2019/01/04 by Boris Rubin, Rubin, Boris, Yingzhan Wang +1
Medicine · Mathematics · Engineering · #Medical Imaging Techniques and Applications #Mathematical Analysis and Transform Methods #Advanced Numerical Analysis Techniques
paper · pdf · doi:10.48550/arxiv.1901.01150
We study a Radon-like transform that takes functions on the Grassmannian of j-dimensional affine planes in \Bbb R n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The case j=0 gives the classical Radon-John k-plane transform. For any j and k, our transform has a mixed structure combining the k-plane transform and the dual j-plane transform. The main results include action of such transforms on rotation invariant functions, sharp existence conditions, intertwining properties, connection with Riesz potentials and inversion formulas in a large class of functions. The consideration is inspired by the previous works of F. Gonzalez and S. Helgason who studied the case j+k=n-1, n odd, on smooth compactly supported functions.