Methods of integral geometry and recovering a function with compact support from its projections in unknown directions
The author discusses a method for recovering a function \(d(x_ 1,x_ 2)\) with compact support describing the distribution density of the matter of a planar particle from given projections in unknown directions: A planar asymmetric particle is rotated by unknown angles \(\phi_ 1,...,\phi_ n\) around the origin as its center of mass and projected onto the line \(x_ 2=0\). The problem is to find the relative angles \(\phi_ k-\phi_ 1\), \(k=2(1)n\). Let be \(d_{\phi}(\vec x):=d(X_{\phi}\vec x)\) with the rotation matrix \(D_{\phi}\) and \(M_ k(\phi):\int d_{\phi}(x_ 1)x_ 1^ kdx_ 1\) the kth moment. All arguments of recovering the angles (n\(\geq 7)\) are based on the fact that \(M_ k(\phi)\) is a homogeneous trigonometric polynomial of degree k determinated by its values at \((k+1)\) points. So one can recover the (s,t)th moments of \(d(x_ 1,x_ 2)\) with \(s+t\leq k\) and approximate \(d(x_ 1,x_ 2)\) by a polynomial. The result may be applied in tomography.
scientific article; zbMATH DE number 4067569