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| Catch a Wavelet 2003 |
| The 2003 edition of Catch a Wavelet was organized by Michael Liebling (BIG). |
| 10 Jun 2003 |
Complex Wavelets
Brigitte Forster, Biomedical Imaging Group, EPFL
We present a new family of complex wavelet bases which generate dyadic multiresolution analyses. The phase of the resulting complex wavelet transforms gives information on the high-frequency details in a signal or image and can be used for enhancement.
Our construction approach is based on B-splines of fractional degree. A further parameter is introduced in making the degree complex. This leads to complex multiresolution bases which are tunable in regularity and phase. Moreover, the associated basic scaling functions and wavelets converge to Gabor functions. Thus these new bases are approximately optimally time-frequency localized in the sense of Heisenberg.
This is a joint work with T. Blu and M. Unser. |
| 27 May 2003 |
Wavelets in fMRI
Dimitri Van De Ville, Biomedical Imaging Group, EPFL
Functional magnetic resonance imaging (fMRI) is a fast-developing technique for studying physiological processes in the brains of conscious human subjects. Changes in neuronal activity causes regional hemodynamic adjustments, generating blood oxygenation-level dependent (BOLD) signal changes that can be measured by an MR scanner. Volumes acquired for functional analysis have a rather poor signal-to-noise ratio and suffer from several other degradations (e.g., movements artifacts). SPM (Statistical parametric map) is a widely recognized software package for the analysis of fMRI experiments (Wellcome Department of Cognitive Neurology, UCL, London). One of its key characteristics is the application of a Gaussian smoothing kernel to increase the SNR and improve the detection results. SPM's statistical inference phase uses random field theory to model the smoothing procedure. From the other hand, also wavelet-based techniques have been proposed in literature. Basically, they perform the statistical inference on the coefficients obtained by after the analysis step of a (spatial) discrete wavelet transform. As a clear advantage, they don't require to smooth the data and as such can potentially provide better resolution of detected zones.
In this talk, we will first briefly introduce the key features of fMRI data analysis such as the experiment setup and the "general linear model" (GLM). For an honest comparison, we will apply the GLM (incorporating a model for the hemodynamic response function) to several techniques: a simple spatial t-test, the SPM method, and the wavelet-based approach. Next, we show a remarkable link between SPM's method and the wavelet-based approaches. Nevertheless, current wavelet-based techniques suffer from an important shortcoming: the interpretation of the results in the spatial domain, after performing a statistical test in the wavelet domain, remains difficult and is based on an ad-hoc approach. We will propose a novel way to handle this difficulty by joining statistical inference and approximation theory. |
| 13 May 2003 |
Steerable Filter Banks: Oversampling, Quantization and Rotation Invariance
Dr. Baltasar Beferull-Lozano, First Assistant, EPFL-IC-LCAV
In this talk, we study signal representation with structured overcomplete 2D filter banks in l^{2}(Z^2) which are steerable under rotation, focusing on the interplay between oversampling and quantization, and the properties of rotation invariance. The main property of these transforms is that an arbitrary orientation filtering of an image can be obtained by linear combination of outputs from a few fixed basic filters.
In the first part, we analyze angular oversampling in the presence of quantization. We define two ''consistency'' constraints, one due to the steerability property and the other one due to the quantization itself, and make use of them in order to increase the accuracy in the representation with the number of orientations by using two main techniques in conjunction with Lie theory: a) Projection on Convex Sets (POCS), where accuracy is attained by performing projections on the (convex) sets defined by the constraints, and b) Linear programming principles, where accuracy is obtained by intersecting regions of uncertainty in the angular domain. Our results show both an energy localization in angle and a coding gain for low rates as we increase the number of orientations in the representation.
In the second part, we consider the problem of rotation invariance for the application of distributed texture image classification where the feature encoder and the classifier are physically apart and thus features are quantized and compressed before being transmitted. We define energy-based texture features which are steerable under rotation (steerability in the feature space). We also propose an approach to measure similarity between images that is robust to rotation; images are compared after being angularly aligned in the feature space. The classification process is performed by means of a Decision Tree Classifier where the angular alignment is performed at each node in the tree. Three quantization algorithms are considered and compared: uniform quantization, quantization with optimal rate allocation and classified quantization, the latter being optimized in a rate-distortion-complexity sense. Our results of retrieval performance versus rate show a clear gain with respect to a wavelet transform (as an example, for the same rate, the average retrieval precision is increased from 40% to 65%).
Time permitting, we will describe several lines of current and future work related to shiftable or steerable filter banks (i.e. Multiple-Description for classification, scale shiftability, joint scale-orientation shiftability, quantization design based on a misclassification criterion, etc...).
This work represents approximately 1/3 of my PhD work and is a collaboration with Prof. Antonio Ortega from the Signal and Image Processing Institute (SIPI) at the University of Southern California (USC). |
| 29 Apr 2003 |
Wavelets in Optics: Applications to Spectral Interferometry and Digital Holography
Luc Froehly* and Michael Liebling**, *LOB/BIO-E/EPFL, **LIB/BIO-E/EPFL
In the ever growing menagerie of wavelet transforms, we have chosen two that find an application in optics. As the differences between the many brands of wavelet transforms commonly go unnoticed in the field of optics---the source of confusion probably being the widespread terminology of ``Huygens wavelets'' in the literature---we briefly review some of their differences and requirements.
Widely used as a time-frequency analysis tool, the continuous wavelet transform is at the center of a novel method for 3D imaging. The combination of spectral interferometry and wavelength multiplexing enables to encode and transmit an image over a 1D channel, like for example an optical fiber for a small size endoscope. The 1D signal's continuous wavelet transform, at properly chosen scales, gives direct access to the 3D image.
Specifically designed for the requirements of digital Fresnel holography, Fresnelets are the privileged wavelet bases for performing non-linear filtering and removing unwanted interference terms. The latter arise when the digital hologram is reconstructed and corrupt the result. In the Fresnelet domain, the coefficients associated to the interference terms are separated both spatially and with respect to the frequency bands allowing their targeted suppression. |
| 15 Apr 2003 |
Fusion of multifocal image series using wavelets
Cédric Vonesch, SSC-EPFL and Télécom Paris
More than ever, optical systems are key elements in genetical and biological research. However, when observing a relatively thick object, one can rarely achieve perfect focusing over the whole sample; some regions of the resulting image are blured due to physical limitations. We will describe an algorithm which performs the fusion of series of images taken at different focal planes, in order to obtain one single image that is entirely focused. After explaining the simple idea behind this computational method, we will then extend it to several other applications in the field of biology, such as automated volume measurement or 3D reconstruction of the observed object. |
| 01 Apr 2003 |
Multi-directional Two-dimensional Wavelet Bases and Frames with Low Complexity
Vladan Velisavljevic, LCAV, EPFL
Applications of the wavelet transform in image processing are most frequently based on a separable transform. The basis functions are the outer products of the one-dimensional corresponding bases. Such a method is simple in design and computations. However, the standard method applies the transform in a low number of directions.
We try to keep the simplicity of the standard method but also to add more directions in the transform. In this way a simple multi-directional method is built. While creating multi-directional frames is quite straightforward, multi-directional bases are a bit more complex owing to the subsampling issue. The lattice theory lights up a path of a valid iterative subsampling procedure.
Outperforming results were attained in denoising and non-linear approximation of images. Work is in progress on some non-separable more efficient solutions. |
| 18 Mar 2003 |
Multiresolution Moment Filters
Michael Sühling, Biomedical Imaging Group, EPFL
We introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales. When the window function satisfies a two-scale relation, we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet-like algorithm. We show that B-splines are well suited window functions because, in addition to being refinable, they are positive, symmetric, separable, and very nearly isotropic (gaussian shape).
As an application of these local moments, we present a multi-scale motion estimation algorithm that estimates cardiac motion from sequences of echocardiograms. |
Catch a Wavelet 2001 |
| The 2001 edition of Catch a Wavelet was organized by Minh Do (LCAV), Pierre Vandergheynst (LTS) and Arrate Muñoz (BIG). |
| 21 Jun 2001 |
The spline foundation of wavelet theory
Michael Unser, Biomedical Imaging Group, EPFL
Joint work with Thierry Blu. We present a series of arguments that suggest that splines play a truly fundamental role in wavelet theory. First, we show that any scaling function (or wavelet ) can be expressed as the convolution between a (fractional) B-spline and a distribution. The B-spline component is entirely responsible for the following key wavelet properties: reproduction of polynomials, vanishing moments, and order of approximation. It also accounts for most of the smoothness (regularity) of the basis functions. Second, we prove that any scaling function can be expanded as a sum of harmonic splines; these play essentially the same role here as the Fourier exponentials do for periodic signals. This harmonic expansion provides an explicit time domain representation of scaling functions and wavelets; it also explains their fractal nature. Remarkably, truncating the expansion preserves the essential multiresolution property (two scale relation). Keeping the first term alone yields a fractional spline approximation that captures most of the important wavelet features; e.g., its general shape and smoothness. In the final part, we briefly introduce the new concept of fractional wavelets which extends the conventional theory to non-integer orders. We discuss the main properties of these wavelets and present a fast implementation based on the FFT. We illustrate the method with 2D demonstration software in JAVA. |
| 10 May 2001 |
An Improvement of the Walsh Functions
James Byrnes, University of Massachusetts, Boston
A complete orthonormal set (i.e., a basis) for the space of finite energy functions on a bounded interval, all of which yield low crest factor arrays, will be constructed. The construction, when combined with wavelet -like dilations and translations, immediately gives a function sequence which is optimal with respect to the uncertainty principal for infinite orthonormal subsets of the space of finite energy functions on the real line, confirming a conjecture of H.S. Shapiro. The new basis also satisfies all of the fundamental properties of the Walsh functions. In addition, all of the basis functions correspond to quadrature mirror filters. Applications to burst noise reduction and radar will be indicated and demonstrated. |
| 01 May 2001 |
Dyadic Warped Wavelets
Gianpaolo Evangelista, Audiovisual Communications Laboratory, EPFL
Dyadic wavelet bases have a frequency resolution of one octave. In many applications, e.g., audio, higher or flexible frequency resolution is desired in order to adapt the transform to perceptual or physical characteristics. One way to achieve this is to perform a transformation of the frequency axis, also known as frequency warping. Globally frequency warped wavelets were introduced by Baraniuk and Jones (1993). One can show that arbitrary scale a<1 wavelets can be obtained by means of an a-homogeneous warping map. However, global warping is hard to implement. One-to-one frequency warping of discrete-time can be performed in a unique way by means of Laguerre maps. These form a closed one-parameter family constraining the world of realizable maps. In this talk we generalize the concept of multiresolution approximation to include generalized unitary shift operators obtained by allpass filtering. This naturally leads to the warped multiresolution approximation (WMRA), based on which we construct scale a wavelets obtained by iterated warping with Laguerre maps. The interaction of these maps with upsampling operators is crucial to the extension of the world of realizable maps. The warped form of the two-scale equation naturally leads to warped quadrature mirror filter banks, with minor constraints on the warping map. Vice-versa, by iterating the warped filter bank one can obtain genuine scale a wavelets. The construction requires investigation on the convergence of the iterated warping map. In spite of the complexity of the iterated map, very simple results are obtained. The problem can be viewed as the solution of an eigenvalue equation for the composition operator. This leads to Schröder's equation on the real line. An inner unit disk version of the problem had been solved, for uniform maps, by Königs and was revisited by J. Shapiro (1993). We extended the theory to parametric maps on the real line. Armed with these tools one can define continuous-time wavelets with arbitrary frequency resolution, whose computation can be realized in signal processing structures. |
| 05 Apr 2001 |
Fresnelets---wavelets for digital holography
Michael Liebling, Biomedical Imaging Group
Digital holography is an imaging method in which a hologram is recorded with a CCD-camera and reconstructed numerically. After a brief recall of holography, resp. digital holography, I will present a new class of wavelet bases---Fresnelets---which is obtained by applying the Fresnel transform operator to a wavelet basis of L_2. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing the digital holograms. |
| 15 Mar 2001 |
On the arithmetic complexity of the lifting scheme
Julien Reichel, Visiowave
The Lifting Scheme (LS) is a very efficient implementation of the Discrete Wavelet transforms (DWT). This is due to the exploitation of the redundancy between the low and high-pass filters of the DWT. Using this property the DWT can be computed with less arithmetic operations than conventional filter bank implementations. This talk will presents the arithmetic gain due to the usage of the LS. It will be shown that, contrary to what was believed in the literature, the gain can reach a factor four for large filters. |
Catch a Wavelet 2000 |
| The 2000 edition of Catch a Wavelet was organized by Minh Do (LCAV) and Arrate Muñoz (BIG). |
| 11 Jul 2000 |
Title not available.
Pietro Polotti,
Abstract not available. |
| 27 Jun 2000 |
Title not available.
Arrate Muñoz, Biomedical Imaging Group
Abstract not available. |
| 13 Jun 2000 |
An exact method for computing the Area Moments of Wavelet and Spline curves
Mathews Jacob, Biomedical Imaging Group
A method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis, will be presented. Using Green's Theorem, we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product. The multidimensional filter coefficients are precomputed exactly as the solution of a two-scale relation. To demonstrate the performance improvement of the new method, we compare it with existing methods such as pixel-based approaches and approximation of the region by a polygon . We also propose an alternate scheme when the scaling function is sinc(x). |
| 16 May 2000 |
Texture modeling for coding applications
Gloria Menegaz, LTS
The model-based approach to coding is derived from the insights about the art of communicating pertaining to social sciences. These disclose a new perspective in compression philosophy, based on the redefinition of the role of information relevance. If, in general, the goal of a coding system is to represent the information by eliminating the mathematical redundancy, the availability of a priori information about the source suggests a more general approach, where the goal is getting rid of all kind of irrelevance, either physical or conceptual. The basic idea of model-based coding is to summarize the whole available information into a model representing all the a-priori knowledge about the data. The way the idea of model translates in terms of texture features is based on the assumption that as long as visual appearance is what must be preserved by the global system, the real textural information can be replaced by a synthetic one which is perceptually indistinguishable. We propose a texture a multiresolution probabilistic texture modeling technique. The input texture images are treated as probability density estimators from which new textures, with similar appearance and structural properties, can be sampled. The synthesis method basically consists in sampling successive spatial frequency bands from the input texture, conditioned on the similar joint occurrence of features at all lower spatial frequencies. The technique that we propose is based on the exploitation of continuous and discrete wavelet transforms as texture descriptors. It admits an efficient implementation algorithm while providing synthesis results comparable with the other state-of-the-art methods. Particularly, the DWT-based technique that we have designed only requires a discrete wavelet transform on the input (model) texture and can be implemented using only integer numbers. This makes it quite appealing in view of the implementation on a device, due to its low complexity. Some results illustrate the performance on some natural and computer-generated textures which are usually taken as reference. |
| 02 May 2000 |
Generalized sampling: a variational approach
Jan Kybic, Biomedical Imaging Group
We consider the problem of reconstructing a multidimensional and multivariate function f from the discretely and irregularly sampled responses of q linear shift-invariant filters. Unlike traditional approaches which reconstruct the function in some signal space V (e.g., the space of band-limited functions), our reconstruction is optimal in the sense of a~plausibility criterion J. We start by the interpolation problem where the reconstructed function f is consistent with the measures. The approximation problem leads to the same class of solutions. There is no band-limiting restriction for the input signals. We present a brief review of existing reconstruction techniques, formulate a general framework of the variational approach, show the reconstruction formula and apply it to several practical examples. We explain the general theory, including a generalized interpolation with radial basis functions, in a concise and easy-to-follow way, emphasizing the motivation and practical aspects. |
| 18 Apr 2000 |
Image denoising for non-Gaussian noise
Sylvain Sardy, DMA
We consider the problem of estimating a signal sampled with noise at equispaced locations, for instance an image. By convenience many denoising techniques assume that the noise is Gaussian. While Gaussian noise is often a good approximation to the true noise process, it is sometimes inappropriate for certain applications and consequently the denoising techniques lose their good predictive performance.
Waveshrink and Basis Pursuit are two wavelet-based denoising techniques which, under the assumption of Gaussian noise, are well suited to estimate spatially inhomogeneous signals both from a practical and theoretical point of view. Our goal is to generalize Basis Pursuit to handle a wider class of noise distributions while retaining the good predictive performance of a wavelet-based estimator.
We define two estimators: Robust Basis Pursuit and Generalized Basis Pursuit. We propose an algorithm to solve the corresponding optimization problems, propose an automatic ``universal'' selection of the smoothing parameter, and illustrate the new estimators on real data. |
| 04 Apr 2000 |
Integer wavelet transform for lossy to lossless embedded image coding
Julien Reichel, LTS
The use of the Discrete Wavelet Transform (DWT) for embedded lossy image compression is now well established. One of the possible implementations of the DWT is the Lifting Scheme (LS). Because perfect reconstruction is granted by the structure of the LS, non-linear transforms can be introduced, allowing efficient lossless compression as well. A particular subset of the non-linear transformations that is of interest is the one relying on linear filters followed by a rounding operation. It is commonly referred to as the Integer Wavelet Transform (IWT). It is a an interesting alternative to the DWT because their rate-distortion performances are close and the differences can be predicted. This speech investigate this topic in a theoretical framework and presents a model of the degradations caused by the use of the IWT instead of the DWT for lossy compression. The model is then verified using images compressed by the well-known EZW [1] algorithm. Experiments are also conducted to measure the difference in terms of bitrate and in terms of visual quality. This allows to fully understand the impact of IWT when applied to lossy image compression. |
| 21 Mar 2000 |
Wavelet-based texture characterization with application to content-based image retrieval
Minh Do, LCAV
We first review different wavelet-based techniques in modeling texture images. Those are based on statistical characterization of wavelet coefficients including marginal distributions, joint statistics and embedded hidden Markov models. The effectiveness of each model is demonstrated via examples of synthesizing natural texture images.
In the second part, we focus on the use of texture information for searching, browsing and retrieving images from a large database. For many existing methods, it is unclear on how to define similarity functions on extracted features; usually simple norm-based distances together with heuristic normalization are employed. We show that it is more natural to consider the image retrieval problem in the statistical framework where the two related tasks, feature extraction and similarity measurement, can be jointly considered in a coherent manner. The new wavelet-based texture retrieval method relies on the accuracy in modeling of the marginal distribution of wavelet coefficients using generalized Gaussian density (GGD) and the existence of the closed form of Kullback-Leibler distance between GGD's. Experimental results indicate that the new method significantly improves retrieval rates, e.g. from 65% to 77%, against traditional ones while it has comparable levels of computational complexity. |
Catch a Wavelet 1998-1999 |
| The 1998-99 edition of Catch a Wavelet was organized by Pina Marziliano (LCAV). |
| 07 Jul 1999 |
Irregular Sampling in Approximation Subspaces
Pina Marziliano, LCAV
We describe the irregular sampling problem for discrete-time finite signals in Fourier and wavelet subspaces. The problem is expressed as a linear system of equations which can be solved directly or iteratively. The existence of the solution depends on the rank of the matrix associated to the system. The standard iterative algorithm for reconstructing signals in Fourier subspaces is known as the Papoulis Gerchberg algorithm. Nonlinear approximation of a signal regardless of the subspace results in a smaller approximation error than a linear approximation. This is the motivation for developing the PG algorithm in nonlinear approximation subspaces. This variant makes use of the information obtained from the nonlinear approximation of the signal. We compare the reconstruction speeds of the PG algorithm on linearly and nonlinearly approximated signals in both Fourier and wavelet subspaces. |
| 02 Jun 1999 |
Wavelet Projections for Volume Rendering
Stefan Horbelt, Biomedical Imaging Group and LCAV
We extended Gross's method of volume wavelet rendering by computing splats via an orthogonal projection operator. The method decomposes the volume data into a wavelet pyramid representation in the B-spline domain. The splats of the basis functions are approximated on a multiresolution grid. Using least square approximation ensures the smallest possible error for a given sampling step size. The approximation error on the grid is derived as a function of the sampling step h. Choosing at each step the appropriate wavelet space and spatial resolution produces the smallest possible filters. Our approach reduces the number of computations and allows full control of the image quality. |
| 12 May 1999 |
Projecting onto wavelet-like subspaces with non-integer scale changes: Application to high quality image resizing
Arrate Muñoz, Biomedical Imaging Group
We will present an approach that uses concepts from wavelet theory to project images in subspaces with arbitrary scaling factors. What makes it feasible is a new finite difference method that allows an exact computation of the required inner products for analysis functions that are B-splines of any degree n. The method works for both reduction and magnification of images with an arbitrary scaling factor. When the scale parameter is a power of two, the method is equivalent to a wavelet processing because splines satisfy a two-scale difference equation. Our motivation for this algorithm is high quality image re-sizing. Among the possible applications related to medical imaging we highlight: * Adapting resolution (normalization): For example, making Magnetic Resonance images isotropic (they are typically acquired with a fine within-slice and coarse accross-slice resolution). * Zooming with non-integer factors, which is useful to focus on details for diagnosis purposes. * Constructing pyramids for multi-scaling processing (with scale parameters that are not powers of two) Another relevant application is the electronic publishing and multimedia which requires high-quality pictures (with arbitrary scaling factors). |
| 21 Apr 1999 |
Affine coherent states on the sphere and other manifolds
Pierre Vandergheynst, LTS
Coherent States (CS), a tool widely used in mathematical and theoretical physics to construct atomic decompositions based on the symmetries of a given physical system, have recently renewed the interest for abstract harmonic analysis and representation theory of Lie groups and their homogeneous spaces. In this talk, we will first review the definition and contruction of generalized CS as well as the link between the CS-map and the celebrated continuous wavelet and Gabor expansions. We will then use this formalism to build CS that are indexed by points on a homogeneous space of the conformal group of the (n-d) sphere. These particular states are translated and dilated copies of a wavelet-like function and the associated CS-map behaves quite like a (continuous) wavelet transform. The geometrical picture will be finally completed by an approximation theorem that relates fine scale analysis on the sphere to fine scale wavelet coefficients in the tangent plane. Extension to other manifolds will also be discussed. |
| 10 Mar 1999 |
Image retrieval by structural content
Sushil Bhatthacharjee, LTS
Images represent unstructured information, and cannot be adequately indexed in traditional database systems. Consequently, many research groups around the world have started exploring techniques for content-based image indexing and retrieval.
In this talk I will present the design of an image-retrieval (IR) system that uses local image-features to index images. The system is designed to administer a heterogeneous collection of images. The conceptual similarity measure used to compare two images is the number of small image-patches the images have in common. The patches to be compared are chosen using a 2D continuous wavelet that is a low-level corner detector. Each patch is characterized by a set of Gaussian-derivative filter-responses evaluated at the corresponding feature-point. The responses of 'n' such filters for one region are organized in a n-D vector, referred to as a 'token'. These tokens are further quantized to select 'indexing-terms' which are used to describe each image. The images are thus described by a vector of weights of indexing-terms, and the retrieval function is based on vector-products. By design, the image-description is invariant to rotation and translation. The proposed IR system also supports query-refinement using the classical relevance feedback approach. |
| 10 Feb 1999 |
The dream of a statistician---or why minimax and related procedures appear
Olivier Renaud, DMA
This talk will give the basic idea of a good statistical procedure and will show what kind of question it tries to answer. First in a very simple case of estimation of the center of a distribution, and then on more complicated models like smoothing techniques, amongst which wavelets. We will see that standard methods of estimation are really sensitive to model specification and can induce completely wrong decisions. This lead statisticians to find robust and minimax procedures that protect against any kind of departure from the "idealized hypothesis". Wavelet thresholding is well known to be minimax, but not from all departures. We will investigate this difference and explain why wavelet thresholding is not enough to protect from all kind of deviations. |
| 20 Jan 1999 |
Balancing order and some other discrete-time properties of multiwavelets
Jérôme Lebrun, LCAV
No abstract available. |
| 09 Dec 1998 |
L2 Approximation Error of Wavelet Expansions: Exact asymptotic results, sharp upper bounds
Thierry Blu, Biomedical Imaging Group
A general Fourier-based method is presented that provides an accurate prediction of the approximation error over wavelet-like subspaces, as a function of the sampling step T=2^j. The formalism applies to an extended class of convolution-based signal approximation techniques, which includes interpolation, generalized sampling with prefiltering, and the projectors encountered in wavelet theory. It is shown how to predict the L^2-approximation error, by integrating the spectrum of the function to approximate---not necessarily bandlimited---against a frequency kernel E(\omega) that characterizes the approximation operator. This approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function. The error prediction is exact for stationary processes, as well as for bandlimited signals. Then, the results are applied to approximation spaces that satisfy the requirements of a dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities involved can be computed explicitly as a function of the refinement filter. This is in particular true for the asymptotic expansion of the approximation error for biorthonormal wavelets, as the scale tends to zero. Moreover it is shown how to compute sharp, asymptotically optimal upper bounds for the least-squares approximation error. As an application, the exact asymptotic developments and upper bounds for B-splines and Daubechies scaling functions are obtained. Thanks to these explicit expressions, the improvement that can be obtained by using B-splines instead of Daubechies wavelets is quantified: it follows that, asymptotically when the approximation order (or the number of regularity factors in the refinement filter) tends to infinity, the same accuracy (i.e., approximation error) is attained by using a spline approximation space sampled \pi-times more coarsely than a Daubechies approximation space (on le saura!!!). |
| 25 Nov 1998 |
Wavelets and approximation theory
Group discussion,
No abstract available. |
| 04 Nov 1998 |
Wavelets in audio applications
Gianpaolo Evangelista, LCAV
No abstract available. |
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