Efficient Convolution Kernels for Computerized Tomography

Three concepts are presented: 1) Extended kernels: The Ramachandran-Lakshminarayanan convolution kernel has one zero between each non-zero value in the spatial domain. By extending the kernel in Fourier space, it is shown that each extension leads to two additional zeros in the spatial domain of the convolution kernel, thus decreasing the required number of multiplications necessary for convolution. Any known kernel can be extended and in the limit of extension a simple backprojection reconstruction is obtained. 2) Binary kernels: A technique for generating binary approximations to any convolution kernel is described. Excluding the central element, all other elements of the kernel are approximated by an even power of two thus, multiplications are replaced by shift operations in the convolution procedure. 3) Recursive convolution: It is shown how additions can be saved by using a recursive formulation which generates new elements in the convolution procedure utilizing only a few summation steps. Results from both extended kernels and their binary approximations are described for simulated phantoms and ultrasound data obtained from breast scans of patients.