Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples

Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.

Sinusoidal Single-Pixel Imaging Based on Fourier Positive–Negative Intensity Correlation

Single-pixel imaging techniques extend the time dimension to reconstruct a target scene in the spatial domain based on single-pixel detectors. Structured light illumination modulates the target scene by utilizing multi-pattern projection, and the reflected or transmitted light is measured by a single-pixel detector as total intensity. To reduce the imaging time and capture high-quality images with a single-pixel imaging technique, orthogonal patterns have been used instead of random patterns in recent years. The most representative among them are Hadamard patterns and Fourier sinusoidal patterns. Here, we present an alternative Fourier single-pixel imaging technique that can reconstruct high-quality images with an intensity correlation algorithm using acquired Fourier positive–negative images. We use the Fourier matrix to generate sinusoidal and phase-shifting sinusoid-modulated structural illumination patterns, which correspond to Fourier negative imaging and positive imaging, respectively. The proposed technique can obtain two centrosymmetric images in the intermediate imaging course. A high-quality image is reconstructed by applying intensity correlation to the negative and positive images for phase compensation. We performed simulations and experiments, which obtained high-quality images, demonstrating the feasibility of the methods. The proposed technique has the potential to image under sub-sampling conditions.

Compression and Recovery of 3D Broad-Leaved Tree Point Clouds Based on Compressed Sensing

The terrestrial laser scanner (TLS) has been widely used in forest inventories. However, with increasing precision of TLS, storing and transmitting tree point clouds become more challenging. In this paper, a novel compressed sensing (CS) scheme for broad-leaved tree point clouds is proposed by analyzing and comparing different sparse bases, observation matrices, and reconstruction algorithms. Our scheme starts by eliminating outliers and simplifying point clouds with statistical filtering and voxel filtering. The scheme then applies Haar sparse basis to thin the coordinate data based on the characteristics of the broad-leaved tree point clouds. An observation procedure down-samples the point clouds with the partial Fourier matrix. The regularized orthogonal matching pursuit algorithm (ROMP) finally reconstructs the original point clouds. The experimental results illustrate that the proposed scheme can preserve morphological attributes of the broad-leaved tree within a range of relative error: 0.0010%–3.3937%, and robustly extend to plot-level within a range of mean square error (MSE): 0.0063–0.2245.

Defect and Equivalence of Unitary Matrices. The Fourier Case. Part II

Consider the real space 𝔻U of directions one can move in from a unitary N × N matrix U without disturbing its unitarity and the moduli of its entries in the first order. dimℝ (𝔻U) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov’s theory where 𝔻U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator ℐU on 𝕄N, and where D(U) is the multiplicity of 1 in the spectrum of ℐU. This characterisation allows us to establish the dependence of D(U(1) ⊗ … ⊗ U(r)) on D(U(k))’s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime number, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of ℐU we study the ‘permute and enphase’ symmetries and equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is published as two papers — the first part [1] and the current second one.

On the existence of complex Hadamard submatrices of the Fourier matrices

We use a theorem of Lam and Leung to prove that a submatrix of a Fourier matrix cannot be Hadamard for particular cases when the dimension of the submatrix does not divide the dimension of the Fourier matrix. We also make some observations on the trace-spectrum relationship of dephased Hadamard matrices of low dimension.

Defect and Equivalence of Unitary Matrices. The Fourier Case. Part I

Consider the real space 𝔻U of directions one can move in from a unitary N × N matrix U without disturbing its unitarity and the moduli of its entries in the first order. dimℝ (𝔻U) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov’s theory where 𝔻U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator IU on 𝕄N, and where D(U) is the multiplicity of 1 in the spectrum of IU. This characterization allows us to establish the dependence of D(U(1) ⊗ … ⊗U(r)) on D(U(k))’s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of IU we study the ‘permute and enphase’ symmetries and the equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is divided in two parts — the present one and the second appearing in the next issue of OSID [1].

Multisource encoding and decoding using the signal apparition technique

Signal apparition is a method for encoding sources in simultaneous multisource seismic acquisition and decoding the multisource response of the earth into its single-source responses. For [Formula: see text] sources, encoding is performed by applying periodic sequences of period [Formula: see text] to each of the sources along source lines. Decoding is achieved in the wavenumber domain for each frequency by solving an [Formula: see text] linear system of equations. The system’s matrix is the product of a Fourier matrix and an encoding matrix, the latter containing the information of the codes. The solution of the system is unique when the encoding matrix is invertible. When the encoding sequences consist of time delays applied to sources’ firing times, the determinant of the encoding matrix becomes a polynomial. A unique solution to decoding then exists if the roots of the polynomial avoid the unit circle. Periodic time-shift sequences for two, three, four, and six sources are discussed. A model example of simultaneous four-source data acquisition illustrates the performance of the encoding/decoding technique for the spatially nonaliased case.

Matrix Fourier Transforms for Consistent Mathematical Models

We create a matrix integral transforms method; it allows us to describe analytically the consistent mathematical models. An explicit constructions for direct and inverse Fourier matrix transforms with discontinuous coefficients are established. We introduce special types of Fourier matrix transforms: matrix cosine transforms, matrix sine transforms, and matrix transforms with piecewise trigonometric kernels. The integral transforms of such kinds are used for problems solving of mathematical physics in homogeneous and piecewise homogeneous media. Analytical solution of iterated heat conduction equation is obtained. Stress produced in the elastic semi-infinite solid by pressure is obtained in the integral form.