The notion of a linogram corresponds to the notion of a sinogram in the conventional representation of projection data for image reconstruction. In the sinogram, points which correspond to rays through a fixed point in the cross section to be reconstructed all fall on a sinusoidal curve. In the linogram, however, these points fall on a straight line. The implementation of a novel image reconstruction method using this property is discussed. The implementation is of order N (2) log N, where N is proportional to the number of pixels on a side of the reconstruction region. It is demonstrated that the performance of the algorithm is superior to that of the filter backprojection method in computational speed on realistic size problems and is equivalent to filtered backprojection in accuracy of reconstruction.