Multiresolution Analyses Research Articles

Dual wavelets associated with nonuniform MRA

A generalization of Mallats classical multiresolution analysis, based on thetheory of spectral pairs, was considered in two articles by Gabardo and Nashed. In thissetting, the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair and the associated dilation is aneven positive integer related to the given spectral pair. In this paper, we construct dualwavelets which are associated with Nonuniform Multiresolution Analysis. We show thatif the translates of the scaling functions of two multiresolution analyses are biorthogonal,then the associated wavelet families are also biorthogonal. Under mild assumptions onthe scaling functions and the wavelets, we also show that the wavelets generate Rieszbases

Generalized nonuniform multiresolution analyses

Generalized nonuniform multiresolution analyses

Invariant sets for QMF functions

A quadrature mirror filter (QMF) function can be considered as the transition function for a Markov process on the unit interval. The QMF functions that generate scaling functions for multiresolution analyses are then distinguished by properties of their invariant sets. By characterizing these sets, we answer in the affirmative a question raised by Gundy.

Efficient Gradient‐Domain Compositing Using an Approximate Curl‐free Wavelet Projection

AbstractGradient‐domain compositing has been widely used to create a seamless composite with gradient close to a composite gradient field generated from one or more registered images. The key to this problem is to solve a Poisson equation, whose unknown variables can reach the size of the composite if no region of interest is drawn explicitly, thus making both the time and memory cost expensive in processing multi‐megapixel images. In this paper, we propose an approximate projection method based on biorthogonal Multiresolution Analyses (MRA) to solve the Poisson equation. Unlike previous Poisson equation solvers which try to converge to the accurate solution with iterative algorithms, we use biorthogonal compactly supported curl‐free wavelets as the fundamental bases to approximately project the composite gradient field onto a curl‐free vector space. Then, the composite can be efficiently recovered by applying a fast inverse wavelet transform. Considering an n‐pixel composite, our method only requires 2n of memory for all vector fields and is more efficient than state‐of‐the‐art methods while achieving almost identical results. Specifically, experiments show that our method gains a 5× speedup over the streaming multigrid in certain cases.

A multiscale collocation method for fractional differential problems

We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale collocation method are proved and some numerical results are shown.

Application of wavelet decomposition in time-series forecasting

Observed time series data can exhibit different components, such as trends, seasonality, and jumps, which are characterized by different coefficients in their respective data generating processes. Therefore, fitting a given time series model to aggregated data can be time consuming and may lead to a loss of forecasting accuracy. In this paper, coefficients for variable components in estimations are generated based on wavelet-based multiresolution analyses. Thus, the accuracy of forecasts based on aggregate data should be improved because the constraint of equality among the model coefficients for all data components is relaxed.

Parallel Statistical Multiresolution Estimation for Image Reconstruction

We show that a careful parallelization of statistical multiresolution estimation (SMRE) improves the phase reconstruction in X-ray near-field holography. The central step in, and the computationally most expensive part of, SMRE methods is Dykstra's algorithm. It projects a given vector onto the intersection of convex sets. We discuss its implementation on NVIDIA's compute unified device architecture (CUDA). Compared to a CPU implementation parallelized with OpenMP, our CUDA implementation is up to one order of magnitude faster. Our results show that a careful parallelization of Dykstra's algorithm enables its use in large-scale statistical multiresolution analyses.

MULTIRESOLUTION ANALYSES ON QUASILATTICES

We derive relations between geometric means of the Fourier moduli of a refinable distribution and of a related polynomial. We use Pisot-Vijayaraghavan numbers to construct families of one dimension quasilattices and multiresolution analyses spanned by distributions that are refinable with respect to dilation by the PV numbers and translation by quasilattice points. We conjecture that scalar valued refinable distributions are never integrable, construct piecewise constant vector valued refinable functions, and discuss multidimensional extensions.

Identification of Copy Number Aberrations in Breast Cancer Subtypes Using Persistence Topology.

DNA copy number aberrations (CNAs) are of biological and medical interest because they help identify regulatory mechanisms underlying tumor initiation and evolution. Identification of tumor-driving CNAs (driver CNAs) however remains a challenging task, because they are frequently hidden by CNAs that are the product of random events that take place during tumor evolution. Experimental detection of CNAs is commonly accomplished through array comparative genomic hybridization (aCGH) assays followed by supervised and/or unsupervised statistical methods that combine the segmented profiles of all patients to identify driver CNAs. Here, we extend a previously-presented supervised algorithm for the identification of CNAs that is based on a topological representation of the data. Our method associates a two-dimensional (2D) point cloud with each aCGH profile and generates a sequence of simplicial complexes, mathematical objects that generalize the concept of a graph. This representation of the data permits segmenting the data at different resolutions and identifying CNAs by interrogating the topological properties of these simplicial complexes. We tested our approach on a published dataset with the goal of identifying specific breast cancer CNAs associated with specific molecular subtypes. Identification of CNAs associated with each subtype was performed by analyzing each subtype separately from the others and by taking the rest of the subtypes as the control. Our results found a new amplification in 11q at the location of the progesterone receptor in the Luminal A subtype. Aberrations in the Luminal B subtype were found only upon removal of the basal-like subtype from the control set. Under those conditions, all regions found in the original publication, except for 17q, were confirmed; all aberrations, except those in chromosome arms 8q and 12q were confirmed in the basal-like subtype. These two chromosome arms, however, were detected only upon removal of three patients with exceedingly large copy number values. More importantly, we detected 10 and 21 additional regions in the Luminal B and basal-like subtypes, respectively. Most of the additional regions were either validated on an independent dataset and/or using GISTIC. Furthermore, we found three new CNAs in the basal-like subtype: a combination of gains and losses in 1p, a gain in 2p and a loss in 14q. Based on these results, we suggest that topological approaches that incorporate multiresolution analyses and that interrogate topological properties of the data can help in the identification of copy number changes in cancer.

A Probabilistic Methodology to Determine Elastic Acceleration Response Spectra for Pulse-Type Records through Multi-Resolution Analyses

This article focuses on the characterizations of pulse-type ground motions which are mainly caused by rupture directivity. Multi-resolution analysis is employed to decompose 53 typical pulse-type records which are selected from 12 large earthquakes into a series of ground motion components. A methodology for deriving design spectra, which could incorporate the special effects caused by pulse-type records is proposed based on the bi-normalized response spectra of ground motion components, and is named as Near-Fault Response Spectrum (NFRS). Analysis results indicate that NFRS might be a reliable candidate for the code-based design spectrum especially for structures constructed close to active fault.

Isometric Operators on Hilbert Spaces and the Classical Wold Decomposition for Stationary Time Series

The Wold Theorem plays a fundamental role in the decomposition of weakly stationary time series. From an empirical point of view, it enables to identify uncorrelated shocks, for instance in macroeconomic and financial time series. More theoretically, the decomposition of weakly stationary stochastic processes can be seen as a special case of the Abstract Wold Theorem that allows to decompose Hilbert spaces by using isometric operators. In this work we explain this link in detail by using the Hilbert space spanned by a weakly stationary time series and the lag operator as isometry. Our methodology, fully based on operator theory, provides some tools that are useful for discovering new Wold-type decompositions of stochastic processes, in which the involved isometry is no more the lag operator. Examples may be given by rescaling operators inherited from multiresolution analyses of time series, which may deliver a persistence-based decomposition.

Multiresolution analysis for compactly supported interpolating tensor product wavelets

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C0(ℝn, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝn vanishing at infinity and the function spaces Cu(ℝn, K) consisting of bounded and uniformly continuous functions on ℝn. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.

Extension of Bessel sequences to oblique dual frame sequences and the minimal projection

An extension of two Bessel sequences to oblique dual frame sequences and its applications to shift-invariant spaces are considered. The best-known situation where this kind of extension is necessary is the construction of a pair of biorthogonal multiresolution analyses, where two generating sets whose shifts are only assumed to be Bessel sequences are given. This extension naturally leads to consideration of the ‘minimal projection’ extending two closed subspaces. The existence or non-existence of the minimal projection is discussed.

Advancements in multiresolution analysis

SummaryMaterials have a hierarchical nature, deriving often their most useful properties from microscale or nanoscale constituents. Multiresolution analysis, a generalized continuum mechanics‐based theory, uses extra degrees of freedom to account for an arbitrary number of these nested length scales. In the past, multiresolution analyses have focused mostly on metal alloys. While this article addresses recent advances in image‐based multiresolution analysis of metal alloys, it also highlights extensions to multiresolution theory for modeling of bone mechanics and multiresolution analysis of polymers and polymer nanocomposites. A strong link between molecular dynamics simulations and macroscale multiresolution analyses is shown for both polymers and polymer nanocomposites. The forthcoming work is greatly indebted to the pioneering advances of Ted Belytschko in many areas of computational mechanics; his influence on our work and on the field of finite elements as a whole is substantial Copyright © 2014 John Wiley & Sons, Ltd.

Image Watermark Based on Extended Shearlet and Insertion Using the Largest Information Entropy on Horizontal Cone

Extended discrete shearlet provides a directional multiresolution decomposition. It has been mathematically shown that extended discrete shearlet is a more efficient representation for the signals containing distributed discontinuities such as edges, compared to discrete wavelet. Multiresolution analyses such as curvelet and ridgelet share similar properties, yet their directional representations are significantly different from that of extended discrete shearlet. Taking advantage of the unique properties of directional representation of extended discrete shearlet, we develop an image watermark algorithm based on the largest information entropy. In proposed algorithm, firstly, 1-level extended discrete shearlet transform decomposes the test image into directional components on horizontal cone; each directional component reflects directional features and textured features differently. Next, the directional component whose information entropy is the highest is selected to carry watermark. Compared with related algorithms based on DWT and DCT, the proposed algorithm tends to obtain preferable invisibility when it is robust against common attacks.

On Relating Interpolatory Wavelets to Interpolatory Scaling Functions in Multiresolution Analyses

Construction of interpolatory wavelets is an important topic in discrete signal processing. In classical wavelet sampling theories, interpolatory wavelets are constructed from Riesz bases in wavelet spaces. Since analytical expressions for such Riesz bases are generally complex or unavailable, it has been difficult to obtain suitable interpolatory wavelets in practice. In this paper, interpolatory scaling functions are used to determine and construct interpolatory wavelets. We first show that there may not exist interpolatory wavelets even when interpolatory scaling functions exist. Then, an inequality in terms of interpolatory scaling functions, denoted as the two-scale condition, is given as a necessary and sufficient condition for existence of interpolatory wavelets. Finally, based on the two-scale condition, a filter bank is constructed for obtaining interpolatory wavelets directly from interpolatory scaling functions. In examples, our theorems are applied to some typical wavelet spaces, demonstrating our construction algorithm for interpolatory wavelets.

Evaluation of visual impact of multiple image characteristics observed in edge-grain patterns

Twenty distinct computer-generated softwood-like monochrome edge-grain patterns were printed on matte paper. Four image characteristics including mean growth ring width, growth ring width fluctuation, growth ring contrast, and shading contrast were embedded in the patterns. These image characteristics were expressed numerically by line profile and multiresolution contrast analyses. Five visual impressions, varied, agreeable, vague, showy, and natural, of these patterns on 30 subjects were evaluated by ranking. To determine the importance of each image characteristics on the impressions, a multiple regression analysis was conducted by attributing a criterion variable to an impression and predictor variables to all four image characteristics. This analysis revealed the impacts of the four image characteristics on each impression quantitatively. The results suggested that shading contrast was one of the useful indicators to evaluate surface harmony or homogeneity related to varied impression. Although growth ring width fluctuation had been considered as an important visual feature of wood, its influence on the impressions was extremely small compared with the other three characteristics.

Multiresolution Equivalence and Path Connectedness

An equivalence relation between multiresolution analyses was first introduced in 1996; an analogous definition for generalized multiresolution analyses was given in 2010. This article describes the relationship between the two notions and shows that both types of equivalence classes are path connected in an operator-theoretic sense. The GMRA paths are restricted to canonical GMRAs, and it is shown that whenever two MRAs in L 2(ℝ) are equivalent, the GMRA path construction between their corresponding canonical GMRAs yields the natural analog of the MRA path. Examples are provided of GMRA paths that are distinct from MRA paths.

Multitemporal Analysis of Coastal Erosion Based on Multisource Satellite Images, Ponta Negra Beach, Natal City, Northeastern Brazil

This paper describes the use of moderate and high-resolution remote sensing images integrated with a Geographic Information System to evaluate the sedimentary balance and morphological changes seen on the sandy beach of Ponta Negra in northeastern Brazil. High-energy forces such as wind, tides, waves, and currents cause widespread erosion and morphodynamic instability along the shoreline. This investigation uses diverse images from multitemporal and multiresolution analyses and a Post-Processing Kinematic Global Positioning System (PPK GPS) survey as the primary resources to study the short and long-term effects of shoreline changes and the processes controlling the persistent horizontal and vertical erosion on the beach. We used moderate-resolution remote sensing images from 1973 to 2012 and high-resolution images from 2003 to 2011. Between 1973 and 2012 a total area of 121,494.52 m2 was affected by erosion and 37,266.58 m2 was affected by accretion. Shoreline progradation rates of 4.2 m/year and recession rates of -3.7 m/year were estimated for the period between 1973 and 1986 and mean erosion rates of 1.5 m/year were estimated from 1986 to 2012. Digital Elevation Model (DEM) comparison resulting from different PPK GPS surveys showed a negative sediment balance of approximately -3,261.94 m3 in a plot of 12,500 m2 between May and September 2012. The coast has also been facing a drastically increased human influence in recent decades. In summary, this article aims to contribute information to support the mitigation of coastal erosion and to protect seaside infrastructure.

SMOOTH PARSEVAL FRAMES FOR L2(ℝ) AND GENERALIZATIONS TO L2(ℝd)

Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in L2(ℝd) which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over ℝd, d > 1. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Han's proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in [Formula: see text] which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.