A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential Radon transform using natural pixels

A new algorithm to correct for constant attenuation in SPECT is derived from the singular value decomposition (SVD) of the exponential Radon transform. The new algorithm is based on the assumption that a continuous image can be obtained by backprojecting the discrete array q which is the least squares solution to Mq=p, where p is the array of discrete measurements and the matrix nil represents the composite operator of the backprojection operator A_μ* followed by the projection operator A_μ. In the authors´ implementation, a singular value decomposition of M is used to solve the equation Mq=p, and the final image is obtained by sampling the backprojection of the solution q at a discrete array of points. Analytical expressions are given to calculate the matrix elements of M which are integrals of exponential factors over the overlapped area of two projection strip functions (natural pixels). An eigen analysis of the exponential Radon transform is compared with that of the Radon transform. The condition number of the spectrum increases with increased attenuation coefficient which correlates with the increase in statistical error propagation seen in clinical images obtained with low energy radionuclides. Computer simulations show an improvement in the SVD method over the convolution backprojection method when the projection data is corrupted with noise