Multipole-Motivated Reduced-State Estimation

This paper discusses efficient solutions to large-scale two-dimensional estimation problems, using reducedstate methods motivated by the multipole method of mathematical physics. The work is mainly exploratory, building on past efforts in multiscale statistical signal modeling and estimation. We will illustrate applications to the estimation of Markov random field textures, with the motivation and goal the estimation of remotely-sensed fields. 1. I N T R O D U C T I O N The statistical estimation of large, global scale, two-dimensional remote sensing problems and even modestly-sized three-dimensional problems presents tremendous and pertinent challenges: heightened environmental awareness and concerns have led to an explosion in the quantity of remotely-sensed data, much of which contains irregular gaps and nonritationary underlying fields. The origin of the difficulty in producing statistical estimates is simple. Methods such as nested dissection[4, 51 or multiscale estimation[l] are all based on recursive divide-and-conquer: a subset of the random field is found, such that conditioned on this subset the remaining portions of the field can be processed independently. For example, the four quadrants of a first-order Markov random field can be decorrelated by conditioning on the boundary pixels, shown in Figure 1. So whereas a single pixel can decorrelate i;he two halves of a one-dimensional process, a column of pixels is required for a 2D field, and a whole plane of pixels in three dimensions. Thus for an n x n x ... hypercube of voxels in d dimensions, the computational effort to ~i~~~~ 1: ~~~~~l~ sampled boundaries which conditionally decorrelate the four quadrants of a first-order Markov random field. The research of this paper was supported in part by the Natural Science & Engineering Research Council of Canada, and by the Office of Naval Research under Grant N0014-91-J-1004. Figure 2: A reduced-state approximation to Figure ’’ 0-8186-8821-1/98 $10.00