A uniform reconstruction formula in integral geometry

A new method for analytic inversion of Radon type integral transforms is proposed. Key words: Regular hypersurface family, Funk-Radon transform, Principal value integral, Reconstruction, Hyperbolic algebraic curve MSC 53C65 44A12 65R10 1 Introduction We present a uniform reconstruction method for a class of geometric integral transforms for submanifolds of codimension 1. The reconstruction does not include summation of an in…nite series and looks like a standard inversion of the Radon transform. We specify this method for classical and new acquisition geometries. The condition of regularity is necessary for an inversion operator to be bounded in a Sobolev space scale, but it is not su¢ cient. Existence of an exact reconstruction formula depends on vanishing of some singular integrals of rational forms on a sphere. In §8 we discuss reconstruction for families of spheres. This subject is in focus of recent research, see surveys of related results in [13],[12],[15]. 2 Geometry and integrals Let X and be smooth n dimensional manifolds where n > 1; let Z be a smooth closed hypersurface in X and p : Z ! X; : Z ! be natural projections. We suppose that there exists a real smooth function in X (called generating function) such that Z = f(x; ) ; (x; ) = 0g and dx 6= 0 on Z. Suppose that (i) The map has rank n and the mapping P : N (Z)! T (X) is a local di¤eomorphism. Here, N (Z) denotes the conormal bundle of Z and P (x; ; x; ) = (x; x) 2 T (X) : It follows that the set Z ( ) = 1 ( ) = fx; (x; ) = 0g is for any 2 a smooth hypersurface in X; and for any point x 2 X and for any tangent hyperplane h Tx (X) there is a locally unique hypersurface Z ( ) through x tangent to h: 1 Proposition 2.1 For an arbitrary generating function property (i) is equivalent to the condition det (dx;td ; ) 6= 0 where (x; t; ; ) = t (x; ) ; t; 2 R; t > 0 for any local coordinate system x1; :::; xn in X and any local coordinate system 1; :::; n in : For a proof see [18], Proposition 1.1. De…nition. We call a generating function regular if it satis…es conditions (i) and (ii) there are no conjugate points, that is the equations (x; ) = (y; ) and d (x; ) = d (y; ) are ful…lled for no x 6= y 2 X; 2 . We assume further that X is an open set in an Euclidean space E; let dV be the volume form and dS be a hypersurface element in E. Consider the integral M f ( ) = Z ( (x; )) fdV = c = Z