Anisotropic Wavelet Research Articles

Applying anisotropic wavelet shrinkage to image segmentation

Applying anisotropic wavelet shrinkage to image segmentation

Anisotropic Meshless Frames on ℝ n

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of ℝn. The wavelets we construct are C∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay.

Direct Numerical Simulation of Turbulence Using Divergence-Free Wavelets

We present a numerical method based on divergence-free wavelets to solve the incompressible Navier–Stokes equations. We introduce a new scheme which uses anisotropic (or generalized) divergence-free wavelets and which needs only fast wavelet transform algorithms. We prove its stability and show convincing numerical experiments.

Anisotropic Wavelet-Based Image Nearness Measure

The problem considered in this article is how to solve the image correspondence problem in cases where it is important to measure changes in the contour, position, and spatial orientation of bounded regions. This article introduces a computational intelligence approach to the solution of this problem with anisotropic (direction dependent) wavelets and a tolerance near set approach to detecting similarities in pairs of images. Near sets are a recent generalization of rough sets introduced by Z. Pawlak during the early 1980s. Near sets resulted from a study of the perceptual basis for rough sets. Pairs of sets containing objects with similar descriptions are known as near sets. The proposed wavelet-based image nearness measure is compared with F. Hausdorff and P. Mahalanobis image distance measures. The results of three wavelet-based image resemblance measures for several well-known images, are given. A direct benefit of this research is an effective means of grouping together (classifying) images that corr...

Structure-Oriented Multidirectional Wiener Filter for Denoising of Image and Video Signals

In this letter, we propose a structure-oriented multidirectional Wiener filter to reduce additive white Gaussian noise in image and video signals. A local activity profile based on second derivatives is used to restrict filtering to homogeneous directions to combat blurring. The proposed filter improves the Wiener estimate of denoised pixels to reduce the residual blurring of the conventional Wiener filter while achieving higher noise-reduction gains of up to 5.6 dB peak signal-to-noise-ratio (PSNR). The parameters of the proposed filter (block size, shape and coefficients) are adapted to image structure and noise level for optimization with respect to noise-reduction gain and structure preservation. The effectiveness of the proposed method is shown using both the PSNR and the modulation transfer function calculated for a range of spatial frequencies to measure the degradation in contrast due to blurring. Our results show that the proposed method achieves a higher contrast transfer ratio than the conventional Wiener filter indicating improved preservation of high frequency content. We also show the performance of the proposed filter relative to reference anisotropic diffusion and wavelet methods.

Image Coding Using Dual-Tree Discrete Wavelet Transform

In this paper, we explore the application of 2-D dual-tree discrete wavelet transform (DDWT), which is a directional and redundant transform, for image coding. Three methods for sparsifying DDWT coefficients, i.e., matching pursuit, basis pursuit, and noise shaping, are compared. We found that noise shaping achieves the best nonlinear approximation efficiency with the lowest computational complexity. The interscale, intersubband, and intrasubband dependency among the DDWT coefficients are analyzed. Three subband coding methods, i.e., SPIHT, EBCOT, and TCE, are evaluated for coding DDWT coefficients. Experimental results show that TCE has the best performance. In spite of the redundancy of the transform, our DDWT _ TCE scheme outperforms JPEG2000 up to 0.70 dB at low bit rates and is comparable to JPEG2000 at high bit rates. The DDWT _TCE scheme also outperforms two other image coders that are based on directional filter banks. To further improve coding efficiency, we extend the DDWT to an anisotropic dual-tree discrete wavelet packets (ADDWP), which incorporates adaptive and anisotropic decomposition into DDWT. The ADDWP subbands are coded with TCE coder. Experimental results show that ADDWP _ TCE provides up to 1.47 dB improvement over the DDWT _TCE scheme, outperforming JPEG2000 up to 2.00 dB. Reconstructed images of our coding schemes are visually more appealing compared with DWT-based coding schemes thanks to the directionality of wavelets.

Multiple CT-reconstructions for locally adaptive anisotropic wavelet denoising

The signal-to-noise ratio in computed tomography (CT) data should be improved by using adaptive noise estimation for level-dependent threshold determination in the wavelet domain. The projection data measured in CT and, thus, the slices reconstructed from these data are noisy. For a reliable diagnosis and subsequent image processing, like segmentation, the ratio between relevant tissue contrasts and the noise amplitude must be sufficiently large. By separate reconstructions from disjoint subsets of projections, e.g. even and odd numbered projections, two CT volumes can be computed, which only differ with respect to noise. We show that these images allow a position and orientation adaptive noise estimation for level-dependent threshold determination in the wavelet domain. The computed thresholds are applied to the averaged wavelet coefficients of the input data. The final result contains data from the complete set of projections, but shows approximately 50% improvement in signal-to-noise ratio. The proposed noise reduction method adapts itself to the noise power in the images and allows for the reduction of spatially varying and oriented noise.

Despeckle Filtering Algorithms and Software for Ultrasound Imaging

Abstract It is well-known that speckle is a multiplicative noise that degrades image quality and the visual evaluation in ultrasound imaging. This necessitates the need for robust despeckling techniques for both routine clinical practice and teleconsultation. The goal for this book is to introduce the theoretical background (equations), the algorithmic steps, and the MATLAB™ code for the following group of despeckle filters: linear filtering, nonlinear filtering, anisotropic diffusion filtering and wavelet filtering. The book proposes a comparative evaluation framework of these despeckle filters based on texture analysis, image quality evaluation metrics, and visual evaluation by medical experts, in the assessment of cardiovascular ultrasound images recorded from the carotid artery. The results of our work presented in this book, suggest that the linear local statistics filter DsFlsmv, gave the best performance, followed by the nonlinear geometric filter DsFgf4d, and the linear homogeneous mask area filte...

An adapted group-dilation anisotropic multifractal formalism for functions

Anisotropic phenomena can be observed almost everywhere in nature. This makes them important subjects for theoretical and experimental studies. In this work, we focus on the study of anisotropic quasi-self-similar signals. It holds that the classical multifractal formalism in all its formulations does not hold for this class. We then use an homogeneous norm introduced by Calderon and Torchinsky to check the validity of an adapted anisotropic multifractal formalism for these signals. Our techniques are based on group theory combined with wavelet characterization of anisotropic function spaces. We then show the efficiency of anisotropic wavelets in detecting singularities.

Methods for the anisotropic wavelet packet transform

Anisotropic wavelet packets present a flexible transform with interesting properties and applications. While certain aspects of this transform have been investigated in conjunction with applications, this paper aims at providing a basic theoretical framework for working with anisotropic wavelet packets. Random decompositions are developed which have distributions with different average decomposition depths and degrees of anisotropy. They can be used in cryptographic applications or to test other algorithms. For the uniform distribution, it is necessary to determine the number of possible bases for all decomposition depths. A best basis algorithm for anisotropic decompositions is developed. A graph theoretical representation of the anisotropic decomposition structure is presented, which is unique for each decomposition and, thus, free of redundancy, which is important for compression purposes. A compression algorithm based on these techniques is developed and tested on random decompositions.

Potentials of Gaussians and approximate wavelets

AbstractWe derive new formulas for harmonic, diffraction, elastic, and hydrodynamic potentials acting on anisotropic Gaussians and approximate wavelets. These formulas can be used to construct accurate cubature formulas for these potentials. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Anisotropic wavelet bases and thresholding

AbstractWe consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces $ H^a_p $ for p > 0, which are known to be equal to Lp for p > 1. In particular, we characterize those functions that provide a specific convergence rate by intrinsic smoothness properties. To this end we apply restricted nonlinear approximation, see [3], which is a generalization of n ‐term approximation in which a weight function is used to control the terms of the approximations. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

BestN-term approximation in electronic structure calculations. II. Jastrow factors

We present a novel application of best N-term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions near electron-electron and electron-nuclear cusps. Based on Nitsche's characterization of best N-term approximation spaces , we prove that for q>1 and with respect to a certain class of anisotropic wavelet tensor product bases. Computational arguments are given in favour of this specific class compared to other possible tensor product bases. Finally, we compare the approximation properties of wavelet bases with standard Gaussian-type basis sets frequently used in quantum chemistry.

On the Number of Rectangular Tilings

Adaptive multiscale representations via quadtree splitting and two-dimensional (2-D) wavelet packets, which amount to space and frequency decompositions, respectively, are powerful concepts that have been widely used in applications. These schemes are direct extensions of their one-dimensional counterparts, in particular, by coupling of the two dimensions and restricting to only one possible further partition of each block into four subblocks. In this paper, we consider more flexible schemes that exploit more variations of multidimensional data structure. In the meantime, we restrict to tree-based decompositions that are amenable to fast algorithms and have low indexing cost. Examples of these decomposition schemes are anisotropic wavelet packets, dyadic rectangular tilings, separate dimension decompositions, and general rectangular tilings. We compute the numbers of possible decompositions for each of these schemes. We also give bounds for some of these numbers. These results show that the new rectangular tiling schemes lead to much larger sets of 2-D space and frequency decompositions than the commonly-used quadtree-based schemes, therefore bearing the potential to obtain better representation for a given image.

Characterization of permeability anisotropy using wavelet analysis

Wavelet analysis is an image analysis technique that can extract local information at multiple scales. Because of this capability, wavelet analysis can be used to identify dominant scales in statistically heterogeneous and anisotropic random fields. We develop and test a wavelet analysis method for identifying dominant scales and orientations in permeability fields and for identifying boundaries between regions with different dominant orientations. We evaluate three different wavelets (fully anisotropic Morlet wavelet, Mexican hat wavelet, and Cauchy wavelet) and show that the Morlet wavelet is the most effective of these three wavelets in identifying dominant orientations. We also investigate the use of several different quantitative wavelet measures in identifying dominant scales and orientations in permeability fields. The technique is demonstrated using both a synthetic data set with known characteristics and a laboratory‐collected permeability data set from Massillon sandstone.

Model-Based Automated Extraction of Microtubules From Electron Tomography Volume

We propose a model-based automated approach to extracting microtubules from noisy electron tomography volume. Our approach consists of volume enhancement, microtubule localization, and boundary segmentation to exploit the unique geometric and photometric properties of microtubules. The enhancement starts with an anisotropic invariant wavelet transform to enhance the microtubules globally, followed by a three-dimensional (3-D) tube-enhancing filter based on Weingarten matrix to further accentuate the tubular structures locally. The enhancement ends with a modified coherence-enhancing diffusion to complete the interruptions along the microtubules. The microtubules are then localized with a centerline extraction algorithm adapted for tubular objects. To perform segmentation, we novelly modify and extend active shape model method. We first use 3-D local surface enhancement to characterize the microtubule boundary and improve shape searching by relating the boundary strength with the weight matrix of the searching error. We then integrate the active shape model with Kalman filtering to utilize the longitudinal smoothness along the microtubules. The segmentation improved in this way is robust against missing boundaries and outliers that are often present in the tomography volume. Experimental results demonstrate that our automated method produces results close to those by manual process and uses only a fraction of the time of the latter.

Automated extraction of fine features of kinetochore microtubules and plus-ends from electron tomography volume

Kinetochore microtubules (KMTs) and the associated plus-ends have been areas of intense investigation in both cell biology and molecular medicine. Though electron tomography opens up new possibilities in understanding their function by imaging their high-resolution structures, the interpretation of the acquired data remains an obstacle because of the complex and cluttered cellular environment. As a result, practical segmentation of the electron tomography data has been dominated by manual operation, which is time consuming and subjective. In this paper, we propose a model-based automated approach to extracting KMTs and the associated plus-ends with a coarse-to-fine scale scheme consisting of volume preprocessing, microtubule segmentation and plus-end tracing. In volume preprocessing, we first apply an anisotropic invariant wavelet transform and a tube-enhancing filter to enhance the microtubules at coarse level for localization. This is followed with a surface-enhancing filter to accentuate the fine microtubule boundary features. The microtubule body is then segmented using a modified active shape model method. Starting from the segmented microtubule body, the plus-ends are extracted with a probabilistic tracing method improved with rectangular window based feature detection and the integration of multiple cues. Experimental results demonstrate that our automated method produces results comparable to manual segmentation but using only a fraction of the manual segmentation time.

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗q on certain quasi-Banach spaces. We prove that the approximation spaces Aα q(L2) and Aα q(H1) equal tensor products of Besov spaces Bα q(Lq), e.g., Aα q(L2([0,1]d)) = Bα q(Lq([0,1])) ⊗q · ⊗q Bα q · ·(Lq([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.

BestN-term approximation in electronic structure calculations I. One-electron reduced density matrix

We discuss best N -term approximation spaces for one-electron wavefunctions and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted spaces of wavelet coefficients to proof that both and ρ are in for all with . Our proof is based on the assumption that the possess an asymptotic smoothness property at the electron-nuclear cusps.

Non‐Gaussianity Due to Possible Residual Foreground Signals inWilkinson Microwave Anistropy ProbeFirst‐Year Data Using Spherical Wavelet Approaches

We perform multiscale non-Gaussianity detection and localization to the Wilkinson Microwave Anisotropy Probe (WMAP) first-year data in both wavelet and real spaces. Such an analysis is facilitated by spherical wavelet transform and inverse transform techniques developed by the YAWtb team. Skewness and kurtosis as test statistics are calculated on scales from about 1° to 30° on the sky, as well as toward different directions using anisotropic spherical Morlet wavelets (SMW). A maximum deviation from Gaussian simulations with a right tail probability of ~99.9% is detected at an angular scale of ~12° at an azimuthal orientation of ~0° on the sky. In addition, some significant non-Gaussian spots have been identified and localized in real space from both the combined Q-V-W map recommended by the WMAP team and the Tegmark foreground-cleaned map. Systematic effects due to beams and noise can be rejected as the source of this non-Gaussianity. Several tests show that residual foreground contamination may significantly contribute to this non-Gaussian feature. It is thus still premature to do more precise tests on the non-Gaussianity of the intrinsic CMB fluctuations before we can identify the origin of these foreground signals, understand their nature, and finally remove them from the CMB maps completely.