Space-frequency Localization Research Articles

Shapes of Uncertainty in Spectral Graph Theory

We present a flexible framework for uncertainty principles in spectral graph theory. In this framework, general filter functions modeling the spatial and spectral localization of a graph signal can be incorporated. It merges several existing uncertainty relations on graphs, among others the Landau-Pollak principle describing the joint admissibility region of two projection operators, and uncertainty relations based on spectral and spatial spreads. Using theoretical and computational aspects of the numerical range of matrices, we are able to characterize and illustrate the shapes of the uncertainty curves and to study the space-frequency localization of signals inside the admissibility regions.

Dual Watermarking for Color Images: A New Image Copyright Protection Model based on the Fusion of Successive and Segmented Watermarking

Watermarking has been suggested as a generic technique to solve various problems associated with topics in the areas of digital rights management and multimedia security. Most of the early methods were based on single watermark embedding, but there are great limitations when single watermark embedding algorithms are tried into practical applications. The multiple watermarks are intended to convey different information at the same time. Current approaches rely on adding many watermarks in different bands or channels by means of scaling factor and embedding locations that are mainly defined by experts. This brought many challenges in achieving equilibrium between security, robustness, and quality. The aim of this work is to introduce an intelligent dual watermarking model for colour images that ensure image security for copyright protection (dual watermarking for single purpose). To optimize the dual watermarking requirements, the proposed model that employs both successive (re-) and segmented watermarking techniques is to be implemented with a multi-purpose evolutionary algorithm. Genetic algorithm is adopted to determine the embedding locations and scaling factors for different watermarks according to the features of each host image to balance between imperceptibility and robustness. The wavelet transform is utilized for salient features extraction because of its excellent space-frequency localization of salient image features. In addition, the suggested model encrypts the watermarks with the aid of Walsh transform; so that it is difficult to handle the watermarks even after being extracted by the attackers. Experimental results show that the proposed model is more robust against common image manipulation attacks in terms of PSNR and NCC.

Evaluation of genome similarities using the non-decimated wavelet transform.

The wavelets have become increasingly popular in the field of bioinformatics due to their capacity in multiresolution analysis and space-frequency localization; the latter particularity is acquired due to a moving window that runs through the analyzed space. As a feature, they have a better ability to capture hidden components of biological data and an efficient link between biological systems and the mathematical objects used to describe them. The decomposition of signals/sequences at different levels of resolution allows obtaining distinct characteristics in each level. The energy (variance) obtained at each level provides a new set of information that can be used to search similarities between sequences. We show that the behavior of GC-content sequence can be succinctly described regarding the non-decimated wavelet transform, and we indicate how this characterization can be used to improve clustering of the similar strains of the genome of the Mycobacterium tuberculosis, having a very efficient level of detail. The clustering analysis using the energy obtained at each level of the analyzed sequences was essential to verify the dissimilarity of the sequences.

Wavelet-Compressed Representation of Landscapes for Hydrologic and Geomorphologic Applications

The availability of high-resolution digital elevation data (submeter resolution) from LiDAR has increased dramatically over the past few years. As a result, the efficient storage and transmission of those large data sets and their use for geomorphic feature extraction and hydrologic/environmental modeling are becoming a scientific challenge. This letter explores the use of multiresolution wavelet analysis for compression of LiDAR digital elevation data sets. The compression takes advantage of the fact that, in most landscapes, neighboring pixels are correlated and thus contain some redundant information. The space–frequency localization of the wavelet filters allows one to preserve detailed high-resolution features where needed while representing the rest of the landscape at lower resolution. We explore a lossy compression methodology based on biorthogonal wavelets and demonstrate that, by keeping only approximately 10% of the original information (data compression ratio ∼94%), the reconstructed landscapes retain most of the information of relevance to geomorphologic applications, such as the ability to accurately extract channel networks for environmental flux routing, as well as to identify geomorphic process transition from the curvature–slope and slope–distance relationships.

Multi-resolution analysis of nanocrystalline ITO thin films

Indium tin oxide (ITO) thin films were prepared by the electron beam evaporation method. The films were annealed at different temperatures. The surface texture of ITO films before and after annealing has been characterized through analyzing the surface profile of AFM images using the wavelet transform (WT) method. Continuous wavelet transform (CWT) coefficients as a function of the scale factor indicate that it is possible to distinguish the ‘smooth’ or ‘rough’ areas of the surface profile. The central advantages of the WT method compared to the existing signal processing methods are its space-frequency localization and multi-scale analysis of the roughness features.

In situ investigation of the 3D mechanical microstructure at nanoscale: nano-CT imaging method of local small region in large scale sample.

To investigate the local micro-/nanoscale region in a large scale sample, an image reconstruction method for nanometer computed tomography (nano-CT) was proposed in this paper. In the algorithm, wavelets were used to localize the filtered-backprojection (FBP) algorithm because of its space-frequency localization property. After the implementation of the algorithm, two simulation local reconstruction experiments were performed to confirm its effectiveness. Three evaluation criteria were used in the experiments to judge the quality of the reconstructed images. The experimental results showed that the algorithm proposed in this paper performed best because (1) the quality of its results had improved 20%–30% compared to the results of FBP and 10%–30% compared to the results of another wavelet algorithm; (2) the new algorithm was stable under different circumstances. Besides, an actual reconstruction experiment was performed using real projection data that had been collected in a CT experiment. Two-dimensional (2D) and three-dimensional (3D) images of the sample were reconstructed. The microstructure of the sample could be clearly observed in the reconstructed images. Since much attention has been directed towards the nano-CT technique to investigate the microstructure of materials, this new wavelet-based local tomography algorithm could be considered as a meaningful effort.

The method of separation for evolutionary spectral density estimation of multi-variate and multi-dimensional non-stationary stochastic processes

The method of separation can be used as a non-parametric estimation technique, especially suitable for evolutionary spectral density functions of uniformly modulated and strongly narrow-band stochastic processes. The paper at hand provides a consistent derivation of method of separation based spectrum estimation for the general multi-variate and multi-dimensional case. The validity of the method is demonstrated by benchmark tests with uniformly modulated spectra, for which convergence to the analytical solution is demonstrated. The key advantage of the method of separation is the minimization of spectral dispersion due to optimum time- or space–frequency localization. This is illustrated by the calibration of multi-dimensional and multi-variate geometric imperfection models from strongly narrow-band measurements in I-beams and cylindrical shells. Finally, the application of the method of separation based estimates for the stochastic buckling analysis of the example structures is briefly discussed.

SPACE-FREQUENCY LOCALIZATION AS BIVARIATE MOTHER WAVELETS SELECTING CRITERION FOR HYPERANALYTIC BAYESIAN IMAGE DENOISING

Throughout recent years, many wavelet transforms (WTs) were used in digital image processing: the discrete WT (DWT), the stationary WT (SWT) or the hyperanalytic WT (HWT). All these transforms have in common a feature, the mother wavelets (MW). A great number of MWs was already proposed in literature. The purpose of this paper is the selection of MW for hyperanalytic Bayesian image denoising on the basis of its space-frequency localization. The MW with the same space-frequency localization as the elements of the input image gives the better results. Some procedures for the evaluation of the space-frequency localization of MWs and input images are proposed and applied to optimize the results obtained by the simulations of denoising, indicating the most appropriate MW.

Multiscale analysis of geomorphological and geological features in high resolution digital elevation models using the wavelet transform

At the end of the 1990s the emergence of high resolution (1 m) digital elevation models (DEMs) settled the context of high precision geomorphological analysis. These new elevation models permitted to reveal structures that remained heretofore undetectable. Earth scientists henceforth benefit from a source of data with a textural detail that was never attained before. Despite its richness, this information must be treated efficiently to extract features helping geomorphologists to analyze the processes occurring at the surface. Such processes are complex, localized and naturally multiscale. Recently, space-frequential descriptors as wavelets have been proposed successfully for the analysis of DEMs. The wavelet transform is widely used in image processing since it allows us to decompose a signal into a weighted sum of atoms with joint space-frequency localization. Such a decomposition facilitates a coherent navigation from scale to scale, but also permits to detect heretofore undiscerned phenomena at different scales. This is appealing in geomorphology, where structural components, related to a given phenomenon, are well determined in these sub-spaces specific to the scale continuum. In this paper, we propose a filtering procedure of the wavelet decomposition as an approach for the analysis of geomorphological multiscale structures. This filtering procedure enhances the high pass information contained at each scale. The proposed bottom–up approach is applied here to a case study to detect geomorphological structural elements in a valley of the Swiss Jura. It demonstrates that the proposed filtering procedure is an efficient tool for geomorphological multiscale generalization.

Wavelet Reconstruction of Geologic Facies From Nonlinear Dynamic Flow Measurements

The discrete wavelet transform (DWT) that is widely used in compressing natural images is considered for an effective representation of the geological facies in subsurface flow and transport inverse modeling problems. The inference of the heterogeneous hydraulic rock properties from the scattered dynamic measurements of the flow rates and pressures is a frequently encountered ill-posed inverse problem in subsurface characterization. To better pose this inverse problem, the original grid-based description of the spatial facies maps is replaced with a small number of DWT coefficients that are estimated from indirect nonlinear dynamic measurements. The compressed description of the facies in the wavelet domain after removing the unresolvable high-frequency components leads to an inverse problem with fewer parameters to resolve and improved geologic facies continuity. The main difficulty in the application of the DWT to inverse problems is the lack of sufficient data to resolve higher frequency detail coefficients. Prior information and sensitivity of the flow response to variation in the DWT coefficients are used to infer the location and value of the significant DWT coefficients. The results suggest that the large-scale geologic facies description that control the global flow pattern can be successfully inferred from the dynamic measurements in a reduced wavelet domain. While the flow data may contain information about significant DWT coefficients, a limited observability in ill-posed inverse problems may not allow the identification of these coefficients and the corresponding local spatial features. Therefore, an effective exploitation of the space-frequency localization advantage of the wavelets over the Fourier bases may not be available in solving ill-posed inverse problems.

A Robust Blind Image Watermarking Method Using Local Maximum Amplitude Wavelet Coefficient Quantization

In this paper, an innovative blind watermarking algorithm has been proposed for imagery applications. This algorithm has used the coefficients of the discrete wavelet transform of the host image in the form of super trees to embed the predefined binary watermark in the host image. In this scheme, a pseudo random sequence is generated to determine the exact wavelet super trees used for embedding procedure. In the next step, after choosing the maximum and second maximum amplitude coefficients of each super tree, the distance vector between two coefficients is computed. For embedding bit zero of the specified watermark, the values of the distance vector elements are decreased, while for embedding bit 1, those values will be increased based on the proposed formulas. The experimental results show that the proposed algorithm has significant robustness against image processing attacks, especially JPEG compression and also the PSNR value for the watermarked images generated by the proposed method is more than 42 dB. The watermark bits may be inserted into the host image either in spatial or in the transform domain. It is shown that hiding information in the transform domain will lead to a more robust watermarking system against most of the attacks (6). Among these methods, methods based on discrete wavelet transform (DWT) are of great renown. This is due to specific features of space-frequency localization, multi-resolution display, linear computational complexity and the excellent modeling related to Human Vision System

On the Construction of Discrete Directional Wavelets: HDWT, Space-Frequency Localization, and Redundancy Factor

We address the issue of constructing directional wavelet bases. After considering orthonormal directional wavelets whose Fourier transforms are indicator functions, we give a construction of directional wavelets with fast decay that is based on an hexagonal filter bank tree. An implementation for squarely sampled images and numerical results are presented. Then we discuss the frequency localization of directional wavelet bases. We analyze the incompatibility between a proper localization and the nonredundancy constraint, and show that the nonpermissibility condition can be extended to general wavelet bases that are not necessarily generated by a filter bank tree. At last, we show that there exist directional wavelet tight frames that are well localized and have a redundancy factor arbitrarily close to 1.

Advanced modeling of visual information processing: A multi-resolution directional-oriented image transform based on Gaussian derivatives

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 includes some important properties of human vision such as the Gaussian derivative model for early vision proposed by Young [The Gaussian derivative theory of spatial vision: analysis of cortical cell receptive field line-weighting profiles, General Motors Res. Labs. Rep. 4920, 1986]. The image transform that we propose in this work uses analysis operators similar to those of 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. [H.R. Wilson, J.R. Bergen, A four mechanism model for spatial vision. Vision Res. 19 (1979) 19–32). (3) In the case of two-dimensional (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 problems of image coding, noise reduction and image fusion. Practical considerations are also of concern.

Decorrugation and removal of directional trends of magnetic fields by the wavelet transform: application to archaeological areas

ABSTRACTThe problem of removing directional trends frequently occurs in the processing of magnetic data and also in the subsequent steps of data interpretation. The so‐called corrugations are typical directional trends occurring in levelled data, which may be removed in several ways. Classical techniques are based on high‐pass filtering of the data and successively filtering these transformed data with directional cosine filters. Other linear features are due to real sources, such as pipelines in shallow surveys or dike swarms in regional surveys. They should, nevertheless, be considered as noise, due to the fact that their effect is strong and tends to hide the field features related to structures of more interest. We deal with both kinds of problem, presenting the results of a study in an archaeological area of southern Italy. Decorrugation of magnetic field anomalies is performed using a method based on the excellent space–frequency localization properties of wavelet bases, allowing a very sharp filtering of the field along a selected direction. We compare this technique with the classical one in a synthetic case and find that the wavelet decorrugation is simpler and produces low distortion maps. Besides the field decorrugation, the wavelet approach was also shown to be useful in the subsequent enhancement of the measured field. In fact, we show that the wavelet analysis offers a unique framework where various filtering problems (directional, isotropic, global or local as well) may be easily solved. As regards the archaeological case, strong noisy effects from elongated sources (pipelines) were successfully removed in a sharp and local way.

Multiscale seismic tomography

Summary Seismic traveltime tomography is commonly discretized by a truncated expansion of the pursued model in terms of chosen basis functions. Whether parametrization affects the actual resolving power of a given data set as well as the robustness of the resulting earth model has long been seriously debated. From the perspective of the model resolution, however, there is one important aspect of the parametrization issue of seismic tomography that has yet to be systematically explored, that is, the space–frequency localization of a chosen parametrization. In fact, the two most common parametrizations tend to enforce resolution in each of their own particular domains. Namely, parametrization in terms of spherical harmonics with global support tends to emphasize spectral resolution while sacrificing the spatial resolution, whereas the compactly supported pixels tend to behave in the opposite manner. Some of the significant discrepancies among tomographic models are very likely to be manifestations of this effect, when dealing with data sets with non-uniform sampling. With an example of the tomographic inversion for the lateral shear wave heterogeneity of the D″ layer using S–SKS traveltimes, we demonstrate an alternative parametrization in terms of the multiresolution representation of the pursued model function. Unlike previous attempts of multiscale inversion that invoke pixels with variable sizes, or overlay several layers of tessellation with different grid intervals, our formulation invokes biorthogonal generalized Harr wavelets on a sphere. We show that multiresolution representation can be constructed very easily from an existing block-based discretization. A natural scale hierarchy of the pursued model structure constrained by the resolving power of the given sampling is embedded within the solution obtained. It provides a natural regularization scheme based on the actual ray-path sampling and is thus free from a priori prejudices intrinsic to most regularization schemes. Unlike solutions obtained through spherical harmonics or spherical blocks that tend to collapse structures onto ray paths, our parametrization imposes regionally varying Nyquist limits, that is, robustly resolvable local wavelength bands within the obtained solution.

A Haar Wavelet Approach to Compressed Image Quality Measurement

The traditional mean-squared-error and peak-signal-to-noise-ratio error measures are mainly focused on the pixel-by-pixel difference between the original and compressed images. Such metrics are improper for subjective quality assessment, since human perception is very sensitive to specific correlations between adjacent pixels. In this work, we explore the Haar wavelet to model the space–frequency localization property of human visual system (HVS) responses. It is shown that the physical contrast in different resolutions can be easily represented in terms of wavelet coefficients. By analyzing and modeling several visual mechanisms of the HVS with the Haar transform, we develop a new subjective fidelity measure which is more consistent with human observation experience.

Morphological characterisation of engineered surfaces by wavelet transform

This work introduces a wavelet transform technique based on the detection of the multi-scale view of the components of a signal . The main advantages of wavelet transform over the existing signal processing techniques are its space-frequency localization and multi-scale analysis of roughness and waviness motifs. After an extensive review of the mathematical and signal processing fundamentals of the wavelet technique, numerical implementations are carried out to explore the potential applications of wavelet transform and its inverse transform multi-scale analysis of roughness and local morphological characterization by the detection of the roughness singularities and 21) motif size of an engineered surface.

A methodology for the multi-resolution simulation of grinding wheel surface

A new analysis method, multi-resolution analysis which is based on wavelet transform is proposed in the study of a grinding wheel surface. The main advantages of the wavelet transform over the traditional signal processing techniques are its space-frequency localization and multiscale view of the components of the signal. Using wavelet transformations, we can effectively eliminate the measurement noise in the wheel measurement process, and decompose the measured profile according to different scales. The wheel surface can be reconstructed from the multi-resolution profile using the combination of different scales according to practical needs, and particular to this paper, the characteristic wavelength of the wheel can be obtained from the multi-resolution study of the wheel. The characteristic wavelength associated with their characteristic grain can be used in the prediction of ground surface roughness. The wavelet transformation method can also be used in the data compression of the measurement data, which provides and efficient method for the modeling of the grinding wheel surfaces. The analysis and simulation results can be applied to the grinding process simulation and grinding process control.

Combined Spherical Harmonic and Wavelet Expansion—A Future Concept in Earth's Gravitational Determination

The basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade-off between “space localization” on the sphere and “frequency localization” in terms of spherical harmonics is described in form of an “uncertainty principle.” A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts ofP(acket)-scale discretized andD(aubechies)-scale discretized wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (C0) (like Abel–Poisson or Gauß–Weierstraß operators) lead in canonical way to pyramid algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the “low frequency parts” of a physical quantity by spherical harmonics and the “high frequency parts” by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today's physical geodesy, viz. the determination of the high frequency parts of the earth's gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion.

Multi-scale analysis of engineering surfaces

Conventional surface characterization techniques involving random process analysis are limited in characterizing multi-scale surface features relevant to manufacturing processes and functions. This paper introduces a novel technique for multi-scale characterization of engineering surfaces by applying wavelet transform. The main advantages of wavelet transform over other existing signal processing techniques are its space-frequency localization and multi-scale view of the components of a signal. Utilizing these properties of wavelet transform, we can effectively apply multi-channel filter banks to the surface data and link the manufacturing and functional aspects of a surface with its multi-scale features. Surfaces produced by typical manufacturing processes are analyzed using wavelet transform, and the usefulness of wavelet transform in the multi-scale analysis of engineering surfaces is demonstrated.