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General Inner Approximation of Vector-valued Functions
International audienceThis paper addresses the problem of evaluating a subset of the range of a vector-valued function. It is based on a work by Goldsztejn and Jaulin which provides methods based on interval analysis to address this problem when the dimension of the domain and co-domain of the function are equal. This paper extends this result to vector-valued functions with domain and co-domain of different dimensions. This extension requires the knowledge of the rank of the Jacobian function on the whole domain. This leads to the sub-problem of extracting an interval sub-matrix of maximum rank from a given interval matrix. Three different techniques leading to approximate solutions of this extraction are proposed and compared
Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space
In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial
frequency domain (k-space), typically by time-consuming line-by-line scanning
on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of
data using multiple receivers (parallel imaging), and by using more efficient
non-Cartesian sampling schemes. As shown here, reconstruction from samples at
arbitrary locations can be understood as approximation of vector-valued
functions from the acquired samples and formulated using a Reproducing Kernel
Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial
sensitivities of the receive coils. This establishes a formal connection
between approximation theory and parallel imaging. Theoretical tools from
approximation theory can then be used to understand reconstruction in k-space
and to extend the analysis of the effects of samples selection beyond the
traditional g-factor noise analysis to both noise amplification and
approximation errors. This is demonstrated with numerical examples.Comment: 28 pages, 7 figure
Multiresolution approximation of the vector fields on T^3
Multiresolution approximation (MRA) of the vector fields on T^3 is studied.
We introduced in the Fourier space a triad of vector fields called helical
vectors which derived from the spherical coordinate system basis. Utilizing the
helical vectors, we proved the orthogonal decomposition of L^2(T^3) which is a
synthesis of the Hodge decomposition of the differential 1- or 2-form on T^3
and the Beltrami decomposition that decompose the space of solenoidal vector
fields into the eigenspaces of curl operator. In the course of proof, a general
construction procedure of the divergence-free orthonormal complete basis from
the basis of scalar function space is presented. Applying this procedure to MRA
of L^2(T^3), we discussed the MRA of vector fields on T^3 and the analyticity
and regularity of vector wavelets. It is conjectured that the solenoidal
wavelet basis must break r-regular condition, i.e. some wavelet functions
cannot be rapidly decreasing function because of the inevitable singularities
of helical vectors. The localization property and spatial structure of
solenoidal wavelets derived from the Littlewood-Paley type MRA (Meyer's
wavelet) are also investigated numerically.Comment: LaTeX, 33 Pages, 3 figures. submitted to J. Math. Phy
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