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The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . 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 ..."
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Cited by 541 (16 self)
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. 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, inplace 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...
SPIRAL: Code Generation for DSP Transforms
 PROCEEDINGS OF THE IEEE SPECIAL ISSUE ON PROGRAM GENERATION, OPTIMIZATION, AND ADAPTATION
"... 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 performancecritical domain of linear digital signal proces ..."
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Cited by 212 (39 self)
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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 performancecritical 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 domainspecific mathematical structure of transform algorithms to implement a feedbackdriven 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.
Sparse Geometric Image Representations with Bandelets
, 2004
"... 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 ..."
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Cited by 196 (4 self)
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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.
An overview of the JPEG2000 still image compression standard
 Signal Processing: Image Communication
, 2002
"... 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 154447ITUT Recommendation T.800) that is being issued in six pa ..."
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Cited by 111 (0 self)
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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 154447ITUT 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
Nonlinear wavelet transforms for image coding via lifting
, 2003
"... 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 al ..."
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Cited by 105 (3 self)
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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 realworld images demonstrate the promise of our techniques.
Fast Multiplierless Approximations of the DCT with the Lifting Scheme
 IEEE Trans. on Signal Processing
, 2001
"... 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 rotationbased ..."
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Cited by 72 (11 self)
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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 rotationbased 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 liftingbased 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 WalshHadamard transform and the DCT. The corresponding 2D binDCT allows a 16bit implementation, enables lossless compression, and maintai...
Lossless image compression using integer to integer wavelet transforms
 In Proc. of IEEE International Conference on Image Processing
, 1997
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Hybrid representations for audiophonic signal encoding
 Signal Processing
, 2002
"... Abstract. We discuss in this paper a new approach for signal models in the context of audio signal encoding. The method is based upon hybrid models featuring transient, tonal and stochastic components in the signal. Contrary to several existing approaches, our method does not rely on any prior segm ..."
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Cited by 64 (19 self)
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Abstract. We discuss in this paper a new approach for signal models in the context of audio signal encoding. The method is based upon hybrid models featuring transient, tonal and stochastic components in the signal. Contrary to several existing approaches, our method does not rely on any prior segmentation of the signal. The three components are estimated and encoded using a strategy very much in the spirit of transform coding. While the details of the method described here are taylored to audio signals, the general strategy should also apply to other types of signals exhibiting significantly different features, for example images. 1.
Multiresolution signal decomposition schemes. Part 1: Linear and morphological pyramids
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2000
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