We investigate central issues such as invertibility, stability, synchronization, and frequency characteristics for nonlinear wavelet transforms built using the lifting framework. The nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework. We also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels. Preliminary compression results for model and real-world images demonstrate the promise of our techniques.

A hybrid mesh is a multiresolution surface representation that combines advantages from regular and irregular meshes. Irregular operations allow a hybrid mesh to change topology throughout the hierarchy and approximate detailed features at multiple scales. A preponderance of regular refinements allows for efficient data-structures and processing algorithms. We provide a user driven procedure for creating a hybrid mesh from scanned geometry and present a progressive hybrid mesh compression algorithm.

A basic element of Digital Geometry Processing algorithms is the establishment of a smooth parameterization for a given model. In this paper we propose an algorithm which establishes parameterizations for a set of models. The parameterizations are called consistent because they share the same base domain and respect features. They give immediate correspondences between models and allow remeshes with the same connectivity. Such remeshes form the basis for a large class of algorithms, including principal component analysis, wavelet transforms, detail and texture transfer between models, and n-way shape blending. We demonstrate the versatility of our algorithm with a number of examples. 1

We present an algorithm based on lattice reduction for the fast decoding of diagonal differential modulation across multiple antenna. While the complexity of the maximum-likelihood (ML) algorithm is exponential both in the number of antenna and the rate, the complexity of our approximate lattice algorithm is polynomial in the number of antennas and the rate. We show that the error performance of our lattice algorithm is very close to the ML algorithm.

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels.

Normal meshes are new fundamental surface descriptions inspired by differential geometry. A normal mesh is a multiresolution mesh where each level can be written as a normal offset from a coarser version. Hence the mesh can be stored with a single float per vertex. We present an algorithm to approximate any surface arbitrarily closely with a normal semi-regular mesh. Normal meshes can be useful in numerous applications such as compression, filtering, rendering, texturing, and modeling.

We propose a new progressive compression scheme for arbitrary topology, highly detailed and densely sampled meshes arising from geometry scanning. We observe that meshes consist of three distinct components: geometry, parameter, and connectivity information. The latter two do not contribute to the reduction of error in a compression setting. Using semi-regular meshes, parameter and connectivity information can be virtually eliminated. Coupled with semi-regular wavelet transforms, zerotree coding, and subdivision based reconstruction we see improvements in error by a factor four (12dB) compared to other progressive coding schemes.

We construct signal constellations for differential transmission with multiple basestation antennas. The signals are derived using the theory of fixed-point-free groups and are especially suitable for mobile cellular applications because they do not require the handset to have more than one antenna or to know the time-varying propagation environment. Yet we achieve full transmitter diversity and excellent performance gains over a single-antenna system.

We present a framework for differential modulation with multiple antennas across a continuously fading channel, where neither the transmitter nor the receiver knows the fading coefficients. The framework can be seen as a natural extension of standard differential phase-shift keying commonly used in single-antenna unknown-channel systems. We show how our differential framework links the unknown-channel system with a known-channel system, and we develop performance design criteria. As a special ease, we introduce a class of diagonal signals where only one antenna is active at any time, and demonstrate how these signals may be used to achieve full transmitter diversity and low probability of error.

We build discrete-time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The associated scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integer-to-integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. While we concentrate on the discrete-time case, some discussion of convergence and stability issues together with examples is given.

We propose a systematic method for creating constellations of unitary space-time signals for multiple-antenna communication links. Unitary space-time signals, which are orthonormal in time across the antennas, have been shown to be well-tailored to a Rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients. The signals can achieve low probability of error by exploiting multiple-antenna diversity. Because the fading coefficients are not known, the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum-Euclidean-distance norm. Our construction begins with the first signal in the constellation-an oblong complex-valued matrix whose columns are orthonormal-and systematically produces the remaining signals by successively rotating this signal in a high-dimensional complex space. This construction easily produces large constellations of high-dimensional signals. We demonstrate its efficacy through examples involving one, two, and three transmitter antennas.

We generalize basic signal processing tools such as downsampling, upsampling, and filters to irregular connectivity triangle meshes.This is accomplished through the design of a non-uniform relaxation procedure whose weights depend on the geometry and we show its superiority over existing schemes whose weights depend only on connectivity.This is combined with known mesh simplification methods to build subdivision and pyramid algorithms.We demonstrate the power of these algorithms through a number of application examples including smoothing, enhancement, editing, and texture mapping.

We present a new method for user controlled morphing of two \nhomeomorphic triangle meshes of arbitrary topology. In particular we focus on the problem of establishing a correspondence map between source and target meshes. Our method employs the MAPS algorithm to parameterize both meshes over simple base domains and an additional harmonic map bringing the latter into correspondence. \nTo control the mapping the user specifies any number of \nfeature pairs, which control the parameterizations produced by the MAPS algorithm. Additional controls are provided through a direct manipulation interface allowing the user to tune the mapping between the base domains. We give several examples of æsthetically pleasing morphs which can be created in this manner with little user input. Additionally we demonstrate examples of temporal \nand spatial control over the morph.

Summary form only given. Image compression relies on efficient representations of images, and within smooth image regions, the wavelet transform provides such a representation. However, near edges, wavelet coefficients decay slowly and are expensive to code. We focus on improving the transform by incorporating adaptivity. Construction of nonlinear filter banks has been discussed, but the question of how to utilize the nonlinearities remained. We answer this question by describing our transform via lifting. Lifting provides a spatial domain framework for the wavelet transform. In the lifting formalism, wavelet coefficients are seen as prediction residuals from a linear prediction operation. Wavelet coefficients are large near edges because the linear predictors are built to interpolate low order polynomials. Our goal is to avoid this problem by adapting the predictor based on local image properties. In smooth regions of the image, we use high order polynomial predictors. We adaptively reduce the prediction order to avoid attempting to predict values across discontinuities.

An irregular connectivity mesh representative of a surface having an arbitrary topology is processed to generate a parameterization which maps points in a coarse base domain to points in the mesh. An illustrative embodiment uses a multi-level mesh simplification process in conjunction with conformal mapping to efficiently construct a parameterization of a mesh comprising a large number of triangles over a base domain comprising a smaller number of triangles. The parameterization in this embodiment corresponds to the inverse of function mapping each point in the original mesh to one of the triangles of the base domain, such that the original mesh can be reconstructed from the base domain and the parameterization. The mapping function is generated as a combination of a number of sub-functions, each of which relates data points in a mesh of one level in a simplification hierarchy to data points in a mesh of the next coarser level of the simplification hierarchy. The parameterization can also be used to construct, from the original irregular connectivity mesh, an adaptive remesh having a regular connectivity which is substantially easier to process than the original mesh.

Invertible wavelet transforms that map integers to integers are important for lossless representations. We present an approach to build integer to integer wavelet transforms based upon the idea of factoring wavelet transforms into lifting steps. This allows the construction of an integer version of every wavelet transform. We demonstrate the use of these transforms in lossless image compression.

We describe a multiresolution representation for meshes based on subdivision, which is a natural extension of the existing patch-based surface representations. Combining subdivision and the smoothing algorithms of Taubin [26] allows us to construct a set of algorithms for interactive multiresolution editing of complex hierarchical meshes of arbitrary topology. The simplicity of the underlying algorithms for refinement and coarsification enables us to make them local and adaptive, thereby considerably improving their efficiency. We have built a scalable interactive multiresolution editing system based on such algorithms.

Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating schemes since they match the original data exactly, and are crucial for fast multiresolution and wavelet techniques. Dyn, Gregory, and Levin introduced the Butterfly scheme [17], which yields C¹ surfaces in the topologically regular setting. Unfortunately it exhibits undesirable artifacts in the case of an irregular topology. We examine these failures and derive an improved scheme, which retains the simplicity of the Butterfly scheme, is interpolating, and results in smoother surfaces.

The author looks ahead to see what the future can bring to wavelet research. He tries to find a common denominator for "wavelets" and identifies promising research directions and challenging problems.

The periodic Bar-David (1974) sampling theorem provides an implicit representation of band limited signals using their crossings with a cosine function to form a multiplicative representation involving a Riesz product. The cosine crossings form a unique and stable representation of the signal. We incorporate the wavelet transform into the cosine crossing representation and show that they may be used to compactly represent overcomplete signal expansions.

Article Spherical wavelets: efficiently representing functions on the sphere Share on Authors: Peter Schröder Department of Mathematics, University of South Carolina Department of Mathematics, University of South CarolinaView Profile , Wim Sweldens Department of Mathematics, Department of Computer Science, Katholieke Universiteit Leuven, Belgium Department of Mathematics, Department of Computer Science, Katholieke Universiteit Leuven, BelgiumView Profile Authors Info & Claims SIGGRAPH '95: Proceedings of the 22nd annual conference on Computer graphics and interactive techniquesSeptember 1995 Pages 161–172https://doi.org/10.1145/218380.218439Online:15 September 1995Publication History 338citation2,133DownloadsMetricsTotal Citations338Total Downloads2,133Last 12 Months44Last 6 weeks8 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access