@article {488, title = {Fractal estimation using models on multiscale trees}, journal = {IEEE Transactions on Signal Processing}, volume = {44}, year = {1996}, pages = {1297 - 1300}, abstract = {

We estimate the Hurst parameter H of fractional Brownian motion (or, by extension, the fractal exponent phi; of stochastic processes having 1/f phi;-like spectra) by applying a multiresolution framework. This framework admits an efficient likelihood function evaluation, allowing us to compute the maximum likelihood estimate of this fractal parameter with relative ease. In addition to yielding results that compare well with other proposed methods, and in contrast with other approaches, our method is directly applicable with, at most, very simple modification in a variety of other contexts including fractal estimation given irregularly sampled data or nonstationary measurement noise and the estimation of fractal parameters for 2-D random fields

}, keywords = {2-D random fields, Brownian motion, fractal estimation, fractal parameter, fractals, fractional Brownian motion, Hurst parameter, irregularly sampled data, likelihood function evaluation, maximum likelihood estimate, maximum likelihood estimation, multiresolution framework, nonstationary measurement noise, random noise, random processes, signal resolution, signal sampling, stochastic processes, trees (mathematics)}, issn = {1053-587X}, doi = {http://dx.doi.org/10.1109/78.502347}, author = {P Fieguth and A S. Willsky} } @inproceedings {453, title = {Multiresolution model development for overlapping trees via canonical correlation analysis}, booktitle = {International Conference on Image Processing}, year = {1995}, abstract = {

A class of multiscale stochastic models has been introduced in which Gaussian random processes are described by scale-recursive dynamics that are indexed by the nodes of a tree. One of the primary reasons the framework is useful is that it leads to an extremely fast, statistically optimal algorithm for least-squares estimation in the context of 2-D images. We refine this approach to estimation by eliminating the visually distracting blockiness that has been observed in the previous work. We eliminate the blockiness by discarding the standard assumption that distinct nodes at a given level of our tree correspond to disjoint portions of the image domain; as a consequence of this simple idea, a given image pixel may now correspond to several tree nodes. We develop tools for systematically building overlapping-tree multiscale representations of prespecified statistics, and we develop a corresponding estimation algorithm for this processes. In this way, we achieve nearly optimal estimation results, we generate corresponding error covariance information, and we eliminate blockiness without sacrificing the resolution of fine-scale detail

}, keywords = {canonical correlation analysis, correlation methods, covariance analysis, error analysis, error covariance information, estimation algorithm, estimation theory, fast statistically optimal algorithm, Gaussian processes, Gaussian random processes, image domain, image pixel, image representation, image resolution, multiresolution model, multiscale representations, multiscale stochastic models, optimal estimation results, overlapping trees, random processes, scale recursive dynamics, statistics, tree nodes, trees (mathematics)}, doi = {http://dx.doi.org/10.1109/ICIP.1995.529035}, author = {P Fieguth and W W. Irving and A S. Willsky} } @inproceedings {459, title = {Multiresolution stochastic imaging of satellite oceanographic altimetric data}, booktitle = {Image Processing, 1994. Proceedings. ICIP-94., IEEE International Conference}, year = {1994}, abstract = {

Large data assimilation problems present a number of challenges, many of which the authors\&$\#$39; multiscale estimation framework is capable of addressing. They demonstrate the application of their estimation framework to an oceanographic data assimilation problem of considerable interest: the smoothing of Topex/Poseidon ocean altimetric data. they produce surface estimates and error statistics on a 512 times;512 grid in one minute on a Sparc-10. Using a generic surface reconstruction problem they demonstrate alternative uses of the multiscale framework: higher order state models, combining estimates from an ensemble of trees, and overlapping tree models

}, keywords = {error statistics, geophysical signal processing, higher order state models, image reconstruction, large data assimilation problems, multiresolution stochastic imaging, multiscale estimation framework, multiscale framework, oceanographic techniques, overlapping tree models, radar altimetry, radar imaging, remote sensing by radar, satellite oceanographic altimetric data, smoothing methods, state estimation, stochastic processes, surface estimates, surface reconstruction problem, Topex/Poseidon ocean altimetric data, trees, trees (mathematics)}, doi = {http://dx.doi.org/10.1109/ICIP.1994.413519}, author = {P Fieguth and A S. Willsky and W C. Karl} } @inproceedings {460, title = {Multiresolution stochastic imaging of satellite oceanographic altimetric data}, booktitle = {Image Processing, 1994. Proceedings. ICIP-94., IEEE International Conference}, year = {1994}, abstract = {

Large data assimilation problems present a number of challenges, many of which the authors\&$\#$39; multiscale estimation framework is capable of addressing. They demonstrate the application of their estimation framework to an oceanographic data assimilation problem of considerable interest: the smoothing of Topex/Poseidon ocean altimetric data. they produce surface estimates and error statistics on a 512 times;512 grid in one minute on a Sparc-10. Using a generic surface reconstruction problem they demonstrate alternative uses of the multiscale framework: higher order state models, combining estimates from an ensemble of trees, and overlapping tree models

}, keywords = {error statistics, geophysical signal processing, higher order state models, image reconstruction, large data assimilation problems, multiresolution stochastic imaging, multiscale estimation framework, multiscale framework, oceanographic techniques, overlapping tree models, radar altimetry, radar imaging, remote sensing by radar, satellite oceanographic altimetric data, smoothing methods, state estimation, stochastic processes, surface estimates, surface reconstruction problem, Topex/Poseidon ocean altimetric data, trees, trees (mathematics)}, doi = {http://dx.doi.org/10.1109/ICIP.1994.413519}, author = {P Fieguth and A S. Willsky and W C. Karl} }