THE INTERIOR RADON-TRANSFORM

The interior Radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured. This problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the X-ray dose. The corresponding operator R(I) is investigated as a mapping between weighted L2-spaces. The main result is a singular value decomposition (SVD) for this operator for functions of unbounded support in R2. The proof is based on the construction of intertwining differential operators. The techniques used are unified in the sense that SVDs for other Radon transforms with rotational symmetry can easily be derived in the same way. Consistency conditions are also obtained for R(I) for functions of bounded and unbounded support. Many of the results generalize to higher dimensions; they are stated whenever they follow directly from the two-dimensional case.