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Consider the nonparametric curve estimation problem. Wavelet shrinkage methods are applied to the data in the wavelet domain for noise reduction. After denoising, the function can be estimated by w...

The paper is devoted to the new model for image decomposition, that splits an image f into three components u+v+w, where u a piecewise-smooth or the "cartoon" component, v a texture component and w the noise part in variational approach. This decomposition model in fact incorporates the advantages of two preceding models: the second-order total variation minimization of Rudin-Osher-Fatemi (ROF2), and wavelet shrinkage for oscillatory functions. This decomposition model is presented as an extension of the three components decomposition algorithm of Aujol et al. in [Dual norms and image decomposition models, Int. J. Comput. Vis. 63(1)(2005) 85–104]. It also continues the idea introduced previously by authors in [Denoising 3D medical images using a second order variational model and wavelet shrinkage, Imag. Anal. and Rec., Lecture Notes in Computer Science, 7325(2012), 138-145], for two components decomposition model. The ROF2 model was first proposed by Bergounioux et al. in [A second-order model for image denoising, Set-Valued Anal. and Var. Anal., 18(2010), 277-306], it is an improved regularization method to overcome the undesirable staircasing effect. The wavelet shrinkage is well combined to separate the oscillating part due to texture from that due to noise. Experimental results validate the proposed algorithm and demonstrate that the image decomposition model presents effective and comparable performance to other previous models.

In this paper, a novel algorithm based on the local projection noise reduction approach is applied to smooth noise for strongly contaminated chaotic oscillators. Specifically, one-dimensional time series are embedded into a high dimensional phase space and the noise level is defined through orthogonal projections of the data points within the neighbourhood of the reference point onto linear subspaces. The current vector of the phase space is denoised by performing two-dimensional discrete stationary wavelet transform (SWT)-based filtering in the neighbourhood of the phase point. Numerical results show that our algorithm effectively recovers continuous-time chaotic signals in heavy-noise environments and outperforms the classical local projection noise reduction approach for simulated data from the Rössler system and Duffing oscillator at signal-to-noise ratios (SNRs) from 15 to 0 dB, either for the real world data – human breath time series.

The method of detrended fluctuation analysis (DFA) is useful in revealing the extent of long-range dependence, it has successfully been applied to different fields of interest. In this paper we proposed a smoothed detrended fluctuation analysis method based on the principle of wavelet shrinkage. The procedure is illustrated and compared with the DFA method by Monte Carlo simulations on fractional Gaussian noise models.

Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given function f and is widely used in data compression, signal processing and statistics, etc. For Calderon-Zygmund operators T, it is interesting to construct estimator of Tf, based on wavelet shrinkage estimator of f. With the help of a representation of operators on wavelets, due to Beylkin et al., an estimator of Tf is presented in this paper. The almost everywhere convergence and norm convergence of the proposed estimators are established.

In this paper, the generalized Bayes estimator of elliptical distribution parameter’s under asymmetric Linex error loss function is considered. The new shrinkage generalized Bayes estimator by applying wavelet transformation is investigated. We develop admissibility and minimaxity of shrinkage estimator on multivariate normal distribution.We present the simulation in order to test validity of purpose estimator.

In Medical diagnosis operations such as feature extraction and object recognition will play the key role. These tasks will become difficult if the images are corrupted with noise. So the development of effective algorithms for noise removal became an important research area in present days. Developing Image denoising algorithms is a difficult task since fine details in a medical image embedding diagnostic information should not be destroyed during noise removal. Many of the wavelet based denoising algorithms use DWT (Discrete Wavelet Transform) in the decomposition stage are suffering from shift variance and lack of directionality. To overcome this in this paper we are proposing the denoising method which uses dual tree complex wavelet transform to decompose the image and shrinkage operation to eliminate the noise from the noisy image. In the shrinkage step we used semi-soft and stein thresholding operators along with traditional hard and soft thresholding operators and verified the suitability of dual tree complex wavelet transform for the denoising of medical images. The results proved that the denoised image using DTCWT (Dual Tree Complex Wavelet Transform) have a better balance between smoothness and accuracy than the DWT and less redundant than UDWT (Undecimated Wavelet Transform). We used the SSIM (Structural similarity index measure) along with PSNR (Peak signal to noise ratio) to assess the quality of denoised images.

By substituting an anisotropic diffusion operator for the isotropic Laplace operator in the directed diffusion equation, adding two different coefficients in the two diffusion terms, and choosing the image denoised by soft wavelet shrinkage as the initial approximate image, an improved directed diffusion equation model is proposed. Experiments show that the new model can solve the problem that edges and details will be rapidly blurred during the diffusion process in the directed diffusion equation.

In this paper, we introduce a novel hybrid variational model which gen- eralizes the classical total variation method and the wavelet shrinkage method. An alternating minimization direction algorithm is then employed. We also prove that it converges strongly to the minimizer of the proposed hybrid model. Finally, some nu- merical examples illustrate clearly that the new model outperforms the standard total variation method and wavelet shrinkage method as it recovers better image details and avoids the Gibbs oscillations. AMS subject classifications: 90C25, 52A41, 28E99, 62P35

Cross-validation has been successfully used in various areas of statistics. However, it has not been used much in wavelet shrinkage estimation because fast wavelet methods cannot be applied to deleted data. In this paper, we show this problem can be avoided by using a fast imputation of data. This allows level-dependent cross- validation which is attractive to data with different sparseness. The proposed methods can be easily extended to higher dimensional problem such as image. Results from simulation and examples demonstrate the promising empirical properties of the procedure. In particular, the methods proposed in this work provide outstanding results for non-Gaussian noises because cross-validation is not based on normality assumptions.

This paper considers the problem of selecting a robust threshold of wavelet shrinkage. Previous approaches reported in literature to handle the presence of outliers mainly focus on developing a robust procedure for a given threshold; this is related to solving a nontrivial optimization problem. The drawback of this approach is that the selection of a robust threshold, which is crucial for the resulting fit is ignored. This paper points out that the best fit can be achieved by a robust wavelet shrinkage with a robust threshold. We propose data-driven selection methods for a robust threshold. These approaches are based on a coupling of classical wavelet thresholding rules with pseudo data. The concept of pseudo data has influenced the implementation of the proposed methods, and provides a fast and efficient algorithm. Results from a simulation study and a real example demonstrate the promising empirical properties of the proposed approaches.

In this paper, we present a new class of iterative regularization methods in the setting of Besov spaces, which can be seen as generalizations of J. Xu's method. By incorporating translation invariant wavelet transform, minimizers of the new methods can be understood as the alternative to translation invariant wavelet shrinkage with weight that is dependent on the wavelet decomposition scale and the Besov smooth order. And we generalize the iterative regularization methods to a new class of nonlinear inverse scale spaces with scale and Besov smooth order dependent weight. The numerical results show an excellent denoising effect and improvement over J. Xu's method.

We consider the denoising of a function (an image or a signal) containing smooth regions and edges. Classical ways to solve this problem are variational methods and shrinkage of a representation of the data in a basis or a frame. We propose a method which combines the advantages of both approaches. Following the wavelet shrinkage method of Donoho and Johnstone, we set to zero all frame coefficients with respect to a reasonable threshold. The shrunk frame representation involves both large coefficients corresponding to noise (outliers) and some coefficients, erroneously set to zero, leading to Gibbs-like oscillations in the estimate. We design a specialized (nonsmooth) objective function allowing all these coefficients to be selectively restored, without modifying the other coefficients which are nearly faithful, using regularization in the domain of the restored function. We analyze the well-posedness and the main properties of this objective function. We also propose an approximation of this method which...

The core of the wavelet approach to nonparametric regression is thresholding of wavelet coefficients. This paper reviews a cross-validation method for the selection of the thresholding value in wavelet shrinkage of Oh, Kim, and Lee (2006), and introduces the R package CVThresh implementing details of the calculations for the procedures. This procedure is implemented by coupling a conventional cross-validation with a fast imputation method, so that it overcomes a limitation of data length, a power of 2. It can be easily applied to the classical leave-one-out cross-validation and K-fold cross-validation. Since the procedure is computationally fast, a level-dependent cross-validation can be developed for wavelet shrinkage of data with various sparseness according to levels.

Consider an asymmetric Linex loss. We provide the soft wavelet shrinkage estimation of a Bayesian interpretation under such a loss.

The current work addresses topics related to signal filtering via wavelet shrinkage. Wavelets techniques applied to signal filtering go back to the 80’s and an overview about the main results in the area are presented, as well as the most recent contributions, concerning applications and methodological aspects involved. A short discussion about wavelets and a detailed presentation about the threshold operation are given. The main methods to define the threshold values are also described and their performances are tested through numerical simulations for 1D and 2D data. The presented results highlight the potential of associating wavelet methods to signal denoising techniques.

In this paper, we propose a new patch-based image denoising algorithm using graph signal processing. The concept of this algorithm is to take advantage of the redundancy of the BM3D transform and the edge preservation property of graph-based image processing. More specifically, we collect similar patches in the image, and construct a graph by connecting obtained patches. Then we apply a graph wavelet filter bank on graph signals to attenuate additive white gaussian noise by shrinking derived coefficients. We apply our proposed algorithm to depth map denoising. The experimental results demonstrate significant performance gains for the edge preservation and the noise reduction.

The Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.

We propose methods of both slow and rapid post-processing of signals for erasure of artifacts that arise in the process of thresholding and quantization. We use wavelets as tools to define constraints and variational functionals as measures of complexity of signals. The methods come from analyses of different possibilities of blending variational calculus and wavelet multiresolution in ways that appear to be natural.

Wavelet shrinkage estimators are obtained by applying a shrinkage rule on the empirical wavelet coefficients. Such simple estimators are now well explored and widely used in wavelet-based nonparametrics. Results of Tao (1996, Appl. Comput. Harmon. Anal.3, 384–387) demonstrated that hard and soft thresholding shrinkage estimators absolutely converge almost everywhere to the original function when the threshold value goes to zero. Such natural and intuitive behavior of threshold estimators is expected, yet this result does not translate to the Fourier expansions. In this paper we show that almost everywhere convergence of shrinkage estimators holds for a range of shrinkage rules, not necessarily thresholding, subject to some mild technical conditions. Comments about the norm convergence are provided as well.

Two-dimensional element distributions generated by scanning secondary ion mass spectrometry (SIMS) are characterised by Poisson statistics of small integer values, specially when the concentration of the measured element is in the sub-ppm range. To achieve high signal-to-noise ratios extremely long measurement time is needed. Because of the signal fluctuations from one measurement point to the next, structures even larger than the resolution of the instrument may not be detectable if the differences in concentration are small. This paper reports the application of a wavelet shrinkage algorithm for de-noising of images following Poisson distribution. In reconstructions of SIMS images resulting from this algorithm the noise is significantly suppressed without great loss of lateral resolution.

During ambulatory monitoring, it is sometimes required to record an electroencephalogram (EEG) and an electrocardiogram (ECG) simultaneously. It would be ideal if both EEG and ECG could be obtained with one measurement. Here, we introduce an algorithm that combines the wavelet shrinkage and variable cosine window operation to separate the EEG and ECG components from an EEG signal recorded with a noncephalic reference (NCR). Evaluation using simulated data and actual measured data showed that accurate frequency analysis of EEG and an R-R detection-based heart rate analysis were feasible with our proposed algorithm, which improved the signal-averaging based algorithm so that ECG components containing ectopic beats can be applied.

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.

We propose a new algorithm for denoising of multivariate function values given at scattered points in $${\mathbb{R}^{d}}$$ . The method is based on the one-dimensional wavelet transform that is applied along suitably chosen path vectors at each transform level. The idea can be seen as a generalization of the relaxed easy path wavelet transform by Plonka (Multiscale Model Simul 7:1474–1496, 2009) to the case of multivariate scattered data. The choice of the path vectors is crucial for the success of the algorithm. We propose two adaptive path constructions that take the distribution of the scattered points as well as the corresponding function values into account. Further, we present some theoretical results on the wavelet transform along path vectors in order to indicate that the wavelet shrinkage along path vectors can really remove noise. The numerical results show the efficiency of the proposed denoising method. peerReviewed

Total variation (TV) is a powerful method that brings great benefit for edge-preserving regularization. Despite being widely employed in image processing, it has restricted applicability for 1-D signal processing since piecewise-constant signals form a rather limited model for many applications. Here we generalize conventional TV in 1-D by extending the derivative operator, which is within the regularization term, to any linear differential operator. This provides flexibility for tailoring the approach to the presence of nontrivial linear systems and for different types of driving signals such as spike-like, piecewise-constant, and so on. Conventional TV remains a special case of this general framework. We illustrate the feasibility of the method by considering a nontrivial linear system and different types of driving signals.

This paper proposes Empirical Mode Decomposition (EMD) followed by wavelet shrinkage denoising in hyperspectral image classification to improve classification accuracy. EMD decomposes signals into several Intrinsic Mode Functions (IMFs) and a final residue. In this paper, firstly, EMD is applied to each hyperspectral image band separately to obtain the IMFs of all image bands. Then, the first IMF of each band is applied to wavelet shrinkage denoising, as this IMF includes all local high spatial frequency components. The sums of lower order IMFs are then used to reconstruct hyperspectral image bands that are used as new features for classification. Support Vector Machine (SVM) based classification is used as classification approach in this paper. Experimental results show the effectiveness of the proposed approach.

During ambulatory monitoring, it is often required to record the electroencephalogram (EEG) and the electrocardiogram (ECG) simultaneously. It would be ideal if both EEG and ECG can be obtained with one measurement. We introduce an algorithm combining the wavelet shrinkage and signal averaging techniques to extract the EEG and ECG components from an EEG lead signal to a noncephalic reference (NCR). The evaluation using simulation data and measured data showed that the normalized power spectrum unvaried in all frequency bands for the EEG components, and the sensitivity and specificity of R-wave detection for the ECG component were nearly 100%.

The measure technology of low-field pulsed NMR (nuclear magnetic resonance) is a new technology, which has great potential development. It can be widely used in industry and agriculture for product detection as a lossless method. The most important work of NMR measurement is the multi-exponential inversion of NMR signal based on the peak information. But the signal is interfused by noise during the measurement. It is important to reduce the noise while keeping the peaks well. The effective wavelet shrinkage method based on SWT (Stationary Wavelet Transform) is firstly introduced in this paper. All wavelet coefficients at the same scale are shrinked with the same threshold in the traditional algorithm, which is not appropriate for NMR signal because the SNR of NMR signal varies in different position. In view of this, the adaptive thresholds are introduced for NMR signal denoising. Different coefficients at the same wavelet scale are shrinked with different thresholds which are self-adaptive to SNR in our algorithm. The novel method is applied to the low-field pulsed NMR experimental equipment developed by ourselves. The experimental results indicate that our method can keep the peaks and edges while restraining noise well.

We consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces 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) We consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces 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)

The present paper investigates theoretical performance of various Bayesian wavelet shrinkage rules in a nonparametric regression model with i.i.d. errors which are not necessarily normally distributed. The main purpose is comparison of various Bayesian models in terms of their frequentist asymptotic optimality in Sobolev and Besov spaces. We establish a relationship between hyperparameters, verify that the majority of Bayesian models studied so far achieve theoretical optimality, state which Bayesian models cannot achieve optimal convergence rate and explain why it happens., Comment: Published at http://dx.doi.org/10.1214/009053606000000128 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

International audience; Among image restoration literature, there are mainly two kinds of approach. One is based on a process over image wavelet coefficients, as wavelet shrinkage for denoising. The other one is based on a process over image gradient. In order to get an edge-preserving regularization, one usually assume that the image belongs to the space of functions of Bounded Variation (BV). An energy is minimized, composed of an observation term and the Total Variation (TV) of the image. Recent contributions try to mix both types of method. In this spirit, the goal of this paper is to define a unified framework including together wavelet methods and energy minimization as TV. In fact, for denoising purpose, it is already shown that wavelet soft-thresholding is equivalent to choose the regularization term as the norm of the Besov space $B^{11}_1$. In the present work, this equivalence result is extended to the case of deconvolution problem. We propose a general functional to minimize, which includes the TV minimization, wavelet coefficients regularization, mixed (TV+wavelet) regularization or more general terms. Moreover we give a projection-based algorithm to compute the solution. The convergence of the algorithm is also stated. We show that the decomposition of an image over a dictionary of elementary shapes (atoms) is also included in the proposed framework. So we give a new algorithm to solve this difficult problem, known as Basis Pursuit. We also show numerical results of image deconvolution using TV, wavelets, or TV+wavelets regularization terms.

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.

Most two-dimensional methods for wavelet shrinkage are efficient for edge-preserving image denoising, but they suffer from poor rotation invariance. We address this problem by designing novel shrinkage rules that are derived from rotationally invariant nonlinear diffusion filters. The resulting Haar wavelet shrinkage methods are computationally inexpensive and they offer substantially improved rotation invariance.

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of (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 shrinkage denoising.

We consider a model problem of recovering a function $f(x_1,x_2)$ from noisy Radon data. The function $f$ to be recovered is assumed smooth apart from a discontinuity along a $C^2$ curve, that is, an edge. We use the continuum white-noise model, with noise level $\varepsilon$. Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level $\varepsilon$ only as $O(\varepsilon^{1/2})$ as $\varepsilon\to 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(\varepsilon^{2/3})$. However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain. We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\varepsilon^{4/5})$ as noise level $\varepsilon\to 0$. This rate of convergence holds uniformly over a class of functions which are $C^2$ except for discontinuities along $C^2$ curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example.

The still image compression standard which is being developed under the name of JPEG-2000 (Section 2) is a technological challenge which relies on some advances in pure mathematics. This interaction between image processing and functional analysis also benefits partial differential equations.

This chapter reviews the construction of wavelet thresholding (shrinkage) estimators by regularization. Both penalty and maximum a posterori approaches are presented and their relation is discussed.

The procedures of Donoho, Johnstone, Kerkyacharian and Picard [DJKP] estimate functions by inverting thresholded wavelet transform coefficients of the data. The choice of threshold is crucial to the success of the method and is currently subject to an intense research effort. We describe how we have applied the statistical technique of cross-validation to choose a threshold and we present results that indicate that its performance for correlated data. Finally, to illustrate the techniques, we apply various wavelet-based estimation methods to some noisy one- and two-dimensional signals and display the results.

Nonlinear diffusion 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 or multiscale. In this paper we show that one step of (stabilized) explicit discretization of nonlinear diffusion can be expressed in terms of tight frame shrinkage on a single spatial level or multiscale. We prove that our scheme permits larger steps while having more choices of shrinkage functions. Numerical examples demonstrate the behavior of our scheme for one or two scales.

A new adaptive shrinkage de-noising algorithm for images based on the neighbourhood characteristics is presented. This algorithm determines the shrinkage thresholds according to the neighbouring coefficients and the noise level. The experimental results show a great improvement with respect to the standard wavelet shrinkage algorithms.

We study a class of numerical schemes for nonlinear diffusion filtering that offers insights on the design of novel wavelet shrinkage rules for isotropic and anisotropic image enhancement. These schemes utilise analytical or semi-analytical solutions to dynamical systems that result from space-discrete nonlinear diffusion filtering on minimalistic images with 2 × 2 pixels. We call them locally analytic schemes (LAS) and locally semi-analytic schemes (LSAS), respectively. They can be motivated from discrete energy functionals, offer sharp edges due to their locality, are very simple to implement because of their explicit nature, and enjoy unconditional absolute stability. They are applicable to singular nonlinear diffusion filters such as TV flow, to bounded nonlinear diffusion filters of Perona–Malik type, and to tensor-driven anisotropic methods such as edge-enhancing or coherence-enhancing diffusion filtering. The fact that these schemes use processes within 2 × 2 -pixel blocks allows to connect them to shift-invariant Haar wavelet shrinkage on a single scale. This interpretation leads to novel shrinkage rules for two- and higher-dimensional images that are scalar-, vector- or tensor-valued. Unlike classical shrinkage strategies they employ a diffusion-inspired coupling of the wavelet channels that guarantees an approximation with an excellent degree of rotation invariance. By extending these schemes from a single scale to a multi-scale setting, we end up at hybrid methods that demonstrate the possibility to realise the effects of the most sophisticated diffusion filters within a fairly simplistic wavelet setting that requires only Haar wavelets in conjunction with coupled shrinkage rules.