Incremental Fourier interpolation of 2-D fractional Brownian motion

This paper presents a new method to interpolate two-dimensional fractional Brownian motion (fBm), fBm interpolation can be used in multimedia applications such as landscape synthesis or zooming into a synthetic scene, where the objective is to generate an fBm field that passes through a sparse set of known points. The fBm interpolation problem differs from standard image interpolation because noise must be added to the interpolated points to obtain an interpolated image with the proper second-order statistics. Our interpolation method is based on the first-order increments of both the original fBm and interpolated fBm. These increments are stationary and yield interpolation equations with a Toeplitz-block-Toeplitz structure which can be approximated by a circulant-block-circulant matrix. By taking advantage of fast Fourier transform, the computational complexity is O(N²log₂N) for N×N image interpolation. Simulation shows this method achieves good second-order statistics, even for small-size images