FAST B-SPLINE TRANSFORMS FOR CONTINUOUS IMAGE REPRESENTATION AND INTERPOLATION

This correspondence describes efficient algorithms for the continuous representation of a discrete signal in terms of B-splines (direct B-spline transform), and for interpolative signal reconstruction (indirect B-spline transform) with an expansion factor m. Expressions for the z-transforms of the sampled B-spline functions are determined and a convolution property of these kernels is established. It is shown that both the direct and indirect spline transforms involve linear operators that are translation invariant and are implemented efficiently by linear filtering. Fast computational algorithms based on the recursive implementation of these filters are proposed. A B-spline interpolator can also be characterized in terms of its transfer function and its global impulse response (cardinal spline of order n). The case of the cubic spline is treated in greater detail. The present approach is compared with previous methods that are reexamined from a critical point of view. We conclude that, contrary to the claims of several authors, B-spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner.