to transfer light to a cluster of receiving polygons. Since the largest subset needed for any particular cluster is much smaller than the total size of the environment, these subset computations can be performed in much less memory than can classical or hierarchical radiosity. The computation is then ordered for further efficiency. Careful ordering of ene...