Radon Transform Second Edition

Abstract

It was proved by J. Radon in 1917 that a di erentiable function on R3 can be determined explicitly by means of its integrals over the planes in R3. Let J(!; p) denote the integral of f over the hyperplane hx; !i = p, ! denoting a unit vector and h ; i the inner product. Then f(x) = 1 82Lx Z S2 J(!; h!; xi) d! ; where L is the Laplacian on R3 and d! the area element on the sphere S2 (cf. Theorem 3.1). We now observe that the formula above has built in a remarkable duality: rst one integrates over the set of points in a hyperplane, then one integrates over the set of hyperplanes passing through a given point. This suggests considering the transforms f ! b f; ' ! ' de ned below. The formula has another interesting feature. For a xed ! the integrand x ! J(!; h!; xi) is a plane wave, that is a function constant on each plane perpendicular to !. Ignoring the Laplacian the formula gives a continuous decomposition of f into plane waves. Since a plane wave amounts to a function of just one variable (along the normal to the planes) this decomposition can sometimes reduce a problem for R3 to a similar problem for R. This principle has been particularly useful in the theory of partial di erential equations. The analog of the formula above for the line integrals is of importance in radiography where the objective is the description of a density function by means of certain line integrals. In this chapter we discuss relationships between a function on Rn and its integrals over k-dimensional planes in Rn. The case k = n 1 will be the one of primary interest. We shall occasionally use some facts about Fourier transforms and distributions. This material will be developed in sucient detail in Chapter V so the treatment should be self-contained. Following Schwartz [1966] we denote by E(Rn) and D(Rn), respectively, the space of complex-valued C1 functions (respectively C1 functions of compact support) on Rn. The space S(Rn) of rapidly decreasing functions on Rn is de ned in connection with (6) below. Cm(Rn) denotes the space of m times continuously di erentiable functions. We write C(Rn) for C0(Rn), the space of continuous function on Rn. For a manifold M, Cm(M) (and C(M)) is de ned similarly and we write D(M) for C1 c (M) and E(M) for C1(M).