Hermitian Product Research Articles

Coherent change detection of fine traces based on multi-angle SAR observations

Coherent change detection (CCD) is an effective method to detect subtle scene changes that occur between temporal synthetic aperture radar (SAR) observations. Most coherence estimators are obtained from a Hermitian product based on local statistics. Increasing the number of samples in the local window can improve the estimation bias, but cause the loss of the estimated images spatial resolution. The limitations of these estimators lead to unclear contour of the disturbed region, and even the omission of fine change targets. In this paper, a CCD approach is proposed to detect fine scene changes from multi-temporal and multi-angle SAR image pairs. Multi-angle CCD estimator can improve the contrast between the change target and the background clutter by jointly accumulating singleangle alternative estimator results without further loss of image resolution. The sensitivity of detection performance to image quantity and angle interval is analyzed. Theoretical analysis and experimental results verify the performance of the proposed algorithm.

PolSAR Ship Detection Based on Azimuth Sublook Polarimetric Covariance Matrix

Polarimetric synthetic aperture radar (PolSAR) is an important tool for marine remote sensing. In the past two decades, subaperture decomposition is considered as an alternative method to extract ship targets, especially in complex conditions where the detection based on intensity performs not well. The introduction of polarization information can enhance the detection ability of radar. The combination of sublook techniques and polarization will lead to more precise target extraction. The first methods jointly using information of polarization and sublook is the polarimetric internal Hermitian product (Pol-IHP), which nowadays has been corrected as polarimetric Hermitian internal product (Pol-HIP). However, it is limited by the number of sublooks and recently developed as generalized multi-sublooks correlation (GMC). In this study, a novel sublook polarimetric covariance matrix (SPCM) is established, based on which a sublook-polarimetric whitening filter (Sub-PWF) is further proposed for ship detection. The SPCM represents a given pixel using a high-dimensional covariance matrix which utilizes the spectral correlation in imaging progress and the scattering characteristics of different polarization channels. The superiority of Sub-PWF based on SPCM is validated via the measured single look complex (SLC) PolSAR data from RadarSAT-2 (RS-2) and GaoFen-3 (GF-3). Results show this new method can improve the target to clutter ratio (TCR) and reduce the clutter fluctuation. The ROC curves in both small and large scenes proves that Sub-PWF can reach ideal detection performance.

Harmonic analysis for rank-1 randomised Horn problems

The randomised Horn problem, in both its additive and multiplicative versions, has recently drawn an increasing interest. Especially, closed analytical results have been found for the rank-1 perturbation of sums of Hermitian matrices and products of unitary matrices. We will generalise these results to rank-1 perturbations for products of positive-definite Hermitian matrices and prove the other results in a new unified way. Our ideas work along harmonic analysis for matrix groups via spherical transforms that have been successfully applied in products of random matrices in the past years. In order to achieve the unified derivation of all three cases, we define the spherical transform on the unitary group and prove its invertibility.

Implementing zonal harmonics with the Fueter principle

By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on a constant function. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expressions for zonal harmonics starting from the 2 and 3 dimensional cases. It turns out that all zonal harmonics in any dimension are related to the real part of powers of the standard Hermitian product in C. At the end we compare formulas, obtaining interesting equalities involving the real part of positive and negative powers of the standard Hermitian product.In the two appendices we show how our computations are optimal compared to direct ones.

The Design of a Trimodal Broadside Antenna Element for Compact Massive MIMO Arrays: Utilizing the Theory of Characteristic Modes

Massive multiple-input, multiple-output (MIMO) arrays are becoming critical elements in cellular base station infrastructure. We utilize the theory of characteristic modes (TCM) to design a novel trimodal broadside antenna element that is suitable for constructing compact massive MIMO arrays. The proposed antenna element consists of three ports, formed by reviewing an existing compact two-port Y-shaped patch antenna from a TCM perspective. Using this perspective, the Y-shaped antenna is modified into a snowflake-shaped patch antenna and excited by three ports via capacitive coupling. The advantage of this design approach is that the size of the three-port antenna is approximately the same as a conventional dual-polarized patch antenna, hence allowing 50% more antenna elements for the same array aperture. By cutting the ground plane into a hexagonal shape, a multiple of the proposed three-port canonical antennas are concatenated together to form 21- and 102-port massive MIMO arrays with all radiation patterns pointing in the broadside direction. Key performance characteristics of the massive MIMO arrays, i.e., the Hermitian product of the simulated massive MIMO channel and mutual coupling, validate the effectiveness of the compact design.

Formula omitted]-linear codes over [formula omitted]-algebras

Huffman (2013) [12] studied Fq-linear codes over Fqm and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative Fq-algebra. An Fq-linear code over S of length n is an Fq-submodule of Sn. In this paper, we study Fq-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over Fq-algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of Fq-linear codes over finite commutative graded Fq-algebras.

Orthogonality of super‐Jack polynomials and a Hilbert space interpretation of deformed Calogero–Moser–Sutherland operators

We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials $SP_\lambda((z_1,\ldots,z_n),(w_1,\ldots,w_m);\theta)$ with respect to a natural positive semi-definite, but degenerate, Hermitian product $\langle\cdot,\cdot\rangle_{n,m}^\prime$. In case $m=0$ (or $n=0$), our product reduces to Macdonald's well-known inner product $\langle\cdot,\cdot\rangle_n^\prime$, and we recover his corresponding orthogonality results for the Jack polynomials $P_\lambda((z_1,\ldots,z_n);\theta)$. From our main results, we readily infer that the kernel of $\langle\cdot,\cdot\rangle_{n,m}^\prime$ is spanned by the super-Jack polynomials indexed by a partition $\lambda$ not containing the $m\times n$ rectangle $(m^n)$. As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type $A(n-1,m-1)$.

A comparison of the scalar Hermitian product versus the complex product for obtaining electron densities and Fukui functions from complex wavefunctions for the temporary anion resonance states of Be−, Mg− and Ca−

ABSTRACTDensities and Fukui functions using the complex product, c-product (f|g) and the scalar Hermitian product <f|g> are compared for two 2P resonance states of Be−, the 2P resonance state of Mg− and the 2D resonance state of Ca−. Use of the c-product has been shown to be necessary when analysing the complex eigenfunctions of a complex-scaled Hamiltonian. The c-product produces complex electron densities from complex wavefunctions, but only the real part is used for the Fukui functions. The real part of the density matrix obtained using the c-product doesn’t satisfy the Pauli principle, and can produce negative regions in the electron densities. The Fukui functions and densities obtained using the c-product compare well with the Fukui functions and densities obtained using the scalar Hermitian product.

Enumeration of self-dual cyclic codes of some specific lengths over finite fields

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length [Formula: see text] over a finite field if and only if [Formula: see text] and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for self-dual cyclic codes of length [Formula: see text] over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are positive integers. Some number theoretical tools are established. Based on these results, alternative formulas and efficient algorithms to determine the number of self-dual cyclic codes of such lengths are provided.

Quantum stabilizer codes construction from Hermitian self-orthogonal codes over GF(4)

In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.

Method of B0 mapping with magnitude-based correction for bipolar two-point Dixon cardiac MRI.

The conventional two-point (2pt) Dixon technique explicitly estimates B0 map by performing phase unwrapping. When signal loss, phase singularity, artifacts, or spatially isolated regions corrupt the measured phase images, this unwrapping-based technique will face difficulty. This work aims to improve the reliability of B0 mapping by performing unwrapping error correction. To detect the unwrapping-caused phase errors, we determined a magnitude-based fat/water mask and used it as reference to identify pixels being mismatched by the phase-based mask, which was derived from the B0-corrected phase term of the Hermitian product between echoes. Then, we corrected the afore-determined phase error on a region-by-region basis. We tested the developed method with nine patients' data, and the results were compared with a well-established region-growing technique. By adding the step to correct unwrapping-caused error, we improved the robustness of B0 mapping, resulting in better fat-water separation when compared with the conventional 2pt and the phasor-based region-growing techniques. We showed the feasibility of B0 mapping with bipolar 2pt human cardiac data. The software is freely available to the scientific community. Magn Reson Med 78:1862-1869, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Quantum discriminant analysis for dimensionality reduction and classification

We present quantum algorithms to efficiently perform discriminant analysis for dimensionality reduction and classification over an exponentially large input data set. Compared with the best-known classical algorithms, the quantum algorithms show an exponential speedup in both the number of training vectors M and the feature space dimension N. We generalize the previous quantum algorithm for solving systems of linear equations (2009 Phys. Rev. Lett. 103 150502) to efficiently implement a Hermitian chain product of k trace-normalized N ×N Hermitian positive-semidefinite matrices with time complexity of . Using this result, we perform linear as well as nonlinear Fisher discriminant analysis for dimensionality reduction over M vectors, each in an N-dimensional feature space, in time , where ϵ denotes the tolerance error, and p is the number of principal projection directions desired. We also present a quantum discriminant analysis algorithm for data classification with time complexity .

Complex Exceptional Orthogonal Polynomials and Quasi-invariance

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1, \dots, \lambda_n)$ is a partition. G\'omez-Ullate et al showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of $\mathbb C[x]$ satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schr\"odinger operators, is also presented.

Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{CP}^2$

We look at lattices in Iso+(H2R)Iso+(HR2), the group of orientation preserving isometries of the real hyperbolic plane. We study their geometry and dynamics when they act on CP2CP2 via the natural embedding of SO+(2,1)↪SU(2,1)⊂SL(3,C)SO+(2,1)↪SU(2,1)⊂SL(3,C). We use the Hermitian cross product in C2,1C2,1 introduced by Bill Goldman, to determine the topology of the Kulkarni limit set ΛKulΛKul of these lattices, and show that in all cases its complement ΩKulΩKul has three connected components, each being a disc bundle over H2RHR2. We get that ΩKulΩKul coincides with the equicontinuity region for the action on CP2CP2. Also, it is the largest set in CP2CP2 where the action is properly discontinuous and it is a complete Kobayashi hyperbolic space. As a byproduct we get that these lattices provide the first known examples of discrete subgroups of SL(3,C)SL(3,C) whose Kulkarni region of discontinuity in CP2CP2 has exactly three connected components, a fact that does not appear in complex dimension 11 (where it is known that the region of discontinuity of a Kleinian group acting on CP1CP1 has 00, 11, 22 or infinitely many connected components).

Hermitian realizations of κ-Minkowski space–time

General realizations, star products and plane waves for κ-Minkowski space–time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.

Inter-echo variance as a weighting factor for multi-channel combination in multi-echo acquisition for local frequency shift mapping.

To develop and evaluate a local frequency shift (LFS) mapping method specifically designed for multi-echo acquisitions and multi-channel receive coils. The proposed method uses the pixel-by-pixel inter-echo variance (IEV) as a weighting factor during channel-combination. Five healthy volunteers were scanned at 7 T. The IEV-weighted method was quantitatively compared to established (adaptive and Hermitian product) channel-combination methods with respect to IEV of LFS over the entire brain. In all experiments, the IEV-weighted method generated LFS maps free of artifacts caused by unwrapping errors. Based on measurements of the inter-echo frequency variance throughout the whole brain, the IEV-weighted method produced the lowest variation and the best contrast at the edge of the brain. The primary finding of the present study is that accurate LFS maps are achievable if the data from each channel is processed independently prior to combination followed by a weighted combination using IEV as the weighting term. The software is freely available to the scientific community.

PLANCHEREL–ROTACH FORMULAE FOR AVERAGE CHARACTERISTIC POLYNOMIALS OF PRODUCTS OF GINIBRE RANDOM MATRICES AND THE FUSS–CATALAN DISTRIBUTION

Formulae of Plancherel–Rotach type are established for the average characteristic polynomials of Hermitian products of rectangular Ginibre random matrices on the region of zeros. These polynomials form a general class of multiple orthogonal hypergeometric polynomials generalizing the classical Laguerre polynomials. The proofs are based on a multivariate version of the complex method of saddle points. After suitable rescaling the asymptotic zero distributions for the polynomials are studied and shown to coincide with the Fuss–Catalan distributions. Moreover, introducing appropriate coordinates, elementary and explicit characterizations are derived for the densities as well as for the distribution functions of the Fuss–Catalan distributions of general order.

A new construction of Lagrangians in the complex Euclidean plane in terms of planar curves

We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane C2 by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their position and tangent vectors.Among this family, we characterize minimal, constant mean curvature, Hamiltonian stationary, solitons for mean curvature flow and Willmore surfaces in terms of simple properties of the curvatures of the generating curves. As an application, we provide explicitly conformal parametrizations of known and new examples of these classes of Lagrangians in C2.

An Arithmetic Poisson Formula for the multi-variate resultant

In these pages we compute the expectation of several functions of multi-variate complex polynomials. The common thread of all our outcomes is the basic technique used in their proofs. The used techniques combine essentially the unitary invariance of Bombieri–Weyl’s Hermitian product and some elementary Integral Geometry. Using different combinations of these techniques we compute the expectation of the logarithm of the absolute value of an affine polynomial and we compute the expected value of Akatsuka Zeta Mahler’s measure. As main consequences of these results and techniques, we show a probabilistic answer to question (d) in Armentano and Shub (2012) [2], concerning the complexity of one point homotopy, and an Arithmetic Poisson Formula for the multi-variate resultant. These two last statements and bounds are related to the complexity of algorithms for polynomial equation solving.

A Hierarchical Ship Detection Scheme for High-Resolution SAR Images

This paper presents a new hierarchical scheme for detecting ships from high-resolution synthetic aperture radar (SAR) images. The scheme consists of two stages: detection and discrimination. In the detection stage, the existing internal Hermitian product is extended to obtain a new detector. The new detector makes a combined use of the complex coherence among more than two subapertures and the intensity of each subaperture. When the subaperture number is increased, the target/clutter contrast is shown to be improved. Ship candidates are obtained by applying a threshold. Ship discrimination is performed by using one-class classification. The covariance descriptor, developed by Tuzel in 2006, is introduced to SAR ship discrimination as the feature. The traditional one-class quadratic discriminator is used as the discriminator. After this stage, most false alarms are rejected, and the real ship targets in the candidates are maintained. The effectiveness of the proposed scheme is verified using RADARSAT-2 data. Experimental results show that the proposed scheme can detect most ship targets in the image and few false alarms occur.