263 research outputs found
A Spectral Graph Uncertainty Principle
The spectral theory of graphs provides a bridge between classical signal
processing and the nascent field of graph signal processing. In this paper, a
spectral graph analogy to Heisenberg's celebrated uncertainty principle is
developed. Just as the classical result provides a tradeoff between signal
localization in time and frequency, this result provides a fundamental tradeoff
between a signal's localization on a graph and in its spectral domain. Using
the eigenvectors of the graph Laplacian as a surrogate Fourier basis,
quantitative definitions of graph and spectral "spreads" are given, and a
complete characterization of the feasibility region of these two quantities is
developed. In particular, the lower boundary of the region, referred to as the
uncertainty curve, is shown to be achieved by eigenvectors associated with the
smallest eigenvalues of an affine family of matrices. The convexity of the
uncertainty curve allows it to be found to within by a fast
approximation algorithm requiring typically sparse
eigenvalue evaluations. Closed-form expressions for the uncertainty curves for
some special classes of graphs are derived, and an accurate analytical
approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random
graphs is developed. These theoretical results are validated by numerical
experiments, which also reveal an intriguing connection between diffusion
processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure
One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures
In this review we provide a rigorous and self-contained presentation of
one-body reduced density-matrix (1RDM) functional theory. We do so for the case
of a finite basis set, where density-functional theory (DFT) implicitly becomes
a 1RDM functional theory. To avoid non-uniqueness issues we consider the case
of fermionic and bosonic systems at elevated temperature and variable particle
number, i.e, a grand-canonical ensemble. For the fermionic case the Fock space
is finite-dimensional due to the Pauli principle and we can provide a rigorous
1RDM functional theory relatively straightforwardly. For the bosonic case,
where arbitrarily many particles can occupy a single state, the Fock space is
infinite-dimensional and mathematical subtleties (not every hermitian
Hamiltonian is self-adjoint, expectation values can become infinite, and not
every self-adjoint Hamiltonian has a Gibbs state) make it necessary to impose
restrictions on the allowed Hamiltonians and external non-local potentials. For
simple conditions on the interaction of the bosons a rigorous 1RDM functional
theory can be established, where we exploit the fact that due to the finite
one-particle space all 1RDMs are finite-dimensional. We also discuss the
problems arising from 1RDM functional theory as well as DFT formulated for an
infinite-dimensional one-particle space.Comment: 55 pages, 7 figure
High rank tensor and spherical harmonic models for diffusion MRI processing
Diffusion tensor imaging (DTI) is a non-invasive quantitative method of characterizing tissue micro-structure. Diffusion imaging attempts to characterize the manner by which the water molecules within a particular location move within a given amount of time. Measurement of the diffusion tensor (D) within a voxel allows a macroscopic voxel-averaged description of fiber structure, orientation and fully quantitative evaluation of the microstructural features of healthy and diseased tissue.;The rank two tensor model is incapable of resolving multiple fiber orientations within an individual voxel. This shortcoming of single tensor model stems from the fact that the tensor possesses only a single orientational maximum. Several authors reported this non-mono-exponential behavior for the diffusion-induced attenuation in brain tissue in water and N-Acetyl Aspartate (NAA) signals, that is why the Multi-Tensor, Higher Rank Tensor and Orientation Distribution Function (ODF) were introduced.;Using the higher rank tensor, we will propose a scheme for tensor field interpolation which is inspired by subdivision surfaces in computer graphics. The method applies to Cartesian tensors of all ranks and imposes smoothness on the interpolated field by constraining the divergence and curl of the tensor field. Results demonstrate that the subdivision scheme can better preserve anisotropicity and interpolate rotations than some other interpolation methods. As one of the most important applications of DTI, fiber tractography was implemented to study the shape geometry changes. Based on the divergence and curl measurement, we will introduce new scalar measures that are sensitive to behaviors such as fiber bending and fanning.;Based on the ODF analysis, a new anisotropy measure that has the ability to describe multi-fiber heterogeneity while remaining rotationally invariant, will be introduced, which is a problem with many other anisotropy measures defined using the ODF. The performance of this novel measure is demonstrated for data with varying Signal to Noise Ratio (SNR), and different material characteristics
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
Discrete Harmonic Analysis. Representations, Number Theory, Expanders and the Fourier Transform
This self-contained book introduces readers to discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform on finite groups and finite fields, as well as their noncommutative versions. It also features applications to number theory, graph theory, and representation theory of finite groups. Beginning with elementary material on algebra and number theory, the book then delves into advanced topics from the frontiers of current research, including spectral analysis of the DFT, spectral graph theory and expanders, representation theory of finite groups and multiplicity-free triples, Tao's uncertainty principle for cyclic groups, harmonic analysis on GL(2,Fq), and applications of the Heisenberg group to DFT and FFT. With numerous examples, figures, and over 160 exercises to aid understanding, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science
Wavefield modeling and signal processing for sensor arrays of arbitrary geometry
Sensor arrays and related signal processing methods are key technologies in many areas of engineering including wireless communication systems, radar and sonar as well as in biomedical applications. Sensor arrays are a collection of sensors that are placed at distinct locations in order to sense physical phenomena or synthesize wavefields. Spatial processing from the multichannel output of the sensor array is a typical task. Such processing is useful in areas including wireless communications, radar, surveillance and indoor positioning.
In this dissertation, fundamental theory and practical methods of wavefield modeling for radio-frequency array processing applications are developed. Also, computationally-efficient high-resolution and optimal signal processing methods for sensor arrays of arbitrary geometry are proposed. Methods for taking into account array nonidealities are introduced as well. Numerical results illustrating the performance of the proposed methods are given using real-world antenna arrays.
Wavefield modeling and manifold separation for vector-fields such as completely polarized electromagnetic wavefields and polarization sensitive arrays are proposed. Wavefield modeling is used for writing the array output in terms of two independent parts, namely the sampling matrix depending on the employed array including nonidealities and the coefficient vector depending on the wavefield. The superexponentially decaying property of the sampling matrix for polarization sensitive arrays is established. Two estimators of the sampling matrix from calibration measurements are proposed and their statistical properties are established.
The array processing methods developed in this dissertation concentrate on polarimetric beamforming as well as on high-resolution and optimal azimuth, elevation and polarization parameter estimation. The proposed methods take into account array nonidealities such as mutual coupling, cross-polarization effects and mounting platform reflections. Computationally-efficient solutions based on polynomial rooting techniques and fast Fourier transform are achieved without restricting the proposed methods to regular array geometries. A novel expression for the Cramér-Rao bound in array processing that is tight for real-world arrays with nonidealities in the asymptotic regime is also proposed.
A relationship between spherical harmonics and 2-D Fourier basis, called equivalence matrix, is established. A novel fast spherical harmonic transform is proposed, and a one-to-one mapping between spherical harmonic and 2-D Fourier spectra is found. Improvements to the minimum number of samples on the sphere that are needed in order to avoid aliasing are also proposed
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