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In this paper, we introduce the localization operators associated to the higher-dimensional Shearlet transform HDST introduced earlier in Dahlke et al. (The continuous shearlet transform in arbitrary space dimensions, Philipps-Universität Marburg, (2008), J Fourier Anal Appl 16:340–364, 2010). We prove that they are in the trace class S 1 and we give a trace formula for them. Typical examples of localization operators are presented. Knowing the fact that the study of the time-frequency analysis is both theoretically interesting and practically useful, the second aim of this article is to explore two more aspects of the time-frequency analysis associated with the HDST including the spectral analysis associated with the concentration operators and the scalogram. [ABSTRACT FROM AUTHOR]

Image compression is an important stage in picture processing since it reduces the data extent and promptness of image diffusion and storage, whereas image reconstruction helps to recover the original information that was communicated. Wavelets are commonly cited as a novel technique for image compression, although the production of waves proceeding smooth areas with the image remains unsatisfactory. Stockwell transformations have been recently entered the arena for image compression and reconstruction operations. As a result, a new technique for image compression based on the improved Stockwell transform is proposed. The discrete cosine transforms, which involves bandwidth partitioning is also investigated in this work to verify its experimental results. Wavelet-based techniques such as multilevel Haar wavelet, generic multiwavelet transform, Shearlet transform, and Stockwell transforms were examined in this paper. The MATLAB technical computing language is utilized in this work to implement the existing approaches as well as the suggested improved Stockwell transform. The standard images mostly used in digital image processing applications, such as Lena, Cameraman and Barbara are investigated in this work. To evaluate the approaches, quality constraints such as mean square error (MSE), normalized cross-correlation (NCC), structural content (SC), peak noise ratio, average difference (AD), normalized absolute error (NAE) and maximum difference are computed and provided in tabular and graphical representations. [ABSTRACT FROM AUTHOR]

The biological cross-sectional images majorly consist of closed-loop structures, which are suitable to be represented by the second-order shearlet system with curvature (Bendlet). In this study, an adaptive filter method for preserving textures in the bendlet domain is proposed. The Bendlet system represents the original image as an image feature database based on image size and Bendlet parameters. This database can be divided into image high-frequency and low-frequency sub-bands separately. The low-frequency sub-bands adequately represent the closed-loop structure of the cross-sectional images and the high-frequency sub-bands accurately represent the detailed textural features of the images, which reflect the characteristics of Bendlet and can be effectively distinguished from the Shearlet system. The proposed method takes full advantage of this feature, then selects the appropriate thresholds based on the images' texture distribution characteristics in the database to eliminate noise. The locust slice images are taken as an example to test the proposed method. The experimental results show that the proposed method can significantly eliminate the low-level Gaussian noise and protect the image information compared with other popular denoising algorithms. The PSNR and SSIM obtained are better than other methods. The proposed algorithm can be effectively applied to other biological cross-sectional images.

Multi-focus image fusion plays an important role in the application of computer vision. In the process of image fusion, there may be blurring and information loss, so it is our goal to obtain high-definition and information-rich fusion images. In this paper, a novel multi-focus image fusion method via local energy and sparse representation in the shearlet domain is proposed. The source images are decomposed into low- and high-frequency sub-bands according to the shearlet transform. The low-frequency sub-bands are fused by sparse representation, and the high-frequency sub-bands are fused by local energy. The inverse shearlet transform is used to reconstruct the fused image. The Lytro dataset with 20 pairs of images is used to verify the proposed method, and 8 state-of-the-art fusion methods and 8 metrics are used for comparison. According to the experimental results, our method can generate good performance for multi-focus image fusion. [ABSTRACT FROM AUTHOR]

In computer vision, we must handle with the various structural aspects of image or video data. The texture is one of the most important aspects of this type of data, which is utilised to identify objects or regions of interest in an image. As imaging conditions change, textures inside actual images significantly change in brightness, contrast, size, and skew. To recognise textures in real-world images, a similarity measure that is invariant to these features must be used. Existing recognition techniques did not perform well due to issues such as illumination, scale, and subject rotation. To address this issue, invariant feature representation methods are being developed to generate features that are insensitive to such variations. In this paper, we proposed a robust hybrid feature descriptor and predicted the faces under illumination, scale, and pose variations using an optimum multi-kernel support vector machine. Additionally, the suggested robust hybrid feature descriptor is enhanced by combining a hybrid transform composed of discrete wavelet and discrete shearlet transforms with some image statistical textural data. The proposed face recognition system is implemented in MATLAB, and analysed using various parameters to show proposed methods improved performance as compared to the state of the art methods. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the continuous quaternion shearlet transform on R 2 to R 2 d , called the multivariate two sided continuous quaternion shearlet transform. Using the two sided quaternion Fourier transform, we derive several important properties such as (reconstruction formula, plancherel’s formula, etc.). We present several example of the multivariate two sided continuous quaternion shearlet transform. We apply the multivariate two sided continuous quaternion shearlet transform properties and the two sided quaternion Fourier transform to establish the Heisenberg uncertainty principle. Last we study the multivariate two sided continuous quaternion shearlet transform on subset of finite measures. [ABSTRACT FROM AUTHOR]

Recently, Lessig et al. [Appl Comput Harmon Anal. 2019;46:384-399] introduced the notion of bendlets; a shearlet-like system that is based on anisotropic scaling, translation, shearing and bending of a compactly supported generator. The theoretical framework of the bendlet transform is yet to be explored exclusively. Taking this opportunity, our aim is to investigate the mathematical properties of the bendlet transform such as the Rayleigh theorem, inversion formula, characterization of range and the pointwise convergence of the inversion formula. In continuation, we obtain the Heisenberg-type uncertainty principle and the Pitt's inequality for the bendlet transform. Subsequently, we employ the bendlet transform for obtaining the solutions of the wave and Laplace equations. Finally, we introduce the notion of quaternionic bendlet transform and also study its fundamental properties. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The GF-3 satellite is China’s first self-developed active imaging C-band multi-polarization synthetic aperture radar (SAR) satellite with complete intellectual property rights, which is widely used in various fields. Among them, the detection and recognition of banklines of GF-3 SAR image has very important application value for map matching, ship navigation, water environment monitoring and other fields. However, due to the coherent imaging mechanism, the GF-3 SAR image has obvious speckle, which affects the interpretation of the image seriously. Based on the excellent multi-scale, directionality and the optimal sparsity of the shearlet, a bankline detection algorithm based on shearlet is proposed. Firstly, we use non-local means filter to preprocess GF-3 SAR image, so as to reduce the interference of speckle on bankline detection. Secondly, shearlet is used to detect the bankline of the image. Finally, morphological processing is used to refine the bankline and further eliminate the false bankline caused by the speckle, so as to obtain the ideal bankline detection results. Experimental results show that the proposed method can effectively overcome the interference of speckle, and can detect the bankline information of GF-3 SAR image completely and smoothly.

GF-3 satellite is China's first C-band multi-polarized synthetic aperture radar (SAR) satellite with the 1-meter resolution, which has been widely used in various fields. Road detection for GF-3 SAR images is an important part of the application of GF-3, especially in fields of map update, target recognition, and image matching. However, speckle appears in GF-3 SAR images due to coherent imaging system and it hinders the interpretation of images seriously. Especially the detection of weak roads under strong speckle background becomes extremely difficult. As a representative of multiscale geometric analysis (MGA) tool, shearlet has the optimal sparse representation feature and strong directional orientation, which can effectively capture edge and other anisotropic feature information, and can accurately describe the sparse characteristics of GF-3 SAR images. Based on shearlet, a method for detecting weak roads under strong speckle interference is proposed. Firstly, the Frost filter is used for despeckling. Secondly, shearlet is used for road detection. Finally, morphological operations are adopted to obtain the final result. Road detection experiments on various types of GF-3 SAR images demonstrate that, the proposed method can effectively overcome the interference of speckle, and completely and smoothly detect road information, which is very suitable for the detection of weak roads under strong speckle interference of GF-3 SAR images.

Melanoma or skin cancer is the most dangerous and deadliest disease. As the incidence and mortality rate of skin cancer increases worldwide, an automated skin cancer detection/classification system is required for early detection and prevention of skin cancer. In this study, a Hybrid Artificial Intelligence Model (HAIM) is designed for skin cancer classification. It uses diverse multi-directional representation systems for feature extraction and an efficient Exponentially Weighted and Heaped Multi-Layer Perceptron (EWHMLP) for the classification. Though the wavelet transform is a powerful tool for signal and image processing, it is unable to detect the intermediate dimensional structures of a medical image. Thus the proposed HAIM uses Curvelet (CurT), Contourlet (ConT) and Shearlet (SheT) transforms as feature extraction techniques. Though MLP is very flexible and well suitable for the classification problem, the learning of weights is a challenging task. Also, the optimization process does not converge, and the model may not be stable. To overcome these drawbacks, EWHMLP is developed. Results show that the combined qualities of each transform in a hybrid approach provides an accuracy of 98.33% in a multi-class approach on PH2 database. [ABSTRACT FROM AUTHOR]

Motivated by the ideas from two step model and its deformation, we propose a coupling model for MR image reconstruction, based on the advantages of TGV and shearlet regularization terms. By using variable splitting technique, splitting Bregman iteration scheme and alternating minimization method, the proposed model can be decomposed into several subproblems to avoid solving high-order PDEs. The u subproblem can be solved by Cramer's rule and the diagonalization technique of the Fourier transform. The other subproblems can be solved simply by some shrinkage formulas. We also use the Barzilai-Borwein step selection scheme to accelerate these subproblem's solutions. Finally, an ADMM algorithm is proposed to solve the coupling model. The numerical results show that the proposed coupling model and algorithm are feasible and effective. [ABSTRACT FROM AUTHOR]

Quantitative Shapiro's dispersion uncertainty principle and umbrella theorem are proved for the multivariate continuous shearlet transform S H ψ introduced earlier in Dahlke et al. [The continuous shearlet transform in arbitrary space dimensions. Preprint Nr. 2008-7, Philipps-Universität Marburg; 2008; The continuous shearlet transform in arbitrary space dimensions. J Fourier Anal Appl. 2010;16:340–364]. Also, we extend local uncertainty principles for a set of finite measure to S H ψ . [ABSTRACT FROM AUTHOR]

Light field (LF) acquisition devices capture spatial and angular information of a scene. In contrast with traditional cameras, the additional angular information enables novel post-processing applications, such as 3D scene reconstruction, the ability to refocus at different depth planes, and synthetic aperture. In this paper, we present a novel compression scheme for LF data captured using multiple traditional cameras. The input LF views were divided into two groups: key views and decimated views. The key views were compressed using the multi-view extension of high-efficiency video coding (MV-HEVC) scheme, and decimated views were predicted using the shearlet-transform–based prediction (STBP) scheme. Additionally, the residual information of predicted views was also encoded and sent along with the coded stream of key views. The proposed scheme was evaluated over a benchmark multi-camera based LF datasets, demonstrating that incorporating the residual information into the compression scheme increased the overall peak signal to noise ratio (PSNR) by 2 dB. The proposed compression scheme performed significantly better at low bit rates compared to anchor schemes, which have a better level of compression efficiency in high bit-rate scenarios. The sensitivity of the human vision system towards compression artifacts, specifically at low bit rates, favors the proposed compression scheme over anchor schemes. [ABSTRACT FROM AUTHOR]

In this paper, we present some new elements of harmonic analysis related to multivariate continuous shearlet transform introduced earlier in Dahlke et al. (J Fourier Anal Appl 16:340–364, 2010; The continuous shearlet transform in arbitrary space dimensions, Philipps-Universität Marburg, Marburg, 2008). Thus, some results (Parseval's formula, inversion formula, etc.) are established. Next, we prove an analogue of Heisenberg's inequality for shearlet transform. Last, we study shearlet transform on subset of finite measures. [ABSTRACT FROM AUTHOR]

Image retrieval system is an urgent issue for in medicine. In the past, traditional image retrieval system based solely on the label of images and gave limited results. To reduce this disadvantage, the content-based medical image retrieval has been developed. However, this system still has many challenges. In this paper, we proposed a new method for content-based medical image retrieval. The proposed method includes two stages: the offline task and online task in medical image database. In the first stage, we extracted local object features of medical images in shearlet domain. Then, we detect the contour of object in images by active contour model. In the second stage, we make online task for content-based image retrieval in database. Our system receipts a query image and shows the similar in images by similarity comparison with the information collected from the first stage. Experimental results have shown that the proposed method is better than the other methods.

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. [ABSTRACT FROM AUTHOR]

In this article we propose two minimization models for blind deconvolution. In the first model, we use shearlet transform as a regularization term for recovering image. Also total variation method is used as a regularization term for point spread function(PSF). To speed up the process, Fast ADMM approach is exploited. In the second model, shearlet transform is utilized as a regularization term for both image and PSF. [ABSTRACT FROM AUTHOR]

This paper proposes an edge based compressively sensed (CS) geodesic active contour (GAC) model, termed CS-GAC, to ensure faithful edge detection and accurate object segmentation. The motivation behind this paper is that edge information driving the contour evolution can be iteratively obtained by incomplete CS measurements. In each iteration, the CS-GAC is a three-step process including edge detection, active contouring and sparse reconstruction. Instead of working on the final reconstructed images themselves, the evolution of the CS-GAC is driven by a few CS measurements and guided by updatable edge information. The edge information is generated by a complex shearlet transform (CST) based edge map. In the framework, reconstruction and edge detection work alternately. The iterative update property that takes advantages of both edge sparsity and edge detection can largely improve the evolution precision. Numerical experiments show that the CS-GAC can obtain challenging segmentation results in comparisons with the state of the art methods, and has competitive prospects. • Edge information driving the contour evolution is iteratively obtained from CS measurements. • Updating property of the edge indicator takes advantages of both edge sparsity and edge detection. • The complex shearlet transform based edge map has been improved by the iteratively updating mechanism. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the generalization of the homogeneous approximation property (HAP) for a continuous shearlet transform to higher dimensions. First, we give a pointwise convergence result on the inverse shearlet transform in higher dimensions. Second, we show that every pair of admissible shearlets possess the HAP in the sense of L 2 ( R d ) $L^{2}(R^{d})$ . Third, we give a sufficient condition for the pointwise HAP to hold, which depends on both shearlets and functions to be reconstructed.

In this paper, we propose an improved model for image decomposition and denoising based on Shearlet and nonlocal data fidelity term. The model splits an image into three parts: the cartoon component modeled by total variation (TV) space, the texture component modeled by G space, and the noise component modeled by Shearlet smoothness space. We introduce Shearlet smoothness space to model noise component due to the appreciate property of directional sensitivity. We also incorporate nonlocal weight to the data fidelity term of the new model, in order to reduce the drawbacks of TV regularization presenting in the restored image. The experiment results demonstrate that the new model performs better in image decomposition and denoising than previous models and effectively removes the defaults of TV regularization thanks to the nonlocal data fidelity term. [ABSTRACT FROM AUTHOR]

In interferogram filtering,traditional threshold filtering algorithm based on wavelet transform does not consider the statistical properties of SAR's interference phase,and the filtering effect obtained in the low coherence regions is not satisfactory. This paper presents a kind of phase noise filtering algorithm combining the shearlet transform and standard deviation of phase. The algorithm uses the phase standard deviation to correct the filter threshold and improve the filtering effect. In addition, in order to evaluate the filtering effect and to select the appropriate filtering method for the measured data, a local mean square error distribution of the simulated interferogram is proposed as the filtering quality evaluation index. Compared with Goldstein filtering, wavelet filtering, optimal direction fusion filtering and shearlet soft threshold filtering, the results show that the proposed method can not only weaken the noise of interferogram, but also keep the details and avoid the weak filtering in low coherence regions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new synthetic aperture radar (SAR) image detection algorithm based on the de-noising algorithm via the sparse representation and a new morphology edge detector. Firstly, we apply the Shearlet transform to the SAR image to get the sparse representation of it. Then, morphological edge detector with direction is applied to directional sub-band coefficients of the Shearlet which are recovered by the iterative de-noising process. Finally, the completed SAR image edge is obtained by merging each sub-band edge using Dempster-Shafer evidence theory. By completely using the directional sub-bands of the Shearlet transform, the proposed algorithm overcomes the disadvantages of transform detection algorithms which are very unrobust to noise and can also generate inaccurate edges. The experimental results demonstrate the effectiveness and superiority of our proposed algorithm in terms of the edge positioning accuracy, integrity, and the number of false edge points. [ABSTRACT FROM AUTHOR]

As speckle seriously restricts the applications of remote sensing images in many fields, the ability to efficiently and effectively suppress speckle in a coherent imaging system is indispensable. In order to overcome the over-smoothing problem caused by the speckle suppression algorithm based on classical sparse representation, we propose a non-local speckle suppression algorithm that combines the non-local prior knowledge of the image into the sparse representation. The proposed algorithm first applies shearlet to sparsely represent the input image. We then incorporate the non-local priors as constraints into the image sparse representation de-noising problem. The denoised image is obtained by utilizing an alternating minimization algorithm to solve the corresponding constrained de-noising problem. The experimental results show that the proposed algorithm can not only significantly remove speckle noise, but also improve the visual effect and retain the texture information of the image better. [ABSTRACT FROM AUTHOR]

In any image segmentation problem, there exist uncertainties. These uncertainties occur from gray level and spatial ambiguities in an image. As a result, accurate segmentation of text regions from non-text regions (graphics/images) in mixed and complex documents is a fairly difficult problem. In this paper, we propose a novel text region segmentation method based on digital shearlet transform (DST). The method is capable of handling the uncertainties arising in the segmentation process. To capture the anisotropic features of the text regions, the proposed method uses the DST coefficients as input features to a segmentation process block. This block is designed using the neutrosophic set (NS) for management of the uncertainty in the process. The proposed method is experimentally verified extensively and the performance is compared with that of some state-of-the-art techniques both quantitatively and qualitatively using benchmark dataset. [ABSTRACT FROM AUTHOR]

Shearlet transform has been widely applied in related fields due to its admirable properties in image approximation. Image fusion method based on accelerated non-negative matrix factorization (ANMF) and expanded energy of Laplace (EEOL) rules was proposed in this study to integrate the complementary information of medical images with multiple modalities and improve the accuracy of clinical diagnosis and therapy. First, the registered medical images were decomposed into low- and high-frequency sub-band coefficients in shearlet domain. Then, the ANMF rule was used in merging low-frequency coefficients. Next, the visual-contrast-based EEOL rule was adopted in extracting details of source images from high-frequency coefficients. Finally, the ultimate fused image was reconstructed by applying inverse shearlet transform. Experimental results reveal that aside from visual effect, the proposed method achieves the best in three of five criteria and the run time is reduced by 29.21% compared with a method based on non-subsampled contourlet transform (NSCT) in computed tomography (CT)--magnetic resonance imaging (MRI) fusion. Moreover, the proposed method takes the first place in four of five criteria with run time reduced by 48.32% and 24.55% compared with two shearlet-based methods in a MRI--positron emission tomography (PET) case. This study indicates that the proposed method is superior to the selected approaches in visual and statistical evaluation, which is conducive to clinical practice of medical image fusion. [ABSTRACT FROM AUTHOR]