Orthonormal basis

About: Orthonormal basis is a research topic. Over the lifetime, 6014 publications have been published within this topic receiving 174416 citations.

The Cauchy Problem in Kinetic Theory

Abstract: Preface 1. Properties of the Collision Operator. Kinetic Theory, Derivation of the Equations, The Form of the Collision Operator, The Hard Sphere Case, Conservation Laws and the Entropy, Relevance of the Maxwellian, The Jacobian Determinant, The Structure of Collision Invariants, Relationship of the Boltzmann Equation to the Equations of Fluids, References 2. The Boltzmann Equation Near the Vacuum. Invariance of $|x-tv|^2+|x-tu|^2$, Sequences of Approximate Solutions, Satisfaction of the Beginning Condition, Proof that $u=\ell$, Remarks and Related Questions, References 3. The Boltzmann Equation Near the Equilibrium. The Perturbation from Equilibrium, Computation of the Integral Operator, Estimates on the Integral Operator, Properties of $L$, Compactness of $K$, Solution Spaces, An Orthonormal Basis for $N(L)$, Estimates on the Nonlinear Term, Equations for 13 Moments, Computation of the Coefficient Matrices, Compensating Functions, Time Decay Estimates, Time Decay in Other Norms, The Major Theorem, The Relativistic Boltzmann Equation, References 4. The Vlasov--Poisson System. Introduction, Preliminaries and A Priori Estimates, Sketch of the Existence Proof, The Good, the Bad and the Ugly, The Bound on the Velocity Support, Blow-up in the Gravitational Case, References 5. The Vlasov--Maxwell System. Collisionless Plasmas, Control of Large Velocities, Representation of the Fields, Representation of the Derivatives of the Fields, Estimates on the Particle Density, Bounds on the Field, Bounds on the Gradient of the Field, Proof of Existence, References 6. Dilute Collisionless Plasmas. The Small--data Theorem, Outline of the Proof, Characteristics, The Particle Densities, Estimates on the Fields, Estimates on Derivatives of the Fields, References 7. Velocity Averages: Weak Solutions to the Vlasov--Maxwell System. Sketch of the Problem, The Velocity Averaging Smoothing Effect, Convergence of the Current Density, Completion of the Proof, References 8. Convergence of a Particle Method for the Vlasov--Maxwell System. Introduction, The Particle Simulation, The Field Errors, The Particle Errors, Summing the Errors, References Index.

Signal Reconstruction From Noisy Random Projections

Abstract: Recent results show that a relatively small number of random projections of a signal can contain most of its salient information. It follows that if a signal is compressible in some orthonormal basis, then a very accurate reconstruction can be obtained from random projections. This "compressive sampling" approach is extended here to show that signals can be accurately recovered from random projections contaminated with noise. A practical iterative algorithm for signal reconstruction is proposed, and potential applications to coding, analog-digital (A/D) conversion, and remote wireless sensing are discussed

On the Nonorthogonality Problem

Abstract: Publisher Summary This chapter discusses three orthonormalization procedures, such as successive orthonormalization, symmetric orthonormalization, and canonical orthonormalization The simplest way of orthonormalizing a finite set of functions is by the classical Schmidt procedure, in which each member of the set in order is orthogonalized against all the previous members and subsequently normalized In solid-state theory, one could probably construct orthonormal combinations of the atomic orbitals of the system, which would still preserve the natural symmetry In such an approach, it would be necessary to treat the given functions ϕ = {ϕ 1 , ϕ 2 …, ϕ n } simultaneously, on an equivalent basis instead of successively as in the Schmidt procedure In molecular and solid-state theory, there are cases when also the symmetric orthonormalization procedure will break down, depending on the fact that, even if the basis ϕ = {ϕ 1 , ϕ 2 …, ϕ n } is linearly independent from the mathematical point of view, it may be approximately linearly dependent from the computational point of view This phenomenon causes a great many complications and may lead to very misleading results, since the associated secular equations may be almost identically vanishing Unfortunately, it seems as if many of the conventionally used basic systems are strongly affected by approximate linear dependencies In order to systematize this problem, it is convenient to study the metric matrix

Spectral Compression of Mesh Geometry.

Abstract: We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.

A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding

Abstract: This paper introduces a new approach to orthonormal wavelet image denoising. Instead of postulating a statistical model for the wavelet coefficients, we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights. We then minimize an estimate of the mean square error between the clean image and the denoised one. The key point is that we have at our disposal a very accurate, statistically unbiased, MSE estimate-Stein's unbiased risk estimate-that depends on the noisy image alone, not on the clean one. Like the MSE, this estimate is quadratic in the unknown weights, and its minimization amounts to solving a linear system of equations. The existence of this a priori estimate makes it unnecessary to devise a specific statistical model for the wavelet coefficients. Instead, and contrary to the custom in the literature, these coefficients are not considered random any more. We describe an interscale orthonormal wavelet thresholding algorithm based on this new approach and show its near-optimal performance-both regarding quality and CPU requirement-by comparing it with the results of three state-of-the-art nonredundant denoising algorithms on a large set of test images. An interesting fallout of this study is the development of a new, group-delay-based, parent-child prediction in a wavelet dyadic tree