Shearlet Coefficients Research Articles

Study of Shearlet Domain Scale-Angle Adaptive Surface Wave Separation

In the seismic data processing, high energy surface wave overlaps with effective wave, which affects the quality of seismic data imaging. The traditional frequency domain method can effectively remove surface wave, but it will damage the low frequency information of seismic data. In view of this, we proposed a scale-angle adaptive surface wave separation method based on Shearlet sparse transform. The surface wave model obtained by filtering method and the effective wave can decomposed in Shearlet domain, based on the characteristic differences between surface wave and effective wave in frequency, velocity and direction. At different scales and angles, an adaptive surface wave removal operator based on L2 norm of Shearlet coefficient is developed to remove surface wave information from seismic data. The actual seismic data application results show that compared with the traditional surface wave suppression method, the proposed method can thoroughly separate surface wave and effective wave, and obtain high quality denoising seismic data.

Shearlet-based intensity classification loop filter for video coding

The new Versatile Video Coding standard (VVC) exhibits a significantly higher coding efficiency in comparison to its predecessor High Efficiency Video Coding (HEVC). Since the finalization of VVC in July 2020 many new in-loop compression tools were suggested, especially with data-driven approaches. The question arises if non-data-driven in-loop filtering approaches can further increase coding efficiency beyond the capability of VVC. In this paper we approach the task of filtering images corrupted by quantization noise from the perspective of applied harmonic analysis, an area of applied mathematics focusing on the efficient representation, analysis and encoding of data. The shearlet-based intensity classification loop filter (SCLF) is a novel non-data-driven approach using an overcomplete and sparsifying transform, the shearlet transform. The basic idea is to achieve a signal-noise separation by applying a shearlet transform. Shearlets can identify important structures of natural images such as edges in the sparse transform domain. Each shearlet transform coefficient is classified into different classes. All coefficients with similar intensities are grouped into the same class, which eventually gives a partition of the set of all shearlet coefficients. This separates important information from noise and ensures an accurate reconstruction of original information by performing an inverse shearlet transform combined with Wiener filtering separately for each class. SCLF effectively removes compression artifacts and therefore restores subjective visual quality. Simulation results show that by adding SCLF to the in-loop stage yet further average bit rate reductions of up to 1.5% are achieved for JVET common test sequences.

Analysis of directional higher order jump discontinuities with trigonometric shearlets

<p style='text-indent:20px;'>In a recent article, we showed that trigonometric shearlets are able to detect directional step discontinuities along edges of periodic characteristic functions. In this paper, we extend these results to bivariate periodic functions which have jump discontinuities in higher order directional derivatives along edges. In order to prove suitable upper and lower bounds for the shearlet coefficients, we need to generalize the results about localization- and orientation-dependent decay properties of the corresponding inner products of trigonometric shearlets and the underlying periodic functions.</p>

Deep learning with multiresolution handcrafted features for brain MRI segmentation

The segmentation of magnetic resonance (MR) images is a crucial task for creating pseudo computed tomography (CT) images which are used to achieve positron emission tomography (PET) attenuation correction. One of the main challenges of creating pseudo CT images is the difficulty to obtain an accurate segmentation of the bone tissue in brain MR images. Deep convolutional neural networks (CNNs) have been widely and efficiently applied to perform MR image segmentation. The aim of this work is to propose a segmentation approach that combines multiresolution handcrafted features with CNN-based features to add directional properties and enrich the set of features to perform segmentation. The main objective is to efficiently segment the brain into three tissue classes: bone, soft tissue, and air. The proposed method combines non subsampled Contourlet (NSCT) and non subsampled Shearlet (NSST) coefficients with CNN’s features using different mechanisms. The entropy value is calculated to select the most useful coefficients and reduce the input’s dimensionality. The segmentation results are evaluated using fifty clinical brain MR and CT images by calculating the precision, recall, dice similarity coefficient (DSC), and Jaccard similarity coefficient (JSC). The results are also compared to other methods reported in the literature. The DSC of the bone class is improved from 0.6179 ± 0.0006 to 0.6416 ± 0.0006. The addition of multiresolution features of NSCT and NSST with CNN’s features demonstrates promising results. Moreover, NSST coefficients provide more useful information than NSCT coefficients.

An Automated Brain Image Analysis System for Brain Cancer using Shearlets

In this paper, an Automated Brain Image Analysis (ABIA) system that classifies the Magnetic Resonance Imaging (MRI) of human brain is presented. The classification of MRI images into normal or low grade or high grade plays a vital role for the early diagnosis. The Non-Subsampled Shearlet Transform (NSST) that captures more visual information than conventional wavelet transforms is employed for feature extraction. As the feature space of NSST is very high, a statistical t-test is applied to select the dominant directional sub-bands at each level of NSST decomposition based on sub-band energies. A combination of features that includes Gray Level Co-occurrence Matrix (GLCM) based features, Histograms of Positive Shearlet Coefficients (HPSC), and Histograms of Negative Shearlet Coefficients (HNSC) are estimated. The combined feature set is utilized in the classification phase where a hybrid approach is designed with three classifiers; k-Nearest Neighbor (kNN), Naive Bayes (NB) and Support Vector Machine (SVM) classifiers. The output of individual trained classifiers for a testing input is hybridized to take a final decision. The quantitative results of ABIA system on Repository of Molecular Brain Neoplasia Data (REMBRANDT) database show the overall improved performance in comparison with a single classifier model with accuracy of 99% for normal/abnormal classification and 98% for low and high risk classification.

Two simple shearlet-based inverses for the multidimensional Radon and John transforms

We use well-known intertwining relations of integral operators between the Euclidean space and the manifold of planes/lines to develop new analytical inverses of the Radon/John transforms. The integral operators are convolutions with shearlets as kernels. The shearlets are smooth compactly supported and away from the origin in the Fourier domain, therefore the operators are bounded. We show a series of relations in the form of shearlet coefficients, convolutions and inner products in different spaces. This framework yields potential applications since there is a natural translation from the continuous to the discrete theory of shearlets. We also develop a new biorthogonal shearlet decomposition for the 3D Radon transform inversion. This shades light on the differences between our method of integral (bounded) operators and the biorthogonal curvelet/shearlet decomposition of differential (unbounded) operators.

A Novel Gaussian-Copula modeling for image despeckling in the shearlet domain

Image denoising, degraded by speckle noise, has turned into a prevalent problem in image processing. Speckle noise provides a granular perspective into the image and makes it difficult to have a clear interpretation of the singular and non-singular points. This paper proposes a novel statistical approach based on Gaussian Copula modeling in the Shearlet domain. The proposed multi-dimensional Minimum Mean Square Error (MMSE) processor consists of two key components. First, Bi Parameter Cauchy Gaussian Mixture model (BCGM) as the marginal distribution of shearlet coefficients is employed. Second, the joint-prior distribution modeling is formed, based on proposing the Gaussian copula, to model the dependency of the target coefficient respect to its neighbors. The closed form mathematical expression of proposed multi dimensional MMSE processor has some computational advantages, such as parallel computing. It will be shown that the proposed processor has adaptive behavior, meaning that, working non-linearly according to the estimated noise variance on each scale and direction of the shratlet transform. This behavior confirms that the designed processor is not sensitive to initial parameter settings, unlike the state-of-the-art filters in this area.

Sparse-Promoting 3-D Airborne Electromagnetic Inversion Based on Shearlet Transform

The conventional, L2-norm-based, regularization term in electromagnetic (EM) inversions implements smooth constraints on model complexity in the space domain, which can smoothen the boundaries of complex underground structures. To improve the resolution of 3-D frequency-domain airborne EM (AEM) inversions, we propose a new algorithm for sparse-regularized inversion based on the shearlet transform. Unlike traditional methods that invert the model parameters in the space domain, we first transform the 3-D resistivity model into the frequency domain and then invert the sparse coefficients using an L1-norm measure to ensure the sparseness of the solution. Finally, we transform the shearlet coefficients back to the space domain to update the model. The shearlet transform has inherent multiscale and multidirectional properties, making it capable of effectively extracting complex geometries such as curved boundaries. We adopt the finite-difference method and the iteratively reweighted least-squares scheme for our 3-D AEM modeling and inversions and apply the “moving footprint” technique to speed up the inversion. Tests using synthetic data show that sparse-regularized inversion based on the shearlet transform can obtain more-focused inversion results than conventional smoothness-constrained inversions based on the L2-norm. Tests using field survey data also reveal that the new method can achieve more realistic underground structures.

Gammadion binary pattern of Shearlet coefficients (GBPSC): An illumination-invariant heterogeneous face descriptor

• Usage of Shearlet coefficients is proposed to cancel illumination and noise variations. • Novel Gammadion Binary Pattern is proposed to extract out local micro level features. • GBPSC descriptor is devised to capture modality-invariant patterns of facial features. • GBPSC along with CNN outperforms other competing methods even with a small dataset. This paper presents a novel face image descriptor called Gammadion Binary Pattern of Shearlet Coefficients (GBPSC) for illumination and noise invariant, homogeneous and heterogeneous face recognition. Exploiting the energy concentration property of the Digital Shearlet Transform, an efficient illumination and noise invariant feature extractor has been devised. Finally, inspired by the Gammadion structure, a robust multi-directional local binary pattern named Gammadion Binary Pattern (GBP) has been proposed. GBP is applied on the previously extracted illumination and noise invariant feature map to generate the GBPSC images. Recognition results on Extended Yale B and TUFTS dataset indicate the primacy of the proposed scheme in terms of common feature representation under varying illumination, and modality. Furthermore, the merger of the proposed GBPSC and Convolutional Neural Network (CNN) consistently outperforms other state-of-the art methods.

Deep Shearlet Residual Learning Network for Single Image Super-Resolution.

Recently, the residual learning strategy has been integrated into the convolutional neural network (CNN) for single image super-resolution (SISR), where the CNN is trained to estimate the residual images. Recognizing that a residual image usually consists of high-frequency details and exhibits cartoon-like characteristics, in this paper, we propose a deep shearlet residual learning network (DSRLN) to estimate the residual images based on the shearlet transform. The proposed network is trained in the shearlet transform-domain which provides an optimal sparse approximation of the cartoon-like image. Specifically, to address the large statistical variation among the shearlet coefficients, a dual-path training strategy and a data weighting technique are proposed. Extensive evaluations on general natural image datasets as well as remote sensing image datasets show that the proposed DSRLN scheme achieves close results in PSNR to the state-of-the-art deep learning methods, using much less network parameters.

WSMN: An optimized multipurpose blind watermarking in Shearlet domain using MLP and NSGA-II

Digital watermarking is a remarkable issue in the field of information security to avoid the misuse of images in multimedia networks. Although access to unauthorized persons can be prevented through cryptography, it cannot be simultaneously used for copyright protection or content authentication with the preservation of image integrity. Hence, this paper presents an optimized multipurpose blind watermarking in the Shearlet domain with the help of smart algorithms, including MLP and NSGA-II. In this method, four copies of the robust copyright logo are embedded in the approximate coefficients of Shearlet by using an effective quantization technique. Furthermore, an embedded random sequence as a semi-fragile authentication mark is effectively extracted from details by the neural network. Due to performing an effective optimization algorithm for selecting optimum embedding thresholds and also distinguishing the texture of blocks, the imperceptibility and robustness have been preserved. The experimental results reveal the superiority of scheme with regard to the quality of watermarked images and robustness against hybrid attacks over other state-of-the-art schemes. The average PSNR and SSIM of the dual watermarked images are 38 dB and 0.95, respectively; Besides, it can effectively extract the copyright logo and locates forgery regions under severe attacks with satisfactory accuracy.

Texture features in the Shearlet domain for histopathological image classification

BackgroundA various number of imaging modalities are available (e.g., magnetic resonance, x-ray, ultrasound, and biopsy) where each modality can reveal different structural aspects of tissues. However, the analysis of histological slide images that are captured using a biopsy is considered the gold standard to determine whether cancer exists. Furthermore, it can reveal the stage of cancer. Therefore, supervised machine learning can be used to classify histopathological tissues. Several computational techniques have been proposed to study histopathological images with varying levels of success. Often handcrafted techniques based on texture analysis are proposed to classify histopathological tissues which can be used with supervised machine learning.MethodsIn this paper, we construct a novel feature space to automate the classification of tissues in histology images. Our feature representation is to integrate various features sets into a new texture feature representation. All of our descriptors are computed in the complex Shearlet domain. With complex coefficients, we investigate not only the use of magnitude coefficients, but also study the effectiveness of incorporating the relative phase (RP) coefficients to create the input feature vector. In our study, four texture-based descriptors are extracted from the Shearlet coefficients: co-occurrence texture features, Local Binary Patterns, Local Oriented Statistic Information Booster, and segmentation-based Fractal Texture Analysis. Each set of these attributes captures significant local and global statistics. Therefore, we study them individually, but additionally integrate them to boost the accuracy of classifying the histopathology tissues while being fed to classical classifiers. To tackle the problem of high-dimensionality, our proposed feature space is reduced using principal component analysis. In our study, we use two classifiers to indicate the success of our proposed feature representation: Support Vector Machine (SVM) and Decision Tree Bagger (DTB).ResultsOur feature representation delivered high performance when used on four public datasets. As such, the best achieved accuracy: multi-class Kather (i.e., 92.56%), BreakHis (i.e., 91.73%), Epistroma (i.e., 98.04%), Warwick-QU (i.e., 96.29%).ConclusionsOur proposed method in the Shearlet domain for the classification of histopathological images proved to be effective when it was investigated on four different datasets that exhibit different levels of complexity.

Numerical solutions of the wave equation using time-independent shearlet coefficients

In this paper, we present a new function (a shearlet) and calculate shearlet coefficients which yield a solution to the wave equation. In our method, the effect of the parameter time which makes the calculations complicated, is reduced.

An Image Denoising Method Based on BM4D and GAN in 3D Shearlet Domain

To overcome the disadvantages of the traditional block-matching-based image denoising method, an image denoising method based on block matching with 4D filtering (BM4D) in the 3D shearlet transform domain and a generative adversarial network is proposed. Firstly, the contaminated images are decomposed to get the shearlet coefficients; then, an improved 3D block-matching algorithm is proposed in the hard threshold and wiener filtering stage to get the latent clean images; the final clean images can be obtained by training the latent clean images via a generative adversarial network (GAN).Taking the peak signal-to-noise ratio (PSNR), structural similarity (SSIM for short) of image, and edge-preserving index (EPI for short) as the evaluation criteria, experimental results demonstrate that the proposed method can not only effectively remove image noise in high noisy environment, but also effectively improve the visual effect of the images.

Segmentation‐based MAP despeckling of medical ultrasound images in shearlet domain based on normal inverse Gaussian distribution

Speckle noise is an undesirable phenomenon that exhibits granular pattern which reduces the diagnostic capability of clinical ultrasound (US) images. In this study, a new speckle denoising method based on modelling of detail band shearlet coefficients of log-transformed US images is presented. In each detail shearlet subband, coefficients corresponding to signal and speckle noise are modelled as normal inverse Gaussian and Gaussian priors, respectively. These coefficients, based on image local statistics are segmented into different regions of heterogeneity viz. homogeneous, heterogeneous and strongly heterogeneous regions, respectively, so as to control over smoothing of denoised images. Then, using the prior distributions maximum a posteriori (MAP) estimation is performed over all regions of detail bands except those regions that represent strongly heterogeneous coefficients. For better performance, an adaptive weight function is also used in the MAP expression which reduces the loss of feature information. Experimentation is done on noise-free synthetic kidney and foetus US images and a set of 60 real US images. The results are presented for objective and subjective quality assessment of the proposed method and five other methods for speckle denoising. The potential of the proposed method in comparison to other methods can easily be ascertained from the obtained denoising results.

Self-Supervised Light Field Reconstruction Using Shearlet Transform and Cycle Consistency

The image-based rendering approach using Shearlet Transform (ST) is one of the state-of-the-art Densely-Sampled Light Field (DSLF) reconstruction methods. It reconstructs Epipolar-Plane Images (EPIs) in image domain via an iterative regularization algorithm restoring their coefficients in shearlet domain. Consequently, the ST method tends to be slow because of the time spent on domain transformations for dozens of iterations. To overcome this limitation, this letter proposes a novel self-supervised DSLF reconstruction method, CycleST, which applies ST and cycle consistency to DSLF reconstruction. Specifically, CycleST is composed of an encoder-decoder network and a residual learning strategy that restore the shearlet coefficients of densely-sampled EPIs using EPI reconstruction and cycle consistency losses. Besides, CycleST is a self-supervised approach that can be trained solely on Sparsely-Sampled Light Fields (SSLFs) with small disparity ranges ($\leqslant$ 8 pixels). Experimental results of DSLF reconstruction on SSLFs with large disparity ranges (16 - 32 pixels) from two challenging real-world light field datasets demonstrate the effectiveness and efficiency of the proposed CycleST method. Furthermore, CycleST achieves ~ 9x speedup over ST, at least.

Use of the shearlet energy entropy and of the support vector machine classifier to process weak microseismic and desert seismic signals

Low-amplitude signal detection is a key procedure in borehole microseismic and desert seismic exploration. Usually, signals are difficult to detect due to their low amplitude and noise contamination. To solve this problem, we propose a method combining shearlet energy entropy with a support vector machine (SVM) to detect low-amplitude signals. In the proposed method, the signal feature is extracted using shearlet energy entropy. The signal is more sparsely represented in the shearlet domain because of the multi-scale and multi-direction characteristic of the shearlet transform, which favours signal feature extraction. Furthermore, in calculating shearlet energy entropy, we use the correlation of shearlet coefficients to enhance the difference between signal and noise in the shearlet domain. Shearlet energy entropy makes the SVM achieve a more accurate classification result compared with other traditional features such as amplitude and energy. The results of synthetic and field data show that our method is more effective than the STA/LTA and the convolutional neural network for low-amplitude microseismic signal and desert seismic signal detection.

Identification of Colon Cancer Using Multi-Scale Feature Fusion Convolutional Neural Network Based on Shearlet Transform

Colon cancer identification is of great significance in medical diagnosis. Real-time, objective and accurate inspection results will facilitate medical professionals to explore symptomatic treatment promptly. However, the existing methods depend on hand-crafted features which require extensive professional expertise and long inspection period. Therefore, we propose a multi-scale feature fusion convolutional neural network (MFF-CNN) based on shearlet transform to identify histopathological image of colon cancer. The characteristic of the framework is the shearlet coefficients of histopathological image in multiple decomposition scales were extracted as supplementary feature which were also fed to the network with the original pathological image. After feature learning and feature fusion, the MFF-CNN based on shearlet transform can achieve the identification accuracy of 96% and average F-1 score of 0.9594 for colorectal adenocarcinoma epithelium (TUM) and normal colon mucosa (NORM). The false negative rate and false positive rate can be reduced to 5.5% and 2.5%, respectively. The superior performance of the network opens a new perspectives for real-time, objective and accurate diagnosis of cancer.

Skin Cancer Diagnostic using Machine Learning Techniques - Shearlet Transform and Naïve Bayes Classifier

Development of abnormal cells in the skin is known as skin cancer or melanoma, which can spread other parts of the body. Melanoma rarely occurs in eye, mouth and intestines. In this study, the classification of melanoma using shearlet transform coefficients and naïve Bayes classifier is discussed. The melanoma images are decomposed by the shearlet transform. Then, from the shearlet coefficients, predefined number of (50, 75 and 100) coefficients are selected from the decomposed subbands. The selected subband coefficients are directly applied to the naïve Bayes classifier. Performance of skin cancer classification system is measured in terms of accuracy. Results show that a better classification accuracy of 90.5 % is achieved at 3rd level with 100 coefficients of shearlet transform and naïve Bayes classifier for skin image classification system.

A Novel Despeckling Method for Medical Ultrasound Images Based on the Nonsubsampled Shearlet and Guided Filter

Ultrasound diagnostic techniques are widely used in medical clinical diagnostics. However, the presence of speckle noise in the ultrasound imaging process reduces the image quality and creates inconvenience to the physician during clinical diagnosis. The ability to reduce the influence of speckle noise has important significance therefore in medical ultrasound image diagnosis. This paper offers a solution. It proposes a novel despeckling method based on nonsubsampled shearlet transformation and a guided filter. First, a nonsubsampled Laplacian pyramid filter is used to decompose the noisy image thus decomposing the image into high-frequency and low-frequency subbands. Under the direction of the non-sampling filter bank, a high-frequency subband multi-directional decomposition is obtained. Next, based on the threshold function and the correlation of the shearlet coefficients in the transformation domain, an improved threshold shrinkage algorithm is proposed to perform the threshold shrinkage processing on the shearlet coefficients of the high-frequency subbands. Finally, the low-frequency subbands in the transformation domain are processed by the guided filter, and a denoised ultrasonic image is obtained by the inverse transformation of the shearlet. So as to verify the effectiveness of the proposed method, experiments were conducted, and the results were compared to those of other existing denoising filters. These showed the proposed method performs more effectively at denoising and delivers clearer image detail.