We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We develop a methodology to combine wavelets together these new systems to perform noise removal by exploiting all these systems simultaneously. The results of the combined reconstruction exhibits clear advantages over any individual system alone. For example, the residual error contains essentially no visually intelligible structure: no structure is lost in the reconstruction.

In this paper, we propose an image restoration algorithm based on state-of-the-art wavelet domain statistical models. We present an efficient method to estimate the model parameters from the observations, and solve the restoration problem in orthonormal and translation--invariant (TI) wavelet domains. Substantial improvements over previous wavelet-based restoration methods are obtained. The use of a TI wavelet transform further enhances the restoration performance. We study the improvement from the viewpoint of Bayesian estimation theory and show that replacing an estimator with its TI version will reduce the expected risk if the signal and the degradation model are stationary.

This paper deals with a restoration (both denoising and deblurring) method. For instance in the case of denoising, this latter is only a small modification from the usual wavelet thresholding. However, it has the significant advantage to allow the use of several bases in such a way that we select what is considered as information by a basis or another basis or another basis, and so on for as many bases as we want. The computational cost of the method is mainly the computation of the coordinates of the signal (or image) in the bases.

In this paper speckle reduction is approached as a Wiener-like filtering performed in the wavelet domain by means of an adaptive shrinkage of the detail coefficients of an undecimated decomposition. The amplitude of each coefficient is divided by the variance ratio of the noisy coefficient to the noise-free one. All the above quantities are analytically calculated from the speckled image, the speckle variance, and the wavelet filters. On the test image Lenna corrupted by synthetic speckle, the proposed method outperforms Kuan's LLMMSE filtering by almost 3 dB SNR. Experiments carried out on true and synthetic speckled images demonstrate that the visual quality of the results is excellent in terms of both background smoothing and preservation of edge sharpness and textures. The absence of decimation in the wavelet decomposition avoids the typical ringing impairments produced by critically-subsampled wavelet-based denoising.

Orthogonal frequency division multiplexing (OFDM) has gained considerable interest as an efficient technology for high-date-data transmission over wireless channels. Standard OFDM systems are associated with a rectangular grid in the time-frequency plane. However such a setup is in general not optimum for pulse shaping OFDM systems for doubly dispersive channels. We introduce lattice-OFDM systems (LOFDM), which are OFDM systems constructed with respect to general lattices in the time-frequency plane. We show how to design optimum pulse shaping LOFDM system. Our analysis is based on results from Heisenberg groups, Gabor frames, and sphere coverings. Numerical simulations confirm that LOFDM systems outperform ordinary OFDM systems with regard to robustness against intersymbol interference and interchannel interference.

In tomographic medical devices such as SPECT or PET cameras, image reconstruction is an unstable inverse problem, due to the presence of additive noise. A new family of regularization methods for reconstruction, based on a thresholding procedure in wavelet and wavelet packet decompositions, is studied. This approach is based on the fact that the decompositions provide a near-diagonalization of the inverse Radon transform and of the prior information on medical images. An optimal wavelet packet decomposition is adaptively chosen for the specific image to be restored. Corresponding algorithms have been developed for both 2-D and full 3-D reconstruction. These procedures are fast, non-iterative, flexible, and their performance outperforms Filtered Back-Projection and iterative procedures such as OS-EM.

We build a feature vector that can be used for content-based image retrieval of grayscale images of objects against a background of texture. The feature vector is based on moment invariants of detail coefficients produced by the lifting scheme. The prediction filters in this scheme are chosen adaptively: low order (small stencils) near edges and high order elsewhere. The aim is to retrieve similar images of an object irrespective of translation, rotation, reflection or resizing of the object, light conditions and the background texture. We present preliminary results.

We propose a novel approach for scattered data smoothing based on second generation wavelets. This wavelet transform automatically adapts to the irregularity of the grid. Our implementation also pays attention to numerical stability, a crucial property in estimation procedures. The wavelet coefficients are shrunk either with simple soft-thresholding or with an empirical Bayesian estimation.

The Lifting Scheme (LS) is a very efficient implementation of the Discrete Wavelet transform (DWT). In this work we compute the arithmetic gain realized when the LS is used instead of conventional filter banks. It is shown that contrary to was was presented in the original work from Sweldens a gain of four is possible. However the LS should be used with care as it can increase the memory bandwidth. Some implementations are presented together with their impact on the bandwidth.

We will review the latest developments concerning uniform tight frames and their applications to signal processing.

Given a window function which generates a Gabor (resp. Wavelet) frame. We consider the best approximation by those window functions that generate normalized tight (or just tight) frames. Using a parameterizations of window functions by certain class of operators in the von Neumann algebras associated with shift operators in time and frequency over certain lattices, we are able to prove that for any window function of a Gabor frame, there exists a unique window function which generates a tight Gabor frame and is the best approximation (among all the tight Gabor frames) for the given window function. More generally, we show that this is true for any frame induced by a projective unitary representation for a group. However, this result is not valid for wavelet frames. We will provide a restricted approximation result for semi-orthogonal wavelet frames.

We introduce the concept of frames of translates and we characterize the countable families of vectors generating such frames. In this context we generalize the concept of the Grammian introduced by Ron and Shen. We apply this characterization to study Generalized frame MRAs of L²(R). We also provide the characterization of frame multiwavelet sets associated with GFMRAs of L²(R) and we present examples of GFMRAs.

The key element in the design of fast algorithms in numerical analysis and signal processing is the selection of an efficient approximation for the functions and operators involved. In this talk we will consider approximations using wavelet and multiwavelet bases as well as a new type of approximation for bandlimited functions using exponentials obtained via Generalized Gaussian quadratures. Analytically, the latter approximation corresponds to using the basis of the Prolate Spheroidal Wave functions. We will briefly comment on the future development of approximation techniques and the corresponding fast algorithms.

The singular functions for the problem of recovering a time limited function from its Fourier transform in a certain band of frequencies are given, in the simplest case, by the prolate spheroidal wave functions. We explore a number of issues related to this problem including the effective computation of the corresponding Slepian functions for a polar cap on the surface of the Earth. The same method would work for a region on the sphere bounded by two parallels.

What happens in a persons brain when they see a picture of somebody they recognize? Recognition takes place almost immediately. But where in the brain does it take place? Functional Magnetic Resonance Imaging (fMRI) is a technique which can be used to study mental activity in the brain. However as currently used, the temporal resolution of fMRI studies are too slow to answer such questions. To increase its usefulness, new methods of speeding up fMRI studies must be introduced. In this paper we discuss a method which improves the time resolution in fMRI. Using prior knowledge of the region of interest (ROI) and the time constraints we wish to obtain, the method tailors the k-space (Fourier space) sampling region and creates a matching prolate spheroidal wave function filter in order to maximize the energy concentration in the ROI. The method enables one, at high time resolution, to study the total activity over a pre-defined region of the brain. Thereby giving the opportunity to study the change in mental activity that occurs in that region, when a specific task is performed. This is a problem that, besides having a clear medical interest, also involves interesting mathematical and statistical aspects, especially in Fourier and time series analysis.

We have previously reported experimental results that directly connect speech and hearing and lead to the concept of a universal warping function. In this paper we report further experiments based on a large database collected by Hillenbrand et al. These new results further validate the concept of a universal warping function.

In this paper we present topology aspects in non- localization results. A well-known such no-go result in the Balian-Low theorem that states the generator of a Weyl- Heisenberg Riesz basis cannot be well time-frequency localized. More general, the statement applies to multi WH Riesz bases, or super frames as well. These results turn out to be connected to non-triviality of a complex vector bundle. Another class of problem is related to optimality of coherent approximations of stochastic signals. More specific, for a given deficit ((alpha) (beta) >1), find the best Riesz sequence generator optimal to respect to the mean square approximation error. A topological obstruction turns out to be responsible for ill-localization of the optimal generator.

In this work we analyze the existence of single scaling functions embedded in a multiresolution analysis structure generated by a multiscaling function. Particularly, we consider the case of spline functions.

We give an explicit expression for the transform of a signal in an arbitrary representation which has first been filtered in another representation. Using this formula we connect the work of Cohen for obtaining convolution and correlation theorems in arbitrary representations with the work of Lindsey and Suter for partitioning the space of integral transforms.

An approach for contrast enhancement utilizing multi-scale analysis is introduced. Sub-band coefficients were modified by the method of adaptive histogram equalization. To achieve optimal contrast enhancement, the sizes of sub-regions were chosen with consideration to the support of the analysis filters. The enhanced images provided subtle details of tissues that are only visible with tedious contrast/brightness windowing methods currently used in clinical reading. We present results on chest CT data, which shows significant improvement over existing state-of-the-art methods: unsharp masking, adaptive histogram equalization (AHE), and the contrast limited adaptive histogram equalization (CLAHE). A systematic study on 109 clinical chest CT images by three radiologists suggests the promise of this method in terms of both interpretation time and diagnostic performance on different pathological cases. In addition, radiologists observed no noticeable artifacts or amplification of noise that usually appears in traditional adaptive histogram equalization and its variations.

This work provides a new approach to estimate the parameters of a semi-parametric generalized linear model in the wavelet domain. The method is illustrated with the problem of detecting significant changes in fMRI signals that are correlated to a stimulus time course. The fMRI signal is described as the sum of two effects: a smooth trend and the response to the stimulus. The trend belongs to a subspace spanned by large scale wavelets. We have developed a scale space regression that permits to carry out the regression in the wavelet domain while omitting the scales that are contaminated by the trend. Experiments with fMRI data demonstrate that our approach can infer and remove drifts that cannot be adequately represented with low degree polynomials. Our approach results in a noticeable improvement by reducing the false positive rate and increasing the true positive rate.

The Continuous Wavelet Transform (CWT) is an effective way to analyze nonstationary signals and to localize and characterize singularities. Fast algorithms have already been developed to compute the CWT at integer time points and dyadic or integer scales. We propose here a new method that is based on a B-spline expansion of both the signal and the analysis wavelet and that allows the CWT computation at arbitrary scales. Its complexity is O(N), where N represents the size of the input signal; in other words, the cost is independent of the scale factor. Moreover, the algorithm lends itself well to a parallel implementation.

This paper describes the use of moving-block expansions in Slepian sequences (discrete prolate spheroidal sequences) as a method to generate precision complex demodulators for data analysis. Such filters are used to separate the annual cycle from the low-frequency trend in climate date, to isolate individual modes in helioseismology data, etc., so statistical efficiency and reliability is required. In particular one must combine the conflicting requirements of isolating a narrow frequency band with having only a short data series available. For such uses, the concentration properties of the Slepian sequences give optimum protection against signals at frequencies other than the band of interest. However, like other orthogonal expansions, the standard inverse suffers from Gibb's ripples and similar effects. Here, the usual orthogonal series expansion is replaced with a partial inverse-theory reconstruction with a smoothness constraint. Defining a set of polynomials orthogonal with respect to inner products with the Slepian sequences allows construction of a sequence of projection operators with variable smoothness properties.

In this paper we report on the performance of Lemarie uniwavelets and biwavelets for solving the ill-posed inverse problem of recovering the derivative of a noisy signal. The noise under consideration can be white gaussian or other noises like Tukey noise. The denoising procedures utilized are wavelet and biwavelet Efromovich-Pinsker (EP) estimators, which have been shown to be universal for both estimating the function and its derivative.

In this paper, we show the existence of third-order approximation order preserving prefilters for multiwavelets. We develop the necessary conditions needed to satisfy the approximation criteria and give a constructive proof of their existence. Additionally, we give an example prefilter for the Chui-Lian multiwavlet and implement the prefilter/multiwavelet in an image compression scheme. The compression results are then compared against the standard wavelets of Haar, D4, and D6.

Multiresolution structures are important in applications, but they are also useful for analyzing properties of associated wavelets. Given a nonorthogonal (multi-) wavelet in a Hilbert space, we construct a core subspace. Subsequently, the dilates of the core subspace defines a ladder of nested subspaces. Of fundamental importance are two questions: 1) when is the core subspace shift invariant; and if yes, then 2) when is the core subspace generated by shifts of a single vector, i.e. there exists a scaling vector. If the wavelet generates a Riesz basis then the answer to question 1) is yes if and only if the wavelet is a biorthogonal wavelet. Additionally, if the wavelet generates a tight frame of arbitrary frame constant, then the core subspace is shift invariant. Question 1) is still open in case the wavelet generates a non-tight frame. We also present some known results to question 2) and provide some preliminary improvements. Our analysis here arises from investigating the dimension function and the multiplicity function of a wavelet. These two functions agree if the wavelet is orthogonal. Finally, we discuss how these questions are important for considering linear perturbation of wavelets. Utilizing the idea of the local commutant of a unitary system developed by Dai and Larson, we show that nearly all linear perturbations of two orthonormal wavelets form a Riesz wavelet. If in fact these wavelets correspond to a von Neumann algebra in the local commutant of a base wavelet, then the interpolated wavelet is biorthogonal. Moreover, we demonstrate that in this case the interpolated wavelets have a scaling vector if the base wavelet has a scaling vector.

We describe a method for adapting local shift-invariant bases to non-uniform grids via what we call a squeeze map. When the shift-invariant basis is orthogonal there is a squeeze map such that the nonuniform basis is orthogonal and has the same smoothness and same approximation order as the shift-invariant basis. When the smoothness or approximation order is large enough the squeeze map is uniquely determined and may be calculated locally in terms of the ratios of adjacent intervals. Therefore a basis may be rapidly generated for a given grid. Furthermore local changes in a grid (for example knot insertion or deletion) only affect a few of the basis functions. When starting with a refinable scaling vector the squeeze map machinery gives a procedure for generating orthogonal wavelets on semi-regular grids (that is, an arbitrary non-uniform coarse space with uniform refinements) with the same polynomial reproduction and smoothness as the shift-invariant space.

The main drive behind the use of data compression in seismic data is the very large size of seismic data acquired. Some of the most recent acquired marine seismic data sets exceed 10 Tbytes, and in fact there are currently seismic surveys planned with a volume of around 120 Tbytes. Nevertheless, seismic data are quite different from the typical images used in image processing and multimedia applications. Some of their major differences are the data dynamic range exceeding 100 dB in theory, very often it is data with extensive oscillatory nature, the x and y directions represent different physical meaning, and there is significant amount of coherent noise which is often present in seismic data. The objective of this paper is to achieve higher compression ratio, than achieved with the wavelet/uniform quantization/Huffman coding family of compression schemes, with a comparable level of residual noise. The goal is to achieve above 40dB in the decompressed seismic data sets. One of the conclusions is that adaptive multiscale local cosine transform with different windows sizes performs well on all the seismic data sets and outperforms the other methods from the SNR point of view. Comparison with other methods (old and new) are given in the full paper. The main conclusion is that multidimensional adaptive multiscale local cosine transform with different windows sizes perform well on all the seismic data sets and outperforms other methods from the SNR point of view. Special emphasis was given to achieve faster processing speed which is another critical issue that is examined in the paper. Some of these algorithms are also suitable for multimedia type compression.

Exploiting the geometrical interpretation of the Haar transform as a rotation of the coordinate system by an angle (pi) /4, a new Haar-like transform is proposed which rotates the co-ordinate system by an angle which depends on the data. This data dependent transform gives better compression ratios for both signals and images than the Haar transform.

Compact support is undoubtedly one of the wavelet properties that is given the greatest weight both in theory and applications. It is usually believed to be essential for two main reasons : (1) to have fast numerical algorithms, and (2) to have good time or space localization properties. Here, we argue that this constraint is unnecessarily restrictive and that fast algorithms and good localization can also be achieved with non-compactly supported basis functions. By dropping the compact support requirement, one gains in flexibility. This opens up new perspectives such as fractional wavelets whose key parameters (order, regularity, etc...) are tunable in a continuous fashion. To make our point, we draw an analogy with the closely related task of image interpolation. This is an area where it was believed until very recently that interpolators should be designed to be compactly supported for best results. Today, there is compelling evidence that non-compactly supported interpolators (such as splines, and others) provide the best cost/performance tradeoff.

In this work, a multi-channel model for image representation is derived based on the scale-space theory. This model is inspired in biological insights and it includes some important properties of human vision such as the Gaussian derivative model for early vision proposed by Young. The image transform that we propose in this work uses similar analysis operators as the Hermite transform at multiple scales, but the synthesis scheme of our approach integrates the responses of all channels at different scales. The advantages of this scheme are: 1) both analysis and synthesis operators are Gaussian derivatives. This allows for simplicity during implementation. 2) The operator functions possess better space-frequency localization, and it is possible to separate adjacent scales one octave apart, according to Wilson's results on human vision channels. 3) In the case of 2-D signals, it is easy to analyze local orientations at different scales. A discrete approximation is also derived from an asymptotic relation between the Gaussian derivatives and the discrete binomial filters. We show in this work how the proposed transform can be applied to the problem of image coding. Practical considerations are also of concern.

We present a library of biorthogonal wavelet transforms and the related library of biorthogonal symmetric waveforms. For the construction we use interpolatory, quasiinterpolatory and smoothing splines with finite masks (local splines). With this base we designed a set of perfect reconstruction infinite and finite impulse response filter banks with linear phase property. The construction is performed in a lifting manner. The developed technique allows to construct wavelet transforms with arbitrary prescribed properties such as the number of vanishing moments, shape of wavelets, and frequency resolution. Moreover, the transforms contain some scalar control parameters which enable their flexible tuning in either time or frequency domains. The transforms are implemented in a fast way. The transforms, which are based on interpolatory splines, are implemented through recursive filtering. We present encouraging results towards image compression.

We present a coder that yields good quality images at very high compression rates. It performs embedded coding and can carry out both lossy and lossless compression, properties which are suitable for progressive transmission. It is based on an integer to integer wavelet transform, and uses augmented zerotrees with a hybrid technique that incorporates bitplane coding as well.

The discrete wavelet transform was introduced as a linear operator. It works on signals that are modeled as functions from the integers into the real or complex numbers. Since many signals have finite function values, a linear discrete wavelet transform over a finite ring has been proposed recently. Another recent development is the research of nonlinear wavelet transforms triggered by the introduction of Sweldens' lifting scheme. This paper builds on these developments and defines an essentially nonlinear translation invariant discrete wavelet transform that works on signals that are functions from the integers into any finite set. As only discrete arithmetic is needed, such transforms can be calculated very time efficiently. The basic properties of these generalized discrete wavelet transforms are given along with explicit examples.

We present a new class of wavelet bases---Fresnelets---which is obtained by applying the Fresnel transform operator to a wavelet basis of L₂. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing optically generated holograms recorded on CCD-arrays. We first investigate the multiresolution properties (translation, dilation) of the Fresnel transform that are needed to construct our new wavelet. We derive a Heisenberg-like uncertainty relation that links the localization of the Fresnelets with that of the original wavelet basis. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. We conclude that the Fresnel B-splines are particularly well suited for processing holograms because they tend to be well localized in both domains.

We present a generic approach that identifies and differentiates among signals for wide range of problems. Originally our algorithm was developed to detect the presence of a specific vehicle belonging to a certain class via the analysis of the acoustic signals emitted while it is moving. A crucial factor in having a successful detection (no false alarm) is to construct signatures built from characteristic features that enable to discriminate between the class of interest and the residual information such as background. We construct the signatures of certain classes by the distribution of the energies among blocks which consist of wavelet packet coefficients. We developed an efficient procedure for adaptive selection of the characteristic blocks. We modified the CART algorithm in order to utilize it to be a decision unit in our scheme. However, this technology, which has many algorithmic variations, can be used to solve a wide range of classification and detection problems which are based on acoustic processing and, more generally, for classification and detection of signals which have near-periodic structure. We present results of successful application of the properly modified algorithm to detection of early symptoms of arterial hypertension in children via real-time analysis of pulse signals.

Conventional radars and communications systems employ waveforms with a set of bandwidth constraints for a given application. Unfortunately, for many applications, such generic waveforms do not employ the optimal use of bandwidth or energy to accomplish the mission of the user due to electromagnetic clutter and noise. We therefore suggest an alternate approach that uses a matching pursuits algorithm in conjunction with two types of detection statistics to design the optimal waveform for the application. We then will demonstrate how this approach can maximize the matched filter detection performance and band-width allocation in a radar and communications example with a high interference environment.

Wavelet-based image denoising is very attractive to analysis and synthesis functions. It enables us to divide a complicated function into several simpler ones and study them individually. In this paper, we present a new image-denoising algorithm based on multiresolution local contrast entropy of the wavelength coefficients. We introduce a new adaptive threshold estimation depending on the probability distribution of the noise in the wavelet coefficients. This threshold enables the proposed algorithm to be adaptive to unknown smoothness of the denoising images. The experiment confirms that the proposed approach is cable of achieving good results for additive white Gaussian noise.

Wavelet signal processing has demonstrated remarkable capabilities in reducing noise, achieving better resolution through edge detection and increasing data transmission by means of data compression. While wavelets are digital, another field, Optical Phase Conjugation (OPC), is analog and has been applied to similar problems: signal and image distortion reduction and optical data storage. Wavelets have been applied to optical solitons, laser beam diagnostics, diode laser arrays, interferometry and optical correlators. Wavelet signal processing will be applied to Optical Phase Conjugation to examine laser beam interaction in nonlinear crystals and remove distortion from input and output laser beams.

In this paper, a novel optical correlation system on the basis of wavelet packet theory and the mechanism of volume holographic associative storage is proposed for image recognition. Through the wavelet packet transform, a set of best eigen-images, which are regarded as the reference images for recognition in the associative correlation, are extracted from the training images, and then stored into a volume holographic crystal using the two-wave mixing volume holographic storage technique. When any image for identification is input into the crystal which means a correlator, angularly separated beams with different light intensities are obtained simultaneously. They represent the optical correlation results between the input and the set of eigen-images, and can be applied for the classification and recognition. This process takes the advantages of both the agility of wavelet packet transform and the high degree of parallelism of the photorefractive correlator. Theoretical analysis of this process is presented, and experimental results are given.

The Local Fourier Transform (LFT) provides a nice tool for concentrating both a signal and its Fourier transform. But there are certain properties of this algorithm that make it unattractive for various applications. In this paper, some of these disadvantages are explored, and a new approach to localized Fourier analysis is proposed, the continuous boundary local Fourier transform (CBLFT), which attempts to correct some of these shortcomings. Results ranging from segmentation to representation cost to compression are also presented.

This paper discusses wavelet thresholding in smoothing from non-equispaced, noisy data in one dimension. To deal with the irregularity of the grid we use so called second generation wavelets, based on the lifting scheme. We explain that a good numerical condition is an absolute requisite for successful thresholding. If this condition is not satisfied the output signal can show an arbitrary bias. We examine the nature and origin of stability problems in second generation wavelet transforms. The investigation concentrates on lifting with interpolating prediction, but the conclusions are extendible. The stability problem is a cumulated effect of the three successive steps in a lifting scheme: split, predict and update. The paper proposes three ways to stabilize the second generation wavelet transform. The first is a change in update and reduces the influence of the previous steps. The second is a change in prediction and operates on the interval boundaries. The third is a change in splitting procedure and concentrates on the irregularity of the data points. Illustrations show that reconstruction from thresholded coefficients with this stabilized second generation wavelet transform leads to smooth and close fits.

It is well-known to waveleticians that refinable functions exhibit subtle relationships between their approximation order and smoothness properties. We show how one can exploit this phenomenon to construct Hermite subdivision schemes with optimal smoothness but suboptimal approximation order for a given support size of the subdivision mask. The construction method considered here is based on a blend of the theory of subdivision schemes and computational techniques in non-smooth optimization. Our construction method produces schemes which are much smoother than those constructed based on optimizing approximation orders. We discuss also several interesting bivariate Hermite schemes, with appealing symmetry property, and illustrate how they can be applied to build interpolating subdivision surfaces.

This paper considers reversible transforms which are used in wavelet compression according to nowadays JPEG2000 standard. Original data decomposition in a form of integer wavelet transformation realized in subband decomposition scheme is optimized by design and selection of the most effective transforms. Lifting scheme is used to construct new biorthogonal symmetric wavelets. Number and distribution of vanishing moments, subband coding gain, associated filter length, computational complexity and number of lifting steps were mainly analyzed in the optimization of designed transforms. Coming from many tests of compression efficiency evaluation in JPEG2000 standardization process, the best selected transforms have been compared to designed ones to conclude the most efficient for compression wavelet bases and their important features. Certain new transforms overcome all other in both phases of lossy-to-lossless compression (e.g. up to 0.5 dB of PSNR for 0.5 bpp in comparison to the state-of-art transforms of JPEG2000 compression, and up to 3dB over 5/3 standard reversible transform). Moreover, the lossy compression efficiency of proposed reversible wavelets is comparable to reference irreversible wavelets potential in several cases. The highest improvement over that reference PSNR values is close to 1.2 dB.

This paper combines the use of wavelet decompositions and Zernike polynomial approximations to extract aberration coefficients associated to an interferogram. Zernike polynomials are well known to represent aberration components of a wave-front. Polynomial approximation properties on a discrete mesh after an orthogonalization process via Gram-Schmidt decompositions are very useful to straightforward estimate aberration coefficients. It is shown that decomposition of interferograms into wavelet domains can reduce the number of computations without a significant effect on the estimated aberration coefficients amplitudes if full size interferograms were considered. Haar wavelets because of their non-overlapping and time localization properties appear to be well suited for this application. Aberration coefficients can be computed from multi resolution decompositions schemes and 2-D Zernike polynomial approximations on coarser scales, providing the means to reduce computational complexity on such calculations.

High resolution multidimensional image data yield huge datasets. For compression and analysis, 2D approaches are often used, neglecting the information coherence in higher dimensions, which can be exploited for improved compression. We designed a wavelet compression algorithm suited for data of arbitrary dimensions, and assessed its ability for compression of 4D medical images. Basically, separable wavelet transforms are done in each dimension, followed by quantization and standard coding. Results were compared with conventional 2D wavelet. We found that in 4D heart images, this algorithm allowed high compression ratios, preserving diagnostically important image features. For similar image quality, compression ratios using the 3D/4D approaches were typically much higher (2-4 times per added dimension) than with the 2D approach. For low-resolution images created with the requirement to keep predefined key diagnostic information (contractile function of the heart), compression ratios up to 2000 could be achieved. Thus, higher-dimensional wavelet compression is feasible, and by exploitation of data coherence in higher image dimensions allows much higher compression than comparable 2D approaches. The proven applicability of this approach to multidimensional medical imaging has important implications especially for the fields of image storage and transmission and, specifically, for the emerging field of telemedicine.

We present here new results in Wavelet Multi-Resolution Analysis (W-MRA) applied to shape recognition in automatic vehicle driving applications. Different types of shapes have to be recognized in this framework. They pertain to most of the objects entering the sensors field of a car. These objects can be road signs, lane separation lines, moving or static obstacles, other automotive vehicles, or visual beacons. The recognition process must be invariant to global, affine or not, transformations which are : rotation, translation and scaling. It also has to be invariant to more local, elastic, deformations like the perspective (in particular with wide angle camera lenses), and also like deformations due to environmental conditions (weather : rain, mist, light reverberation) or optical and electrical signal noises. To demonstrate our method, an initial shape, with a known contour, is compared to the same contour altered by rotation, translation, scaling and perspective. The curvature computed for each contour point is used as a main criterion in the shape matching process. The original part of this work is to use wavelet descriptors, generated with a fast orthonormal W-MRA, rather than Fourier descriptors, in order to provide a multi-resolution description of the contour to be analyzed. In such way, the intrinsic spatial localization property of wavelet descriptors can be used and the recognition process can be speeded up. The most important part of this work is to demonstrate the potential performance of Wavelet-MRA in this application of shape recognition.