Multidimensional Integral Inversion, with Applications in Shape Reconstruction

Abstract

In shape reconstruction, the celebrated Fourier slice theorem plays an essential role. It allows one to reconstruct the shape of a quite general object from the knowledge of its Radon transform [S. Helgason, The Radon Transform, or when it defines a quadrature domain in the complex plane [B. Gustafsson, C. He, P. Milanfar, and M. Putinar, Inverse Problems, 16 (2000), pp. 1053-1070], its shape can also be reconstructed from the knowledge of its moments. Essential tools in the solution of the latter inverse problem are quadrature rules and formal orthogonal polynomials.In this paper we show how shape reconstruction from the knowledge of moments can also be realized in the case of general compact objects, not only in two but also in higher dimensions. To this end we use a less-known homogeneous Padé slice property. Again integral transforms-in our case the multivariate Stieltjes transform and univariate Markov transform-formal orthogonal polynomials in the form of Padé denominators, and multidimensional integration formulas or cubature rules play an essential role.We emphasize that the new technique is applicable in all higher dimensions and illustrate it through the reconstruction of several two-and three-dimensional objects.

Region and Contour Identification of Physical Objects

“…we write down the homogeneous accuracy-through-order conditions [11]. When f (t, s, u) in (13) is the characteristic function of the object A, appropriately scaled, and a number of moments c ijk of A is given, then:…”

“…Replacing c i in (4) by C i (cos φ cos θ, cos φ sin θ, sin φ), parameterizes the Hadamard polynomial p (m−1) m (z) by θ and φ. We however don't want to burden the notation for p [11]. When f (t, s, u) in (13) is the characteristic function of the object A, appropriately scaled, and a number of moments c ijk of A is given, then:…”

“…Besides reconstructing three-dimensional objects A the above technique can also be used for the reconstruction of non-polygonal 2D shapes. Several examples of both 2D and 3D reconstructed objects can be found in [11].…”

A new rational approximation technique based on transformational high dimensional model representation

“…They can be used in model-reduction of multidimensional linear shift-invariant recursive systems [3], [23], computing packet loss probabilities in multiplexer models [14] and shape reconstruction [15]. In this paper, we construct a new multivariate rational approximation method and discuss its advantages and disadvantageṡ by comparing with the Padé technique, which is at the moment the leading rational approximation technique.…”

Bounding the support of a measure from its marginal moments

“…But so far, there is no inversion algorithm from moments for ndimensional shapes. However, more recently Cuyt et al [4] have shown how to approximately recover numerically an unknown density f defined on a compact region of R n , from the only knowledge of its moments. So when f is the indicator function of a compact set A ⊂ R n one may thus recover the shape of A with good accuracy, based on moment information only.…”