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Wavelet Families Of Increasing Order In Arbitrary Dimensions
, 1997
"... . 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 ..."
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Cited by 61 (0 self)
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. 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, inplace 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...
SecondGeneration Wavelet Collocation Method for the Solution of Partial Differential Equations
 Journal of Computational Physics
, 2000
"... this paper we demonstrate the algorithm for one particular choice of secondgeneration wavelets, namely lifted interpolating wavelets on an interval with uniform (regular) sampling. The main advantage of using secondgeneration wavelets is that wavelets can be custom designed for complex domains ..."
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Cited by 61 (17 self)
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this paper we demonstrate the algorithm for one particular choice of secondgeneration wavelets, namely lifted interpolating wavelets on an interval with uniform (regular) sampling. The main advantage of using secondgeneration wavelets is that wavelets can be custom designed for complex domains and irregular sampling. Thus, the strength of the new method is that it can be easily extended to the whole class of secondgeneration wavelets, leaving the freedom and flexibility to choose the wavelet basis depending on the application
Fast adaptive wavelet packet image compression
 IEEE Transactions on Image Processing
, 2000
"... Abstract—Wavelets are illsuited to represent oscillatory patterns: rapid variations of intensity can only be described by the small scale wavelet coefficients, which are often quantized to zero, even at high bit rates. Our goal in this paper is to provide a fast numerical implementation of the best ..."
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Cited by 60 (18 self)
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Abstract—Wavelets are illsuited to represent oscillatory patterns: rapid variations of intensity can only be described by the small scale wavelet coefficients, which are often quantized to zero, even at high bit rates. Our goal in this paper is to provide a fast numerical implementation of the best wavelet packet algorithm [1] in order to demonstrate that an advantage can be gained by constructing a basis adapted to a target image. Emphasis in this paper has been placed on developing algorithms that are computationally efficient. We developed a new fast twodimensional (2D) convolutiondecimation algorithm with factorized nonseparable 2D filters. The algorithm is four times faster than a standard convolutiondecimation. An extensive evaluation of the algorithm was performed on a large class of textured images. Because of its ability to reproduce textures so well, the wavelet packet coder significantly out performs one of the best wavelet coder [2] on images such as Barbara and fingerprints, both visually and in term of PSNR. Index Terms—Adaptive transform, best basis, image compression, ladder structure, wavelet packet. I.
37 other authors
 Science
, 1995
"... Symmetric extension is explored as a means for constructing nonexpansive reversible integertointeger (ITI) wavelet transforms for finitelength signals. Two families of reversible ITI wavelet transforms are introduced, and their constituent transforms are shown to be compatible with symmetric exte ..."
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Cited by 59 (1 self)
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Symmetric extension is explored as a means for constructing nonexpansive reversible integertointeger (ITI) wavelet transforms for finitelength signals. Two families of reversible ITI wavelet transforms are introduced, and their constituent transforms are shown to be compatible with symmetric extension. One of these families is then studied in detail, and several interesting results concerning its member transforms are presented. In addition, some new reversible ITI structures are derived that are useful in conjunction with techniques like symmetric extension. Lastly, the relationship between symmetric extension and perliftingstep extension is explored, and some new results are obtained in this regard.
ThreeDimensional Embedded Subband Coding with Optimized Truncation (3D ESCOT)
 3D ESCOT)”, Applied and Computational Harmonic Analysis10
, 2001
"... This paper presents an efficient video coding algorithm: Threedimensional embedded subband coding with optimized truncation (3D ESCOT), in which coefficients in different subbands are independently coded using fractional bitplane coding and candidate truncation points are formed at the end of ..."
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Cited by 58 (20 self)
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This paper presents an efficient video coding algorithm: Threedimensional embedded subband coding with optimized truncation (3D ESCOT), in which coefficients in different subbands are independently coded using fractional bitplane coding and candidate truncation points are formed at the end of each fractional bitplane. A ratedistortion 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, memoryconstrained temporal wavelet transforms are applied along entire motion threads. Blockbased motion threading is implemented in conjunction with 3D ESCOT in a real video coder. Extension of 3D ESCOT to objectbased coding is also addressed. Experiments demonstrate that 3D ESCOT outperforms MPEG4 for most test sequences at the same bit rate. # 2001 Academic Press 1.
A multiscale framework for compressive sensing of video
 in Proc. Picture Coding Symposium (PCS
, 2009
"... Compressive Sensing (CS) allows the highly efficient acquisition of many signals that could be difficult to capture or encode using conventional methods. From a relatively small number of random measurements, a highdimensional signal can be recovered if it has a sparse or nearsparse representation ..."
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Cited by 55 (9 self)
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Compressive Sensing (CS) allows the highly efficient acquisition of many signals that could be difficult to capture or encode using conventional methods. From a relatively small number of random measurements, a highdimensional signal can be recovered if it has a sparse or nearsparse representation in a basis known to the decoder. In this paper, we consider the application of CS to video signals in order to lessen the sensing and compression burdens in single and multicamera imaging systems. In standard video compression, motion compensation and estimation techniques have led to improved sparse representations that are more easily compressible; we adapt these techniques for the problem of CS recovery. Using a coarsetofine reconstruction algorithm, we alternate between the tasks of motion estimation and motioncompensated waveletdomain signal recovery. We demonstrate that our algorithm allows the recovery of video sequences from fewer measurements than either framebyframe or interframe difference recovery methods. 1.
Optimal multiple description transform coding of Gaussian vectors
 In Proc. IEEE Data Compr. Conf
, 1998
"... Includes minor corrections. Multiple description coding (MDC) is source coding for multiple channels such that a decoder which receives an arbitrary subset of the channels may produce a useful reconstruction. Orchard et al. [1] proposed a transform coding method for MDC of pairs of independent Gaus ..."
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Cited by 54 (12 self)
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Includes minor corrections. Multiple description coding (MDC) is source coding for multiple channels such that a decoder which receives an arbitrary subset of the channels may produce a useful reconstruction. Orchard et al. [1] proposed a transform coding method for MDC of pairs of independent Gaussian random variables. This paper provides a general framework which extends multiple description transform coding (MDTC) to any number of variables and expands the set of transforms which are considered. Analysis of the general case is provided, which can be used to numerically design optimal MDTC systems. The case of two variables sent over two channels is analytically optimized in the most general setting where channel failures need not have equal probability or be independent. It is shown that when channel failures are equally probable and independent, the transforms used in [1] are in the optimal set, but many other choices are possible. A cascade structure is presented which facilitates lowcomplexity design, coding, and decoding for a system with a large number of variables. 1
Multiresolution green’s function methods for interactive simulation of largescale elastostatic objects
 ACM Trans. Graph
, 2003
"... This thesis presents a framework for lowlatency interactive simulation of linear elastostatic models and other systems associated with linear elliptic partial differention equations. This approach makes it feasible to interactively simulate largescale physical models. Linearity is exploited by for ..."
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Cited by 53 (12 self)
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This thesis presents a framework for lowlatency interactive simulation of linear elastostatic models and other systems associated with linear elliptic partial differention equations. This approach makes it feasible to interactively simulate largescale physical models. Linearity is exploited by formulating the boundary value problem (BVP) solution in terms of Green’s functions (GFs) which may be precomputed to provide speed and cheap lookup operations. Runtime BVPs are solved using a collection of Capacitance Matrix Algorithms (CMAs) based on the ShermanMorrisonWoodbury formula. Temporal coherence is exploited by caching and reusing, as well as sequentially updating, previous capacitance matrix inverses. Multiresolution enhancements make it practical to simulate and store very large models. Efficient compressed representations of precomputed GFs are obtained using secondgeneration wavelets defined on surfaces. Fast inverse wavelet transforms allow fast summation methods to be used to accelerate runtime BVP solution. Wavelet GF compression factors are directly related to interactive simulation speedup, and examples are provided with
The BinDCT: Fast Multiplierless Approximation of The Dct
 IEEE SIGNAL PROCESSING LETTERS, VOL. 7, NO. 6, JUNE 2000 141
, 2000
"... This paper presents a family of fast biorthogonal block transforms called binDCT that can be implemented using only shift and add operations. The transform is based on a VLSIfriendly lattice structure that robustly enforces both linear phase and perfect reconstruction properties. The lattice coeffi ..."
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Cited by 44 (5 self)
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This paper presents a family of fast biorthogonal block transforms called binDCT that can be implemented using only shift and add operations. The transform is based on a VLSIfriendly lattice structure that robustly enforces both linear phase and perfect reconstruction properties. The lattice coefficients are parameterized as a series of dyadic lifting steps providing fast, efficient, inplace computation of the transform coefficients as well as the ability to map integers to integers. The new 8 8 transforms all approximate the popular 8 8 DCT closely, attaining a coding gain range of 8.778.82 dB, despite requiring as low as 14 shifts and 31 additions per eight input samples. Application of the binDCT in both lossy and lossless image coding yields very competitive results compared to the performance of the original floatingpoint DCT.
Directionadaptive discrete wavelet transform for image compression
, 2007
"... We propose a directionadaptive DWT (DADWT) that locally adapts the filtering directions to image content based on directional lifting. With the adaptive transform, energy compaction is improved for sharp image features. A mathematical analysis based on an anisotropic statistical image model is pr ..."
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Cited by 44 (2 self)
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We propose a directionadaptive DWT (DADWT) that locally adapts the filtering directions to image content based on directional lifting. With the adaptive transform, energy compaction is improved for sharp image features. A mathematical analysis based on an anisotropic statistical image model is presented to quantify the theoretical gain achieved by adapting the filtering directions. The analysis indicates that the proposed DADWT is more effective than other liftingbased approaches. Experimental results report a gain of up to 2.5 dB in PSNR over the conventional DWT for typical test images. Subjectively, the reconstruction from the DADWT better represents the structure in the image and is visually more pleasing.