. 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...

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This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image grey levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms, so that the resulting bandelet basis produces a minimum distortion. Comparisons are made with wavelet image compression and noise removal algorithms.

Abstract — Fast changing, increasingly complex, and diverse computing platforms pose central problems in scientific computing: How to achieve, with reasonable effort, portable optimal performance? We present SPIRAL that considers this problem for the performance-critical domain of linear digital signal processing (DSP) transforms. For a specified transform, SPIRAL automatically generates high performance code that is tuned to the given platform. SPIRAL formulates the tuning as an optimization problem, and exploits the domain-specific mathematical structure of transform algorithms to implement a feedback-driven optimizer. Similar to a human expert, for a specified transform, SPIRAL “intelligently ” generates and explores algorithmic and implementation choices to find the best match to the computer’s microarchitecture. The “intelligence” is provided by search and learning techniques that exploit the structure of the algorithm and implementation space to guide the exploration and optimization. SPIRAL generates high performance code for a broad set of DSP transforms including the discrete Fourier transform, other trigonometric transforms, filter transforms, and discrete wavelet transforms. Experimental results show that the code generated by SPIRAL competes with, and sometimes outperforms, the best available human tuned transform library code. Index Terms — library generation, code optimization, adaptation, automatic performance tuning, high performance computing, linear signal transform, discrete Fourier transform, FFT, discrete cosine transform, wavelet, filter, search, learning, genetic and evolutionary algorithm, Markov decision process I.

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.

In 1996, the JPEGcommittee began to investigate possibilities for a new still image compression standard to serve current and future applications. This initiative, which was named JPEG2000, has resulted in a comprehensive standard (ISO 154447ITU-T Recommendation T.800) that is being issued in six parts. Part 1, in the same vein as the JPEG baseline system, is aimed at minimal complexity and maximal interchange and was issued as an International Standard at the end of 2000. Parts 2–6 define extensions to both the compression technology and the file format and are currently in various stages of development. In this paper, a technical description of Part 1 of the JPEG2000 standard is provided, and the rationale behind the selected technologies is explained. Although the JPEG2000 standard only specifies the decoder and the codesteam syntax, the discussion will span both encoder and decoder issues to provide a better

This paper presents an efficient video coding algorithm: Three-dimensional embedded subband coding with optimized truncation (3-D ESCOT), in which coefficients in different subbands are independently coded using fractional bit-plane coding and candidate truncation points are formed at the end of each fractional bit-plane. A rate-distortion optimized truncation scheme is used to multiplex all subband bitstreams together into a layered one. A novel motion threading technique is proposed to form threads along the motion trajectories in a scene. For efficient coding of motion threads, memory-constrained temporal wavelet transforms are applied along entire motion threads. Block-based motion threading is implemented in conjunction with 3-D ESCOT in a real video coder. Extension of 3-D ESCOT to object-based coding is also addressed. Experiments demonstrate that 3-D ESCOT outperforms MPEG-4 for most test sequences at the same bit rate. # 2001 Academic Press 1.

Invertible wavelet transforms that map integers to integers are important for lossless representations. In this paper, 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. 1 INTRODUCTION High-fidelity images generated from studio-quality video, medical images, seismic data, satellite images, and images scanned from manuscripts for preservation purposes typically demand lossless encoding. Yet, the huge datasize prohibits fast distribution of data. There is thus a need to seek encoding methods that can support storage and transmission of images at a spectrum of resolutions and encoding fidelities, from lossy to lossless, for progressive delivery and for different end-users' needs. In recent years, wavelet transforms have been successfully used for ...

In this paper, we present the design, implementation and application of several families of fast multiplierless approximations of the discrete cosine transform (DCT) with the lifting scheme, named the binDCT. These binDCT families are derived from Chen's and Loeffler's plane rotation-based factorizations of the DCT matrix, respectively, and the design approach can also be applied to DCT of arbitrary size. Two design approaches are presented. In the first method, an optimization program is de ned, and the multiplierless transform is obtained by approximating its solution with dyadic values. In the second method, a general lifting-based scaled DCT structure is obtained, and the analytical values of all lifting parameters are derived, enabling dyadic approximations with different accuracies. Therefore the binDCT can be tuned to cover the gap between the Walsh-Hadamard transform and the DCT. The corresponding 2-D binDCT allows a 16-bit implementation, enables lossless compression, and maintai...

. We build 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 resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integerto -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. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...