Abstract: Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities constructed upon the coefficients of hyperbolic wavelet decompositions. A multifractal analysis is introduced, that jointly accounts for scale invariance and anisotropy. Its properties are studied in depth.
BibTeX:
@article{Abry_P_2012_PREPRINT_hyperbolic_wtetmaat,
author = {Abry, P. and Clausel, M. and Jaffard, S. and Roux, S. and Vedel, B.},
title = {Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures},
journal = {PREPRINT},
year = {2012}
}
Averbuch, A., Beylkin, G., Coifman, R., Fischer, P. and Israeli, M.
Adaptive Solution of Multidimensional PDEs via Tensor Product Wavelet Decomposition
Intern. J. of Pure and Applied Mathematics Vol. 44(1), pp. 75-115
article
BibTeX:
@article{Averbuch_A_2008_j-inter-j-pure-appl-math_adaptive_smpdetpwd,
author = {A. Averbuch and G. Beylkin and R. Coifman and P. Fischer and M. Israeli},
title = {Adaptive Solution of Multidimensional PDEs via Tensor Product Wavelet Decomposition},
journal = {Intern. J. of Pure and Applied Mathematics},
year = {2008},
volume = {44},
number = {1},
pages = {75--115}
}
Ben Slimane, M. and Ben Braiek, H.
Directional and Anisotropic Regularity and Irregularity Criteria in Triebel Wavelet Bases
Abstract: Many natural mathematical objects, as well as many multi-dimensional signals and images from real physical problems, need to distinguish local directional behaviors (for tracking contours in image processing for example). Using some results of Jaffard and Triebel, we obtain criteria of directional and anisotropic regularities by decay conditions on Triebel anisotropic wavelet coefficients (resp. wavelet leaders).
BibTeX:
@article{BenSlimane_B_2012_j-four-anal-appl_directional_arictwb,
author = {Ben Slimane, Mourad and Ben Braiek, Hnia},
title = {Directional and Anisotropic Regularity and Irregularity Criteria in Triebel Wavelet Bases},
journal = {J. Fourier Anal. Appl.},
publisher = {Birkhäuser Boston},
year = {2012},
volume = {18},
pages = {893--914},
note = {10.1007/s00041-012-9226-5},
url = {http://dx.doi.org/10.1007/s00041-012-9226-5}
}
Abstract: Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties.For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue.We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents.These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.
BibTeX:
@inproceedings{Beylkin_G_1993_p-symp-appl-math_wavelets_fna,
author = {Beylkin, G.},
title = {Wavelets and fast numerical algorithms},
booktitle = {#p-symp-appl-math#},
year = {1993},
volume = {47}
}
Beylkin, G., Coifman, R. and Rokhlin, V.
Fast wavelet transforms and numerical algorithms I
Abstract: A class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N $ N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.
BibTeX:
@article{Beylkin_G_1991_j-comm-pure-appl-math_fast_wtna1,
author = {Beylkin, G. and Coifman, R. and Rokhlin, V.},
title = {Fast wavelet transforms and numerical algorithms I},
journal = {Comm. Pure Appl. Math.},
publisher = {Wiley Subscription Services, Inc., A Wiley Company},
year = {1991},
volume = {44},
number = {2},
pages = {141--183},
url = {http://dx.doi.org/10.1002/cpa.3160440202},
doi = {http://dx.doi.org/10.1002/cpa.3160440202}
}
Dahlke, S., Friedrich, U., Maaß, P., Raasch, T. and Ressel, R.A.
An adaptive wavelet solver for a nonlinear parameter identification problem for a parabolic differential equation with sparsity constraints
Abstract: In this paper, we combine concepts from two different mathematical research topics: adaptive wavelet techniques for well-posed problems and regularization theory for nonlinear inverse problems with sparsity constraints. We are concerned with identifying certain parameters in a parabolic reaction-diffusion equation from measured data. Analytical properties of the related parameter-to-state operator are summarized, which justify the application of an iterated soft shrinkage algorithm for minimizing a Tikhonov functional with sparsity constraints. The forward problem is treated by means of a new adaptive wavelet algorithm which is based on tensor wavelets. In its general form, the underlying PDE describes gene concentrations in embryos at an early state of development. We implemented an algorithm for the related nonlinear parameter identification problem and numerical results are presented for a simplified test equation.
BibTeX:
@article{Dahlke_S_2012_j-inv-ill-posed-problems_adaptive_wsnpippdesc,
author = {Dahlke, Stephan and Friedrich, Ulrich and Maaß, Peter and Raasch, Thorsten and Ressel, Rudolf A.},
title = {An adaptive wavelet solver for a nonlinear parameter identification problem for a parabolic differential equation with sparsity constraints},
journal = {J. Inv. Ill-Posed Problems},
year = {2012}
}
Dahlke, S., Friedrich, U., Maaß, P., Raasch, T. and Ressel, R.A.
An adaptive wavelet method for parameteridentification problems in parabolic partialdifferential equations
@article{Dahlke_S_2011_PREPRINT_adaptive_wmpipppde,
author = {S. Dahlke and U. Friedrich and P. Maaß and T. Raasch and R. A. Ressel},
title = {An adaptive wavelet method for parameteridentification problems in parabolic partialdifferential equations},
journal = {PREPRINT},
year = {2011}
}
Davis, A.B., Marshak, A. and Clothiaux, E.E.
Anisotropic multiresolution analysis in 2D: application to long-range correlations in cloud millimeter-radar fields
Abstract: Taking a wavelet standpoint, we survey on the one hand various approaches to multifractal analysis, as a means of characterizing long-range correlations in data, and on the other hand various ways of statistically measuring anisotropy in 2Dfields. In both instances, we present new and related techniques: (i) a simple multifractal analysis methodology based onDiscrete Wavelet Transforms (DWTs), and (ii) a specific DWT adapted to strongly anisotropic fields sampled on rectangular grids with large aspect ratios. This DWT uses a tensor product of the standard dyadic Haar basis (dividing ratio 2) and a nonstandardiriadic counterpart (dividing ratio 3) which includes the famous "French top-hat" wavelet. The new DWT is amenableto an anisotropic version of Multi-Resolution Analysis (MRA) in image processing where the natural support of the field is2z pixels (vertically) by Y' pixels (horizontally), n being the number of levels in the MRA. The complete 2D basis has onescaling function and five wavelets. The new MRA is used in synthesis mode to generate random multifractal fields thatmimic quite realistically the structure and distribution of boundary-layer clouds even though only a few parameters are used tocontrol statistically the wavelet coefficients of the liquid water density field.
BibTeX:
@inproceedings{Davis_A_1999_p-spie-wa_anisotropic_ma2dalrccmrf,
author = {Davis, A. B. and A. Marshak and Clothiaux, E. E.},
title = {Anisotropic multiresolution analysis in 2D: application to long-range correlations in cloud millimeter-radar fields},
booktitle = {Proc. SPIE Wavelet Applications VI},
year = {1999},
volume = {3723},
pages = {194--207},
doi = {http://dx.doi.org/10.1117/12.342928}
}
Abstract: We study the multivariate approximation by certain partial sums (hyperbolicwavelet sums) of wavelet bases formed by tensor products of univariate wavelets.We characterize spaces of functions which have a prescribed approximation error byhyperbolicwavelet sums in terms of a K-functional and interpolation spaces. The resultsparallel those for hyperbolic trigonometric cross approximation of periodic functions[DPT].
BibTeX:
@article{DeVore_R_1998_j-const-approx_hyperbolic_wa,
author = {DeVore, R. and Konyagin, S. V. and Temlyakov, V. N.},
title = {Hyperbolic wavelet approximation},
journal = {Constructive Approximation},
year = {1998},
volume = {14},
pages = {1--26}
}
IEEE Trans. Instrum. Meas. Vol. 21(2), pp. 494-504
article
Abstract: Compressive sensing (CS) is an emerging approachfor the acquisition of signals having a sparse or compressible representationin some basis.While the CS literature has mostly focusedon problems involving 1-D signals and 2-D images, many importantapplications involve multidimensional signals; the constructionof sparsifying bases and measurement systems for such signalsis complicated by their higher dimensionality. In this paper,wepropose the use of Kronecker product matrices in CS for two purposes.First, such matrices can act as sparsifying bases that jointlymodel the structure present in all of the signal dimensions. Second,such matrices can represent themeasurement protocols used in distributedsettings. Our formulation enables the derivation of analyticalbounds for the sparse approximation of multidimensionalsignals and CS recovery performance, as well as a means of evaluatingnovel distributed measurement schemes.
BibTeX:
@article{Duarte_M_2012_j-ieee-tim_kronecker_cs,
author = {Duarte, M. F. and Baraniuk, R. G.},
title = {Kronecker Compressive Sensing},
journal = {IEEE Trans. Instrum. Meas.},
year = {2012},
volume = {21},
number = {2},
pages = {494--504}
}
Fournier, A.
Wavelets and their Applications in Computer Graphics
Abstract: TBC: The ability of a parallel algorithm to make efficient use of increasing computational resources is known as its scalability. In this paper, we develop four parallel algorithms for the 2-dimensional Discrete Wavelet Transform algorithm (2-D DWT), and derive their scalability properties on Mesh and Hypercube interconnection networks. We consider two versions of the 2-D DWT algorithm, known as the Standard (S) and Non-standard (NS) forms, mapped onto P processors under two data partitioning schemes, namely checkerboard (CP) and stripped (SP) partitioning. The two checkerboard partitioned algorithms M2=?(PlogP) (Non-standard form, NS-CP), and as M2=?(Plog2P) (Standard form, S-CP); while on the store-and-forward-routed (SF-routed) Mesh and Hypercube they are scalable as 3?? (NS-CP), and as 2?? (S-CP), respectively, where M 2 is the number of elements in the input matrix, and ? ? (0,1) is a parameter relating M to the number of desired octaves J as J=??logM? . On the CT-routed Hypercube, scalability of the NS-form algorithms shows similar behavior as on the CT-routed Mesh. The Standard form algorithm with stripped partitioning (S-SP) is scalable on the CT-routed Hypercube as M 2 = ?(P 2), and it is unscalable on the CT-routed Mesh. Although asymptotically the stripped partitioned algorithm S-SP on the CT-routed Hypercube would appear to be inferior to its checkerboard counterpart S-CP, detailed analysis based on the proportionality constants of the isoefficiency function shows that S-SP is actually more efficient than S-CP over a realistic range of machine and problem sizes. A milder form of this result holds on the CT- and SF-routed Mesh, where S-SP would, asymptotically, appear to be altogether unscalable.
Review: Could be applied to TRAN Huy-Quan for processing speed-up, or in scalable data compression, for seismic data
BibTeX:
@article{Fridman_J_1997_j-mult-syst-sp_scalability_2ddwta,
author = {Fridman, J. and Manolakos, E. S.},
title = {On the Scalability of 2-D Discrete Wavelet Transform Algorithms},
journal = {Multidimension. Syst. Signal Process.},
publisher = {Kluwer Academic Publishers},
year = {1997},
volume = {8},
pages = {185--217},
url = {http://dx.doi.org/10.1023/A%3A1008229209464},
doi = {http://dx.doi.org/10.1023/A:1008229209464}
}
Grohs, P.
Tree approximation with anisotropic decompositions
Abstract: In recent years anisotropic transforms like the shearlet or curvelet transform have received a considerable amount of interest due to their ability to efficiently capture anisotropic features in terms of nonlinear N-term approximation. In this paper we study treeapproximation properties of such transforms where the N-term approximant has to satisfy the additional constraint that the set of kept indices possesses a tree structure. The main result of this paper is that for shearlet- and related systems, this additional constraint does not deteriorate the approximation rate. As an application of our results we construct (almost) optimal encoding schemes for cartoon images.
BibTeX:
@article{Grohs_P_2012_j-acha_tree_aad,
author = {Grohs, P.},
title = {Tree approximation with anisotropic decompositions},
journal = {Appl. Comp. Harm. Analysis},
year = {2012},
volume = {33},
pages = {44-57}
}
Abstract: We consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces $ H^a_p $ for $p > 0$, which are known to be equal to $L_p$ for $p >$ 1. In particular, we characterize those functions that provide a specific convergence rate by intrinsic smoothness properties. To this end we apply restricted nonlinear approximation, see [3], which is a generalization of $n$-term approximation in which a weight function is used to control the terms of the approximations.
BibTeX:
@article{Hochmuth_R_2007_j-math-nachr_anisotropic_wbt,
author = {Hochmuth, Reinhard},
title = {Anisotropic wavelet bases and thresholding},
journal = {Math. Nachr.},
publisher = {WILEY-VCH Verlag},
year = {2007},
volume = {280},
number = {5-6},
pages = {523--533},
url = {http://dx.doi.org/10.1002/mana.200410500},
doi = {http://dx.doi.org/10.1002/mana.200410500}
}
Hochmuth, R.
$N$-term Approximation in Anisotropic Function Spaces
Abstract: In $L_2((0, 1)^2)$ infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one-dimensional biorthogonal wavelet bases on the interval $(0, 1)$. Most well-known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.
BibTeX:
@article{Hochmuth_R_2002_j-math-nachr_n-term_aafs,
author = {Hochmuth, Reinhard},
title = {$N$-term Approximation in Anisotropic Function Spaces},
journal = {Math. Nachr.},
publisher = {WILEY-VCH Verlag},
year = {2002},
volume = {244},
number = {1},
pages = {131--149},
url = {http://dx.doi.org/10.1002/1522-2616(200210)244:1<131::AID-MANA131>3.0.CO;2-G},
doi = {http://dx.doi.org/10.1002/1522-2616(200210)244:1}
}
Hochmuth, R.
Wavelet Characterizations for AnisotropicBesov Spaces
Abstract: The goal of this paper is to provide wavelet characterizations for anisotropicBesov spaces. Depending on the anisotropy, appropriate biorthogonal tensorproduct bases are introduced and Jackson and Bernstein estimates are provedfor two-parameter families of finite-dimensional spaces. These estimates leadto characterizations for anisotropic Besov spaces by anisotropy-dependent linearapproximation spaces and lead further on to interpolation and embedding results.Finally, wavelet characterizations for anisotropic Besov spaces with respect to $L_p$-spaceswith $0
BibTeX:
@article{Hochmuth_R_2012_j-acha_wavelet_cabs,
author = {Hochmuth, R.},
title = {Wavelet Characterizations for AnisotropicBesov Spaces},
journal = {Appl. Comp. Harm. Analysis},
year = {2002},
volume = {12},
pages = {179--208},
doi = {http://dx.doi.org/10.1006/acha.2001.0377}
}
Neumann, M.H.
MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES
Abstract: It is well known that multivariate curve estimation under standard (isotropic) smoothness conditions suffers from the ``curse of dimensionality''. This is reflected by rates of convergence that deteriorate seriously in standard asymptotic settings. Better rates of convergence than those corresponding to isotropic smoothness priors are possible if the curve to be estimated has different smoothness properties in different directions and the estimation scheme is capable of making use of a lower complexity in some of the directions. We consider typical cases of anisotropic smoothness classes and explore how appropriate wavelet estimators can exploit such restrictions on the curve that require an adaptation to different smoothness properties in different directions. It turns out that nonlinear thresholding with an anisotropic multivariate wavelet basis leads to optimal rates of convergence under smoothness priors of anisotropic type. We derive asymptotic results in the model ``signal plus Gaussian white noise'', where a decreasing noise level mimics the standard asymptotics with increasing sample size.
BibTeX:
@article{Neumann_M_2000_j-statist-sinica_multivariate_wtafs,
author = {Neumann, M. H.},
title = {MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES},
journal = {Statist. Sinica},
year = {2000},
volume = {10},
pages = {399--431}
}
Neumann, M.H. and von Sachs, R.
Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra
Abstract: We derive minimax rates for estimation in anisotropic smoothness classes. These rates are attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present. As an important application we introduce a new adaptive waveletestimator of the time-dependent spectrum of a locally stationary time series. Using this model which was recently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, that is, how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.
BibTeX:
@article{Neumann_M_1997_j-ann-stat_wavelet_tafcaaees,
author = {M. H. Neumann and von Sachs, R.},
title = {Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra},
journal = {Ann. Stat.},
year = {1997},
volume = {25},
number = {1},
pages = {38--76}
}
Nowak, R.D. and Baraniuk, R.G.
Wavelet-based transformations for nonlinear signal processing
Abstract: Nonlinearities are often encountered in the analysis and processing of real-world signals. We introduce two new structures for nonlinear signal processing. The new structures simplify the analysis, design, and implementation of nonlinear filters and can be applied to obtain more reliable estimates of higher order statistics. Both structures are based on a two-step decomposition consisting of a linear orthogonal signal expansion followed by scalar polynomial transformations of the resulting signal coefficients. Most existing approaches to nonlinear signal processing characterize the nonlinearity in the time domain or frequency domain; in our framework any orthogonal signal expansion can be employed. In fact, there are good reasons for characterizing nonlinearity using more general signal representations like the wavelet expansion. Wavelet expansions often provide very concise signal representations and thereby can simplify subsequent nonlinear analysis and processing. Wavelets also enable local nonlinear analysis and processing in both time and frequency, which can be advantageous in nonstationary problems. Moreover, we show that the wavelet domain offers significant theoretical advantages over classical time or frequency domain approaches to nonlinear signal analysis and processing
BibTeX:
@article{Nowak_R_1999_j-ieee-tsp_wavelet-based_tnsp,
author = {Nowak, R. D. and Baraniuk, R. G.},
title = {Wavelet-based transformations for nonlinear signal processing},
journal = {IEEE Trans. Signal Process.},
year = {1999},
volume = {47},
number = {7},
pages = {1852--1865},
doi = {http://dx.doi.org/10.1109/78.771035}
}
Rosiene, C.P. and Nguyen, T.Q.
Tensor-product wavelet vs. Mallat decomposition: a comparative analysis
Abstract: The two-dimensional tensor product wavelet transform is compared to the Mallat representation for the purpose of data compression. It is shown that the tensor product wavelet transform will always provide a coding gain greater than or equal to that of the Mallat representation. Further, the costs of obtaining the tensor product wavelet transform are outlined
BibTeX:
@inproceedings{Rosiene_C_1999_p-iscas_ten_pwmdca,
author = {Rosiene, C. P. and Nguyen, T. Q.},
title = {Tensor-product wavelet vs. Mallat decomposition: a comparative analysis},
booktitle = {Proc. Int. Symp. Circuits Syst.},
year = {1999},
volume = {3},
pages = {431--434},
doi = {http://dx.doi.org/10.1109/ISCAS.1999.778877}
}
Roux, S., Clausel, M., Vedel, B., Jaffard, S. and Abry, P.
@article{Roux_S_2012_INPREPARATION_wavelet_aai,
author = {Roux, S. and Clausel, M. and Vedel, B. and Jaffard, S. and Abry, P.},
title = {Wavelet analysis for anisotropic images},
journal = {INPREPARATION},
year = {2012}
}
Roux, S.G., Clausel, M., Vedel, B., Jaffard, S. and Abry, P.
Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution
Abstract: Textures in images can often be well modeled using self-similar processes while they may at the same time display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will first be shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform by the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axis. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized, this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a non parametric bootstrap based procedure is described, that provides confidence interval in addition to the estimates themselves and enables to construct an isotropy test procedure, that can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis is illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed.
BibTeX:
@article{Roux_S_2013_PREPRINT_self-similar_atahwtc,
author = {Roux, S. G. and Clausel, M. and Vedel, B. and Jaffard, S. and Abry, P.},
title = {Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution},
journal = {PREPRINT},
year = {2013}
}
Schremmer, C.
Decomposition strategies for wavelet-based image coding
Abstract: The wavelet transform has become the most interesting new algorithm for still image compression. Yet there are many parameters within a wavelet analysis and synthesis which govern the quality of a decoded image. In this paper, we discuss different decomposition strategies of a two-dimensional signal and their implications for the decoded image: a pool of gray-scale images has been wave let-transformed with different settings of the wavelet filter bank, quantization threshold and decomposition method. Contrary to the new standard JPEG-2000, where nonstandard decomposition is implemented, our investigation proposes standard decomposition for low-bitrate coding
BibTeX:
@inproceedings{Schremmer_C_2001_p-isspa_decomposition_swbic,
author = {Schremmer, C.},
title = {Decomposition strategies for wavelet-based image coding},
booktitle = {Signal Processing and its Applications, Sixth International, Symposium on. 2001},
year = {2001},
volume = {2},
pages = {529 --532},
doi = {http://dx.doi.org/10.1109/ISSPA.2001.950197}
}
Proc. Conf. "Function spaces, differential operators and nonlinear analysis", Milovy, 2004, pp. 370-387
inproceedings
Abstract: The paper deals with wavelet characterisations for anisotropicspaces $B_pq^s,alpha$ and $F_pq^s,alpha$ in $R^n$ for all parameters $s$, $p$, $q$ and all anisotropies $.Some applications are given.
BibTeX:
@inproceedings{Triebel_H_2005_PREPRINT_wavelet_bafs,
author = {Triebel, H.},
title = {Wavelet bases in anisotropic function spaces},
booktitle = {Proc. Conf. "Function spaces, differential operators and nonlinear analysis", Milovy, 2004},
year = {2005},
pages = {370--387}
}
Velisavljević, V.
Directionlets: Anisotropic Multi-directional Representation with Separable Filtering
Abstract: Efficient representation of geometrical information in images is very important in many image processing areas, including compression, denoising and feature extraction. However, the design of transforms that can capture these geometrical features and represent them with a sparse description is very challenging. Recently, the separable wavelet transform achieved a great success providing a computationally simple tool and allowing for a sparse representation of images. However, in spite of the success, the efficiency of the representation is limited by the spatial isotropy of the wavelet basis functions built in the horizontal and vertical directions as well as the lack of directionality. One-dimensional discontinuities in images (edges and contours), which are very important elements in visual perception, intersect with too many wavelet basis functions leading to a non-sparse representation. To capture efficiently these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, more flexible multi-directional and anisotropic transforms are required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic multi-directional wavelet transform. The transform retains the separable filtering, subsampling and simplicity of computations and filter design from the standard two-dimensional wavelet transform, unlike in the case of some other existing directional transform constructions (e.g. curvelets, contourlets or edgelets). The corresponding anisotropic basis functions, which we call directionlets, have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for non-linear approximation of images, achieving the decay of mean-square errorO(N -1.55), which, while slower than the optimal rate O(N-2), is much better than O(N-1) achieved with wavelets, but at similar complexity. Owing to critical sampling, directionlets can easily be applied to image compression since it is possible to use Lagrange optimization as opposed to the case of overcomplete expansions. The compression algorithms based on directionlets outperform the methods based on the standard wavelet transform achieving better numerical results and visual quality of the reconstructed images. Moreover, we have adapted image denoising algorithms to be used in conjunction with an undecimated version of directionlets obtaining results that are competitive with the current state-of-the-art image denoising methods while having lower computational complexity.
BibTeX:
@phdthesis{Velisavljevic_V_2005_phd_dir_amdrsf,
author = {Velisavljević, V.},
title = {Directionlets: Anisotropic Multi-directional Representation with Separable Filtering},
school = {EPFL},
year = {2005}
}
Velisavljević, V., Beferull-Lozano, B., Vetterli, M. and Dragotti, P.L.
Directionlets: Anisotropic multi-directional representation with separable filtering
Abstract: In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis functions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N-1.55), which, while slower than the optimal rate O(N-2), is much better than O(N-1) achieved with wavelets, but at similar complexity.
BibTeX:
@article{Velisavljevic_V_2006_tip_dir_amdrsf,
author = {Velisavljević, V. and Beferull-Lozano, B. and Vetterli, M. and Dragotti, P. L.},
title = {Directionlets: Anisotropic multi-directional representation with separable filtering},
journal = {IEEE Trans. Image Process.},
year = {2006},
volume = {15},
number = {7},
pages = {1916--1933},
doi = {http://dx.doi.org/10.1109/TIP.2006.877076}
}
Velisavljević, V., Beferull-Lozano, B., Vetterli, M. and Dragotti, P.L.
Directionlets: anisotropic multi-directional representation with separable filtering
@misc{Velisavljevic_V_2006_pp_dir_amdrsf,
author = {Velisavljević, V. and Beferull-Lozano, B. and Vetterli, M. and Dragotti, P. L.},
title = {Directionlets: anisotropic multi-directional representation with separable filtering},
year = {2006},
note = {Preprint, better reference teVelisavljevic_V_2006_tip_dir_amdrsf}
}
Wegmann, B. and Zetzsche, C.
Efficient image sequence coding by vector quantization of spatiotemporal bandpass outputs
Abstract: A coding scheme for image sequences is designed in analogy to human visual information processing. We propose a feature-specific vector quantization method applied to multi-channel representation of image sequences. The vector quantization combines the corresponding local/momentary amplitude coefficients of a set of three-dimensional analytic band-pass filters being selective for spatiotemporal frequency, orientation, direction and velocity. Motion compensation and decorrelation between successive frames is achieved implicitly by application of a non-rectangular subsampling to the 3D-bandpass outputs. The nonlinear combination of the outputs of filters which are selective for constantly moving one- dimensional (i.e. spatial elongated) image structures allows a classification of the local/momentary signal features with respect to their intrinsic dimensionality. Based on statistical investigations a natural hierarchy of signal features is provided. This is then used to construct an efficient encoding procedure. Thereby, the different sensitivity of the human vision to the various signal features can be easily incorporated. For a first example, all multi- dimensional vectors are mapped to constantly moving 1D-structures.
BibTeX:
@inproceedings{Wegmann_B_1992_p-spie-vcip_efficient_iscvqsbo,
author = {Wegmann, Bernhard and Zetzsche, Christoph},
title = {Efficient image sequence coding by vector quantization of spatiotemporal bandpass outputs},
booktitle = {Proc. SPIE 1818, Visual Communications and Image Processing},
year = {1992},
pages = {1146--1154},
url = {http://dx.doi.org/10.1117/12.131386},
doi = {http://dx.doi.org/10.1117/12.131386}
}
Welk, M., Weickert, J. and Steidl, G.
From Tensor-Driven Diffusion to Anisotropic Wavelet Shrinkage
@inproceedings{Welk_M_2006_p-eccv_ten_ddaws,
author = {Welk, M. and Weickert, J. and Steidl, G.},
title = {From Tensor-Driven Diffusion to Anisotropic Wavelet Shrinkage},
booktitle = {Proc. Eur. Conf. Comput. Vis.},
year = {2006}
}
@phdthesis{Westerink_P_1989_phd_subband_ci,
author = {P. H. Westerink},
title = {Subband coding of images},
school = {Delft University of Technology},
year = {1989}
}
Xu, D. and Do, M.N.
Anisotropic 2D wavelet packets and rectangular tiling: theory and algorithms
Abstract: We propose a new subspace decomposition scheme called anisotropic wavelet packets which broadens the existing definition of 2-D wavelet packets. By allowing arbitrary order of row and column decompositions, this scheme fully considers the adaptivity, which helps find the best bases to represent an image. We also show that the number of candidate tree structures in the anisotropic case is much larger than isotropic case. The greedy algorithm and double-tree algorithm are then presented and experimental results are shown.
BibTeX:
@inproceedings{Xu_D_2003_p-spie-wasip_ani_2dwprtta,
author = {Xu, D. and Do, M. N.},
title = {Anisotropic 2D wavelet packets and rectangular tiling: theory and algorithms},
booktitle = {Proc. SPIE, Wavelets: Appl. Signal Image Process.},
year = {2003},
pages = {619--630},
doi = {http://dx.doi.org/10.1117/12.506601}
}
Zavadsky, V.
Image Approximation by Rectangular Wavelet Transform
Abstract: We study image approximation by a separable wavelet basis $$ 2^k_1x-i)2^k_2y-j), x-i)2^k_2y-j), 2^k_1(x-i)y-j), x-i)y-i),$ where $k_1, k_2 in Z_+; i,jinZ; $$ and ?,? are elements of a standard biorthogonal wavelet basis in L 2 (?). Because k 1 ? k 2 , the supports of the basis elements are rectangles, and the corresponding transform is known as the rectangular wavelet transform . We provide a self-contained proof that if one-dimensional wavelet basis has M dual vanishing moments then the rate of approximation by N coefficients of rectangular wavelet transform is $$ O(N^-M) $$ for functions with mixed derivative of order M in each direction. These results are consistent with optimal approximation rates for such functions. The square wavelet transform yields the approximation rate is $$ O(N^-M/2) $$ for functions with all derivatives of the total order M . Thus, the rectangular wavelet transform can outperform the square one if an image has a mixed derivative. We provide experimental comparison of image approximation which shows that rectangular wavelet transform outperform the square one.