Recovering the moments of a function from its radon-transform projections : necessary and sufficient conditions

The question we wish to address in this paper is the following: To what extent does a limited set of noisefree Radon-Transform projections of a function f(z, y) determine this function? This question has been dealt with in the mathematical literature to some extent. Noteworthy are results due to Volcic [4, 9], Fishburn et al [10], Falconer (3], Gardner (5, 11], Kuba [8] and other references contained therein. These results almost exclusively deal with the case when the function f(z, y) is an indicator function over its domain of definition 0. There has also been some effort in the physics and engineering literature to answer this question. Some noteworthy examples are [17, 181. In this paper, we will prove that one may uniquely recover the first p geometric moments [131 of a bounded, positive function f(z, y), with compact support, from a fixed number p of Radon-Transform [6] projections. We further show that one can not uniquely recover any higher order moments of f(z, y) from such limited information. The importance of this result lies in the fact that it directly shows to what extent a limited number of projections of a function determine the function. This, in essence, is a precise notion of the geometric complexity that a limited set of projections can support. In applications of this result to tomographic reconstruction problems such as in Medical Imaging [1], our result shows, in a quantitative way, how well one can theoretically expect to reconstruct an object being imaged in the absence of noise. From a more abstract viewpoint, it sheds some light on the reconstructability of certain elementary binary objects from a limited number of tomographic projections. 1 Background and Notation The Radon-Transform of a function f(z, y) defined over a compact domain of the plane O is defined by g(t, 9) = J f.(, y)6(t * [Z, y,)dzdy. (1) For every fixed t and 0, g(t, 0) is simply the line-integral of f over 0 in the direction w = [cos(9), sin(9)]T, where 6(t [cos(G), sin(9)] . [z, y]T) is a delta function on a line at angle 0 + (r/2) from the z-axis, and distance t from the origin. This work was supported by the National Science Foundation under Grant 9015281-MIP, the Office of Naval Research under Grant N00014-91-J-1004, the US Army Research Office under Contract DAAL0386-K-0171, and the Clement Vaturi Fellowship in Biomedical Imaging Sciences at MIT. -------· ------·--~~1 Not all functions g(t, 9), however, are Radon Transforms of some f(z, y). Several well-known mathematical properties of the Radon transform known as the consistency relations specify valid 2-D Radon transforms. The Radon transform of some function f(2, y) is constrained to lie in a particular functional subspace of the space of all real-valued functions g(t, ) : 7Z x S1 -7, where S1 is the unit circle. This subspace is characterized by the fact that g must be an even function of t, and that certain coefficients of the Fourier expansion of g must be zero [6, 7]. Let w = [cos(9), sin(0)] denote a unit direction vector, and x = [z, y] a vector in 7Z2 . Using the definition of the Radon Transform we can write g(t, w)tkdt = J tkA f(x)6(t w. x)dxdt = f(x)(w * x)dx (2) -so f fXE7Zg XERx The above identity clearly holds in higher dimensions as well. For our purposes, however, we shall only deal with 1Z2. Assuming that the support, 0, of f(z, y) is contained within the unit disk, we can write the identity (2) as 1 k H(k)(9) -J g(t,O)thkdt = ( ) cosk(0) sini (0)P Ijj (3) j=O where the right hand side is obtained by expanding the term (w * x)k = (cos()2z + sin(9)y)k according to the binomial theorem, and where /kjj are the geometric moments of f(z, y) defined as follows. Pq = X,] f(xy) Pyqdzdy (4) The identity (3) was apparently first discovered by I.M. Gelfand and M.I. Graev in 1961 [19]. Note that the left hand side of (3) is simply the kth order moment of the projection function g(t, 9) for a fixed angle 9, denoted by H(h)(0). Defining the vector of kth order moments of f as (k) = [/k,o, ik1,1 , ... , I O,k] ', (5)