Inversion of the Kontorovich-Lebedev integral transform

Translated from Matematicheskie Zametki, Vol. 51, No. 5, pp. 27–34, May, 1992.     

Inversion formulas for the continuous wavelet transform

The inversion formula for the continuous wavelet transform is usually considered in the weak sense. In the present note we investigate the norm and a.e. convergence of the inversion formula in L _p and Wiener amalgam spaces. The summability of the inversion formula is also considered.     

Inversion formulas for the short-time Fourier transform

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of ƒ converges to ƒ in Wiener amalgam norms, hence also in the L_p sense for L_p functions, and pointwise almost everywhere.     

Inversion of the wavelet transform using Riemannian sums

We study the approximation of the inverse wavelet transform using Riemannian sums. For a large class of wavelet functions, we show that the Riemannian sums converge to the original function as the sampling density tends to infinity. When the analysis and synthesis wavelets are the same, we also give some necessary conditions for the Riemannian sums to be convergent.     

Inversion of the Radon transform with incomplete data

The Radon transform R(p, θ), θ∈S^n?1, p∈?¹, of a compactly supported function f(x) with support in a ball B_a of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on S^n?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data.     

Inversion of the Weyl integral transform and the radon transform on Rn using generalized wavelets

We invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operatorL , >–1/2. Next, we use the connection between radial classical wavelets onR ⁿ and generalized wavelets associated with the Bessel operatorL( _n–2)/2 to derive new inversion formulas for the Radon transform onR ⁿ ,n2.     

Inversion formula for the windowed Fourier transform, II

In this paper, we study the approximation of the inversion of windowed Fourier transforms using Riemannian sums. We show that for certain window functions, the Riemannian sums are well defined on L ^p (?), 1?<?p?<?∞, and tend to the function to be reconstructed as the sampling density tends to infinity.     

Inversion of the exponential X-ray transform. II: Numerics

The exponential X-ray transform arises in single photon emission computed tomography and is defined on functions on the plane by ??_μf(φ,x) = ∫f (x + tφ)e^μt where μ is a constant. In [MMAS(10), 561–574, 1988], we derived analytical formulae for filters K corresponding to a general point spread function E that can be used to invert the exponential X-ray transform via a filtered backprojection algorithm. Here, we use those formulae to derive expressions suitable for numerical computation of the filters corresponding to a specific family of bandlimited point spread functions and give the results of reconstructions of a mathematical phantom using these filters. Also included is an analogue of the Shepp–Logan ellipse theorem, [IEEE Trans. Nucl. Sci. (21), 21–43, 1974], for the exponential X-ray transform.     

Inversion of the exponential X-ray transform. I: Analysis

The exponential X-ray transform arises in single photon emission computed tomography and is defined on functions on ?ⁿ by , where μ is a constant. Approximate inversion, and inversion formulae of filtered back-projection type are derived for this operator in all dimensions. In particular, explicit formulae are given for convolution kernels (filters) K corresponding to a general point spread function E that can be used to invert the exponential X-ray transform via a filtered back-projection algorithm. The results extend and refine work of Tretiak and Metz¹⁷.     

Inversion and structure formulae for a generalized Laplace transform

, D_v(.) , .     

Inversion of the laplace transform based on the Fourier-Legendre expansion of the original function

A theorem on the convergence rate of the Fourier-Legendre expansion of the original is proved. The optimal value of the parameter μ in the change of variable x = e ^?μt is determined.