Wavelet Threshold Method Research Articles

A comparison of Monte Carlo dose calculation denoising techniques

Recent studies have demonstrated that Monte Carlo (MC) denoising techniques can reduce MC radiotherapy dose computation time significantly by preferentially eliminating statistical fluctuations (‘noise’) through smoothing. In this study, we compare new and previously published approaches to MC denoising, including 3D wavelet threshold denoising with sub-band adaptive thresholding, content adaptive mean–median-hybrid (CAMH) filtering, locally adaptive Savitzky–Golay curve-fitting (LASG), anisotropic diffusion (AD) and an iterative reduction of noise (IRON) method formulated as an optimization problem. Several challenging phantom and computed-tomography-based MC dose distributions with varying levels of noise formed the test set. Denoising effectiveness was measured in three ways: by improvements in the mean-square-error (MSE) with respect to a reference (low noise) dose distribution; by the maximum difference from the reference distribution and by the ‘Van Dyk’ pass/fail criteria of either adequate agreement with the reference image in low-gradient regions (within 2% in our case) or, in high-gradient regions, a distance-to-agreement-within-2% of less than 2 mm. Results varied significantly based on the dose test case: greater reductions in MSE were observed for the relatively smoother phantom-based dose distribution (up to a factor of 16 for the LASG algorithm); smaller reductions were seen for an intensity modulated radiation therapy (IMRT) head and neck case (typically, factors of 2–4). Although several algorithms reduced statistical noise for all test geometries, the LASG method had the best MSE reduction for three of the four test geometries, and performed the best for the Van Dyk criteria. However, the wavelet thresholding method performed better for the head and neck IMRT geometry and also decreased the maximum error more effectively than LASG. In almost all cases, the evaluated methods provided acceleration of MC results towards statistically more accurate results.

Filtered wavelet thresholding methods

When working with nonlinear filtering algorithms for image denoising problems, there are two crucial aspects, namely, the choice of the thresholding parameter λ and the use of a proper filter function. Both greatly influence the quality of the resulting denoised image. In this paper we propose two new filters, which are a piecewise quadratic and an exponential function of λ, respectively, arid we show how they can be successfully used instead of the classical Donoho and Johnstone's Soft thresholding filter. We exploit the increased regularity and flexibility of the new filters to improve the quality of the final results. Moreover, we prove that our filtered approximation is a near-minimizer of the functional which has to be minimized to solve the denoising problem. We also show that the quadratic filter, due to its shape, yields good results if we choose λ as the Donoho and Johnstone universal threshold, while the exponential one is more suitable if we use the recently proposed H-curve criterion. Encouraging results in extensive numerical experiments on several test images confirm the effectiveness of our proposal.

ウェーブレット閾値法による壁乱流の特徴抽出(OS11-2ウエーブレット解析の現状と新しい展開)

ウェーブレット閾値法による壁乱流の特徴抽出(OS11-2ウエーブレット解析の現状と新しい展開)

B112 ウェーブレットによる特徴抽出と動的適合型解析

Extraction of coherent structure of turbulence by the wavelet thresholding method is demonstrated. The method is not only a basis of dynamically adaptive algorithm but a powerful tool to improve our understanding of physics of turbulence. The method is applied to steady isotropic homogeneous turbulence filed. It is shown that the wavelet can extract the energetic components with lower degree of freedom while the multi-scale nature of turbulence is not destroyed.

Wavelet thresholding in partially linear models: a computation and simulation

AbstractPartially linear models have a linear part as in the linear regression and a non‐linear part similar to that in the non‐parametric regression. The estimates in Partially Linear Models have been studied previously using traditional smoothing methods such as smoothing spline, kernel and piecewise polynomial smoothers. In this paper, a wavelet thresholding method for estimating the corresponding parameters in Partially Linear Models is presented. Extensive simulation results shows that wavelet smoothing approach is comparable to traditional smoothing methods when their assumptions are satisfied. But wavelet smoothing is often superior when assumptions about the smoothness of the underlying function of non‐parametric part are not satisfied. The computational complexity is linear on the sample size. Copyright © 2003 John Wiley & Sons, Ltd.

Wavelet thresholding for non-necessarily Gaussian noise: idealism

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.

Wavelet Algorithms for High-Resolution Image Reconstruction

High-resolution image reconstruction refers to the reconstruction of high-resolution images from multiple low-resolution, shifted, degraded samples of a true image. In this paper, we analyze this problem from the wavelet point of view. By expressing the true image as a function in ${\cal L}({\Bbb R}^2)$, we derive iterative algorithms which recover the function completely in the ${\cal L}$ sense from the given low-resolution functions. These algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation. We apply wavelet (packet) thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate. Our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the Tikhonov least-squares approach. Extension to super-resolution image reconstruction, where some of the low-resolution images are missing, is also considered.

Wavelet thresholding for some classes of non–Gaussian noise

Wavelet shrinkage and thresholding methods constitute a powerful way to carry out signal denoising, especially when the underlying signal has a sparse wavelet representation. They are computationally fast, and automatically adapt to the smoothness of the signal to be estimated. Nearly minimax properties for simple threshold estimators over a large class of function spaces and for a wide range of loss functions were established in a series of papers by Donoho and Johnstone. The notion behind these wavelet methods is that the unknown function is well approximated by a function with a relatively small proportion of nonzero wavelet coefficients. In this paper, we propose a framework in which this notion of sparseness can be naturally expressed by a Bayesian model for the wavelet coefficients of the underlying signal. Our Bayesian formulation is grounded on the empirical observation that the wavelet coefficients can be summarized adequately by exponential power prior distributions and allows us to establish close connections between wavelet thresholding techniques and Maximum A Posteriori estimation for two classes of noise distributions including heavy–tailed noises. We prove that a great variety of thresholding rules are derived from these MAP criteria. Simulation examples are presented to substantiate the proposed approach.

Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes

This paper presents a comparative study of different thresholding methods for speckle noise reduction in electronic speckle pattern interferometry fringes using several wavelet bases. An approach based on the removal of the wavelet subbands of the transformed image is presented. The performance of this method is analyzed and compared with the results obtained with the denoising techniques that use wavelet shrinkage. It is shown that the wavelet subband removal method reduces speckle noise and maintains image features more effectively than the wavelet shrinkage techniques based on soft and hard thresholding.

Myopotential denoising of ECG signals using wavelet thresholding methods

We present empirical comparisons of several wavelet-denoising methods applied to the problem of removing (denoising) myopotential noise from the observed noisy ECG signal. Namely, we compare the denoising accuracy and robustness of several wavelet thresholding methods (VISU, SURE and soft thresholding) and a new thresholding approach based on Vapnik-Chervonenkis (VC) learning theory. Our findings indicate that the VC-based wavelet approach is superior to the standard thresholding methods in that it achieves: Higher denoising accuracy (in terms of both MSE measure and visual quality) and more robust and compact representation of the denoised signal (i.e., it uses fewer wavelets).

Signal estimation and denoising using VC-theory

Signal denoising is closely related to function estimation from noisy samples. The same problem is also addressed in statistics (non-linear regression) and neural network learning. Vapnik–Chervonenkis (VC) theory has recently emerged as a general theory for estimation of dependencies from finite samples. This theory emphasizes model complexity control according to Structural Risk Minimization (SRM) inductive principle, which considers a nested set of models of increasing complexity (called a structure), and then selects an optimal model complexity providing minimum error for future samples. This paper applies the framework of VC-theory to signal estimation/denoising. There are three factors important for accurate signal estimation from finite samples: (1) the type of (orthogonal) basis functions used (i.e. Fourier basis, wavelets etc.). (2) The choice of a structure, i.e. ordering of the basis functions according to their ‘importance’ for accurate signal estimation. This corresponds to the choice of a ‘structure’ under SRM formulation. (3) Selecting an optimal number of terms (basis functions) from the ordered sequence of basis functions (2), aka model selection or complexity control (in statistics). We propose a methodology for specifying appropriate orderings (2) and an analytic expression for model selection (3) for signal processing applications. We also present empirical comparisons between the proposed methodology and current state-of-the-art wavelet thresholding methods for univariate signals. These comparisons suggest that the prudent choice of a structure (2) and the use of VC-based model selection (3) are critical for accurate signal estimation with finite samples.

Wavelet domain de-noising of time-courses in MR image sequences

Magnetic resonance images acquired with high temporal resolution often exhibit large noise artifacts, which arise from physiological sources as well as from the acquisition hardware. These artifacts can be detrimental to the quality and interpretation of the time-course data in functional MRI studies. A class of wavelet-domain de-noising algorithms estimates the underlying, noise-free signal by thresholding (or ‘shrinking’) the wavelet coefficients, assuming the underlying temporal noise of each pixel is uncorrelated and Gaussian. A Wiener-type shrinkage algorithm is developed in this paper, for de-noising either complex- or magnitude-valued image data sequences. Using the de-correlation properties of the wavelet transform, as elucidated by Johnstone and Silverman, the assumption of i.i.d. Gaussian noise can be abandoned, opening up the possibility of removing colored noise. Both wavelet- and wavelet-packet based algorithms are developed, and the Wiener method is compared to the traditional Hard and Soft wavelet thresholding methods of Donoho and Johnstone. The methods are applied to two types of data sets. In the first, an artificial set of complex-valued images was constructed, in which each pixel has a simulated bimodal time-course. Gaussian noise was added to each of the real and imaginary channels, and the noise removed from the complex image sequence as well as the magnitude image sequence (where the noise is Rician). The bias and variance between the original and restored paradigms was estimated for each method. It was found that the Wiener method gives better balance in bias and variance than either Hard or Soft methods. Furthermore, de-noising magnitude data provides comparable accuracy of the restored images to that obtained from de-noising complex data. In the second data set, an actual in vivo complex image sequence containing unknown physiological and instrumental noise was used. The same bimodal paradigm as in the first data set was added to pixels in a small localized region of interest. For the paradigm investigated here, the smooth Daubechies wavelets provide better de-noising characteristics than the discontinuous Haar wavelets. Also, it was found that wavelet packet de-noising offers no significant improvement over the computationally more efficient wavelet de-noising methods. For the in vivo data, it is desirable that the groups of “activated” time-courses are homogeneous. It was found that the internal homogeneity of the group of time-courses increases when de-noising is applied. This suggests using de-noising as a pre-processing tool for both exploratory and inferential data analysis methods in fMRI.

Spatially adaptive wavelet thresholding with context modeling for image denoising

The method of wavelet thresholding for removing noise, or denoising, has been researched extensively due to its effectiveness and simplicity. Much of the literature has focused on developing the best uniform threshold or best basis selection. However, not much has been done to make the threshold values adaptive to the spatially changing statistics of images. Such adaptivity can improve the wavelet thresholding performance because it allows additional local information of the image (such as the identification of smooth or edge regions) to be incorporated into the algorithm. This work proposes a spatially adaptive wavelet thresholding method based on context modeling, a common technique used in image compression to adapt the coder to changing image characteristics. Each wavelet coefficient is modeled as a random variable of a generalized Gaussian distribution with an unknown parameter. Context modeling is used to estimate the parameter for each coefficient, which is then used to adapt the thresholding strategy. This spatially adaptive thresholding is extended to the overcomplete wavelet expansion, which yields better results than the orthogonal transform. Experimental results show that spatially adaptive wavelet thresholding yields significantly superior image quality and lower MSE than the best uniform thresholding with the original image assumed known.

Process data de-noising using wavelet transform

The recovery of process information from noisy data (de-noising) is studied by investigating the classical solution of the estimation problem first. Next, the effectiveness of wavelet-based algorithms for data recovery is considered. A novel method based on coefficient de-noising according to WienerShrink method of wavelet thresholding is proposed. Simulation results are presented, highlighting the advantages of the de-noising method over the classical approaches based on the mean square error criterion.

Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law

Abstract Using wavelet thresholding methods, we give an adaptive estimator of θ=∫f2, where f is coming from the white noise model. We estimate the random bias term, which is the dominating term in the decomposition of the quadratic error, and state a central limit theorem. This estimator has two advantages: it is centered around θ and the rate does not require the knowledge on the regularity of f. Moreover, using our procedure, we provide explicit confidence intervals and critical regions for parametric tests on θ.

Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors

Research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near-ideal estimation performance in various asymptotic frameworks. However, image processing practice has shown that universal thresholding methods are outperformed by simple Bayesian estimators assuming independent wavelet coefficients and heavy-tailed priors such as generalized Gaussian distributions (GGDs). In this paper, we investigate various connections between shrinkage methods and maximum a posteriori (MAP) estimation using such priors. In particular, we state a simple condition under which MAP estimates are sparse. We also introduce a new family of complexity priors based upon Rissanen's universal prior on integers. One particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length (MDL) principle. We develop analytical expressions for the shrinkage rules implied by GGD and complexity priors. This allows us to show the equivalence between universal hard thresholding, MAP estimation using a very heavy-tailed GGD, and MDL estimation using one of the new complexity priors. Theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis-specifications of the prior-a basic concern in image processing applications.

Miscellanea. Score tests for heteroscedasticity in wavelet regression

SUMMARY We consider two score tests for heteroscedasticity in the errors of a signal-plus-noise model, where the signal is estimated by wavelet thresholding methods. The error variances are assumed to depend on observed covariates, through a parametric relationship of known form. The tests are based on the approaches of Breusch & Pagan (1979) and Koenker (1981). We establish the asymptotic validity of the tests and examine their performance in a simulation study. The Koenker test is found to perform well, in terms of both size and power.

Identification of Low-Dimensional Energy Containing/Flux Transporting Eddy Motion in the Atmospheric Surface Layer Using Wavelet Thresholding Methods

The partitioning of turbulent perturbations into a ‘‘low-dimensional’’ active part responsible for much of the turbulent energy and fluxes and a ‘‘high-dimensional’’ passive part that contributes little to turbulent energy and transport dynamics is investigated using atmospheric surface-layer (ASL) measurements. It is shown that such a partitioning scheme can be achieved by transforming the ASL measurements into a domain that concentrates the low-dimensional part into few coefficients and thus permits a global threshold of the remaining coefficients. In this transformation‐thresholding approach, Fourier rank reduction and orthonormal wavelet and wavelet packet methods are considered. The efficiencies of these three thresholding methods to extract the events responsible for much of the heat and momentum turbulent fluxes are compared for a wide range of atmospheric stability conditions. The intercomparisons are performed in four ways: (i) compression ratios, (ii) energy conservation, (iii) turbulent flux conservation, and (iv) finescale filtering via departures from Kolmogorov’s K41 power laws. For orthonormal wavelet and wavelet packets analysis, wavelet functions with varying time‐frequency localization properties are also considered. The study showed that wavelet and wavelet packet Lorentz thresholding can achieve high compression ratios (98%) with minimal loss in energy (3% loss) and fluxes (4%). However, these compression ratios and energy and flux conservation measures are comparable to the linear Fourier rank reduction method if a Lorentz threshold function is applied to the latter. Finally, it is demonstrated that orthonormal wavelet and wavelet packets thresholding are insensitive to the analyzing wavelet.