Exercise on MaxLik problems By Zdeněk Hradil 1. Radon and inverse Radon transformation

D distribution and our problem is called reconstruc-tion of distribution froӀprojection. Radon transformation The following theorem by Radon shows that image reconstruction from projection is possible: The value of a 2-D function at an arbitrary point is uniquely obtained by the integrals along the lines of all directions passing the point. This theorem guarantees that a 2-D object (equivalent to a transparence distribution) is reconstructed from projections obtained by the rotational scanning shown in the previous section. The Radon transformation shows the relationship be-tween the 2-D object and the projections. Let us con-sider a coordinate system shown in Fig. 2. The func-tion g(s, e) is a projection of f(x, y) on the axis s of e direction. The function g(s, e) is obtained by the inte-gration along the line whose normal vector is in e di-rection. The value g(0, e) is defined that it is obtained by the integration along the line passing the origin of (x, y)-coordinate. Since the points on the line whose normal vector is in e direction and passing the origin of (x, y)-coordinate satisfy! yx = tan (e + /2) = Рcos e sin e, (1) we get! x cos e + y sin e = 0. (2) The integration along the line whose normal vector is in e direction and that passes through the origin of (x, y)-coordinate means the integration of f(x, y) only at the points satisfying Eq. (2), g(0, e) is expressed us-ing the b-function as follows:! g(0, e) = f (x, y) b(x cos e + y sin e)dxd