1676 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 11, NOVEMBER 2001 Multiply-Rooted Multiscale Models for Large-Scale Estimation Paul W. Fieguth, Member, IEEE Abstract—Divide-and-conquer or multiscale techniques have become popular for solving large statistical estimation prob- lems. The methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems, each more straightforward than the original. However this step cannot be carried out for asymptotically large problems since the di- mension of the state grows without bound, leading to problems of computational complexity and numerical stability. In this paper, we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided, and the problem is instead addressed piece-by-piece. The approach possesses promising attributes: it is not a local method—the estimate at every point is based on all measurements; it is numerically stable for problems of arbitrary size; and the approach retains the benefits of the multiscale framework on which it is based: a broad class of statistical models, a stochastic realization theory, an algorithm to calculate statistical likelihoods, and the ability to fuse local and nonlocal measurements. Index Terms—Estimation, interpolation, multiscale methods, re- mote sensing. I. INTRODUCTION T HE statistical estimation of large, global scale, two-dimen- sional (2-D) remote sensing problems and even modestly sized three-dimensional (3-D) problems presents tremendous and pertinent challenges: heightened environmental awareness and concerns have led to an explosion in the quantity of re- motely sensed data, most of which contain irregular gaps and are governed by nonstationary underlying fields [24], [27]. That is, we are interested in extremely large estimation problems having nonstationary prior models. Several approaches we dismiss out of hand: brute-force, re- lying on full matrix inversion, is totally impractical for all but the most modestly sized problems; FFT methods offer an ex- cellent strategy for perfectly stationary problems, however such stationarity is rare in remotely sensed measurements; and local methods compute estimates from a local subset of measure- ments, which does not support data fusion with nonlocal mea- surements, and which may be undesirable for processes having long correlation lengths [2]. Instead, we consider a promising alternative approach to estimation which involves a recursive or hierarchical di- Manuscript received May 20, 1999; revised August 22, 2001. This work was supported in part by the Office of Naval Research under Grant N00019-91-J-1004 and by the Natural Sciences and Engineering Research Council of Canada. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Kannan Ramchandran. The author is with the Department of Systems Design Engineering, University of Waterloo, ON, N2L-3G1, Canada (e-mail: pfieguth@uwaterloo.ca). Publisher Item Identifier S 1057-7149(01)09571-9. Fig. 1. Dense set of boundary pixels, which would conditionally decorrelate the four quadrants of a first-order Markov random field. vide-and-conquer strategy [7], [14], [15]: a subset of the random field is found such that conditioning on it, certain remaining portions of the field can be processed independently. In the context of estimation, this implies that the subset conditionally decorrelates the remaining portions : (1) For example, the four quadrants of a first-order Markov random field [5], [6] can be decorrelated by conditioning on the boundary pixels shown in Fig. 1. More generally, for first-order fields, a single pixel can decorrelate two halves of a one-dimensional (1-D) process, a column of pixels is required for a 2-D field, and a whole plane of pixels in three dimensions. This process of conditional decorrelation can be continued recursively, which gives rise to a tree structure (as sketched in Fig. 5 for a 2-D process). However, the need to conditionally decorrelate a large problem into separate pieces is also the fundamental weakness of divide-and-conquer strategies: the conditional decorrelation step is not possible for problems of arbitrary size (Fig. 4), due to issues of computational complexity and numerical stability. We emphasize that these issues will arise for all (except pathologically trivial) prior statistical models: the statistical degrees of freedom represented by the boundary must be at least proportional to the number of pixels, . Therefore the dimensional of the decorrelating state will grow as , and so matrix conditioning and computational 1057–7149/01$10.00 © 2001 IEEE