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Accurate bearing fault diagnosis is essential for ensuring the health and longevity of mechanical systems. Traditional methods often struggle with the dynamic operating conditions of machinery, including variations in speed, load, and noise. This paper proposes a novel deep learning-based approach for robust bearing fault diagnosis. The method utilizes a combination of Shearlet Transform, Autoencoder, and Softmax Classifier. Vibration signals from healthy and faulty bearings are transformed into 2D image representations, capturing intricate details of the underlying mechanical state. Shearlet Transform is then employed to enhance these images, specifically targeting and amplifying subtle fault signatures, leading to improved diagnostic accuracy. The enhanced images are subsequently fed to an autoencoder, where the encoder compresses the data into a lower-dimensional feature space. These compressed features are then used to train and optimize a Softmax Classifier for effective fault classification. The proposed methodology is evaluated under diverse speed and load conditions, mimicking real-world operating scenarios. The achieved high classification accuracy across various operating points demonstrates the robustness and effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we establish Abelian and Tauberian results that characterize the quasiasymptotic behavior of Lizorkin distributions via the asymptotic behavior of their shearlet transform. [ABSTRACT FROM AUTHOR]

In mineral, environmental, and engineering explorations, we frequently encounter geological bodies with varied sizes, depths, and conductivity contrasts with surround rocks and try to interpret them with single survey data. The conventional three-dimensional (3-D) inversions significantly rely on the size of the grids, which should be smaller than the smallest geological target to achieve a good recovery to anomalous electric conductivity. However, this will create a large amount of unknowns to be solved and cost significant time and memory. In this paper, we present a multi-scale (MS) stochastic inversion scheme based on shearlet transform for airborne electromagnetic (AEM) data. The shearlet possesses the features of multi-direction and multi-scale, allowing it to effectively characterize the underground conductivity distribution in the transformed domain. To address the practical implementation of the method, we use a compressed sensing method in the forward modeling and sensitivity calculation, and employ a preconditioner that accounts for both the sampling rate and gradient noise to achieve a fast stochastic 3-D inversion. By gradually updating the coefficients from the coarse to fine scales, we obtain the multi-scale information on the underground electric conductivity. The synthetic data inversion shows that the proposed MS method can better recover multiple geological bodies with different sizes and depths with less time consumption. Finally, we conduct 3-D inversions of a field dataset acquired from Byneset, Norway. The results show very good agreement with the geological information. [ABSTRACT FROM AUTHOR]

This study introduces a novel multi-channel predictive deconvolution method enhanced by Shearlet-based sparse regularization, aimed at improving the accuracy and stability of subsurface seismic imaging, particularly in offshore wind farm site assessments. Traditional multi-channel predictive deconvolution techniques often struggle with noise interference, limiting their effectiveness. By integrating Shearlet transform into the multi-channel predictive framework, our approach leverages its directional and multiscale properties to enhance sparsity and directionality in seismic data representation. Tests on both synthetic and field data demonstrate that our method not only provides more accurate seismic images but also shows significant resilience to noise, compared to conventional methods. These findings suggest that the proposed technique can substantially improve geological feature identification and has great potential for enhancing the efficiency of seabed surveys in marine renewable energy development. [ABSTRACT FROM AUTHOR]

Defect detection systems based on machine vision have been widely used as an essential part of intelligent manufacturing systems. However, in traditional object detection methods that rely on images as input, differences in defect areas, blurred images, and complex background interference can seriously impair detection accuracy. To meet these challenges, this paper proposed a dual-path neural network based on shearlet transform (STDPNet) by taking advantage of shearlet transform in multi-scale analysis and combining it with the improved object detection algorithm proposed in this paper. First, images are multi-scale and multi-directional decomposed with shearlet transform, and multi-directional sub-band information is input to the detection network instead of image information. Then, this paper proposed a dual-path object detection network for the differences between different frequency bands and introduced a transfer learning strategy between paths to improve the model performance. Finally, the training results on the NEU surface defect public dataset show that the mean average precision of STDPNet achieves 86.81% at a detection speed of 44.45 f/s, which exceeds that of Faster R-CNN by 12%. Experiments on different datasets prove that the accuracy is significantly superior to other models, and the proposed method is more advantageous compared to other models in large, fuzzy, and indistinguishable defect types. [ABSTRACT FROM AUTHOR]

Traditional denoising methods often lose details or edges, such as Gaussian filtering. Shearlet transform isa multi-scale geometric analysis tool which has the advantages of multi-resolution and multi-directivity. Compared with wavelet, curvelet, and contourlet transforms, it can retain more edge details while denoising, and the subjective vision and objective evaluation indexes are better than other multi-scale geometric analysis methods. Deep learning has made great progress in the field of denoising, such as U_Net, DnCNN, FFDNet, and generative adversarial network, and the denoising effect is better than BM3D, the traditional optimal method. Therefore, we propose a random noise suppression network ST-hFFDNet based on non-subsampled shearlet transform (NSST) and improved FFDNet. It combines the advantages of non-subsampled shearlet transform, Huber norm, and FFDNet, and has three characteristics. 1) Shearlet transform is an effective feature extraction tool, which can obtain the high and low frequency features of a signal at different scales and in different directions, so that the network can learn signal and noise features of different scales and directions. 2) The noise level map can improve the noise reduction performance of different noise levels. 3) Huber norm can reduce the sensitivity of the network to abnormal data and improve the robustness of network. The network training process is as follows. 1) BSD500 datasets are enhanced by flipping, rotating, scaling, and cutting. 2) AWGN with noise level ae[0,75] is added to the enhanced datasets to obtain the training datasets. 3) NSST multi-scale and multi-direction decomposition is performed on each pair of samples of the training datasets to obtain highand low-frequency images of different scales and directions. 4) Based on the decomposed high and low frequency images, the ST-hFFDNet network is trained by Adam algorithm. 5) All samples of the training data set are carried out in steps (3) and (4), and the trained model is thus obtained. Simulation experiments and real seismic data denoising show that for low noise, the proposed method is slightly better than NSST, DnCNN, and FFDNet and that it is superior to NSST, DnCNN, and FFDNet for high noise. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the continuous and discrete version of the linear canonical shearlet transform (LCST). The authors begin with the definition of the LCST and then establish a relationship between the linear canonical transform (LCT) and the LCST. Next, the paper derives several basic properties like Parseval's Formula, inversion formula, and the characterization of the transform's range. In addition to the continuous version of the transform, the authors also present a discrete version of the LCST. This discrete version allows for practical implementation and efficient computation of the transform in digital signal processing systems. Lastly, the paper establishes a frame condition for the discrete LCST, which thereby helps in establishing the reconstruction formula for the discrete LCST. The paper ends with a conclusion. [ABSTRACT FROM AUTHOR]

For decades, wavelet theory has attracted interest in several fields in dealing with signals. Nowadays, it is acknowledged that it is not very suitable to face aspects of multidimensional data like singularities and this has led to the development of other mathematical tools. A recent application of wavelet theory is in radiomics, an emerging field aiming to improve diagnostic, prognostic and predictive analysis of various cancer types through the analysis of features extracted from medical images. In this paper, for a radiomics study of prostate cancer with magnetic resonance (MR) images, we apply a similar but more sophisticated tool, namely the shearlet transform which, in contrast to the wavelet transform, allows us to examine variations along more orientations. In particular, we conduct a parallel radiomics analysis based on the two different transformations and highlight a better performance (evaluated in terms of statistical measures) in the use of the shearlet transform (in absolute value). The results achieved suggest taking the shearlet transform into consideration for radiomics studies in other contexts. [ABSTRACT FROM AUTHOR]

Super-resolution techniques are employed to enhance image resolution by reconstructing high-resolution images from one or more low-resolution inputs. Super-resolution is of paramount importance in the context of remote sensing, satellite, aerial, security and surveillance imaging. Super-resolution remote sensing imagery is essential for surveillance and security purposes, enabling authorities to monitor remote or sensitive areas with greater clarity. This study introduces a single-image super-resolution approach for remote sensing images, utilizing deep shearlet residual learning in the shearlet transform domain, and incorporating the Enhanced Deep Super-Resolution network (EDSR). Unlike conventional approaches that estimate residuals between high and low-resolution images, the proposed approach calculates the shearlet coefficients for the desired high-resolution image using the provided low-resolution image instead of estimating a residual image between the high- and low-resolution image. The shearlet transform is chosen for its excellent sparse approximation capabilities. Initially, remote sensing images are transformed into the shearlet domain, which divides the input image into low and high frequencies. The shearlet coefficients are fed into the EDSR network. The high-resolution image is subsequently reconstructed using the inverse shearlet transform. The incorporation of the EDSR network enhances training stability, leading to improved generated images. The experimental results from the Deep Shearlet Residual Learning approach demonstrate its superior performance in remote sensing image recovery, effectively restoring both global topology and local edge detail information, thereby enhancing image quality. Compared to other networks, our proposed approach outperforms the state-of-the-art in terms of image quality, achieving an average peak signal-to-noise ratio of 35 and a structural similarity index measure of approximately 0.9. [ABSTRACT FROM AUTHOR]

Traditional denoising methods often lose details or edges, such as Gaussian filtering. Shearlet transform is a multi-scale geometric analysis tool which has the advantages of multi-resolution and multi-directivity. Compared with wavelet, curvelet, and contourlet transforms, it can retain more edge details while denoising, and the subjective vision and objective evaluation indexes are better than other multi-scale geometric analysis methods. Deep learning has made great progress in the field of denoising, such as U_Net, DnCNN, FFDNet, and generative adversarial network, and the denoising effect is better than BM3D, the traditional optimal method. Therefore, we propose a random noise suppression network ST-hFFDNet based on non-subsampled shearlet transform (NSST) and improved FFDNet. It combines the advantages of non-subsampled shearlet transform, Huber norm, and FFDNet, and has three characteristics. 1) Shearlet transform is an effective feature extraction tool, which can obtain the high and low frequency features of a signal at different scales and in different directions, so that the network can learn signal and noise features of different scales and directions. 2) The noise level map can improve the noise reduction performance of different noise levels. 3) Huber norm can reduce the sensitivity of the network to abnormal data and improve the robustness of network. The network training process is as follows. 1) BSD500 datasets are enhanced by flipping, rotating, scaling, and cutting. 2) AWGN with noise level σ∈[0,75] is added to the enhanced datasets to obtain the training datasets. 3) NSST multi-scale and multi-direction decomposition is performed on each pair of samples of the training datasets to obtain high- and low-frequency images of different scales and directions. 4) Based on the decomposed high and low frequency images, the ST-hFFDNet network is trained by Adam algorithm. 5) All samples of the training data set are carried out in steps (3) and (4), and the trained model is thus obtained. Simulation experiments and real seismic data denoising show that for low noise, the proposed method is slightly better than NSST, DnCNN, and FFDNet and that it is superior to NSST, DnCNN, and FFDNet for high noise.

The key to improving the fusion quality of infrared–visible images is effectively extracting and fusing complementary information such as bright–dark information and saliency details. For this purpose, an improved hybrid multiscale fusion algorithm inspired by non-subsampled shearlet transform (NSST) is proposed. In this algorithm, firstly, the support value transform (SVT) is used instead of the non-subsampled pyramid as the frequency separator to decompose an image into a set of high-frequency support value images and one low-frequency approximate background. These support value images mainly contain the saliency details from the source image. And then, the shearlet transform of NSST is retained to further extract the saliency edges from these support value images. Secondly, to extract the bright–dark details from the low-frequency approximate background, a morphological multiscale top–bottom hat decomposition is constructed. Finally, the extracted information is combined by different rules and the fused image is reconstructed by the corresponding inverse transforms. Experimental results have shown the proposed algorithm has obvious advantages in retaining saliency details and improving image contrast over those state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

Digital pathology refers to image-based environment in which acquisition, extraction and interpretation of pathology information is supported by computational techniques. It has a huge potential to facilitate the diagnostic process, however, big data size and necessity of large storage areas are challenging. Therefore, in this research, Compressed Sensing (CS) scheme is studied with digital pathology images in order to reduce the amount of data for reconstruction. CS requires the sparsity of signals for a successful recovery which means that different sparsifying bases can alter the final performance. Wavelet, Contourlet and Shearlet Transforms are investigated to sparsify the digital pathology images, it is seen that Contourlet Transform is superior. Alternating Direction Method of Multipliers (ADMM) is chosen for reconstruction since it is a robust and fast convex optimization method. Despite the fact that digital pathology images are less sparse than classical images, CS reconstruction is satisfactory, which emphasizes the potential of CS for digital pathology. This study can be pioneering in the field of CS with digital pathology so it can encourage further studies of CS based imaging with different type of microscopes or at different wavelengths. [ABSTRACT FROM AUTHOR]

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Synthetic Aperture Radar (SAR) is widely used for producing high quality imaging of Earth surface due to its capability of image acquisition in all-weather conditions. However, one limitation of SAR image is that image textures and fine details are usually contaminated with multiplicative granular noise named as speckle noise. This paper presents a speckle reduction technique for SAR images based on statistical modelling of detail band shearlet coefficients (SC) in homomorphic environment. Modelling of SC corresponding to noiseless SAR image are carried out as Normal Inverse Gaussian (NIG) distribution while speckle noise SC are modelled as Gaussian distribution. These SC are segmented as heterogeneous, strongly heterogeneous and homogeneous regions depending upon the local statistics of images. Then maximum a posteriori (MAP) estimation is employed over SC that belong to homogenous and heterogenous region category. The performance of proposed method is compared with seven other methods based on objective and subjective quality measures. PSNR and SSIM metrics are used for objective assessment of synthetic images and ENL metric is used for real SAR images. Subjective assessment is carried out by visualizing denoised images obtained from various methods. The comparative result analysis shows that for the proposed method, higher values of PSNR i.e. 26.08 dB, 25.39 dB and 23.82 dB and SSIM i.e. 0.81, 0.69 and 0.61 are obtained for Barbara image at noise variances 0.04, 0.1 and 0.15, respectively as compared to other methods. For other images also results obtained for proposed method are at higher side. Also, ENL for real SAR images show highest average value of 125.91 79.05. Hence, the proposed method signifies its potential in comparison to other seven existing image denoising methods in terms of speckle denoising and edge preservation. [ABSTRACT FROM AUTHOR]

Due to varying imaging principles and complexity of human organ structures, single-modality image can only provide limited information. Multimodality image fusion is the technique which integrates multimodal images into a single image which improves the quality of images by retaining significant features and helps diagnostic imaging practitioners for accurate treatment and evaluation of medical problems. In the current prevailing image fusion techniques presents numerous challenges including the prevelance of fusion artefacts, design complexity and high computational cost. In this paper, a novel multimodal medical image fusion method has been presented to address these problems. The proposed approach is based on combination of guided filter and image statistics in shearlet transform domain. The multimodal images are subjugated to image decomposition using shearlet transform that captures textures information of original images in multidirectional orientations and then decompose these paired images in low-and high-frequency coefficients (i.e. base and detail layers). Then guided filter with high epsilon value is used to obtain weights of original paired images. These weights are then added to the base layer to obtained unified base layers. A guided image filter and image statistics fusion rule is used to fuse base layers to obtain a fused base layer in covariance matrix and Eigen values are computed to figure out the significant pixels in the neighborhood. Similarly, a choose max fusion rule is used to fuse the detail layers for reconstruction. A unified fused base and detail layers are merged together to obtain final fusion result using inverse shearlet transform. The proposed method is evaluated using medical image datasets. Experimental result demonstrates that our proposed algorithm exhibits promising results and outperforms other prevailing fusion techniques. [ABSTRACT FROM AUTHOR]

This study presents a new approach based on shearlet transform for the first time to detect damages, and compare it with the wavelet, Laplacian pyramid, curvelet, and contourlet transforms to specify different types of defects in plate structures. Wavelet and Laplacian pyramid transforms have inferior performance to detect flaws with different multi-directions, such as curves, because of their basic element form, expressing the need for more efficient transforms. Therefore, some transforms, including curvelet and contourlet, have been evaluated so far for improving the performance of traditional transforms. Although these transforms have overcome the deficiencies of previous methods, they have a weakness in detecting several imperfections with various shapes in plate structures —one of the essential requirements that each transform should possess. In this study, we have used the shearlet transform that is used for the first time to detect identification and overcome all previous transform dysfunctionalities. In this regard, these transforms were applied to a four-fixed supported square plate with various defects. The obtained results revealed that the shearlet transform has the premier capability to demonstrate all kinds of damages compared to the other transforms, namely wavelet, Laplacian pyramid, curvelet, and contourlet. Also, the shearlet transform can be considered as an excellent and operational approach to demonstrate different forms of defects. Furthermore, the performance and correctness of the transforms have been verified via the experiment.

Backgorund: Nowadays, everybody's life is dominated by COVID-19, which might have been the source of severe acute respiratory syndrome coronavirus 2. This virus disrupts the lungs first of all. Recently, it has been found that coronavirus may affect the brain. Because all body actions rely on the brain, hence investigating its healthy is an essential item in coronavirus effects. Method: Brain image segmentation can be helpful in the detection of the regions damaged by the effects of coronavirus. Since every image given by photography devices may have noises, therefore, first of all, the brain magnetic resonance angiography (MRA) images must be denoised for best investigation. In the present paper, we have presented the construction of multishearlets based on multiwavelets for the first time and have used them for the purpose of denoising. Multiwavelets have some advantages to wavelets. Therefore, we have used them in the shearlet system to expand the properties of multiwavelets in all directions. After denoising, we have proposed a scheme for the automatic characterization of the initial curve in the active contour model for segmentation. Detecting the initial curve is a challenging task in active contour-based segmentation because detecting an initial curve far from the desired region can lead to unfavorable results. Results: The results show the performance of using multishearlets in detecting affected regions by COVID-19. Using multishearlets has led to the high value of peak signal-to-noise ratio and Structural similarity index measure in comparison with original shearlets. Original shearlets are constructed from wavelets whereas we have constructed multishearlets from multiwavelets. Conclusion: The results show that multishearlets can neutralize the effect of noise in MRA images in a good way rather than shearlets. Moreover, the proposed scheme for segmentation can lead to 0.99 accuracy.

Due to the deficient knowledge of binocular vision properties, how to effectively evaluate stereoscopic images still remains a challenging task. Inspired by multichannel processing of human visual system (HVS), we propose a blind method for stereoscopic image quality assessment (SIQA) by extracting quality related features in sub-bands of the image. First of all, we introduce the shearlet transform to decompose the left- and right-view images into multiple sub-bands content with diverse combinations of scales and orientations, and obtain the combined view based on energy-weighted summation of the corresponding sub-bands of two eye views. Then, natural scene statistics (NSS) of the original left and right images are obtained as quality-sensitive features, followed by extracting NSS features of the sub-bands of left, right and combined views. Moreover, we calculate the gradient similarity between each sub-band pair to denote the asymmetric distortion and disparity information. Finally, all the extracted features are mapped into a quality score by support vector regression (SVR). experimental results on multiple benchmark databases verify the superiority of our method.

Model-based image denoising methods are well suited for use as image processors in remote sensing systems such as satellites due to their well-developed mathematical theory and low computational cost, but these methods can often only deal with a single type of random noise. In this paper, based on the model-based image denoising techniques, a remote sensing image mixed noise denoising algorithm using Gaussian curvature in the image surface and shearlet transform is proposed. The Gaussian curvature filtering (GCF) is used to suppress the salt & pepper noise (SPN) in the image to get the first denoised image. The second denoised image is obtained by processing the coefficient matrices of the shearlet transform decomposition using statistical method and adaptive median filtering (AMF). Finally, the reconstructed image is again optimized by AMF to eliminate the residual SPN. Specially, we propose the goodness-of-fit (GOF) test denoising algorithm in shearlet domain based on the empirical distribution function (EDF) statistics to solve the problem of insufficient denoising by the traditional threshold function. Our method can effectively remove the noise and improve the visibility and usability of remote sensing images. Experimental results show that our proposed method has PSNR and MSE performance improvement compared with related model-based and learning-based methods.

Backgorund: Nowadays, everybody's life is dominated by COVID-19, which might have been the source of severe acute respiratory syndrome coronavirus 2. This virus disrupts the lungs first of all. Recently, it has been found that coronavirus may affect the brain. Because all body actions rely on the brain, hence investigating its healthy is an essential item in coronavirus effects. Method: Brain image segmentation can be helpful in the detection of the regions damaged by the effects of coronavirus. Since every image given by photography devices may have noises, therefore, first of all, the brain magnetic resonance angiography (MRA) images must be denoised for best investigation. In the present paper, we have presented the construction of multishearlets based on multiwavelets for the first time and have used them for the purpose of denoising. Multiwavelets have some advantages to wavelets. Therefore, we have used them in the shearlet system to expand the properties of multiwavelets in all directions. After denoising, we have proposed a scheme for the automatic characterization of the initial curve in the active contour model for segmentation. Detecting the initial curve is a challenging task in active contour-based segmentation because detecting an initial curve far from the desired region can lead to unfavorable results. Results: The results show the performance of using multishearlets in detecting affected regions by COVID-19. Using multishearlets has led to the high value of peak signal-to-noise ratio and Structural similarity index measure in comparison with original shearlets. Original shearlets are constructed from wavelets whereas we have constructed multishearlets from multiwavelets. Conclusion: The results show that multishearlets can neutralize the effect of noise in MRA images in a good way rather than shearlets. Moreover, the proposed scheme for segmentation can lead to 0.99 accuracy. [ABSTRACT FROM AUTHOR]

Due to the fast speed or long imaging distance, the road traffic monitoring image is blurred, and there are some problems such as incomplete, wrong extraction and low contrast when extracting the image. There- fore, a road traffic fuzzy image enhancement method based on multi-scale Retinex is proposed. The image is de- composed into low frequency component and high frequency component by Shearlet transform. Linear mapping is used to map the low-frequency coefficients to a specific interval for normalization, and multi-scale Retinex is used to enhance the low-frequency components. Robust median method and hard threshold shrinkage method are used to remove noise and illumination interference in each scale and direction of high frequency component. The fuzzy contrast and linear membership function are used to fuse the two parts of the image, and finally the enhanced clear image is obtained. The experimental results show that the proposed method can optimize the image visual effect, highlight the image detail features, and has good noise suppression performance. Compared with the genera- tive adversarial network method and the feature pyramid method, the peak signal-to-noise ratio, structural simi- larity, information entropy, average gradient and spatial frequency of the image are better. [ABSTRACT FROM AUTHOR]

This study presents a new approach based on shearlet transform for the first time to detect damages, and compare it with the wavelet, Laplacian pyramid, curvelet, and contourlet transforms to specify different types of defects in plate structures. Wavelet and Laplacian pyramid transforms have inferior performance to detect flaws with different multidirections, such as curves, because of their basic element form, expressing the need for more efficient transforms. Therefore, some transforms, including curvelet and contourlet, have been evaluated so far for improving the performance of traditional transforms. Although these transforms have overcome the deficiencies of previous methods, they have a weakness in detecting several imperfections with various shapes in plate structures --one of the essential requirements that each transform should possess. In this study, we have used the shearlet transform that is used for the first time to detect identification and overcome all previous transform dysfunctionalities. In this regard, these transforms were applied to a four-fixed supported square plate with various defects. The obtained results revealed that the shearlet transform has the premier capability to demonstrate all kinds of damages compared to the other transforms, namely wavelet, Laplacian pyramid, curvelet, and contourlet. Also, the shearlet transform can be considered as an excellent and operational approach to demonstrate different forms of defects. Furthermore, the performance and correctness of the transforms have been verified via the experiment. [ABSTRACT FROM AUTHOR]

We develop a distributional framework for the shearlet transform Sψ:S0(R2)→S(S)${\mathcal {S}}_{\psi }\,{:}\, {\mathcal {S}}_0\big (\mathbb {R}^2\big)\,{\rightarrow }\,{\mathcal {S}}(\mathbb {S})$ and the shearlet synthesis operator Sψt:S(S)→S0(R2)${\mathcal {S}}^t_{\psi }\!: \!{\mathcal {S}}(\mathbb {S})\!\rightarrow \!{\mathcal {S}}_0\big (\mathbb {R}^2\big)$, where S0(R2)${\mathcal {S}}_0\big (\mathbb {R}^2\big)$ is the Lizorkin test function space and S(S)${\mathcal {S}}(\mathbb {S})$ is the space of highly localized test functions on the standard shearlet group S$\mathbb {S}$. These spaces and their duals S0′(R2),S′(S)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)},\, \mathcal {S}^\prime (\mathbb {S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector ψ belongs to S0(R2)${\mathcal {S}}_0\big (\mathbb {R}^2\big)$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from S0′(R2)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)}$ to S′(S)$\mathcal {S}^\prime (\mathbb {S})$ and from S′(S)$\mathcal {S}^\prime (\mathbb {S})$ to S0′(R2)$\mathcal {S}_0^\prime \big (\mathbb {R}^2\big)$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group. [ABSTRACT FROM AUTHOR]

The aim of this study is to introduce a convolution-based two-dimensional fractional shearlet transform in the context of fractional time-frequency analysis. The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, the cone adapted variant of the two-dimensional fractional shearlet transform is also studied in detail. Nevertheless, several coherent examples are presented to facilitate a sound illustration of the concepts. [ABSTRACT FROM AUTHOR]

The Internet of Medical Things (IoMT) has included a new layer for development and smart infrastructure growth in the medical field. Besides, the medical data on IoMT systems are constantly expanding due to the rising peripherals in the health system. This paper introduces a new fusion technique in the shearlet domain to improve existing methods, which may provide medical image fusion in the IoMT system. In this paper, firstly low and high frequencies NSST coefficients are obtained of both input images. Over the low frequency component, a new Multi local extrema (MLE) based decomposition is performed to get more detail features (Coarse and detail layers). Over these MLE features saliency based weighted average is performed using co-occurrence filter to get the enhanced low frequency NSST Coefficients. These enhanced low frequency NSST Coefficients of both input images are fused using the proposed weighted function. In high frequency NSST Coefficients, local type-2 fuzzy entropy-based fusion is performed. Finally, inverse NSST is performed to get the final fused image. The experimental results are evaluated and compared with existing methods by visually and also by performance metrics. After a critical analysis, it was found that the results of the proposed method give better outcomes compared to similar and recent existing schemes. [ABSTRACT FROM AUTHOR]

Synthetic aperture radar (SAR) is an important remote sensing sensor whose application is becoming more and more extensive. Compared with traditional optical sensors, it is not easy to be disturbed by the external environment and has a strong penetration. Limited by its working principles, SAR images are not easily interpreted, and fusing SAR images with optical multispectral images is a good solution to improve the interpretability of SAR images. This paper presents a novel image fusion method based on non-subsampled shearlet transform and activity measure to fuse SAR images with multispectral images, whose aim is to improve the interpretation ability of SAR images easily obtained at any time, rather than producing a fused image containing more information, which is the pursuit of previous fusion methods. Three different sensors, together with different working frequencies, polarization modes and spatial resolution SAR datasets, are used to evaluate the proposed method. Both visual evaluation and statistical analysis are performed, the results show that satisfactory fusion results are achieved through the proposed method and the interpretation ability of SAR images is effectively improved compared with the previous methods. [ABSTRACT FROM AUTHOR]

Image denoising is a challenging task as some information contents like that of edges are generally lost in this process. Wavelet transform is used to denoise a 1-D signal, as it can represent point singularity very well. Thereby, it can retain the sudden changes in the signal, even after removing the noise. But an image is a 2-D signal and singularities in it can occur along a curve. Wavelet transform is not a good tool to handle this situation. Shearlet transform can be used instead of wavelet transform in image denoising. There are many thresholding algorithms for the wavelet transform. One such is VISUShrink, a universal thresholding algorithm. This paper proposes a universal thresholding algorithm for Shearlet transform and compares its performance with that of VISUShrink on symmlet4 and daubechies4 wavelets. The results show that the proposed algorithm performs better in most cases. A maximum of 24% increase in PSNR and 27% increase in SSIM are observed. [ABSTRACT FROM AUTHOR]

One of the primary purposes of seismic stratigraphy is to evaluate the components of seismic layer relationships within a depositional chronology. Prestack seismic images contain a wealth of information, such as variations in the offset and azimuth of a seismic event, and naturally produce higher-resolution seismic facies analysis results than poststack data. However, prestack data usually suffer from potential unreliability issues due to low signal-to-noise ratios. As this is often overlooked, the present facies analysis methods sometimes fail to extract accurate features from prestack images, which inevitably influences the facies analysis results. To address this issue, this article provides a robust data-driven technique for extracting offset-temporal features via shearlet transform-based deep convolution autoencoders (STCAEs). Unlike the present facies analysis in the time domain, STCAE can optimally represent prestack images at multiple scales and directions through the two-dimensional shearlet transform, which preserves fine edges while suppressing noise in prestack images. Subsequently, robust features are extracted from prestack images in a data-driven manner through a contractive convolutional autoencoder network. We compare our method with other advanced methods and demonstrate the advantages of the proposed approach in classifying seismic layers in prestack data. [ABSTRACT FROM AUTHOR]

Seismic exploration is a remote-sensing tool applied in a great many projects for engineering and resource-exploration purposes. Random noise suppression is one of the key steps in seismic-signal processing, especially those with important details and features. The threshold-shrinkage method based on Shearlet transform has been effectively applied in seismic-signal denoising. However, the method usually introduces the boundary effect, which influences the imaging quality. The denoising method of total generalized variation (TGV) is easy to produce `oil painting' effect, but it can effectively suppress the boundary effect. This paper proposes a denoising method based on Shearlet threshold-shrinkage and TGV for making full use of their characteristics, which can recover both edges and fine details much better than the existing regularization methods. First, we use the Shearlet threshold-shrinkage result as the input of TGV to obtain the primary denoising result and the residual profile. Second, we use the interactive iteration of Shearlet threshold-shrinkage and TGV to extract the signals efficiently from the residual profile and perform the effective signals stack continuously. During the processing, the adaptive-weight factor is combined for estimating the optimal denoising result. Last, the final estimated denoising result is obtained when the stopping criterion is met or the maximum number of iterations is reached. The synthetic and field results show that the proposed method can effectively suppress random noise. In addition, it can also remove the boundary effect and `oil painting' effect, which further improves the signal-to-noise ratio (SNR).

Despite the growing growth of technology, handwritten signature has been selected as the first option between biometrics by users. In this paper, a new methodology for offline handwritten signature verification and recognition based on the Shearlet transform and transfer learning is proposed. Since, a large percentage of handwritten signatures are composed of curves and the performance of a signature verification/recognition system is directly related to the edge structures, subbands of shearlet transform of signature images are good candidates for input information to the system. Furthermore, by using transfer learning of some pre-trained models, appropriate features would be extracted. In this study, four pre-trained models have been used: SigNet and SigNet-F (trained on offline signature datasets), VGG16 and VGG19 (trained on ImageNet dataset). Experiments have been conducted using three datasets: UTSig, FUM-PHSD and MCYT-75. Obtained experimental results, in comparison with the literature, verify the effectiveness of the presented method in both signature verification and signature recognition.

Ground-penetrating radar (GPR) has been used in a wide range of shallow detection applications, such as underground geological mapping, highway detection, and hydrogeology survey. In recent years, GPR has been most widely utilized in tunnel geological prediction because it has the advantages of high resolution, intuitionistic results, and fast scanning. In addition, GPR signal is a typical nonstationary and time-varying signal, with its electromagnetic wave exhibiting strong absorption attenuation and dispersion as it propagated in complex surrounding rock. At the same time, the GPR response is often characterized by a weak signal and a strong interference because of numerous system interferences in the tunnel detection environment, which lead to difficulties in data processing and interpretation. Therefore, interference elimination is always a difficult problem when GPR is applied to tunnel geological prediction. In this study, through the introduction of shearlet transform (ST) to GPR signal processing, an adaptive thresholding method is proposed to eliminate random interference on the basis of the energy difference between effective and interference signals in the shearlet domain at different scales and directions. The advantages of this method in random interference removal are verified by forward simulation data. On this basis, the interference signal, as well as its energy proximity and frequency anomaly, common in advanced tunnel geological prediction is taken as an example to illustrate the effect of wavelet transform (WT) on its removal. In this manner, WT and ST are combined to suppress interference. First, WT is used to separate abnormal frequency interference. Then, ST based on the adaptive thresholding method is used to suppress random interference. The results of practical engineering cases of karst cave detection in the field show that the method proposed in this study can remove the interference signal, retain the effective signal, and highlight the abnormal geological area on the basis of the processed waveform stacking diagram to improve the interpretation accuracy of GPR data.

Estimation of rainfall-runoff model parameters in ungauged catchments is of significant importance. The Nash geomorphological instantaneous unit hydrograph (NGIUH) model is widely used to predict runoff in ungauged catchments. The NGIUH model parameters are estimated based on the stream network delineation of the catchment to obtain the stream-order-law ratios. Different methods have been presented to delineate stream networks of catchments based on topographic maps and satellite images using remote sensing (RS) and geographic information system (GIS). In this study, the fractal dimension of the stream network (D) and the fractal dimension of the main river (d) were calculated by wavelet image processing of the stream network images. Shearlet transform was applied to compute the bifurcation ratio (RB). New equations were proposed to estimate the NGIUH parameters based on the fractal analysis of the river network and main river length. The proposed approach was evaluated by computing the flood hydrographs in three catchments of Kasilian, Galazchai and Heng-Chi. Based on results, coefficients of efficiency (CE) were 0.42 and 0.96. The errors in peak discharge estimation were in an acceptable range 0.93–12.91%. [ABSTRACT FROM AUTHOR]

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.