The Experts below are selected from a list of 6408 Experts worldwide ranked by ideXlab platform
James H. Burge - One of the best experts on this subject based on the ideXlab platform.
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Orthonormal curvature polynomials over a unit circle basis Set derived from curvatures of zernike polynomials
Optics Express, 2013Co-Authors: Chunyu Zhao, James H. BurgeAbstract:Zernike polynomials are an Orthonormal Set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an Orthonormal Set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a Set of Orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis Set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.
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Orthonormal vector polynomials in a unit circle application fitting mapping distortions in a null test
Proceedings of SPIE, 2009Co-Authors: Chunyu Zhao, James H. BurgeAbstract:We developed a complete and Orthonormal Set of vector polynomials defined over a unit circle. One application of these vector polynomials is for fitting the mapping distortions in an interferometric null test. This paper discusses the source of the mapping distortions and the approach of fitting the mapping relations, and justifies why the Set of vector polynomials is the appropriate choice for this purpose. Examples are given to show the excellent fitting results with the polynomials.
Chunyu Zhao - One of the best experts on this subject based on the ideXlab platform.
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Orthonormal curvature polynomials over a unit circle basis Set derived from curvatures of zernike polynomials
Optics Express, 2013Co-Authors: Chunyu Zhao, James H. BurgeAbstract:Zernike polynomials are an Orthonormal Set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an Orthonormal Set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a Set of Orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis Set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.
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Orthonormal vector polynomials in a unit circle application fitting mapping distortions in a null test
Proceedings of SPIE, 2009Co-Authors: Chunyu Zhao, James H. BurgeAbstract:We developed a complete and Orthonormal Set of vector polynomials defined over a unit circle. One application of these vector polynomials is for fitting the mapping distortions in an interferometric null test. This paper discusses the source of the mapping distortions and the approach of fitting the mapping relations, and justifies why the Set of vector polynomials is the appropriate choice for this purpose. Examples are given to show the excellent fitting results with the polynomials.
Lukas J Fiderer - One of the best experts on this subject based on the ideXlab platform.
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general expressions for the quantum fisher information matrix with applications to discrete quantum imaging
arXiv: Quantum Physics, 2020Co-Authors: Lukas J Fiderer, Tommaso Tufarelli, Samanta Piano, Gerardo AdessoAbstract:The quantum Fisher information matrix is a central object in multiparameter quantum estimation theory. It is usually challenging to obtain analytical expressions for it because most calculation methods rely on the diagonalization of the density matrix. In this paper, we derive general expressions for the quantum Fisher information matrix which bypass matrix diagonalization and do not require the expansion of operators on an Orthonormal Set of states. Additionally, we can tackle density matrices of arbitrary rank. The methods presented here simplify analytical calculations considerably when, for example, the density matrix is more naturally expressed in terms of non-orthogonal states, such as coherent states. Our derivation relies on two matrix inverses which, in principle, can be evaluated analytically even when the density matrix is not diagonalizable in closed form. We demonstrate the power of our approach by deriving novel results in the timely field of discrete quantum imaging: the estimation of positions and intensities of incoherent point sources. We find analytical expressions for the full estimation problem of two point sources with different intensities, and for specific examples with three point sources. We expect that our method will become standard in quantum metrology.
Lopez A Ariste - One of the best experts on this subject based on the ideXlab platform.
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an Orthonormal Set of stokes profiles
Astronomy and Astrophysics, 2003Co-Authors: J C Del Toro Iniesta, Lopez A AristeAbstract:A family of well-known Orthonormal functions, the Set of Hermite functions, is proposed as a suitable basis for expanding the Stokes profiles of any spectral line. An expansion series thus provides different degrees of approximation to the Stokes spectrum, depending on the number of basis elements used (or on the number of coefficients). Hence, an usually large number of wavelength samples, may be substituted by a few such coefficients, thus reducing considerably the size of data files and the analysis of observable information. Moreover, since the Set of Hermite functions is an universal basis, it promises to help in modern inversion techniques of the radiative transfer equation that infer the solar physical quantities from previously compiled look-up tables or artificial neural networks. These features appear to be particularly important in modern solar applications producing huge amounts of spectropolarimetric data and on near-future, on-line applications aboard spacecrafts.
Kesheng Wu - One of the best experts on this subject based on the ideXlab platform.
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a block orthogonalization procedure with constant synchronization requirements
SIAM Journal on Scientific Computing, 2001Co-Authors: Andreas Stathopoulos, Kesheng WuAbstract:First, we consider the problem of Orthonormalizing skinny (long) matrices. We propose an alternative Orthonormalization method that computes the Orthonormal basis from the right singular vectors of a matrix. Its advantages are that (a) all operations are matrix-matrix multiplications and thus cache efficient, (b) only one synchronization point is required in parallel implementations, and (c) it is typically more stable than classical Gram--Schmidt (GS). Second, we consider the problem of Orthonormalizing a block of vectors against a previously Orthonormal Set of vectors and among itself. We solve this problem by alternating iteratively between a phase of GS and a phase of the new method. We provide error analysis and use it to derive bounds on how accurately the two successive Orthonormalization phases should be performed to minimize total work performed. Our experiments confirm the favorable numerical behavior of the new method and its effectiveness on modern parallel computers.
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a block orthogonalization procedure with constant synchronization requirements
Lawrence Berkeley National Laboratory, 2000Co-Authors: Andreas Stathopoulos, Kesheng WuAbstract:A BLOCK ORTHOGONALIZATION PROCEDURE WITH CONSTANT SYNCHRONIZATION REQUIREMENTS ANDREAS STATHOPOULOS AND KESHENG WU y Abstract. We propose an alternative Orthonormalization method that computes the orthonor- mal basis from the right singular vectors of a matrix. Its advantage are: a all operations are matrix-matrix multiplications and thus cache-e cient, b only one synchronization point is required in parallel implementations, c could be more stable than Gram-Schmidt. In addition, we consider the problem of incremental Orthonormalization where a block of vectors is Orthonormalized against a previously Orthonormal Set of vectors and among itself. We solve this problem by alternating itera- tively between a phase of Gram-Schmidt and a phase of the new method. We provide error analysis and use it to derive bounds on how accurately the two successive Orthonormalization phases should be performed to minimize total work performed. Our experiments con rm the favorable numerical behavior of the new method and its e ectiveness on modern parallel computers. Key words. Gram-Schmidt, orthogonalization, Householder, QR factorization, singular value decomposition, Poincare AMS Subject Classi cation. 65F15 1. Introduction. Computing an Orthonormal basis from a given Set of vectors is a basic computation, common in most scienti c applications. Often, it is also one of the most computationally demanding procedures because the vectors are of large dimension, and because the computation scales as the square of the number of vectors involved. Further, among several Orthonormalization techniques the ones that ensure high accuracy are the more expensive ones. In many applications, Orthonormalization occurs in an incremental fashion, where a new Set of vectors we call this internal Set is orthogonalized against a previously Orthonormal Set of vectors we call this external, and then among themselves. This computation is typical in block Krylov methods, where the Krylov basis is expanded by a block of vectors 12, 11 . It is also typical when certain external orthogonalization constraints have to be applied to the vectors of an iterative method. Locking of converged eigenvectors in eigenvalue iterative methods is such an example 19, 22 . This problem di ers from the classical QR factorization in that the external Set of vectors should not be modi ed. Therefore, a two phase process is required; rst orthogonalizing the internal vectors against the external, and second the internal among themselves. Usually, the number of the internal vectors is much smaller than the external ones, and signi cantly smaller than their dimension. Another important di erence is that the accuracy of the R matrix of the QR factorization is not of pri- mary interest, but rather the Orthonormality of the produced vectors Q . A variety of orthogonalization techniques exist for both phases. For the external phase, Gram- Schmidt GS and its modi ed version MGS are the most competitive choices. For the internal phase, QR factorization using Householder transformations is the most stable, albeit more expensive method 11 . When the number of vectors is signi - cantly smaller than their dimension, MGS or GS with reorthogonalization are usually preferred. Computationally, MGS, GS and Housholder transformations are based on level 1 or level 2 BLAS operations 15, 9, 8 . These basic kernel computations, dot prod- ucts, vector updates and sometimes matrix-vector operations, cannot fully utilize the Department of Computer Science, College of William and Mary, Williamsburg, Virginia 23187- 8795, andreas@cs.wm.edu . y NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720, kwu@lbl.gov .
