Neural networks for point clouds, which respect their natural invariance to permutation and rigid motion, have enjoyed recent success in modeling geometric phenomena, from molecular dynamics to recommender systems. Yet, to date, no architecture with polynomial complexity is known to be complete, that is, able to distinguish between any pair of non-isomorphic point clouds. We fill this theoretical gap by showing that point clouds can be completely determined, up to permutation and rigid motion, by applying the 3-WL graph isomorphism test to the point cloud's centralized Gram matrix. Moreover, we formulate an Euclidean variant of the 2-WL test and show that it is also sufficient to achieve completeness. We then show how our complete Euclidean WL tests can be simulated by an Euclidean graph neural network of moderate size and demonstrate their separation capability on highly symmetrical point clouds.

Abstract Let $$\textbf{p}$$ p be a configuration of n points in $$\mathbb R^d$$ R d for some n and some $$d \ge 2$$ d ≥ 2 . Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing $$\textbf{p}$$ p given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when $$\textbf{p}$$ p will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that $$\textbf{p}$$ p is generic.

Injective multiset functions have a key role in the theoretical study of machine learning on multisets and graphs. Yet, there remains a gap between the provably injective multiset functions considered in theory, which typically rely on polynomial moments, and the multiset functions used in practice, which rely on $\textit{neural moments}$ — whose injectivity on multisets has not been studied to date. In this paper, we bridge this gap by showing that moments of neural networks do define injective multiset functions, provided that an analytic non-polynomial activation is used. The number of moments required by our theory is optimal essentially up to a multiplicative factor of two. To prove this result, we state and prove a $\textit{finite witness theorem}$, which is of independent interest. As a corollary to our main theorem, we derive new approximation results for functions on multisets and measures, and new separation results for graph neural networks. We also provide two negative results: (1) moments of piecewise-linear neural networks cannot be injective multiset functions; and (2) even when moment-based multiset functions are injective, they can never be bi-Lipschitz.

We present a sufficient condition for recovering unique texture and viewpoints from unknown orthographic projections of a flat texture process. We show that four observations are sufficient in general, and we characterize the ambiguous cases. The results are applicable to shape from texture and texture-based structure from motion.

Under the conventional diffuse shading model with unknown directional lighting, the set of quadratic surface shapes that are consistent with the spatial derivatives of intensity at a single image point is a two-dimensional algebraic variety embedded in the five-dimensional space of quadratic shapes. We describe the geometry of this variety, and we introduce a concise feedforward model that computes an explicit, differentiable approximation of the variety from the intensity and its derivatives at any single image point. The result is a parallelizable processor that operates at each image point and produces a lighting-invariant descriptor of the continuous set of compatible surface shapes at the point. We describe two applications of this processor: two-shot uncalibrated photometric stereo and quadratic-surface shape from shading.

Abstract Based on mesh deformation, we present a unified mesh parametrization algorithm for both planar and spherical domains. Our approach can produce intermediate frames from the original meshes to the targets. We derive and define a novel geometric flow: ‘unit normal flow (UNF)’ and prove that if UNF converges, it will deform a surface to a constant mean curvature (CMC) surface, such as planes and spheres. Our method works by deforming meshes of disk topology to planes, and spherical meshes to spheres. Our algorithm is robust, efficient, simple to implement. To demonstrate the robustness and effectiveness of our method, we apply it to hundreds of models of varying complexities. Our experiments show that our algorithm can be a competing alternative approach to other state‐of‐the‐art mesh parametrization methods. The unit normal flow also suggests a potential direction for creating CMC surfaces.

In this paper, we describe a mesh editing system that we implemented that uses a natural stretching and bending energy defined over smooth surfaces. As such, this energy behaves uniformly under various mesh resolutions. All of the elements of our approach already exist in the literature. We hope that our discussions of these energies helps to shed light on the behaviors of these methods and provides a unified discussion of these methods.

We prove that universal second-order rigidity implies universal prestress stability and that triangulated convex polytopes in 3-space (with holes appropriately positioned) are prestress stable.

We introduce a multi-scale framework for low-level vision, where the goal is estimating physical scene values from image data—such as depth from stereo image pairs. The framework uses a dense, overlapping set of image regions at multiple scales and a “local model,” such as a slanted-plane model for stereo disparity, that is expected to be valid piecewise across the visual field. Estimation is cast as optimization over a dichotomous mixture of variables, simultaneously determining which regions are inliers with respect to the local model (binary variables) and the correct co-ordinates in the local model space for each inlying region (continuous variables). When the regions are organized into a multi-scale hierarchy, optimization can occur in an efficient and parallel architecture, where distributed computational units iteratively perform calculations and share information through sparse connections between parents and children. The framework performs well on a standard benchmark for binocular stereo, and it produces a distributional scene representation that is appropriate for combining with higher-level reasoning and other low-level cues.

We develop a framework for extracting a concise representation of the shape information available from diffuse shading in a small image patch. This produces a mid-level scene descriptor, comprised of local shape distributions that are inferred separately at every image patch across multiple scales. The framework is based on a quadratic representation of local shape that, in the absence of noise, has guarantees on recovering accurate local shape and lighting. And when noise is present, the inferred local shape distributions provide useful shape information without over-committing to any particular image explanation. These local shape distributions naturally encode the fact that some smooth diffuse regions are more informative than others, and they enable efficient and robust reconstruction of object-scale shape. Experimental results show that this approach to surface reconstruction compares well against the state-of-art on both synthetic images and captured photographs.

Given an anisotropic Voronoi diagram, we address the fundamental question of when its dual is embedded. We show that, by requiring only that the primal be orphan-free (have connected Voronoi regions), its dual is always guaranteed to be an embedded triangulation. Further, the primal diagram and its dual have properties that parallel those of ordinary Voronoi diagrams: the primal's vertices, edges, and faces are connected, and the dual triangulation has a simple, closed boundary. Additionally, if the underlying metric has bounded anisotropy (ratio of eigenvalues), the dual is guaranteed to triangulate the convex hull of the sites. These results apply to the duals of anisotropic Voronoi diagrams of any set of sites, so long as their Voronoi diagram is orphan-free. By combining this general result with existing conditions for obtaining orphan-free anisotropic Voronoi diagrams, a simple and natural condition for a set of sites to form an embedded anisotropic Delaunay triangulation follows.

Specular flow is the motion field induced on the image plane by the movement of points reflected by a curved, mirror-like surface. This flow provides information about surface shape, and when the camera and surface move as a fixed pair, shape can be recovered by solving linear differential equations along integral curves of flow. Previous analysis has shown that two distinct motions (i.e., two flow fields) are generally sufficient to guarantee a unique solution without externally-provided initial conditions. In this work, we show that we can often succeed with only one flow. The key idea is to exploit the fact that smooth surfaces induce integrability constraints on the surface normal field. We show that this induces a new differential equation that facilitates the propagation of shape information between integral curves of flow, and that combining this equation with known methods often permits the recovery of unique shape from a single specular flow given only a single seed point.

Sensor network localization is an instance of the NP-Hard graph realization problem. Thus, methods used in practice are not guaranteed to find the correct localization, even if it is uniquely determined by the input distances. In this article, we show the following: if the sensors are allowed to wiggle, giving us perturbed distance data, we can apply a novel algorithm to realize arbitrary Generically Globally Rigid graphs (GGR), or certain vertex subsets in non-GGR graphs whose relative positions are fixed (which include vertex sets of GGR subgraphs). And this strategy works in any dimension. In the language of structural rigidity theory, our approach corresponds to calculating the approximate kernel of a generic stress matrix for the given graph and distance data. To make our algorithm suitable for real-world applications, we also present: (i) various techniques for improving the robustness of the algorithm in the presence of measurement noise; (ii) an algorithm for detecting certain subsets of graph vertices whose relative positions are fixed in any generic realization of the graph and robustly localizing these subsets of vertices, (iii) a strategy for reducing the number of measurements needed by the algorithm. We provide simulation results of our algorithm.

We present a novel approach to localization of sensors in a network given a subset of noisy inter-sensor distances. The algorithm is based on “stitching” together local structures by solving an optimization problem requiring the structures to fit together in an “As-Rigid-As-Possible” manner, hence the name ARAP. The local structures consist of reference “patches” and reference triangles, both obtained from inter-sensor distances. We elaborate on the relationship between the ARAP algorithm and other state-of-the-art algorithms, and provide experimental results demonstrating that ARAP is significantly less sensitive to sparse connectivity and measurement noise. We also show how ARAP may be distributed.

When a curved mirror-like surface moves relative to its environment, it induces a motion field-or specular flow- on the image plane that observes it. This specular flow is related to the mirror's shape through a non-linear partial differential equation, and there is interest in understanding when and how this equation can be solved for surface shape. Existing analyses of this `shape from specular flow equation' have focused on closed-form solutions, and while they have yielded insight, their critical reliance on externally-provided initial conditions and/or specific motions makes them difficult to apply in practice. This paper resolves these issues. We show that a suitable reparameterization leads to a linear formulation of the shape from specular flow equation. This formulation radically simplifies the reconstruction process and allows, for example, both motion and shape to be recovered from as few as two specular flows even when no externally-provided initial conditions are available. Our analysis moves us closer to a practical method for recovering shape from specular flow that operates under arbitrary, unknown motions in unknown illumination environments and does not require additional shape information from other sources.

Sensor network localization is an instance of the NP-HARD graph realization problem. Thus, methods used in practice are not guaranteed to find the correct localization, even if it is uniquely determined by the input distances. In this paper, we show the following: if the sensors are allowed to wiggle, giving us perturbed distance data, we can apply a novel algorithm to realize arbitrary generically globally rigid (GGR) graphs (or maximal vertex subsets in non-GGR graphs whose relative positions are fixed). And this algorithm works in any dimension. In the language of structural rigidity theory, our approach corresponds to calculating the approximate kernel of a generic stress matrix for the given graph and distance data. To make our algorithm suitable for real-world application, we present techniques for improving the robustness of the algorithm under noisy measurements, and a strategy for reducing the required number of measurements.

Abstract We present a novel approach to parameterize a mesh with disk topology to the plane in a shape‐preserving manner. Our key contribution is a local/global algorithm, which combines a local mapping of each 3D triangle to the plane, using transformations taken from a restricted set, with a global “stitch” operation of all triangles, involving a sparse linear system. The local transformations can be taken from a variety of families, e.g. similarities or rotations, generating different types of parameterizations. In the first case, the parameterization tries to force each 2D triangle to be an as‐similar‐as‐possible version of its 3D counterpart. This is shown to yield results identical to those of the LSCM algorithm. In the second case, the parameterization tries to force each 2D triangle to be an as‐rigid‐as‐possible version of its 3D counterpart. This approach preserves shape as much as possible. It is simple, effective, and fast, due to pre‐factoring of the linear system involved in the global phase. Experimental results show that our approach provides almost isometric parameterizations and obtains more shape‐preserving results than other state‐of‐the‐art approaches. We present also a more general “hybrid” parameterization model which provides a continuous spectrum of possibilities, controlled by a single parameter. The two cases described above lie at the two ends of the spectrum. We generalize our local/global algorithm to compute these parameterizations. The local phase may also be accelerated by parallelizing the independent computations per triangle.

Motivated by perceptual principles, we derive a new color space in which the associated metric approximates perceived distances and color displacements capture relationships that are robust to spectral changes in illumination. The resulting color space can be used with existing image processing algorithms with little or no change to the methods.

Color constancy is almost exclusively modeled with diagonal transforms. However, the choice of basis under which diagonal transforms are taken is traditionally ad hoc. Attempts to remedy the situation have been hindered by the fact that no joint characterization of the conditions for {sensors, illuminants, reflectances} to support diagonal color constancy has previously been achieved. In this work, we observe that the von Kries compatibility conditions are impositions only on the sensor measurements, not the physical spectra. This allows us to formulate the von Kries compatibility conditions succinctly as rank constraints on an order 3 measurement tensor. Given this, we propose an algorithm that computes a (locally) optimal choice of color basis for diagonal color constancy and compare the results against other proposed choices.

Linear parameterization of 3D meshes with disk topology is usually performed using the method of barycentrie coordinates pioneered by Tutte and Floater. This imposes a convex boundary on the parameterization, which can significantly distort the result. Recently, several methods showed how to relax the convex boundary requirement while still using the barycentric coordinates formulation. However, this relaxation can result in other artifacts in the parameterization. In this paper we explore these methods and give a general recipe for "natural" boundary conditions for the family of so-called "three point" barycentric coordinates. We discuss the shortcomings of these methods and show how they may be rectified using an iterative scheme or a carefully crafted "virtual boundary". Finally, we show how these methods adapt easily to solve the problem of constrained parameterization.

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.

We consider quantum versions of two well-studied models of learning Boolean functions: Angluin's model of exact learning from membership queries and Valiant's probably approximately correct (PAC) model of learning from random examples. For each of these two learning models we establish a polynomial relationship between the number of quantum or classical queries required for learning. These results contrast known results that show that testing black-box functions for various properties, as opposed to learning, can require exponentially more classical queries than quantum queries. We also show that, under a widely held computational hardness assumption (the intractability of factoring Blum integers), there is a class of Boolean functions which is polynomial-time learnable in the quantum version but not the classical version of each learning model. For the model of exact learning from membership queries, we establish a stronger separation by showing that if any one-way function exists, then there is a class of functions which is polynomial-time learnable in the quantum setting but not in the classical setting. Thus, while quantum and classical learning are equally powerful from an information theory perspective, the models are different when viewed from a computational complexity perspective.

Summary form only given. A novel method for lossy compression of the two-dimensional curves is introduced based on the arc-length parameterization. This method has a number of advantages: it is progressive, converges uniformly, and requires the number of the bits proportional to the total arc-length of the curve. The method is applied to the compression of the handwritten letters and scanlines of the natural images.

In this paper we present a new variant of the light field representation that supports improved image reconstruction by accommodating sparse correspondence information. This places our representation somewhere between a pure, two-plane parameterized, light field and a lumigraph representation, with its continuous geometric proxy. Our approach factorises the rays of a light field into one of two separate classes. All rays consistent with a given correspondence are implicitly represented using a new auxiliary data structure, which we call a surface camera (or scam). The remaining rays of the light field are represented using a standard two-plane parameterized light field. We present an efficient rendering algorithm that combines ray samples from scams with those from the light field. The resulting image reconstructions are noticeably improved over that of a pure light field.

Surface geometry is often modeled with irregular triangle meshes. The process of remeshing refers to approximating such geometry using a mesh with (semi)-regular connectivity, which has advantages for many graphics applications. However, current techniques for remeshing arbitrary surfaces create only semi-regular meshes. The original mesh is typically decomposed into a set of disk-like charts, onto which the geometry is parametrized and sampled. In this paper, we propose to remesh an arbitrary surface onto a completely regular structure we call a geometry image. It captures geometry as a simple 2D array of quantized points. Surface signals like normals and colors are stored in similar 2D arrays using the same implicit surface parametrization --- texture coordinates are absent. To create a geometry image, we cut an arbitrary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images can be encoded using traditional image compression algorithms, such as wavelet-based coders.

Motivated by work on quantum black-box query complexity, we consider quantum versions of two well-studied models of learning Boolean functions: Angluin's (1988) model of exact learning from membership queries and Valiant's (1984) Probably Approximately Correct (PAC) model of learning from random examples. For each of these two learning models we establish a polynomial relationship between the number of quantum versus classical queries required for learning. Our results provide an interesting contrast to known results which show that testing black-box functions for various properties can require exponentially more classical queries than quantum queries. We also show that under a widely held computational hardness assumption there is a class of Boolean functions which is polynomial-time learnable in the quantum version but not the classical version of each learning model; thus while quantum and classical learning are equally powerful from an information theory perspective, they are different when viewed from a computational complexity perspective.

Aliasing is an important problem when rendering triangle meshes. Efficient antialiasing techniques such as mipmapping greatly improve the filtering of textures defined over a mesh. A major component of the remaining aliasing occurs along discontinuity edges such as silhouettes, creases, and material boundaries. Framebuffer supersampling is a simple remedy, but 2x2 supersampling leaves behind significant temporal artifacts, while greater supersampling demands even more fill-rate and memory. We present an alternative that focuses effort on discontinuity edges by overdrawing such edges as antialiased lines. Although the idea is simple, several subtleties arise. Visible silhouette edges must be detected efficiently. Discontinuity edges need consistent orientations. They must be blended as they approach the silhouette to avoid popping. Unfortunately, edge blending results in blurriness. Our technique balances these two competing objectives of temporal smoothness and spatial sharpness. Finally, the best results are obtained when discontinuity edges are sorted by depth. Our approach proves surprisingly effective at reducing temporal artifacts commonly referred to as "crawling jaggies," with little added cost.

We describe an image based rendering approach that generalizes many current image based rendering algorithms, including light field rendering and view-dependent texture mapping. In particular, it allows for lumigraph-style rendering from a set of input cameras in arbitrary configurations (i.e., not restricted to a plane or to any specific manifold). In the case of regular and planar input camera positions, our algorithm reduces to a typical lumigraph approach. When presented with fewer cameras and good approximate geometry, our algorithm behaves like view-dependent texture mapping. The algorithm achieves this flexibility because it is designed to meet a set of specific goals that we describe. We demonstrate this flexibility with a variety of examples.

Cellular patterns are all around us, in masonry, tiling, shingles, and many other materials. Such patterns, especially in architectural settings, are influenced by geometric features of the underlying shape. Bricks turn corners, stones frame windows and doorways, and patterns on disconnected portions of a building align to achieve a particular aesthetic goal. We present a strategy for feature-based cellular texturing, where the resulting texture is derived from both patterns of cells and the geometry to which they are applied. As part of this strategy, we perform texturing operations on features in a well-defined order that simplifies the interdependence between cells of adjacent patterns. Occupancy maps are used to indicate which regions of a feature are already occupied by cells of its neighbors, and which regions remain to be textured. We also introduce the notion of a pattern generator — the cellular texturing analogy of a shader used in local illumination — and show how several can be used together to build complex textures. We present results obtained with an implementation of this strategy and discuss details of some example pattern generators.

Given an arbitrary mesh, we present a method to construct a progressive mesh (PM) such that all meshes in the PM sequence share a common texture parametrization. Our method considers two important goals simultaneously. It minimizes texture stretch (small texture distances mapped onto large surface distances) to balance sampling rates over all locations and directions on the surface. It also minimizes texture deviation (“slippage” error based on parametric correspondence) to obtain accurate textured mesh approximations. The method begins by partitioning the mesh into charts using planarity and compactness heuristics. It creates a stretch-minimizing parametrization within each chart, and resizes the charts based on the resulting stretch. Next, it simplifies the mesh while respecting the chart boundaries. The parametrization is re-optimized to reduce both stretch and deviation over the whole PM sequence. Finally, the charts are packed into a texture atlas. We demonstrate using such atlases to sample color and normal maps over several models.

This research further develops the light field and lumigraph image-based rendering methods and extends their utility. We present alternate parameterizations that permit 1) interactive rendering of moderately sampled light fields of scenes with significant, unknown depth variation and 2) low-cost, passive autostereoscopic viewing. Using a dynamic reparameterization, these techniques can be used to interactively render photographic effects such as variable focus and depth-of-field within a light field. The dynamic parameterization is independent of scene geometry and does not require actual or approximate geometry of the scene. We explore the frequency domain and ray-space aspects of dynamic reparameterization, and present an interactive rendering technique that takes advantage of today's commodity rendering hardware.

In this paper, we describe an efficient image-based approach to computing and shading visual hulls from silhouette image data. Our algorithm takes advantage of epipolar geometry and incremental computation to achieve a constant rendering cost per rendered pixel. It does not suffer from the computation complexity, limited resolution, or quantization artifacts of previous volumetric approaches. We demonstrate the use of this algorithm in a real-time virtualized reality application running off a small number of video streams. Keywords: Computer Vision, Image-Based Rendering, Constructive Solid Geometry, Misc. Rendering Algorithms. 1 Introduction Visualizing and navigating within virtual environments composed of both real and synthetic objects has been a long-standing goal of computer graphics. The term "Virtualized Reality^TM", as popularized by Kanade [23], describes a setting where a real-world scene is "captured" by a collection of cameras and then viewed through a virtual camera, a...

Approximating detailed with coarse, texture-mapped meshes results in polygonal silhouettes. To eliminate this artifact, we introduce silhouette clipping, a framework for efficiently clipping the rendering of coarse geometry to the exact silhouette of the original model. The coarse mesh is obtained using progressive hulls, a novel representation with the nesting property required for proper clipping. We describe an improved technique for constructing texture and normal maps over this coarse mesh. Given a perspective view, silhouettes are efficiently extracted from the original mesh using a precomputed search tree. Within the tree, hierarchical culling is achieved using pairs of anchored cones. The extracted silhouette edges are used to set the hardware stencil buffer and alpha buffer, which in turn clip and antialias the rendered coarse geometry. Results demonstrate that silhouette clipping can produce renderings of similar quality to high-resolution meshes in less rendering time.

The distribution of digital video content over computer networks has become commonplace. Unfortunately, most digital video encoding standards do not degrade gracefully in the face of packet losses, which often occur in a bursty fashion. We propose an new video encoding system that scales well with respect to the network's performance and degrades gracefully under packet loss. Our encoder sends packets that consist of a small random subset of pixels distributed throughout a video frame. The receiver places samples in their proper location (through a previously agreed ordering), and applies a reconstruction algorithm on the received samples to produce an image. Each of the packets is independent, and does not depend on the successful transmission of any other packets. Additionally, each packet contains information that is distributed over the entire image. We also apply spatial and temporal optimization to achieve better compression.

In this paper we present a set of efficient image based rendering methods capable of rendering multiple frames per second on a PC. The first method warps Sprites with Depth representing smooth surfaces without the gaps found in other techniques. A second method for more general scenes performs warping from an intermediate representation called a Layered Depth Image (LDI). An LDI is a view of the scene from a single input camera view, but with multiple pixels along each line of sight. The size of the representation grows only linearly with the observed depth complexity in the scene. Moreover, because the LDI data are represented in a single image coordinate system, McMillan's warp ordering algorithm can be successfully adapted. As a result, pixels are drawn in the output image in back-to-front order. No z-buffer is required, so alphacompositing can be done efficiently without depth sorting. This makes splatting an efficient solution to the resampling problem. 1 Introduction Image base...

It was illustrated in 1996 that the light leaving the convex hull of an object (or entering a convex region of empty space) Cm be my characterized by a 4D function over the space" of rays crowing a surface surrounding the object (or surrounding the empty apace) [10, 8].Methods to repre sent this function and quickly render individual images from this representation given an arbitrary cameras were also described.This paper extends the work outlined by Gortler et al [8] by demonstrating a taxonomy of methods to accelerate the rendering process by trading off quality for time.Given the speciiic limitation of a given hardware configuration, we discuss methods to tailor a critical time rendering strategy using these methods.

This paper discusses a new method for capturing the complete appearance of bothsynthetic and real world objects and scenes,representing this information, and then using this representation to render images of the object from new camera positions.Unlike the shape capture process traditionally used in computer vision and the rendering process traditionally used in computer graphics, our approach does not rely on geometric representations.Instead we sample and reconstruct a 4D function.which we call a Lumigraph.The Lumigraph is a subset o f the complete plenoptic fu nction that describes the flow of light at all positions in all directions.With the Lumigraph.new images of the object can be generated very quick]y,independentof the geometric or illumination complexity of the scene or object.The paper discusses a complete working system including the capture of sainples.the construction of the Lumigraph, and thesubsequent rendering of images from this new representation.even i f given accurate geometric models.Quicktime V R [6] was one ofthe first systems to suggest that the traditional modeling/rendering process can beskipped.Instead.

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.

Specifying the motion of an animated linked figure such that it achieves given tasks (e.g., throwing a ball into a basket) and performs the tasks in a realistic fashion (e.g., gracefully, and following physical laws such as gravity) has been an elusive goal for computer animators. The spacetime constraints paradigm has been shown to be a valuable approach to this problem, but it suffers from computational complexity growth as creatures and tasks approach those one would like to animate. The complexity is shown to be, in part, due to the choice of finite basis with which to represent the trajectories of the generalized degrees of freedom. This paper describes new features to the spacetime constraints paradigm to address this problem.

Abstract One important goal of image synthesis research is to accelerate the process of obtaining realistic images using the radiosity method. Two important concepts recently introduced are the general framework of projection methods and the hierarchical radiosity method. Wavelet theory, which explores the space of hierarchical basis functions, offers an elegant framework that unites these two concepts and allows us to more formally understand the hierarchical radiosity method. Wavelet expansions of the radiosity kernel have negligible entries in regions where high frequency/fine detail information is not needed. A sparse system remains if these entries are ignored. This is similar to applying a lossy compression scheme to the form factor matrix. The sparseness of the system allows for asymptotically faster radiosity algorithms by limiting the number of matrix terms that need to be computed. The application of these methods to 3D environments is described in 4 . Due to space limitations in that paper many of the subtleties of the construction could not be explored there. In this paper we discuss some of the mathematical details of wavelet projections and investigate the application of these methods to the radiosity kernel of a flatland environment, where many aspect are easier to visualize.

Radiosity methods have been shown to be an effective means to solve the global illumination problem in Lambertian diffuse environments. These methods approximate the radiosity integral equation by projecting the unknown radiosity function into a set of basis functions with limited support resulting in a set of n linear equations where n is the number of discrete elements in the scene. Classical radiosity methods required the evaluation of n2 interaction coefficients. Efforts to reduce the number of required coefficients without compromising error bounds have focused on raising the order of the basis functions, meshing, accounting for discontinuities, and on developing hierarchical approaches, which have been shown to reduce the required interactions to O(n). In this paper we show that the hierarchical radiosity formulation is an instance of a more general set of methods based on wavelet theory. This general framework offers a unified view of both higher order element approaches to radiosity and the hierarchical radiosity methods. After a discussion of the relevant theory, we discuss a new set of linear time hierarchical algorithms based on wavelets such as the multiwavelet family and a flatlet basis which we introduce. Initial results of experimentation with these basis sets are demonstrated and discussed.