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Abstract: This paper studies the connections between discrete two-dimensional schemes for shift-invariant Haar wavelet shrinkage on one hand, and nonlinear diffusion on the other. We show that using a single iteration on a single scale, the two methods can be made equivalent by the choice of the nonlinearity which controls each method: the shrinkage function, or the diffusivity function, respectively. In the two-dimensional setting, this diffusion–wavelet connection shows an important novelty compared to the one-dimensional framework or compared to classical 2-D wavelet shrinkage: The structure of two-dimensional diffusion filters suggests to use a coupled, synchronised shrinkage of the individual wavelet coefficient channels. This coupling enables to design Haar wavelet filters with good rotation invariance at a low computational cost. Furthermore, by transferring the channel coupling of vector- and matrix-valued nonlinear diffusion filters to the Haar wavelet setting, we obtain well-synchronised shrinkage methods for colour and tensor images. Our experiments show that these filters perform significantly better than conventional shrinkage methods that process all wavelets independently. [Copyright &y& Elsevier]

In this paper we investigate a special class of minimization problems for image denoising and deblurring. We focus on problems with positively one homogeneous penalty terms and show that the minimizer can be given in terms of projections. Especially we apply the results to wavelet shrinkage and TV denoising. Furthermore we present an application to deblurring with a TV penalty term. The approach presented here provides a common view on TV denoising and wavelet shrinkage. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the iterative regularization method and the inverse scale space method, recently developed for total-variation (TV) based image restoration, to wavelet-based image restoration. This continues our earlier joint work with others where we applied these techniques to variational-based image restoration, obtaining significant improvement over the Rudin-Osher-Fatemi TV-based restoration. Here, we apply these techniques to soft shrinkage and obtain the somewhat surprising result that a) the iterative procedure applied to soft shrinkage gives firm shrinkage and converges to hard shrinkage and b) that these procedures enhance the noise-removal capability both theoretically, in the sense of generalized Bregman distance, and for some examples, experimentally, in terms of the signal-to-noise ratio, leaving less signal in the residual. [ABSTRACT FROM AUTHOR]

Abstract: Lateral correlation analyses are carried out in different scales for two sets of full-scale thunderstorm downburst full wind speed time series. These two sets of data were recorded at the TTU field site, Texas, USA, on June 4 and 15, 2002, respectively. The same data sets were also used in the authors’ previous paper [L. Chen, C.W. Letchford, Proper orthogonal decomposition of two vertical profiles of full-scale nonstationary correlated downburst wind speeds, J. Wind Eng. Ind. Aerodyn., 93(3)(2005) 187–266.], which presents a methodology to model downburst vertical profiles. The statistical procedures utilized in [7], such as the POD technique, the wavelet shrinkage for time-varying mean inference, the two-stage weighted moving-average method for time-varying variance inference and the Priestley''s EPSD estimator, are also employed in this paper and their details are thus omitted. While in [7] the authors employed the proper orthogonal decomposition (POD) to reduce the dimensions, POD in this article is employed to measure the degree of the correlation between two deterministic time series. Conventionally, wind speeds are decomposed to a time-varying mean speed and a nonstationary fluctuating speed by virtue of wavelet shrinkage [7]. Time-varying mean speeds are in the largest scale, which are essentially due to translations of storms and radial velocities of spreading flow. It was observed that the time-varying standard deviations [7] of fluctuating speeds are basically driven by time-varying mean speeds other than small-scale turbulence. Therefore, time-varying standard deviations are considered in the medium scale. Finally, the modulated fluctuating speeds are viewed in the smallest scale and due to small-scale turbulence. It is found that (1) the two lateral profiles are fully correlated and almost evenly spatially distributed in the largest and medium scales with more than 96% of the total energy; (2) they are uncorrelated in the smallest scale since the measurement locations are at least 263m apart. [Copyright &y& Elsevier]

Wavelet Shrinkage using DWT has been widely used in de-noising although DWT has a translation variance problem. In this study, we solve this problem by using the translation invariant DWT. For this purpose, we propose a new complex wavelet, the Real-Imaginary Spline Wavelet (RI-Spline wavelet). We also propose the Coherent Dual-Tree algorithm for the RI-Spline wavelet and extend it to the 2-Dimensional. Then we apply this translation invariant RI-Spline wavelet for translation invariant de-noising. Experimental results show that our method, when applied to ECG data, the medical image and the textile surface inspection can obtain better de-noising results than that of conventional Wavelet Shrinkage. [ABSTRACT FROM AUTHOR]

Nonlinear diffusion filtering and wavelet shrinkage are two successfully applied methods for discontinuity preserving denoising of signals and images. Recently, relations between both methods have been established taking into account wavelet shrinkage at one scale. In this paper, we propose a new explicit scheme for nonlinear diffusion which directly incorporates ideas from multiscale Haar wavelet shrinkage. We prove that our scheme permits larger time steps while preserving convergence to the mean signal value. Numerical examples demonstrate the behavior of our scheme for two and three scales. [ABSTRACT FROM AUTHOR]

In this paper, a new noise reduction algorithm is introduced and applied to the problem of denoising hyperspectral imagery. This algorithm resorts to the spectral derivative domain, where the noise level is elevated, and benefits from the dissimilarity of the signal regularity in the spatial and the spectral dimensions of hyperspectral images. The performance of the new algorithm is tested on two different hyperspectral datacubes: an Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) datacube that is acquired in a vegetation-dominated site and a simulated AVIRIS datacube that simulates a geological site. The new algorithm provides signal-to-noise-ratio improvement up to 84.44% and 98.35% in the first and the second datacubes, respectively. [ABSTRACT FROM AUTHOR]

Medical images generally have low contrast and they get complex type of noise due to the use of various devices and applications of various algorithms. However, most of the denoising methods consider only additive noise or some special noise model dependent on their systems and conditions only. Such methods when applied to real medical images yield poor results. The present work proposes a method for denoising of medical images using soft-thresholding in wavelet domain on multiple levels. We have developed a method to compute the threshold values for denoising of medical images, which depend on the median as well as the contrast ratio of the wavelet coefficients and also on the level number. We have performed experiments by adding various proportions of Gaussian, Salt-and-Pepper and Speckle noise, and found that the proposed method performs better for these cases. The method is efficient because the threshold values can be calculated directly and it is adaptive as these values depend on mean, median and standard deviation of wavelet coefficients of the particular image. The proposed method also gives a criterion for level-dependent thresholding. Application of the proposed method to Ultrasound, X-ray and MRI images is demonstrated in experiments. In the present work, we have also done studies on how to select the mother wavelet for a particular problem. [ABSTRACT FROM AUTHOR]

Single-index models have found applications in econometrics and biometrics, where multidimensional regression models are often encountered. This article proposes a non-parametric estimation approach that combines wavelet methods for nonequispaced designs with Bayesian models. We consider a wavelet series expansion of the unknown regression function and set prior distributions for the wavelet coefficients and the other model parameters. To ensure model identifiability, the direction parameter is represented via its polar coordinates. We employ ad hoc hierarchical mixture priors that perform shrinkage on wave let coefficients and use Markov chain Monte Carlo methods for a posteriori inference. We investigate an independence-type Metropolis-Hastings algorithm to produce samples for the direction parameter. Our method leads to simultaneous estimates of the link function and of the index parameters. We present results on both simulated and real data, where we look at comparisons with other methods. [ABSTRACT FROM AUTHOR]

Summary: Ordered subset expectation–maximization (OSEM) method in positron emission tomography (PET) has been very popular recently. It is an iterative algorithm and provides images with superior noise characteristics compared to conventional filtered backprojection (FBP) algorithms. Due to the lack of smoothness in images in OSEM iterations, however, some type of inter-smoothing is required. For this purpose, the smoothing based on the convolution with the Gaussian kernel has been used in clinical PET practices. In this paper, we incorporated a robust wavelet de-noising method into OSEM iterations as an inter-smoothing tool. The proposed wavelet method is based on a hybrid use of the standard wavelet shrinkage and the robust wavelet shrinkage to have edge preserving and robust de-noising simultaneously. The performances of the proposed method were compared with those of the smoothing methods based on the convolution with Gaussian kernel using software phantoms, physical phantoms, and human PET studies. The results demonstrated that the proposed wavelet method provided better spatial resolution characteristic than the smoothing methods based on the Gaussian convolution, while having comparable performance in noise removal. [Copyright &y& Elsevier]

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of a (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet denoising. Moreover, by transferring stability notions from diffusion filtering to wavelet shrinkage, we derive conditions on the shrinkage function that ensure that shift invariant single-level Haar wavelet shrinkage is maximum–minimum stable, monotonicity preserving, and variation diminishing. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a technique for inverse halftoning based on the wavelet domain deconvolution that comprises Fourier-domain followed by wavelet-domain noise suppression, in order to benefit from the advantages of each of them. The proposed algorithm can be formulated as a linear deconvolution problem. In fact, we model such a gray-scale image to be the result of a convolution of the original image with a point spread function (PSF) and a colored noise. Our method performs inverse halftoning by first inverting the model specified convolution operator and then attenuating the residual noise using scalar wavelet-domain shrinkage. Using simulations, we verify that the proposed method is competitive with state-of-the-art inverse halftoning techniques in the mean-square-sense and that has also good visual performance. We illustrate the results with simulations on some examples. [Copyright &y& Elsevier]

Abstract: The paper proposes a novel analysis framework for nonstationary wind speeds, from which accumulated results can potentially lead to or enhance empirical nonstationary wind speed models such as the hybrid one for downbursts in (Eng. Struct. 26 (2004) 619). This framework is motivated by the proper orthogonal decomposition (POD) technique and consists of four major steps: separation, POD, approximation and property inference. In the first step, the original wind speed time histories are separated into their time-varying mean speeds and fluctuating speeds through wavelet shrinkage, which is a promising new tool for smoothing; fluctuating speeds are further expressed as the product of their time-varying standard deviations and normalized fluctuating speeds with unit variance. In the second step, the POD is applied to the time-varying means, standard deviations and normalized fluctuation. In the third step, by properly selecting the number of retained modes, the original time series can be approximated to a numerical model. Finally, wind properties such as velocity vertical profiles and turbulence vertical profiles can be calculated from the model. The POD employed herein is a specific POD for arbitrary multivariate data representation. The normalized fluctuations are characterized by both their power spectral densities (PSD) and evolutionary power spectral densities (EPSD). The concept of EPSD for nonstationary processes is briefly presented and one EPSD estimator is given. Specifically, the PSD/EPSD of an original process is related to the PSD/EPSD of the principal coordinate process obtained by the POD. This framework is applied to two sets of nonstationary full-scale thunderstorm downburst wind speed time series. Many appealing downburst properties are obtained and the benefit of the POD is demonstrated. [Copyright &y& Elsevier]

Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients. [ABSTRACT FROM AUTHOR]

Wavelet shrinkage denoising of the displacement estimates to reduce noise artefacts, especially at high overlaps in elastography, is presented in this paper. Correlated errors in the displacement estimates increase dramatically with an increase in the overlap between the data segments. These increased correlated errors (due to the increased correlation or similarity between consecutive displacement estimates) generate the so-called “worm” artefact in elastography. However, increases in overlap on the order of 90% or higher are essential to improve axial resolution in elastography. The use of wavelet denoising significantly reduces errors in the displacement estimates, thereby reducing the worm artefacts, without compromising on edge (high-frequency or detail) information in the elastogram. Wavelet denoising is a term used to characterize noise rejection by thresholding the wavelet coefficients. Worm artefacts can also be reduced using a low-pass filter; however, low-pass filtering of the displacement estimates does not preserve local information such as abrupt change in slopes, causing the smoothing of edges in the elastograms. Simulation results using the analytic 2-D model of a single inclusion phantom illustrate that wavelet denoising produces elastograms with the closest correspondence to the ideal mechanical strain image. Wavelet denoising applied to experimental data obtained from an in vitro thermal lesion phantom generated using radiofrequency (RF) ablation also illustrates the improvement in the elastogram noise characteristics. (E-mail: tvarghese@wisc.edu) [Copyright &y& Elsevier]

The Wavelet-Domain Projection Pursuit Learning Network (WDPPLN) is proposed for restoring degraded image. The new network combines the advantages of both projection pursuit and wavelet shrinkage. Restoring image is very difficult when little is known about a priori knowledge for multisource degraded factors. WDPPLN successfully resolves this problem by separately processing wavelet coefficients and scale coefficients. Parameters in WDPPLN, which are used to simulate degraded factors, are estimated via WDPPLN training, using scale coefficients. Also, WDPPLN uses soft-threshold of wavelet shrinkage technique to suppress noise in three high frequency subbands. The new method is compared with the traditional methods and the Projection Pursuit Learning Network (PPLN) method. Experimental results demonstrate that it is an effective method for unsupervised restoring degraded image. [ABSTRACT FROM AUTHOR]

This paper proposes a new method for image denoising with edge preservation and enhancement, based on image multi-resolution decomposition by a redundant wavelet transform. At each resolution, the coefficients associated with noise and the coefficients associated with edges are modeled by Gaussians, and a shrinkage function is assembled. The shrinkage functions are combined in consecutive resolutions, and geometric constraints are applied to preserve edges that are not isolated. Within the proposed framework, edge related coefficients may be enhanced and denoised simultaneously. Finally, the inverse wavelet transform is applied to the modified coefficients. This method is adaptive, and performs well for images contaminated by natural and artificial noise. [Copyright &y& Elsevier]

Digital radiography is a popular diagnostic imaging method. Denoising and enhancement have an important potential in obtaining as much easily interpretable diagnostic information as possible with reasonable absorbed doses of ionising radiation. Due to the increasing usage of high resolution and high precision images with a limited number of human experts, the computational efficiency of the denoising and enhancement becomes important. In this paper, a local adaptive image enhancement and simultaneous denoising algorithm for fulfilling the requirements of digital X-ray image enhancement is introduced. The algorithm is based on modification of the wavelet transform coefficients by a pointwise nonlinear transformation and reconstructing the enhanced image from the modified wavelet transform coefficients. The implementation of algorithm in software is simple, quick, and universal. [ABSTRACT FROM AUTHOR]

The modeling of biological systems has now become an essential prerequisite for effective bioprocess design, optimization and analysis. The difficulties present in using conventional techniques to model such a complex system make the application of artificial neural networks (ANNs) to these problems particularly attractive because of their capability for nonlinear mapping and lack of necessity for detailed mechanistic knowledge. However, building a reliable ANN model requires sufficient training data, which may be difficult when data are collected from litre-scale experiments. In this work, a bioconversion (with only limited experimental data) was firstly modeled by a radial basis function (RBF) neural network. Although the model provided a very low variance between experiment and simulation, it tended to result in oscillatory behaviour, which clearly does not reflect the accurate profile of the reaction. In order to overcome this drawback, wavelet shrinkage with biorthogonal filters was used to generate a reconstructed function using the RBF model as a base. The synthesis of N-acetyl-d-neuraminic acid by the enzymatic condensation of pyruvate with N-acetyl-d-mannosamine was used as a case study to show the effectiveness of the approach. The effects of alternative filters and border distortion are also discussed. [Copyright &y& Elsevier]

A new wavelet-based nonparametric estimator is introduced in an effort toapproximate variance functions. The new estimator possesses some superiorqualities that are illustrated through its actual performance in somesimulations. [ABSTRACT FROM AUTHOR]

To avoid the problems of relative overlap and low signal-to-noise ratio (SNR) of segmented three-dimensional (3D) multimodal medical images, which limit the effect of medical image diagnosis, a 3D multimodal medical image segmentation algorithm using reinforcement learning and big data analytics is proposed. Bayesian maximum a posteriori estimation method and improved wavelet threshold function are used to design wavelet shrinkage algorithm to remove high-frequency signal component noise in wavelet domain. The low-frequency signal component is processed by bilateral filtering and the inverse wavelet transform is used to denoise the 3D multimodal medical image. An end-to-end DRD U-Net model based on deep reinforcement learning is constructed. The feature extraction capacity of denoised image segmentation is increased by changing the convolution layer in the traditional reinforcement learning model to the residual module and introducing the multiscale context feature extraction module. The 3D multimodal medical image segmentation is done using the reward and punishment mechanism in the deep learning reinforcement algorithm. In order to verify the effectiveness of 3D multimodal medical image segmentation algorithm, the LIDC-IDRI data set, the SCR data set, and the DeepLesion data set are selected as the experimental data set of this article. The results demonstrate that the algorithm's segmentation effect is effective. When the number of iterations is increased to 250, the structural similarity reaches 98%, the SNR is always maintained between 55 and 60 dB, the training loss is modest, relative overlap and accuracy all exceed 95%, and the overall segmentation performance is superior. Readers will understand how deep reinforcement learning and big data analytics test the effectiveness of 3D multimodal medical image segmentation algorithm., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2022 Gao, Li, Wang and Cai.)

Digital radiography is a popular diagnostic imaging method. Denoising and enhancement have an important potential in obtaining as much easily interpretable diagnostic information as possible with reasonable absorbed doses of ionising radiation. Due to the increasing usage of high resolution and high precision images with a limited number of human experts, the computational efficiency of the denoising and enhancement becomes important. In this paper, a local adaptive image enhancement and simultaneous denoising algorithm for fulfilling the requirements of digital X-ray image enhancement is introduced. The algorithm is based on modification of the wavelet transform coefficients by a pointwise nonlinear transformation and reconstructing the enhanced image from the modified wavelet transform coefficients. The implementation of algorithm in software is simple, quick, and universal.

This letter proposed and demonstrated a wavelet technique to reduce the time domain noise to get submeter spatial resolution in the distributed vibration sensor based on phase optical time domain reflectometry. A new wavelet shrinkage approach allows the distributed vibration measurement of 20-Hz and 8-kHz events to be detected over 1-km sensing length with a 5-ns optical pulse, which is equivalent to 50-cm spatial resolution using the single-mode sensing fiber. [ABSTRACT FROM PUBLISHER]

We present the development and implementation of a novel wavelet shrinkage technique for the retrieval of obscured characteristic resonant signatures in the scattered terahertz (THz) reflectivity of molecular crystals. In this implementation, the wavelet basis functions associated with the absorption features were identified using the second-order total variation of the wavelet coefficients. Additionally, wavelet coefficients at certain scales were modified using the phase function corrections and wavelet hard thresholding. Reconstruction of the original spectra using these modified wavelet coefficients yielded the exact resonant frequencies of the chemicals, which were otherwise unrecognizable in the spectral artifacts of the rough surface scattering. We examined the robustness of this method over controlled levels of rough surface scattering, validated using the Kirchhoff approximation, in spectroscopic targets made from α -lactose monohydrate and 4-aminobenzoic acid (PABA), which have close spectral lines. We successfully retrieved the spectral absorption fingerprints in both specular and off-specular reflection geometries. This technique can be utilized for stand-off material characterization using the THz reflection spectroscopy in uncontrolled environments and potentially can be adopted for other broadband spectroscopic modalities.