Showing papers on "Multiresolution analysis published in 2003"

Numerical Analysis of Wavelet Methods

Abstract: Introduction. Notations. 1. Basic examples. 1.1 Introduction. 1.2 The Haar system. 1.3 The Schauder hierarchical basis. 1.4 Multivariate constructions. 1.5 Adaptive approximation. 1.6 Multilevel preconditioning. 1.7 Conclusions. 1.8 Historical notes. 2. Multiresolution approximation. 2.1 Introduction. 2.2 Multiresolution analysis. 2.3 Refinable functions. 2.4 Subdivision schemes. 2.5 Computing with refinable functions. 2.6 Wavelets and multiscale algorithms. 2.7 Smoothness analysis. 2.8 Polynomial exactness. 2.9 Duality, orthonormality and interpolation. 2.10 Interpolatory and orthonormal wavelets. 2.11 Wavelets and splines. 2.12 Bounded domains and boundary conditions. 2.13 Point values, cell averages, finite elements. 2.14 Conclusions. 2.15 Historical notes. 3. Approximation and smoothness. 3.1 Introduction. 3.2 Function spaces. 3.3 Direct estimates. 3.4 Inverse estimates. 3.5 Interpolation and approximation spaces. 3.6 Characterization of smoothness classes. 3.7 Lp-unstable approximation and 0 1. 3.8 Negative smoothness and Lp-spaces. 3.9 Bounded domains. 3.10 Boundary conditions. 3.11 Multilevel preconditioning. 3.12 Conclusions. 3.13 Historical notes. 4. Adaptivity. 4.1 Introduction. 4.2 Nonlinear approximation in Besov spaces. 4.3 Nonlinear wavelet approximation in Lp. 4.4 Adaptive finite element approximation. 4.5 Other types of nonlinear approximations. 4.6 Adaptive approximation of operators. 4.7 Nonlinear approximation and PDE's. 4.8 Adaptive multiscale processing. 4.9 Adaptive space refinement. 4.10 Conclusions. 4.11 Historical notes. References. Index.

Fast parametric elastic image registration

Abstract: We present an algorithm for fast elastic multidimensional intensity-based image registration with a parametric model of the deformation. It is fully automatic in its default mode of operation. In the case of hard real-world problems, it is capable of accepting expert hints in the form of soft landmark constraints. Much fewer landmarks are needed and the results are far superior compared to pure landmark registration. Particular attention has been paid to the factors influencing the speed of this algorithm. The B-spline deformation model is shown to be computationally more efficient than other alternatives. The algorithm has been successfully used for several two-dimensional (2-D) and three-dimensional (3-D) registration tasks in the medical domain, involving MRI, SPECT, CT, and ultrasound image modalities. We also present experiments in a controlled environment, permitting an exact evaluation of the registration accuracy. Test deformations are generated automatically using a random hierarchical fractional wavelet-based generator.

Frames of subspaces

Abstract: One approach to ease the construction of frames is to first construct local components and then build a global frame from these. In this paper we will show that the study of the relation between a frame and its local components leads to the definition of a frame of subspaces. We introduce this new notion and prove that it provides us with the link we need. It will also turn out that frames of subspaces behave as a generalization of frames. In particular, we can define an analysis, a synthesis and a frame operator for a frame of subspaces, which even yield a reconstruction formula. Also concepts such as completeness, minimality, and exactness are introduced and investigated. We further study several constructions of frames of subspaces, and also of frames and Riesz frames using the theory of frames of subspaces. An important special case are harmonic frames of subspaces which generalize harmonic frames. We show that wavelet subspaces coming from multiresolution analysis belong to this class.

Multiresolution registration of remote sensing imagery by optimization of mutual information using a stochastic gradient

Abstract: Image registration is the process by which we determine a transformation that provides the most accurate match between two images. The search for the matching transformation can be automated with the use of a suitable metric, but it can be very time-consuming and tedious. We introduce a registration algorithm that combines a simple yet powerful search strategy based on a stochastic gradient with two similarity measures, correlation and mutual information, together with a wavelet-based multiresolution pyramid. We limit our study to pairs of images, which are misaligned by rotation and/or translation, and present two main results. First, we demonstrate that, in our application, mutual information may be better suited for sub-pixel registration as it produces consistently sharper optimum peaks than correlation. Then, we show that the stochastic gradient search combined with either measure produces accurate results when applied to synthetic data, as well as to multitemporal or multisensor collections of satellite data. Mutual information is generally found to optimize with one-third the number of iterations required by correlation. Results also show that a multiresolution implementation of the algorithm yields significant improvements in terms of both speed and robustness over a single-resolution implementation.

Fresnelets: new multiresolution wavelet bases for digital holography

Abstract: We propose a construction of new wavelet-like bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays. The starting point is a wavelet basis of L/sub 2/ to which we apply a unitary Fresnel transform. The transformed basis functions are shift-invariant on a level-by-level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the Fresnel domain. We derive a Heisenberg-like uncertainty relation that relates the localization of Fresnelets with that of their associated wavelet basis. According to this criterion, the optimal functions for digital hologram processing turn out to be Gabor (1948) functions, bringing together two separate aspects of the holography inventor's work. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. This special choice of Fresnelets is motivated by their near-optimal localization properties and their approximation characteristics. We then present an efficient multiresolution Fresnel transform algorithm, the Fresnelet transform. This algorithm allows for the reconstruction (backpropagation) of complex scalar waves at several user-defined, wavelength-independent resolutions. Furthermore, when reconstructing numerical holograms, the subband decomposition of the Fresnelet transform naturally separates the image to reconstruct from the unwanted zero-order and twin image terms. This greatly facilitates their suppression. We show results of experiments carried out on both synthetic (simulated) data sets as well as on digitally acquired holograms.

Wavelets and statistical analysis of functional magnetic resonance images of the human brain.

Abstract: Wavelets provide an orthonormal basis for multiresolution analysis and decorrelation or 'whitening' of nonstationary time series and spatial processes. Wavelets are particularly well suited to analysis of biological signals and images, such as human brain imaging data, which often have fractal or scale-invariant properties. We briefly define some key properties of the discrete wavelet transform (DWT) and review its applications to statistical analysis of functional magnetic resonance imaging (fMRI) data. We focus on time series resampling by 'wavestrapping' of wavelet coefficients, methods for efficient linear model estimation in the wavelet domain, and wavelet-based methods for multiple hypothesis testing, all of which are somewhat simplified by the decorrelating property of the DWT.

First- and Higher-Order Correlation Detection Using Wavelet Transforms

Abstract: In order to detect intermittent first- and higher-order correlation between a pair of signals in both time and frequency, a wavelet-based coherence and bicoherence technique was developed. Due to the limited averaging in a time-frequency coherence estimate, spurious correlated pockets were detected due to statistical variance. The introduction of multiresolution, localized integration windows was shown to minimize this effect. A coarse ridge extraction scheme utilizing hard thresholding was then applied to extract meaningful coherence. This thresholding scheme was further enhanced through the use of ''smart'' thresholding maps, which represent the likely statistical noise between uncorrelated simulated signals bearing the same power spectral density and probability-density function as the measured signals. It was demonstrated that the resulting filtered wavelet coherence and bicoherence maps were capable of capturing low levels of first- and higher-order correlation over short time spans despite the presence of ubiquitous leakage and variance errors. Immediate applications of these correlation detection analysis schemes can be found in the areas of bluff body aerodynamics, wave- structure interactions, and seismic response of structures where intermittent correlation between linear and nonlinear processes is of interest. DOI: 10.1061/~ASCE!0733-9399~2003!129:2~188! CE Database keywords: Correlation; Methodology.

Combined line-based architecture for the 5-3 and 9-7 wavelet transform of JPEG2000

Abstract: The wavelet transform is a very promising tool for image compression. In JPEG2000, two filter banks are used, one an integer lossless 5-3 filter, and one a lossy 9-7. A combined architecture for the 5-3 and 9-7 transforms with minimum area is presented. The lifting scheme is used to realize a fast wavelet transform. Two lines are processed at a time. This line-based architecture allows minimum memory requirement and fast calculation. The pipeline and the optimization of the operations provide speed, while the combination of the two transforms in one structure contributes to saving the area. On a VIRTEXE1000-8 FPGA implementation, decoding of 2 pixels per clock cycle can be performed at 110 MHz. Only 19% of the total area of the VIRTEXE1000 is needed. Compared to existing architectures, memory resource and area can be reduced thanks to the proposed solution.

Engineering Surface Analysis With Different Wavelet Bases

Abstract: Traditional surface texture analysis involves filtering surface profiles into different wavelength bands commonly referred to as roughness, waviness and form. The primary motivation in filtering surface profiles is to map each band to the manufacturing process that generated the part and the intended functional performance of the component. Current trends in manufacturing are towards tighter tolerances and higher performance standards that require close monitoring of the process. Thus, there is a need for finer bandwidths for process mapping and functional correlation. Wavelets are becoming increasingly popular tools for filtering profiles in an efficient manner into multiple bands. While they have broadly been demonstrated as having superior performance and capabilities than traditional filtering, fundamental issues such as choice of wavelet bases have remained unad-dressed. The major contribution of this paper is to present the differences between wavelets in terms of the transmission characteristics of the associated filter banks, which is essential for surface analysis. This paper also reviews fundamental mathematics of wavelet theory necessary for applying wavelets to surface texture analysis. Wavelets from two basic categories-orthogonal wavelet bases and biorthogonal wavelet bases are studied. The filter banks corresponding to the wavelets are compared and multiresolution analysis on surface profiles is performed to highlight the applicability of this technique.

Wavelet Analysis and Application

Abstract: With the development of Fourier analysis,wavelet analysis become the concerned question of many subjects.On the basis of conception of wavelet transform,the Mallat algorithm has been introduced and applied on signal analysis and disposal,so a failure signal can been decomposed and restructured.Through analysis of experiment,the high frequency part can reflect the turning point,and the position of fault-point can be gained by the important information,so drew a conclusion that application of wavelet analysis is effective on the aspect of singularity measurement.

Multiresolution quantum chemistry in multiwavelet bases

Abstract: Multiresolution analysis in multiwavelet bases is being investigated as an alternative computational framework for molecular electronic structure calculations. The features that make it attractive include an orthonormal basis, fast algorithms with guaranteed precision and sparse representations of many operators (e.g., Green functions). In this paper, we discuss the multiresolution formulation of quantum chemistry including application to density functional theory and developments that make practical computation in three and higher dimensions.

Multi- spectral image resolution refinement using stationary wavelet transform

Abstract: Presents a pixel-level fusion method to refine the resolution of a multi-spectral image using a high-resolution panchromatic image. Our approach is an adaptation of the ARSIS method which takes into account the higher-order statistical moments of the wavelet coefficients and allow processing of non-dyadic images.

A new algorithm for image indexing and retrieval using wavelet correlogram

Abstract: In this paper, a new algorithm for content-based image indexing and retrieval is presented. The proposed method is based on a combination of multiresolution analysis and color correlation histogram of the image. According to the new algorithm, wavelet coefficients of the image are computed first using Daubechies wavelets. A quantization step is then applied before computing horizontal and vertical color correlograms of the wavelet coefficients. Finally, index vectors are constructed using these wavelet correlograms. The retrieval results obtained by the new method on a 1000 image database demonstrated a significant improvement in effectiveness and efficiency compared to the indexing and retrieval methods based on image color correlogram or wavelet transform.

Sharpening of very high resolution images with spectral distortion minimization

Abstract: This work presents a viable solution to the problem of merging a multispectral image with an arbitrary number of bands with a higher resolution panchromatic observation. The proposed method relies on the generalized Laplacian pyramid, which is a multiscale oversampled structure in which spatial details are mapped on different scales. The goal is to selectively perform spatial-frequencies spectrum substitution from an image to another with the constraint of thoroughly retaining the spectral information of the coarser data. To this end, a vector injection model has been defined: at each pixel, the detail vector to be added is always parallel to the approximation. Furthermore, its components are scaled by factors measuring the ratio of local gains between the multispectral and panchromatic data. Such a model is calculated at a coarser resolution where both types of data are available and extended to the finer resolution by embedding the modulation transfer function of the multispectral scanner into the multiresolution analysis. In this way, the interband structure model can be extended to the higher resolution without the drawback of the poor enhancement occurring when the model assumes MTFs close to be ideal. Results are presented and discussed on very high resolution QuickBird data of an urban area.

Multichannel blind image deconvolution using the Bussgang algorithm: spatial and multiresolution approaches

Abstract: This work extends the Bussgang blind equalization algorithm to the multichannel case with application to image deconvolution problems. We address the restoration of images with poor spatial correlation as well as strongly correlated (natural) images. The spatial nonlinearity employed in the final estimation step of the Bussgang algorithm is developed according to the minimum mean square error criterion in the case of spatially uncorrelated images. For spatially correlated images, the nonlinearity design is rather conducted using a particular wavelet decomposition that, detecting lines, edges, and higher order structures, carries out a task analogous to those of the (preattentive) stage of the human visual system. Experimental results pertaining to restoration of motion blurred text images, out-of-focus spiky images, and blurred natural images are reported.

Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties

Abstract: The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a 4×4 Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.

Shape similarity comparison of 3D models using alpha shapes

Abstract: As the number of in-house and public-domain 3D shape models increase, importance of shape-similarity based search and retrieval for 3D shapes models has increased rapidly. In this paper, we describe our preliminary findings in applying a multiresolution analysis technique to the task of shape similarity comparison of polygon soup models. We used the 3D alpha shapes algorithm to create a multiresolution hierarchy of shapes from the given 3D models. We then applied a (single resolution) shape descriptor to each of the models at multiple resolution levels to derive a multiresolution shape descriptor. According to our evaluation experiments, the retrieval performance of our resolution descriptor outperformed its single resolution counterpart, proving the effectiveness of the basic approach.

Multifocus image fusion in wavelet domain

Abstract: How to select wavelet filters, decomposition levels and fusion schemes is of great concern fur multifocus image fusion in wavelet domain We presented a simple image fusion algorithm based on wavelet transform By wavelet transform, an image can be represented by a low frequency approximation, which contain the average information of the image, and several high frequency details with different scales and directions, which contain the texture or edge feature of the image For the multifocus images, there are some areas unclear in certain source images which correspond to small wavelet coefficients, and clear in other source images which correspond to large coefficients So we simply took the coefficients with greater modulus as the final coefficients to get the fusion image, which contains more details and is clearer in visual and more convenient for image analysis and understanding We also discussed the influence of the wavelet filters, decomposition levels and fusion schemes on the fusion results, compared wavelet based fusion with other multiresolution fusions, such as Laplacian pyramid, gradient pyramid, contrast pyramid and ratio pyramid Some results are obtained, which are of great value for research and experiment in this field

Wavelet thresholding for non-necessarily Gaussian noise: idealism

Abstract: For various types of noise (exponential, normal mixture, compactly supported, ...) wavelet thresholding methods are studied. Problems linked to the existence of optimal thresholds are tackled, and minimaxity properties of the methods also analyzed. A coefficient dependent method for choosing thresholds is also briefly presented.

Video Denoising Using Multiple Class Averaging with Multiresolution

Abstract: This paper presents a non-linear technique for noise reduction in video that is suitable for real-time processing The proposed algorithm automatically adapts to detected levels of detail and motion, but also to the noise level, provided it is short-tail noise, such as Gaussian noise It uses a one-level wavelet decomposition, and performs independent processing in four different bands in the wavelet domain The non-decimated transform is used because it leads to better results for image/video denoising than the decimated transform The results show that from both a PSNR and a visual quality, the proposed filter outperforms the other state of the art filters for different image sequences

Multiresolution analysis in statistical mechanics. II. The wavelet transform as a basis for Monte Carlo simulations on lattices

Abstract: In this paper, we extend our analysis of lattice systems using the wavelet transform to systems for which exact enumeration is impractical. For such systems, we illustrate a wavelet-accelerated Monte Carlo (WAMC) algorithm, which hierarchically coarse-grains a lattice model by computing the probability distribution for successively larger block spins. We demonstrate that although the method perturbs the system by changing its Hamiltonian and by allowing block spins to take on values not permitted for individual spins, the results obtained agree with the analytical results in the preceding paper, and “converge” to exact results obtained in the absence of coarse-graining. Additionally, we show that the decorrelation time for the WAMC is no worse than that of Metropolis Monte Carlo (MMC), and that scaling laws can be constructed from data performed in several short simulations to estimate the results that would be obtained from the original simulation. Although the algorithm is not asymptotically faster than traditional MMC, the new algorithm executes several orders of magnitude faster than a full simulation of the original problem because of its hierarchical design. Consequently, the new method allows for rapid analysis of a phase diagram, allowing computational time to be focused on regions near phase transitions.

Multiresolution analysis as an approach for tool path planning in NC machining

Abstract: Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish cuts in NC machining. In this paper, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting while lower resolution curves are used farther away from the object boundary. Experimental results indicate that wavelet-based tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts.