Flat Integral Transform

Abstract

Let E be a Euclidean space of dimension n. Consider the manifold A (E) of all affine subspaces A ⊂ E. Let dV (A) be the Euclidean volume density in A. The operator

$$f \mapsto Mf\left( A \right) \doteq \leftarrow \int_A {fd} V\left( A \right)\,,\,A \in A\left( E \right)$$

defined for functions f that decrease sufficiently fast at infinity is called a flat integral transform in E. Fix a natural k < n and consider the submanifold A_k(E) of affine subspaces of dimension k. It is an algebraic variety of dimension (k + 1)(n - k). Denote by M _k the restriction of M to A_k (E). If k = n -1 we keep the notation of the Radon transform Rf = M _n-1 f. We have discussed the reconstruction problem for the transform M _n-1 in Chapter 2 and for the operator M ₁ in Chapter 4. The inversion problem for the operator M _k f, k < n - 1 immediately reduces to the case k = n - 1. Indeed, we reconstruct M _k+1 f from M _k f by inversion of the Radon transform in each k + 1-plane in E. On the other hand, the scope of integrals M _k f is redundant for reconstruction off if k < n -1, since dim A_k(E) = (k + 1)(n - k)> n. Therefore there is a large variety of inversion methods for M _k f. To avoid redundancy we state the reconstruction problem as follows: