Tight Frame Research Articles

Synthetic Aperture Radar Image Despeckling Based on Multi-Weighted Sparse Coding.

Synthetic aperture radar (SAR) images are inherently degraded by speckle noise caused by coherent imaging, which may affect the performance of the subsequent image analysis task. To resolve this problem, this article proposes an integrated SAR image despeckling model based on dictionary learning and multi-weighted sparse coding. First, the dictionary is trained by groups composed of similar image patches, which have the same structural features. An effective orthogonal dictionary with high sparse representation ability is realized by introducing a properly tight frame. Furthermore, the data-fidelity term and regularization terms are constrained by weighting factors. The weighted sparse representation model not only fully utilizes the interblock relevance but also reflects the importance of various structural groups in despeckling processing. The proposed model is implemented with fast and effective solving steps that simultaneously perform orthogonal dictionary learning, weight parameter updating, sparse coding, and image reconstruction. The solving steps are designed using the alternative minimization method. Finally, the speckles are further suppressed by iterative regularization methods. In a comparison study with existing methods, our method demonstrated state-of-the-art performance in suppressing speckle noise and protecting the image texture details.

Frames over finite fields: Equiangular lines in orthogonal geometry

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but 18 known SRG parameters on v≤1300 vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension 5.

A Brief Survey of the Graph Wavelet Frame

In recent years, the research of wavelet frames on the graph has become a hot topic in harmonic analysis. In this paper, we mainly introduce the relevant knowledge of the wavelet frames on the graph, including relevant concepts, construction methods, and related theory. Meanwhile, because the construction of graph tight framelets is closely related to the classical wavelet framelets on ℝ, we give a new construction of tight frames on ℝ. Based on the pseudosplines of type II, we derive an MRA tight wavelet frame with three generators ψ1, ψ2, and ψ3 using the oblique extension principle (OEP), which generate a tight wavelet frame in L2(ℝ). We analyze that three wavelet functions have the highest possible order of vanishing moments, which matches the order of the approximation order of the framelet system provided by the refinable function. Moreover, we introduce the construction of the Haar basis for a chain and analyze the global orthogonal bases on a graph . Based on the sequence of framelet generators in L2(ℝ) and the Haar basis for a coarse‐grained chain, the decimated tight framelets on graphs can be constructed. Finally, we analyze the detailed construction process of the wavelet frame on a graph.

Maltsev equal-norm tight frames

A frame in $\mathbb{R}^d$ is a set of $n\geqslant d$ vectors whose linear span coincides with $\mathbb{R}^d$. A frame is said to be equal-norm if the norms of all its vectors are equal. Tight frames enable one to represent vectors in $\mathbb{R}^d$ in the form closest to the representation in an orthonormal basis. Every equal-norm tight frame is a useful tool for constructing efficient computational algorithms. The construction of such frames in $\mathbb{C}^d$ uses the matrix of the discrete Fourier transform, and the first constructions of equal-norm tight frames in $\mathbb{R}^d$ appeared only at the beginning of the 21st century. The present paper shows that Maltsev's note of 1947 was decades ahead of its time and turned out to be missed by the experts in frame theory, and Maltsev should be credited for the world's first design of an equal-norm tight frame in $\mathbb{R}^d$. Our main purpose is to show the historical significance of Maltsev's discovery. We consider his paper from the point of view of the modern theory of frames in finite-dimensional spaces. Using the Naimark projectors and other operator methods, we study important frame-theoretic properties of the Maltsev construction, such as the equality of moduli of pairwise scalar products (equiangularity) and the presence of full spark, that is, the linear independence of any subset of $d$ vectors in the frame.

Image restoration: Structured low rank matrix framework for piecewise smooth functions and beyond

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.

Distributed Matrix Multiplication Based on Frame Quantization for Straggler Mitigation

The number of available worker nodes for distributed computing can be limited and varying in wireless networks with noisy channels. However, the conventional straggler mitigation schemes ensure perfect reconstruction only when the number of worker nodes is more than a predetermined number, called the recovery threshold. When the number of worker nodes is less than the recovery threshold, the conventional schemes show a poor reconstruction performance. To facilitate distributed computing in noisy channels, leveraging the tensor product representation and frame quantization theory, we propose a novel straggler mitigation scheme for distributed matrix multiplication. The proposed scheme offers a reasonable reconstruction performance even when the number of worker nodes is varying and limited. By analyzing the computation error, we figure out the conditions required for encoding and decoding frame design, which suggest using uniform tight frames for encoding and their Moore-Penrose inverse for decoding. The superiority of the proposed scheme to the conventional straggler mitigation schemes when the number of available worker nodes is limited is verified by simulations.

Large Graph Signal Denoising With Application to Differential Privacy

Over the last decade, signal processing on graphs has become a very active area of research. Specifically, the number of applications, for instance in statistical or deep learning, using frames built from graphs, such as wavelets on graphs, has increased significantly. We consider in particular the case of signal denoising on graphs via a data-driven wavelet tight frame methodology. This adaptive approach is based on a threshold calibrated using Stein's unbiased risk estimate adapted to a tight-frame representation. We make it scalable to large graphs using Chebyshev-Jackson polynomial approximations, which allow fast computation of the wavelet coefficients, without the need to compute the Laplacian eigendecomposition. However, the overcomplete nature of the tight-frame, transforms a white noise into a correlated one. As a result, the covariance of the transformed noise appears in the divergence term of the SURE, thus requiring the computation and storage of the frame, which leads to an impractical calculation for large graphs. To estimate such covariance, we develop and analyze a Monte-Carlo strategy, based on the fast transformation of zero mean and unit variance random variables. This new data-driven denoising methodology finds a natural application in differential privacy. A comprehensive performance analysis is carried out on graphs of varying size, from real and simulated data.

Seismic Data Regularization on Nonequispaced Grid via a Joint Sparsity-Promotion Method

The seismic data regularization problem is vital to seismic data processing. We propose a joint sparsity-promotion method based on the compressive sensing theory named the curvelet-data-driven-tight-frame-based sparsity-promoting (CDSP) method. The CDSP method regularizes the seismic data directly on the nonequispaced grid along the spatial dimension. The joint sparsity is exploited in the curvelet and data-driven tight frame transform simultaneously on the projected regular spatial grid. The projection from the nonequispaced grid to the equispaced grid is achieved by the nonequispaced discretized Fourier transform. Comparing with the curvelet-sparsity-promotion-based (CSP) regularization method, CDSP combines the advantage of the predefined curvelet transform and adaptive-learned sparse transform into one single optimization model. An alternative directional method of multipliers (ADMM) is applied to solve the optimization problem. One synthetic and two field examples show that the CDSP method performs better on the preservation of the continuity of events and produces less artifacts than the CSP method.

Data-Driven Time-Frequency Method and Its Application in Detection of Free Gas Beneath a Gas Hydrate Deposit

The time-frequency (TF) analysis method plays a significant role in the detection of natural gas hydrates. As a data-driven method, compressed sensing (CS) has been widely used in the TF methods due to the sparsity of the TF representation. This study proposes a data-driven TF method based on the CS theory and a non-convex regularization. In the implementation, a continuous wavelet transform (CWT) with a generalized beta wavelet (GBW) is formulated as an inverse problem based on the CS theory. The selection of appropriate parameters enables the GBW to match the seismic wavelets better than the widely used Morlet wavelet. The GBW can constitute a tight frame to reduce calculation time, particularly for large-scale field data processing. Additionally, the proposed TF method introduces the generalized minimax concave (GMC) penalty function as a non-convex regularization term. Compared with the classical sparse approximation method with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1} $ </tex-math></inline-formula> regularization, the GMC regularization term can enhance the sparsity in sparse inverse problems and ensure the convexity of sparse inversions. This article also presents an exponentially decreasing threshold scheme to adaptively select the regularization parameters. Three synthetic examples are investigated to demonstrate the performance of the proposed sparse TF representation with GMC regularization. Finally, the proposed TF method’s performance in detecting free gas of gas hydrates is validated using field seismic data obtained from the Blake Ridge.

Location Estimates From Channel State Information via Binary Programming

Recent years have witnessed a marked increase in the efficacy and speed to solution of binary program solvers. Leveraging these developments, we propose converting the sparse modelling problem to a binary program, and we derive the conditions under which such a conversion yields the optimal signal support. Furthermore, we derive an upper bound on this condition, which we can minimize by designing a structured tight frame as the dictionary. We apply these theoretical insights to binary optimization of angle of arrival and time of flight estimation (BOAT) using channel state information (CSI), with the goal of localizing a transmitter in an indoor multipath-rich environment. Finally, we show how a covariance array processing approach, together with the novel binary programming paradigm, results in a system that outperforms two state-of-the-art solutions. Our claims are substantiated using both simulated data and experimental data collected using “off-the-shelf” WiFi IEEE 802.11ac routers.

Sparsity-driven deterministic sampling strategy for coded aperture x-ray computed tomography

The sub-sampling strategies in x-ray computed tomography (CT) have gained importance due to their practical relevance. In this direction of research, also known as coded aperture x-ray computed tomography (CAXCT), both random and deterministic strategies have been proposed in the literature. Of the techniques available, those based on compressive sensing (CS) have recently gained more traction because CS-based ideas efficiently exploit inherent duplication present in the system. The quality of the reconstructed CT images nevertheless depends on the sparse signal recovery properties of the sub-sampled Radon matrices. In the present work, we optimize CAXCT deterministically in such a way that the corresponding sub-sampled Radon matrices remain close to the equiangular tight frames for better numerical behaviour. We show that this optimization, involving the Khatri–Rao product, leads to a non-negative sparse approximation. While comparing and contrasting our method with its existing counterparts, we show that the proposed algorithm is computationally less involved. Finally, we demonstrate efficacy of the proposed deterministic sub-sampling strategy in recovering CT images in both noiseless and noisy cases.

Disadvantages of using Traditional, Interactive and Distance Learning Methods when Studying the Specialty "Pharmacy"

&lt;p&gt;Interactive and remote technologies are being actively introduced into the training process in the specialty “Pharmacy.” In addition to ease of use, they have several advantages and disadvantages for all stakeholders. The purpose of the work is to analyze the disadvantages of traditional, interactive, and remote methods of teaching in the specialty “Pharmacy” at the Far Eastern State Medical University. This is a sociological survey of students of the Faculty of Pharmacy and Biomedicine (higher education), as well as the Medical-Pharmaceutical College (secondary education). Statistical processing was performed using the non-parametric Mann-Whitney, Kruskal–Wallis, Spearman’s rank correlation coefficient. Traditional teaching methods are characterized by the inconvenience of tight time frames. During on-line classes, it is more difficult to use an individual methodology for the student and change the teaching method. The most characteristic shortcomings of distance learning include the lack of motivation and the possibility of violations (writing off, forgery, etc.) by the student. Despite the geographical remoteness of the regions, material, and technical difficulties in switching to distance learning, as a rule, did not arise. The lack of an individual approach and the subjectivity of student assessments in distance and traditional teaching methods are due to the high occupancy of groups and their frequent combination (10-25 people) during the class. The disadvantage of remote methods is the complexity of regular control by the teacher, the lack of secure personality identification systems when sending written works and testing on-line, and the low level of consciousness of future specialists. According to the results of the questionnaire, the least number of shortcomings have interactive teaching methods, implying the active participation of the student himself and an individual approach.&lt;/p&gt;

Sparse inversion-based seismic random noise attenuation via self-paced learning

Seismic random noise reduction is an important task in seismic data processing at the Chinese loess plateau area, which benefits the geologic structure interpretation and further reservoir prediction. The sparse inversion is one of the widely used tools for seismic random noise reduction, which is often solved via the sparse approximation with a regularization term. The ℓ1 norm and total variation (TV) regularization term are two commonly used regularization terms. However, the ℓ1 norm is only a relaxation of the ℓ0 norm, which cannot always provide a sparse result. The TV regularization term may provide unexpected staircase artifacts. To avoid these disadvantages, we proposed a workflow for seismic random noise reduction by using the self-paced learning (SPL) scheme and a sparse representation (i.e. the continuous wavelet transform, CWT) with a mixed norm regularization, which includes the ℓp norm and the TV regularization. In the implementation, the SPL, which is inspired by human cognitive learning, is introduced to avoid the bad minima of the non-convex cost function. The SPL can first select the high signal-to-noise ratio (SNR) seismic data and then gradually select the low SNR seismic data into the proposed workflow. Moreover, the generalized Beta wavelet (GBW) is adopted as the basic wavelet of the CWT to better match for seismic wavelets and then obtain a more localized time-frequency (TF) representation. It should be noted that the GBW can easily constitute a tight frame, which saves the calculation time. Synthetic and field data examples are adopted to demonstrate the effectiveness of the proposed workflow for effectively suppressing seismic random noises and accurately preserving valid seismic reflections.

Towards a classification of incomplete Gabor POVMs in d?>

Every (full) finite Gabor system generated by a unit-norm vector is a finite unit-norm tight frame (FUNTF) and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the corresponding rank 1 matrices form a basis for the space of matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process, we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank 1 matrices generated by a finite Gabor frame. We also use these results to construct some sets of unit vectors in with a relatively smaller number of distinct inner products.

Frames over finite fields: Basic theory and equiangular lines in unitary geometry

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension d=22l+1 over the field F32. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.

A fractional-order regularization with sparsity constraint for blind restoration of images

Blind image deconvolution is one of the most challenging problems in image restoration. Inspired by the work on sparsity constraint and deblurring of blind motion, we propose a model with fractional-order regularization and sparsity constraint for blind restoration and construct split Bregman combining an iterative thresholding algorithm. Fractional-order penalty term in Besov space is expanded by wavelet basis and computed using iterative thresholding algorithm. The regularized terms of blur kernel under tight wavelet frame systems are solved by the split Bregman method. Numerical experiments show that our algorithm can effectively remove different kinds of blur without requiring any prior information of the blur kernels and obtain higher signal-to-noise ratios and lower relative errors. In addition, fractional-order derivative in Besov space can preserve both edges and smoothness better than the integer-order derivative.

CIRCULAR ECONOMY AS A NEW PARADIGM OF INDUSTRY DEVELOP- MENT

The country's sustainable development strategy, as one of the main vectors, provides for the transition to a circular economy, which determines the formation of a new business model based on minimizing the hydrocarbon footprint, using alternative energy sources, extending the service life of goods, and developing sharing relationships, recycling waste for their reuse, focusing factories on the production of products involving their recycling, the transition to biodegradable products. The transition to a closedcycle economy should be made by industrial enterprises in a tight time frame to minimize cross-border taxes for the carbon footprint. On the one hand, this process requires a set of state measures. On the other hand, Russian companies focused on active export activities are already implementing certain principles of the green economy. The article presents the experience of a number of them.

Approximation and localized polynomial frame on conic domains

Highly localized kernels constructed by orthogonal polynomials have been fundamental in the recent development of approximation and computational analysis on the unit sphere, unit ball, and several other regular domains. In this work, we first study homogeneous spaces that are assumed to contain highly localized kernels and establish a framework for approximation and localized tight frames in such spaces, which extends recent works on bounded regular domains. We then show that the framework is applicable to homogeneous spaces defined on bounded conic domains, which consists of conic surfaces and the solid domains bounded by such surfaces and hyperplanes. The highly localized kernels on conic domains require precise estimates that rely on recently discovered addition formulas for orthogonal polynomials with respect to special weight functions on each domain and an intrinsic distance that takes into account the boundary of the domain, the latter is not comparable to the Euclidean distance at around the apex of the cone.The main results provide construction of semi-discrete localized tight frame in weighted L2 norm and characterization of best approximation by polynomials on conic domains. The latter is achieved by using a K-functional, defined via the differential operator that has orthogonal polynomials as eigenfunctions, as well as a modulus of smoothness defined via a multiplier operator that is equivalent to the K-functional. Several intermediate results are of interest in their own right, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities. Moreover, although the highly localizable kernels hold only for special families of weight functions on each domain, many intermediate results are shown to hold for doubling weights defined via the intrinsic distance on the domain.

Optimization of Nonuniform Sensor Placement for Blade Tip Timing Based on Equiangular Tight Frame Theory

Abstract Blade tip timing (BTT) data are usually an under-sampled signal and are vulnerable to noise and sensor failures. In this paper, based on an arbitrary-angle compressed-sensing method and equiangular tight frame theory, combined with a niching microgenetic algorithm, a method for placing BTT sensors is proposed to ensure higher reconstruction accuracy and reliability. If the dimensions of the sensing matrix are moderate, the index range of arrangements with excellent performance in multifrequency signal reconstruction is determined by enumerating all the uniform-distribution extraction placements. A two-parameter search method is then proposed. Reconstruction of a mixed-signal is carried out to verify the asynchronous signal-reconstruction performance. Thus, to achieve a larger frequency multiplication recognition range and probe-installation flexibility, a method for optimizing the BTT sensor placement is proposed. Finally, a finite-element simulation of the signal from an aero-engine fan blade is used to verify the reconstruction ability of the proposed method. The results show that the placement determined by the optimization algorithm can achieve similar or even better performance than the optimal placement under uniform distribution extraction. The proposed sensor-placement optimization method has a high reconstruction success rate and the BTT system is robust. This approach has significant value for engineering applications.

CT Image Reconstruction via Nonlocal Low-Rank Regularization and Data-Driven Tight Frame

X-ray computed tomography (CT) is widely used in medical applications, where many efforts have been made for decades to eliminate artifacts caused by incomplete projection. In this paper, we propose a new CT image reconstruction model based on nonlocal low-rank regularity and data-driven tight frame (NLR-DDTF). Unlike the Spatial-Radon domain data-driven tight frame regularization, the proposed NLR-DDTF model uses an asymmetric treatment for image reconstruction and Radon domain inpainting, which combines the nonlocal low-rank approximation method for spatial domain CT image reconstruction and data-driven tight frame-based regularization for Radon domain image inpainting. An alternative direction minimization algorithm is designed to solve the proposed model. Several numerical experiments and comparisons are provided to illustrate the superior performance of the NLR-DDTF method.