Volume 25, number 3 OPTICS COMMUNICATIONS June 1978 IMPLEMENTATION OF THE INVERSE RADON TRANSFORM BY OPTICAL CONVOLUTION M. NISHIMURA *, D. PSALTIS, F. CAIMI and D. CASASENT Carnegie.Mellon University, Department of Electdcal Engineering, P~'ttsburgh, PA 15213, USA Received 28 December 1977 Revised manuscript received 8 March 1978 The implementation of the inverse radon transformed by optical correlation is described. This method described does not require formation of the derivative of the projections and allows the use of a fixed mask that is both real and positive and is dependent only on the geometry of the recording system. Experimental confirmation of the concept is included. 1. Introduction The inverse radon transform [1 ] describes how recorded back projections can be summed to produce an exact reconstruction of the image slice. This is the key operation used in the computer assisted transaxial (CT) X-ray imaging and other systems in which a re- construction is obtained from projections [2-4]. The form of the reconstruction algorithm we use was for- mulated by Vest and others [5-7]. The hypersurface involved is discussed by Gel'Fand et at. [7]. In section 2, we describe the CT geometry and no- tation to be used and the formulation of the inverse radon transform in a form compatible with its optical realization. This derivation differs from our earlier one [8], in that differential operations are not required and a positive reference mask function is used. In sec- tion 3, the optical system used is described and our ex- perimental results presented. 2. Mathematical formulation The operation of transaxial imagery construction from projections is well known [2,4]. We denote * M. Nishimura is an Associate Professor at Maizuru Technical College, Kyoto, Japan. He was a visiting professor at CMU while this work was performed. the absorption coefficient along a slice z' of the input object by la(x,y). Polar coordinates are generally used in which case we describe the input object or desired reconstruction by/a(r, 0). The collected data from which/~(r, 0) is to be obtained are the projections f at different projection angles ~. These projections are recorded along a line x' (normalized from + 1 to - 1) normal to the direction of projection. We denote these projections by f(x', ¢). They consist of the recorded energy absorption along the line x' that is normal to specify projection angles 0m- The exact reconstruction of/a(r, 0) from these pro- jections is given by 2~-2f f a/(x',0) dx' /a(r, 0) = dO _o. ax' r cos(~-0)-x' (1) The x' integral in (1) must be treated as a principal value. By partial integration, (1) can be written with arguments f(x', cp) and h i [r cos(0-0)-x'], thereby eliminating the derivative in (l). Care must be taken in choosing h 1 proportional to the derivative of the principal value of [r cos(0-0)-x'] -1. Several func- tions satisfying this exist [2]. h 1 is usually bipolar with the positive part approximating a delta function and the negative part proportional to [r cos(0-0)-x'] 2. Our present concern is to demonstrate a novel op- tical implementation of the Radon transform. We thus chose h 1 = [r cos(0-0)-x'] 2, neglect the delta func- 301