Results 1  10
of
18
Efficient multiscale regularization with applications to the computation of optical flow
 IEEE Trans. Image Process
, 1994
"... AbsfruetA new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence. Standard formulations of this problem require the computationally intensive solution of an elliptic partial d ..."
Abstract

Cited by 98 (33 self)
 Add to MetaCart
AbsfruetA new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence. Standard formulations of this problem require the computationally intensive solution of an elliptic partial differential equation that arises from the often used “smoothness constraint” ’yl”. regularization. The interpretation of the smoothness constraint is utilized as a “fractal prior ” to motivate regularization based on a recently introduced class of multiscale stochastic models. The solution of the new problem formulation is computed with an efficient multiscale algorithm. Experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size. The new approach also has a number of other important advantages. Specifically, multiresolution flow field estimates are available, allowing great flexibility in dealing with the tradeoff between resolution and accuracy. Multiscale error covariance information is also available, which is of considerable use in assessing the accuracy of the estimates. In particular, these error statistics can be used as the basis for a rational procedure for determining the spatiallyvarying optimal reconstruction resolution. Furthermore, if there are compelling reasons to insist upon a standard smoothness constraint, our algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation. Finally, the usefulness of our approach should extend to a wide variety of illposed inverse problems in which variational techniques seeking a “smooth ” solution are generally Used. I.
Modeling and estimation of multiresolution stochastic processes
 IEEE TRANS. ON INFORMATION THEORY
, 1992
"... An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As described, a natural framework for developing such a theory ..."
Abstract

Cited by 94 (17 self)
 Add to MetaCart
An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As described, a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it will be seen how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of “ multiscale stationarity ” in which scale plays the role of a timelike variable. Focus is primarily on the investigation of models on homogenous trees. In particular, the elements of a dynamic system theory on trees are described
Multiscale Representations of Markov Random Fields
 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL 41. NO 12. DECEMBER 1993
, 1993
"... Recently, a framework for multiscale stochastic modeling was introduced based on coarsetofine scalerecursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this pap ..."
Abstract

Cited by 85 (27 self)
 Add to MetaCart
Recently, a framework for multiscale stochastic modeling was introduced based on coarsetofine scalerecursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this paper, we show that this model class is also quite rich. In particular, we describe how 1D Markov processes and 2D Markov random fields (MRF’s) can be represented within this framework. The recursive structure of 1D Markov processes makes them simple to analyze, and generally leads to computationally efficient algorithms for statistical inference. On the other hand, 2D MRF’s are well known to be very difficult to analyze due to their noncausal structure, and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification. In contrast, our multiscale representations are based on scalerecursive models and thus lead naturally to scalerecursive algorithms, which can be substantially more efficient computationally than those associated with MRF models. In 1D, the multiscale representation is a generalization of the midpoint deflection construction of Brownian motion. The representation of 2D MRF’s is based on a further generalization to a “midline ” deflection construction. The exact representations of 2D MRF’s are used to motivate a class of multiscale approximate MRF models based on onedimensional wavelet transforms. We demonstrate the use of these latter models in the context of texture representation and, in particular, we show how they can be used as approximations for or alternatives to wellknown MRF texture models.
Multiresolution optimal interpolation and statistical analysis of TOPEX/POSEIDON satellite altimetry
 IEEE Trans. Geosci. Remote Sensing
, 1995
"... Abstruct A recently developed multiresolution estimation framework offers the possibility of highly efficient statistical analysis, interpolation, and smoothing of extremely large data sets in a multiscale fashion. This framework enjoys a number of advantages not shared by other statisticallybased ..."
Abstract

Cited by 58 (33 self)
 Add to MetaCart
Abstruct A recently developed multiresolution estimation framework offers the possibility of highly efficient statistical analysis, interpolation, and smoothing of extremely large data sets in a multiscale fashion. This framework enjoys a number of advantages not shared by other statisticallybased methods. In particular, the algorithms resulting from this framework have complexity that scales only linearly with problem size, yielding constant complexity load per grid point independent of problem size. Furthermore these algorithms directly provide interpolated estimates at multiple resolutions, accompanying error variance statistics of use in assessing resolutionlaccuracy tradeoffs and in detecting statistically significant anomalies, and maximum likelihood estimates of parameters such as spectral power law coefficients. Moreover, the efficiency of these algorithms is completely insensitive to irregularities in the sampling or spatial distribution of measurements and to heterogeneities in measurement errors or model parameters. For these reasons this approach has the potential of being an effective tool in a variety of remote sensing problems. In this paper, we demonstrate a realization of this potential by applying the multiresolution framework to a problem of considerable current interestthe interpolation and statistical analysis of ocean surface data from the TOPEXPOSEIDON altimeter. I.
Image Processing with Multiscale Stochastic Models
, 1993
"... In this thesis, we develop image processing algorithms and applications for a particular class of multiscale stochastic models. First, we provide background on the model class, including a discussion of its relationship to wavelet transforms and the details of a twosweep algorithm for estimation. A ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
In this thesis, we develop image processing algorithms and applications for a particular class of multiscale stochastic models. First, we provide background on the model class, including a discussion of its relationship to wavelet transforms and the details of a twosweep algorithm for estimation. A multiscale model for the error process associated with this algorithm is derived. Next, we illustrate how the multiscale models can be used in the context of regularizing illposed inverse problems and demonstrate the substantial computational savings that such an approach offers. Several novel features of the approach are developed including a technique for choosing the optimal resolution at which to recover the object of interest. Next, we show that this class of models contains other widely used classes of statistical models including 1D Markov processes and 2D Markov random fields, and we propose a class of multiscale models for approximately representing Gaussian Markov random fields...
Multiscale autoregressive models and wavelets
, 1999
"... The multiscale autoregressive (MAR) framework was introduced to support the development of optimal multiscale statistical signal processing. Its power resides in the fast and flexible algorithms to which it leads. While the MAR framework was originally motivated by wavelets, the link between these t ..."
Abstract

Cited by 24 (4 self)
 Add to MetaCart
The multiscale autoregressive (MAR) framework was introduced to support the development of optimal multiscale statistical signal processing. Its power resides in the fast and flexible algorithms to which it leads. While the MAR framework was originally motivated by wavelets, the link between these two worlds has been previously established only in the simple case of the Haar wavelet. The first contribution of this paper is to provide a unification of the MAR framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem. The second contribution of this paper is to develop waveletbased approximate internal MAR models for stochastic processes. This will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients. Taking advantage of the statistical machinery provided by the MAR framework, we will illustrate the application of our models to samplepath generation and estimation from noisy, irregular, and sparse measurements.
Multiscale Methods for the Segmentation and Reconstruction of Signals and Images
 IEEE Trans. Image Processing
, 2000
"... Abstract—This paper addresses the problem of both segmenting and reconstructing a noisy signal or image. The work is motivated by large problems arising in certain scientific applications, such as medical imaging. Two objectives for a segmentation and denoising algorithm are laid out: it should be c ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
Abstract—This paper addresses the problem of both segmenting and reconstructing a noisy signal or image. The work is motivated by large problems arising in certain scientific applications, such as medical imaging. Two objectives for a segmentation and denoising algorithm are laid out: it should be computationally efficient and capable of generating statistics for the errors in the reconstruction and estimates of the boundary locations. The starting point for the development of a suitable algorithm is a variational approach to segmentation [1]. This paper then develops a precise statistical interpretation of a onedimensional (1D) version of this variational approach to segmentation. The 1D algorithm that arises as a result of this analysis is computationally efficient and capable of generating error statistics. A straightforward extension of this algorithm to two dimensions would incorporate recursive procedures for computing estimates of inhomogeneous Gaussian Markov random fields. Such procedures require an unacceptably large number of operations. To meet the objective of developing a computationally efficient algorithm, the use of recently developed multiscale statistical methods is investigated. This results in the development of an algorithm for segmenting and denoising which is not only computationally efficient but also capable of generating error statistics, as desired. Index Terms—Denoising, multiscale statistical models, segmentation. I.
Multiscale methods for the segmentation of images
 M.S. thesis, Mass. Inst. Technol
, 1996
"... at the ..."
2003), Scalerecursive estimation for multisensor Quantitative Precipitation Forecast verification: A preliminary assessment
 J. Geophys. Res
"... [1] Precipitation is a highly heterogeneous process with considerable natural variability at scales ranging from a few meters to several hundreds of kilometers. This process is monitored with a variety of sensors (e.g., rain gauges, radars, and satellites) which provide direct or indirect measuremen ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
[1] Precipitation is a highly heterogeneous process with considerable natural variability at scales ranging from a few meters to several hundreds of kilometers. This process is monitored with a variety of sensors (e.g., rain gauges, radars, and satellites) which provide direct or indirect measurements of precipitation at different scales. At the same time, physically based models at the storm, regional, continental, and global scales are used to predict precipitation and rely on the observed data for model verification, model initialization, and data assimilation. Because of the tremendous scaledependent variability of precipitation fields, merging or comparing observations at different scales, or comparing model outputs at one scale with observations at one or more different scales, is not straightforward. This study explores the use of a recently developed scalerecursive estimation (SRE) framework for the problem of Quantitative Precipitation Forecast (QPF) verification using observations from multiple sensors. SRE can explicitly account for the multiscale variability of precipitation and the scaledependent uncertainty associated with the model output and the multisensor observations. Special emphasis is placed on the specification of the multiscale structure of precipitation under sparse or
Graph structure and recursive estimation of noisy linear relations
 J. Math. Syst., Est., and Contr
, 1912
"... This paper examines estimation problems speci ed by noisy linear relations describing either dynamical models or measurements. Each such problem has a graph structure, which can be exploited to derive recursive estimation algorithms only when the graph is acyclic, i.e., when it is obtained by combin ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
This paper examines estimation problems speci ed by noisy linear relations describing either dynamical models or measurements. Each such problem has a graph structure, which can be exploited to derive recursive estimation algorithms only when the graph is acyclic, i.e., when it is obtained by combining disjoint trees. Aggregation techniques appropriate for reducing an arbitrary graph to an acyclic one are presented. The recursive maximum likelihood estimation procedures that we present are based on two elementary operations, called reduction and extraction, which are used to compress successive observations, and discard unneeded variables. These elementary operations are used to derive ltering and smoothing formulas applicable to both linear and arbitrary trees, which are, in turn applicable to estimation problems in settings ranging from 1D descriptor systems to 2D di erence equations to multiscale statistical models of random elds. These algorithms can be viewed as direct generalizations to a far richer setting of Kalman ltering and both two lter and RauchTungStriebel smoothing for standard causal state space models.