Biorthogonal Decomposition Research Articles

A new very high-order finite-difference method for linear stability analysis and bi-orthogonal decomposition of hypersonic boundary layer flow

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art eN prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) [17] deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) [16] relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003) [22]. Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis.

Unsteady loads on circular cylinder in cross-flow: An observation in natural wind and comparison with wind tunnel experiments

We present the synchronized unsteady wall pressure measurements performed at the circumference of a full scale 35 m circular chimney submitted to natural wind. The atmospheric turbulent boundary layer generates a turbulence intensity of 15 % at the altitude of the measurements and the corresponding Reynolds number is 1.13 106. For the first time, the steady and unsteady pressure distributions around the circumference are presented, despite large difficulties in the measuring process. The analysis of the pressure distribution using the bi-orthogonal decomposition (BOD) is a key factor in the success of the processing. By comparison with wind tunnel similar measurements, with 2D cylinders, the time-averaged pressure distribution is found between the one of a subcritical regime and a supercritical regime, slightly closer to the subcritical regime notably because the base pressure coefficients are similar. The atmospheric turbulence is an essential ingredient for the unsteady pressure response on the chimney, which generates high levels of RMS pressure, much higher than all those measured in wind tunnel, even under the turbulent flow generated by a grid. The analysis by BOD allowed the extraction of a term which is mainly generated by alternate vortex shedding. Despite a very noisy signal, the Strouhal number is determined to be 0.20–0.24. The corresponding shape of the pressure distribution is compared with those found in wind tunnel: it is shown that the full scale field measurements are close, but not identical, to the small scale measurements in wind tunnel on a rough cylinder submitted to a grid generated turbulent flow.

Detail-preserving arterial pulse wave measurement based Biorthogonal wavelet decomposition from remote RGB observations

Extracting arterial pulse wave from the video, also known as Imaging Photoplethysmography (IPPG), has emerged as an efficient health monitoring approach. The detail-preserving arterial pulse wave can significantly improve healthcare efficiency through the provision of physiological and pathological information. However, the accurate measurement of arterial pulse wave by IPPG is still challenged by the fact that the pulsation is weakly expressed at the skin surface and extremely vulnerable to interferences. This work proposes a novel approach to measure the detail-preserving arterial pulse wave via interference isolation from facial video. A skin reflection model is introduced to guide the design of the IPPG measurement system, where the IPPG signal is pre-filtered to obtain the variation related to the pulse wave, and a Biorthogonal wavelet is further employed to distinguish the unique pattern of arterial subtle variation from interferences. Experimental results validate the efficiency of proposed method with a correlation coefficient of 0.92.

Biorthogonal wavelet based entropy feature extraction for identification of maize leaf diseases

Crop disease is considered as a major constraint to both food quality and production. Even in this era of precision agriculture, the lacking of compulsory infrastructure has made rapid identification of crop diseases quite hard in numerous regions around the world. In this paper, we introduced a new method based on biorthogonal wavelet transform (BWT) to identify prime maize leaf diseases. We performed biorthogonal wavelet decomposition and pixel wise morphological operation to segment the maize leaf lesion from input image. For feature extraction, by applying 2-D biorthogonal wavelet transform (BWT) at multiple levels we proposed a novel method to extract colour channel wise wavelet entropy features by investigating discriminatory potential of three different biorthogonal wavelet filters (bior3.3, bior3.5, and bior3.7). The effectiveness of our extracted features were evaluated by employing five different classifiers and obtaining 95.78% overall identification accuracy with 10-fold cross validation. All the materials related our study can be found at: https://github.com/BadhanMazumder/BiorthogonalWavelet_MaizeDiseaseDetection.git.

Biorthogonal decomposition of the disturbance flow field generated by particle impingement on a hypersonic boundary layer

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

Investigation of the Sound Source Regions in Open and Closed Organ Pipes

The airflow in the mouth of an open and closed flue organ pipe of corresponding geometrical proportions is studied. The phase locked particle image velocimetry with subsequent analysis by the biorthogonal decomposition is employed in order to compare the flow mechanisms and related features. The most significant differences lie in the mean velocity distribution and rapidity of the jet lateral motion. Remarks on the pressure estimation from PIV data and its importance for the aeroacoustic source terms are made and a specific example is discussed.

Optical coupled-mode theory for dielectric solids of revolution

We propose a single resonance coupled-mode approach to light scattering by dielectric solids of revolution. By using a biorthogonal decomposition of the $S$ matrix found with the extended boundary condition method we derived all parameters required for application of the temporal coupled-mode theory in a closed form. The proposed approach allows for constructing a frequency-dependent Fano response due to a single resonance after the full-wave solution has been found at a single incident frequency.

Biorthogonal harmonics for the decomposition of gravitational radiation. I. Angular modes, completeness, and the introduction of adjoint-spheroidal harmonics

The estimation of radiative modes is a central problem in gravitational wave theory, with essential applications in signal modeling and data analysis. This problem is complicated by most astrophysically relevant systems not having modes that are analytically tractable. A ubiquitous workaround is to use not modes, but multipole moments defined by spin-weighted spherical harmonics. However, spherical multipole moments are only related to the modes of systems without angular momentum. As a result, they can obscure the underlying physics of astrophysically relevant systems, such as binary black hole merger and ringdown. In such cases, spacetime angular momentum means that radiative modes are not spherical, but spheroidal in nature. Here, we work through various problems related to spheroidal harmonics. We show for the first time that spheroidal harmonics are not only capable of representing arbitrary gravitational wave signals, but that they also possess a kind of orthogonality not used before in general relativity theory. Along the way we present a new class of spin-weighted harmonic functions dubbed ``adjoint-spheroidal'' harmonics. These new functions may be used for the general estimation of spheroidal multipole moments via complete biorthogonal decomposition (in the angular domain). By construction, adjoint-spheroidal harmonics suppress mode-mixing effects known to plague spherical harmonic decomposition; as a result, they better approximate a system's true radiative modes. We discuss potential applications of these results. Lastly, we briefly comment on the challenges posed by the analogous problem with Teukolsky's radial functions.

Optimization of Discrete Wavelet Transform Feature Representation and Hierarchical Classification of G-Protein Coupled Receptor Using Firefly Algorithm and Particle Swarm Optimization

Ineffective protein feature representation poses problems in protein classification in hierarchical structures. Discrete wavelet transform (DWT) is a feature representation method which generates global and local features based on different wavelet families and decomposition levels. To represent protein sequences, the proper wavelet family and decomposition level must be selected. This paper proposed a hybrid optimization method using particle swarm optimization and the firefly algorithm (FAPSO) to choose the suitable wavelet family and decomposition level of wavelet transformation for protein feature representation. The suggested approach improved on the work of earlier researchers who, in most cases, manually selected the wavelet family and level of decomposition based solely on experience and not on data. The paper also applied the virtual class methods to overcome the error propagation problems in hierarchical classification. The effectiveness of the proposed method was tested on a G-Protein Coupled Receptor (GPCR) protein data set consisting of 5 classes at the family level, 38 classes at the subfamily level, and 87 classes at the sub-subfamily level. Based on the result obtained, the most selected wavelet family and decomposition level chosen to represent GPCR classes by FAPSO are Biorthogonal wavelets and decomposition level 1, respectively. The experimental results show that the representation of GPCR protein using the FAPSO algorithm with virtual classes can yield 97.9%, 86.9%, and 81.3% classification accuracy at the family, subfamily, and sub-subfamily levels, respectively. In conclusion, the result shows that the selection of optimized wavelet family and decomposition level by the FAPSO algorithm, and the virtual class method can be potentially used as the feature representation method and a hierarchical classification method for GPCR protein.

Twin Strouhal numbers in pressure loading of circular cylinder at high Reynolds numbers

We present a wind tunnel experiment on a large smooth circular cylinder mounted between walls. The Reynolds number range of interest is [800 000 – 2 170 000]. This low supercritical regime is often encountered in wind engineering applications, especially the excitation of flexible circular structures by vortex shedding. Main measurements are the unsteady wall pressure distributions around the cylinder by means of synchronized pressure taps with high resolution in time and space. By using the bi-orthogonal decomposition of this set of signals, twin Strouhal numbers are detected which correspond to the second and the fourth terms of the decomposition. These terms are found to produce the unsteady lift on the cylinder with a main contribution of the second one for which the associated Strouhal number is close to the one usually found at subcritical Reynolds numbers.

GPCR Protein Feature Representation using Discrete Wavelet Transform and Particle Swarm Optimisation Algorithm

Features play an important role in representing classes in the hierarchy structure, and using unsuitable features will affect classification performance. The discrete wavelet transform (DWT) approach provides the ability to create the appropriate features to represent data. DWT can produce global and local features using different wavelet families and decomposition levels. These two parameters are essential to obtain a suitable representation for classes in the hierarchy structure. This study proposes using a particle swarm optimisation (PSO) algorithm to select the suitable wavelet family and decomposition level for G-protein coupled receptor (GPCR) hierarchical class representation. The results indicate that the PSO algorithm mostly selects Biorthogonal wavelets and decomposition level 2 to represent GPCR protein. Concerning the performance, the proposed method achieved an accuracy of 97.9%, 85.9%, and 77.5% at the family, subfamily, and sub-subfamily levels, respectively.

Classical entanglement of twisted random light propagating through atmospheric turbulence [Invited

We examine the impact of the atmospheric turbulence on a recently discovered type of classical entanglement of partially coherent beams endowed with a twist phase. We derive a compact analytical expression for the Schmidt number of a bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) beam propagating through the turbulent atmosphere. We elucidate conditions for a TGSM source to generate a strongly classically entangled paraxial field over a desired propagation distance in the turbulent atmosphere. Our results will find applications to free-space optical communications and motivate further research into classical entanglement with random light.

On the role of discrete and continuous modes in a cooled high-speed boundary layer flow

The disturbance flow field for a Mach$6$flat-plate boundary layer flow with a wall-to-free-stream temperature ratio of$0.5$is studied using direct numerical simulation (DNS), linear stability theory (LST) and biorthogonal decomposition. In the second-mode instability region, the DNS flow field can be reconstructed using a single LST mode, namely$F^+_1$. However, when the supersonic mode emerges, none of the discrete modes nor the continuous branches alone can precisely match the DNS data. A superposition of the pair of discrete supersonic modes$F^\pm _1$and the slow acoustic continuous spectrum is required to reproduce the disturbance amplitude distributions and the amplification rates observed in the DNS. Another finding is that the supersonic mode$F^-_1$, which should be decaying following LST, in actuality is amplified based on the projected DNS data.

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

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

Tomotok: python package for tomography of tokamak plasma radiation

A python package, called Tomotok, focused on performing tomographic inversion of tokamak plasma radiation is being developed at the Institute of Plasma Physics of the Czech Academy of Sciences. It aims at providing multiple inversion algorithms with an user friendly interface. In order to enable and ease performing tomographic inversion on different devices worldwide, it is planned to publish this software as open source in the near future. In this contribution, the package structure allowing an easy implementation of various tokamak and diagnostic geometries is described and an overview of the package contents is given. Apart from inversion methods, overview of Tomotok auxiliary content is given. The package provides tools for creating simple synthetic diagnostic system. These can be used for testing and benchmarking the code. This includes tools for building geometry matrices that describe the view of detectors using single line of sight approximation and artificial data generators capable of creating simple or hollow Gaussian profiles. The implemented inversion methods cover the minimum Fisher regularisation, biorthogonal decomposition and linear algebraic methods. The implementation of each method is explained, example results obtained by inverting phantom models are presented and discussed. The computation speed of implemented algorithms is benchmarked and compared.

Twist phase and classical entanglement of partially coherent light.

We demonstrate that the presence of a twist phase in a random light beam leads to classical entanglement between phase space degrees of freedom of the beam. We find analytically the bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) source and quantify its entanglement by evaluating the Schmidt number of the decomposition. We show that (i) classical entanglement of a TGSM source vanishes concurrently with the twist in the fully coherent limit and (ii) entanglement dramatically increases as the source coherence level decreases. We also show that the discovered type of classical entanglement of a Gaussian Wigner function does not degrade on beam propagation in free space.

ПИРАМИДАЛЬНАЯ СХЕМА ПОСТРОЕНИЯ БИОРТОГОНАЛЬНЫХ ВЕЙВЛЕТ-КОДОВ НАД КОНЕЧНЫМИ ПОЛЯМИ

The existence of a biorthogonal decomposition of the space V of dimension n over the field GF(q) is constructively proved, namely, two representations of it are obtained as direct sums of subspaces V = W0⊕W1⊕. . .⊕WJ⊕VJ and V = W˜0⊕W˜1⊕. . .⊕W˜J⊕V˜J ,such that at the j-th level of the decomposition, for 0 < j 6 J, Vj−1 = Vj⊕Wj , V˜j−1 == V˜j ⊕ W˜j , the subspace Vj is orthogonal to W˜j , and the subspace Wj is orthogonal to V˜j . The partition of the space at the j-th level is made with the help of pairs of level filters (hj, gj) and (h˜j, g˜j), for the construction of which the corresponding algorithms have been developed and theoretically proved. A new family of biorthogonal wavelet codes is built on the basis of the multilevel wavelet decomposition scheme with coding rate 2−L, where L is the number of used decomposition levels, and examples of such codes are given.

Extraction of main elements from a TEM composite laser pattern

We examined the atypical formation and superposition of transverse electromagnetic modes (TEM) of high order in a commercial He-Ne laser (@543.5 nm). The competence of the dominant modes exhibits a periodic and stable dynamic, uncommon in industrialized systems. Also, the intensity shows a smooth transition. The origin of the phenomenon is only a product of intra-cavity diffractive components. We found the promotors are a stable optical cavity but quasi-degenerated, with large transversal section (the Fresnel number is irregular), and a breaking the axial symmetry (producing astigmatism). The analysis of the average intensity patterns, and a correlation algorithm — via bi-orthogonal decomposition based on Hermite-Gauss polynomials — expose the five principal TEM elements. These results could be meaningful and inspiring both in courses of optics, laser engineering as in image processing.

A simple shearlet-based 2D Radon inversion with an application to computed tomography

We find a new and simple inversion formula for the 2D Radon transform (RT) with a straight use of the shearlet system and of well-known properties of the RT. Since the continuum theory of shearlets has a natural translation to the discrete theory, we also obtain a computable algorithm that recovers a digital image from noisy samples of the 2D Radon transform which also preserves edges. A very well-known RT inversion in the applied harmonic analysis community is the biorthogonal curvelet decomposition (BCD). The BCD uses an intertwining relation of differential (unbounded) operators between functions in Euclidean and Radon domains. Hence the BCD is ill-posed since the inverse is unstable in the presence of noise. In contrast, our inversion method makes use of an intertwining relation of integral transformations with very smooth kernels and compact support away from the origin in the Fourier domain, i.e. bounded operators. Therefore, we obtain, at least, the same asymptotic behavior of mean-square error as the BCD (and its shearlet version) for the class of cartoon-like functions. Numerical simulations show that our inverse surpasses, by far, the inverse based on the BCD. Our algorithm uses only fast transformations.

Flow field induced by a dielectric barrier discharge plasma actuator analyzed with bi–orthogonal decomposition

A typical dielectric barrier discharge (DBD) plasma actuator generates complex periodic flow structures in burst mode. Efficient use of these actuators depends on clear understanding of the relationship between the operational parameters of the actuator and flow structure. The present study reports the temporal and spatial evolution of these flow structures by utilizing the bi-orthogonal decomposition (BOD) technique. The flow induced by the actuator is captured using a time resolved particle image velocimetry (2D–2C–TR–PIV) system. The BOD of flow field is carried out using instantaneous velocity field data. The DBD plasma actuator is operated at different combinations of duty cycle, α (50% ≤ α ≤ 90%), and burst frequency, fb (10 Hz ≤ fb ≤ 90 Hz). The modal energy content is used to characterize the flow field as a function of operating variables, i.e., α and fb of the actuation signal. The mean mode of the decomposition successfully approximates the time averaged behavior of the induced flow field. The mean mode energy level increases with the increase in both α and fb with a more pronounced effect observed as a function of fb. The coherent structures are located close to the near wall at high burst frequency. The non-dimensional entropy decreases with the increase in both α and fb with a more pronounced effect of fb than that of α. The decrease in entropy value indicates that space–time complexities are reduced at higher burst frequency. The topos of higher order modes reveal the presence of coherent structures that grow in time and convect along the wall like a train of vortices. The chronos of mode 2 and mode 3 is locked in with respect to the burst frequency. However, the chronos of mode 4 and mode 5 shows frequency doubling at lower burst frequency actuation and frequency halving at higher burst frequency actuation. The entropy value or space–time complexity of flow structures generated by DBD plasma actuator is related to the nonlinear vortex interaction mechanism, i.e., period doubling and period halving of chronos.