3D Image Reconstruction from Compton camera data

In this paper, we address analytically and numerically the inversion of the integral transform (cone or Compton transform) that maps a function on R to its integrals over conical surfaces. It arises in a variety of imaging techniques, e.g. in astronomy, optical imaging, and homeland security imaging, especially when the so called Compton cameras are involved. Several inversion formulas are developed and implemented numerically in 3D (the much simpler 2D case was considered in a previous publication). Introduction In this paper, we address analytic and numerical aspects of inversion of the integral transform that maps a function on R to its integral over conical surfaces (with main concentration on the 3D case, while the much simpler 2D case was treated in [37]). It arises in a variety of imaging techniques, e.g. in optical imaging [12], but most prominently when the so called Compton cameras are used, e.g. in astronomy, SPECT medical imaging [11,33], as well as in homeland security imaging [1,2,19,39]. We will call it cone or Compton transform (in 2D, the names V-line transform and broken ray transform are also used), ∗Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA, e-mail: [email protected] †Same department, e-mail: [email protected]