#### DMCA

## Multiresolution markov models for signal and image processing (2002)

### Cached

### Download Links

- [sensorweb.mit.edu]
- [ssg.mit.edu]
- [ssg.mit.edu]
- [ssg.mit.edu]
- [www.iro.umontreal.ca]
- [www.iro.umontreal.ca]
- CiteULike

### Other Repositories/Bibliography

Venue: | Proceedings of the IEEE |

Citations: | 153 - 17 self |

### Citations

11978 | Maximum likelihood from incomplete data via the EM algorithm.
- Dempster, Laird, et al.
- 1977
(Show Context)
Citation Context ...dden Markov tree models illustrated in Example 6 and developed in detail in [59], [80], [261], [281], and [282], a very effective approach to parameter estimation involves the use of the EM algorithm =-=[97]-=-. The employment of EM requires the specification of the so-called complete data, which includes not only the actual measured data but also some additional hidden variables, which, if available, make ... |

2730 | Atomic decomposition by basis pursuit, - Chen, Donoho, et al. - 1998 |

2604 | Ten Lectures on Wavelets. - Daubechies - 1992 |

2114 |
Exactly Solved Models in Statistical Mechanics
- Baxter
- 1982
(Show Context)
Citation Context ...pecified in terms of the distribution at the root node and the parent–child transition distributions for every node . Such models have a long history, extending back to studies in statistical physics =-=[26]-=-, dynamic programming [32], artificial intelligence and other investigations of graphical models [7], [89], [128], [169], [267], [294], [295], and signal and image processing [42], [58], [59], [80], [... |

1634 |
Spatial interaction and the statistical analysis of lattice systems.
- BESAG
- 1974
(Show Context)
Citation Context ...ard Markov processes in time, with Markov random fields (MRFs) and with the large class 1398 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 8, AUGUST 2002of Bayes’ nets, belief networks, and graphical models =-=[35]-=-, [36], [89], [108], [123], [128], [143], [168]–[170], [197], [204], [236], [267], [294], [295], [302], [337], [339], [357]. It is the exploitation of this Markovian property that leads to the efficie... |

1388 | The Laplacian Pyramid as a Compact Image Code,”
- Burt, Adelson
- 1983
(Show Context)
Citation Context ...t ranges of spatial or spatio–temporal scales [31], [112], [198], [219], [220], [352], [354]. Studies of large classes of natural imagery also show characteristic variability at multiple scales [46], =-=[47]-=-, [140], [157], [218], [243], [250], [261], [268], [281], [297]–[300], [333], as do mathematical models of self-similar or fractal processes [288] such as fractional Brownian motion (fBm) [30], [83], ... |

1248 | On the statistical analysis of dirty pictures,” - Besag - 1986 |

1083 |
Optimal filtering.
- Anderson, Moore
- 1979
(Show Context)
Citation Context ...on itself as follows: (16) which is nothing more than the generalization of the usual Lyapunov equation for the evolution of the state covariance of temporal state space systems driven by white noise =-=[12]-=-, [174], [182]. Note that this computation directly produces the diagonal blocks of the overall covariance matrix for , and the total complexity of this calculation is . The quadratic dependence on th... |

1038 |
Statistic for Long-Memory Processes
- Beran
- 1994
(Show Context)
Citation Context ...s [46], [47], [140], [157], [218], [243], [250], [261], [268], [281], [297]–[300], [333], as do mathematical models of self-similar or fractal processes [288] such as fractional Brownian motion (fBm) =-=[30]-=-, [83], [116], [232], [313], motivating examinations of the properties of the wavelet transforms of such signals and images [69], [83], [102], [114], [117], [154], [176], [191], [235], [273], [293], [... |

881 | Approximating discrete probability distributions with dependence trees.
- Chow, Liu
- 1968
(Show Context)
Citation Context ...ated to the representation of phenomena at different scales and spatial locations. Nevertheless, it is worth noting that the topic of identifying the structure of the tree has received some attention =-=[64]-=-, [177], [230], [238], [239], [303], mostly in fields other than signal and image processing. Perhaps the best known work in this area is that of Chow and Liu [64]. The idea in this work is that we ar... |

719 |
The computational complexity of probabilistic inference using Bayesian belief networks.
- Cooper
- 1990
(Show Context)
Citation Context ...ze , i.e., that grows exponentially with , and explicit computation of projections of this distribution corresponding to particular marginals or joints has been shown to be NP-Hard for general graphs =-=[76]-=-. However, for an MR process on a tree, represents a generalization of a Markov chain, and computations of marginals at all nodes can be computed by a coarse-to-fine tree recursion generalizing the us... |

684 |
A Multigrid Tutorial.
- Briggs
- 1987
(Show Context)
Citation Context ...ion of large systems of equations [e.g., representing discretizations of partial differential equations (PDEs)]. Multigrid methods WILLSKY: MR MARKOV MODELS FOR SIGNAL AND IMAGE PROCESSING 1397[44], =-=[45]-=-, [109], [190], [319] represent one class of examples in which coarser (and hence computationally simpler) versions of a problem are used to guide (and thus accelerate) the solution of finer versions,... |

676 | Entropy-based algorithms for best basis selection,
- Coifman, Wickerhauser
- 1992
(Show Context)
Citation Context ...els first introduced in Example 1 and discussed further later in this section and in Section VI-B. Finally, there is also a substantial body of work on so-called adaptive representations (e.g., [52], =-=[73]-=-, [192], [193], [228], and [229]) using entire families or “dictionaries” of bases, which taken together generally form vastly overcomplete sets. The objective in each of these methods is to select on... |

541 |
Fast Wavelet Transforms and Numerical Algorithms I,
- Beylkin, Coifman, et al.
- 1991
(Show Context)
Citation Context ...], [256], [280] approximate the effects of distant parts of a random field with coarser aggregate values, providing substantial computational gains for many problems. Similarly, wavelet-based methods =-=[37]-=-, [38], [89], [95], [215], [228], [247], [264], [276], [286], [329], [335], [361] provide potentially significant speed-ups for a variety of computationally intensive problems. (a) B. Our Starting Poi... |

415 | Wavelet-based statistical signal processing using hidden Markov models,”
- Crouse, Nowak, et al.
- 1998
(Show Context)
Citation Context ...ariables may simply play the role of capturing the intrinsic memory in the signals that are observed or of primary interest. The models we describe also have close ties to hidden Markov models (HMMs) =-=[80]-=-, [222], [261], [265], [272], [281], [302], in which the hidden variables may represent higher level descriptors which we wish to estimate, as in speech analysis, image segmentation, and higher level ... |

359 | The Generalized Distributive Law,”
- Aji, McEliece
- 2000
(Show Context)
Citation Context ...for every node . Such models have a long history, extending back to studies in statistical physics [26], dynamic programming [32], artificial intelligence and other investigations of graphical models =-=[7]-=-, [89], [128], [169], [267], [294], [295], and signal and image processing [42], [58], [59], [80], [175], [199], [213], [261], [281], [283]. Later in this paper we will illustrate examples of such mod... |

308 | A multiscale random field model for Bayesian image segmentation.
- Bouman, Shapiro
- 1994
(Show Context)
Citation Context ...[265], [272], [281], [302], in which the hidden variables may represent higher level descriptors which we wish to estimate, as in speech analysis, image segmentation, and higher level vision problems =-=[42]-=-, [53], [59], [175], [179], [180], [183], [199], [283], [323]. Whatever the nature of the variables defined on such a tree, there is one critical property that they must satisfy, namely, that collecti... |

302 | Tractable inference for complex stochastic processes.
- Boyen, Koller
- 1998
(Show Context)
Citation Context ...., processes on grids such as in Fig. 14 in which one of the two independent variables is time. The idea of propagating approximate graphical models in time is a topic of significant current interest =-=[43]-=-, [134], [254], and we refer the readers to these references for details. We note in particular that in [43] the authors confront a problem of considerable concern not only for DBNs but for the approx... |

301 | A generalized Gaussian image model for edge-preserving MAP estimation,”
- Bouman, Sauer
- 1993
(Show Context)
Citation Context ... for example, [132] and [234]), approaches that use non-Gaussian models in order to better capture the “heavy tail” nature of imagery (for example, the generalized Gaussian models studied in depth in =-=[41]-=-) and an array of procedures using wavelet transforms (e.g., [2], [57]–[59], [68], [80], [104], [192], [193], [261], [281], [301], [330], and [333]). For this latter set of methods, the general idea i... |

276 | Multiresolution sampling procedure for analysis and synthesis of texture images.
- Debonet
- 1997
(Show Context)
Citation Context ...omplex graph that does not yield a junction tree or cutset tree model with acceptably small state dimension. Another very interesting approach to MR modeling for image processing is that developed in =-=[90]-=-–[92]. The basic idea behind this approach is quite simple. Given a sample image, we form an MR pyramid by performing an MR decomposition of the image—the specific decomposition used in [91] is an ove... |

262 | Wavelet thresholding via a Bayesian approach.
- Abramovich, Sapatinas, et al.
- 1998
(Show Context)
Citation Context ...models in order to better capture the “heavy tail” nature of imagery (for example, the generalized Gaussian models studied in depth in [41]) and an array of procedures using wavelet transforms (e.g., =-=[2]-=-, [57]–[59], [68], [80], [104], [192], [193], [261], [281], [301], [330], and [333]). For this latter set of methods, the general idea is to exploit the localization properties of wavelets to allow mu... |

251 |
Multilevel adaptive solutions to boundary value problems”.
- Brandt
- 1977
(Show Context)
Citation Context ... solution of large systems of equations [e.g., representing discretizations of partial differential equations (PDEs)]. Multigrid methods WILLSKY: MR MARKOV MODELS FOR SIGNAL AND IMAGE PROCESSING 1397=-=[44]-=-, [45], [109], [190], [319] represent one class of examples in which coarser (and hence computationally simpler) versions of a problem are used to guide (and thus accelerate) the solution of finer ver... |

238 | Image compression via joint statistical characterization in the wavelet domain.
- Buccigrossi, Simoncelli
- 1999
(Show Context)
Citation Context ...er vast ranges of spatial or spatio–temporal scales [31], [112], [198], [219], [220], [352], [354]. Studies of large classes of natural imagery also show characteristic variability at multiple scales =-=[46]-=-, [47], [140], [157], [218], [243], [250], [261], [268], [281], [297]–[300], [333], as do mathematical models of self-similar or fractal processes [288] such as fractional Brownian motion (fBm) [30], ... |

227 | Ill-posed problems in early vision.
- Bertero, Poggio, et al.
- 1988
(Show Context)
Citation Context ...of computer vision, namely that of reconstructing surfaces from regular or irregularly sampled measurements of surface height and/or of the normal to the surface (as in the shape-from-shading problem =-=[34]-=-, [53], [152]). One well-known approach to reconstruction problems such as this involves the use of a variational formulation. In particular, let denote the 2-D planar region over which the surface is... |

224 | Spatially Adaptive Wavelet Thresholding with Context Modeling for Image denoising”,
- Chang, Yu, et al.
- 2000
(Show Context)
Citation Context ... in this paragraph corresponds to performing nonlinear operations on individual wavelet coefficients. Among the algorithms that result from such models are so-called wavelet shrinkage algorithms [2], =-=[49]-=-, [57], [68], [104], [192], [193], [251], [301], [330]. As described in [80], such independent mixture models result in very simple nonlinear operations on individual wavelet coefficients for optimal ... |

206 |
Adaptive Bayesian wavelet shrinkage,
- Chipman, Kolaczyk, et al.
- 1997
(Show Context)
Citation Context ...s in order to better capture the “heavy tail” nature of imagery (for example, the generalized Gaussian models studied in depth in [41]) and an array of procedures using wavelet transforms (e.g., [2], =-=[57]-=-–[59], [68], [80], [104], [192], [193], [261], [281], [301], [330], and [333]). For this latter set of methods, the general idea is to exploit the localization properties of wavelets to allow much eas... |

186 |
Nonserial Dynamic Programming.
- Bertele, Brioschi
- 1972
(Show Context)
Citation Context ...istribution at the root node and the parent–child transition distributions for every node . Such models have a long history, extending back to studies in statistical physics [26], dynamic programming =-=[32]-=-, artificial intelligence and other investigations of graphical models [7], [89], [128], [169], [267], [294], [295], and signal and image processing [42], [58], [59], [80], [175], [199], [213], [261],... |

184 |
Translation invariant denoising,"
- Coifman, Donoho
- 1995
(Show Context)
Citation Context ...e are several approaches to dealing with this, including those described in Section VI and also in Section VII. What was used to produce the results in Fig. 9 is the same simple method used by others =-=[72]-=-, [270], and [298], namely, averaging the estimation results using several different tree models, each of which is shifted slightly with respect to the others, so that the overall average smoothes out... |

176 | Digital Image Restoration. - Andrews, Hunt - 1977 |

149 | Multiple resolution segmentation of textured images,
- Bouman, Liu
- 1991
(Show Context)
Citation Context ...annealing for (d) their solution or leading to suboptimal methods such as iterated conditional mode (ICM) [36]. These problems have led a variety of authors to consider MR algorithms and models [14], =-=[40]-=-, [42], [48], [53], [58], [59], [135], [144], [179], [180]. We describe how some of these methods fall directly into the framework on which we focus and how others relate to it. F. Multisensor Fusion ... |

146 |
Classification of texture using Gaussian Markov Random Fields,
- Chellappa, Chatterjee
- 1985
(Show Context)
Citation Context ...Another problem of importance in computer vision and in other image processing applications is that of texture discrimination. One well-known class of statistical texture models is that based on MRFs =-=[50]-=-, [71], [178], [233]. For example, Fig. 4 shows two synthetic MRF textures, one modeling pigskin and one sand. The problem of discriminating textures such as these given noisy measurements is a standa... |

142 | Multiple shrinkage and subset selection in wavelets,
- Clyde, Parmigiani, et al.
- 1998
(Show Context)
Citation Context ...to better capture the “heavy tail” nature of imagery (for example, the generalized Gaussian models studied in depth in [41]) and an array of procedures using wavelet transforms (e.g., [2], [57]–[59], =-=[68]-=-, [80], [104], [192], [193], [261], [281], [301], [330], and [333]). For this latter set of methods, the general idea is to exploit the localization properties of wavelets to allow much easier and mor... |

136 | Multiscale recursive estimation, data fusion, and regularization,”
- Chou, Willsky, et al.
- 1994
(Show Context)
Citation Context ...lected variables. To be sure, more efficient methods can be devised that exploit the structure of particular graphs, but it is for trees that we obtain especially simple and scalable algorithms [23], =-=[60]-=-, [61], [225], [226] for such computations. In particular, consider the linear model (6), where is a white noise process, with covariance , independent of the state at the root node whose covariance w... |

131 | Texture mixing and texture movie synthesis using statistical learning.
- Bar-Joseph, El-Yaniv, et al.
- 2001
(Show Context)
Citation Context ...vector of coefficients, denoted by , sensitive to variations in different directions at the location and scale corresponding to node . From this one image sample—or perhaps from a small set of images =-=[20]-=-—we wish to learn non-Gaussian, nonlinear, coarse-to-fine statistical dynamics. In particular, what is done in [20] and [90]–[92] is to use nonparametric density estimation methods to estimate both th... |

119 | Wavelet decomposition approaches to statistical inverse problems.
- ABRAMOVITCH, SILVERMAN
- 1998
(Show Context)
Citation Context ...inverse problems, introduced and popularized by Donoho [105], uses a variation, known as the wavelet–vaguelette decomposition (WVD). As shown in [105] and in other references in this area (see, e.g., =-=[3]-=-, [187], and [206]), WVDs can be designed for important classes of inverse problems (including tomography). Such decompositions correspond to using an orthogonal wavelet decomposition for the random f... |

108 | Multiscale image segmentation, using wavelet-domain hidden markov models.
- Choi, Baraniuk
- 2001
(Show Context)
Citation Context ...ion [292], optical flow estimation [10], [223], surface reconstruction [111], texture classification [225], WILLSKY: MR MARKOV MODELS FOR SIGNAL AND IMAGE PROCESSING 1399and image segmentation [42], =-=[58]-=-, [199], [212], [324], to name a few); higher level recognition and vision problems [183], [323]; photon-limited imaging [188], [261], [263], [322]; network traffic modeling [279]; oceanographic, atmo... |

107 | Modeling and estimation of multiresolution stochastic processes - Basseville - 1992 |

105 | Markovian representation of stochastic processes by canonical variables, - Akaike - 1975 |

100 |
Applications of a general propagation algorithm for probabilistic expert systems,
- Dawid
- 1992
(Show Context)
Citation Context ...80] approximate the effects of distant parts of a random field with coarser aggregate values, providing substantial computational gains for many problems. Similarly, wavelet-based methods [37], [38], =-=[89]-=-, [95], [215], [228], [247], [264], [276], [286], [329], [335], [361] provide potentially significant speed-ups for a variety of computationally intensive problems. (a) B. Our Starting Point A key cha... |

99 |
Classifcation of Rotated and Scaled Textured Images using Gaussian Markov Random Field Models,”
- Cohen, Fan, et al.
- 1991
(Show Context)
Citation Context ...r problem of importance in computer vision and in other image processing applications is that of texture discrimination. One well-known class of statistical texture models is that based on MRFs [50], =-=[71]-=-, [178], [233]. For example, Fig. 4 shows two synthetic MRF textures, one modeling pigskin and one sand. The problem of discriminating textures such as these given noisy measurements is a standard hyp... |

88 |
A "Segmentation and Estimation of Image Region Properties Through Cooperative Hierarchical Computation."
- Burt, Hong, et al.
- 1981
(Show Context)
Citation Context ...r (d) their solution or leading to suboptimal methods such as iterated conditional mode (ICM) [36]. These problems have led a variety of authors to consider MR algorithms and models [14], [40], [42], =-=[48]-=-, [53], [58], [59], [135], [144], [179], [180]. We describe how some of these methods fall directly into the framework on which we focus and how others relate to it. F. Multisensor Fusion for Groundwa... |

81 | A Non-parametric MultiScale Statistical Model for Natural Images,”
- Bonet, Viola
- 1998
(Show Context)
Citation Context ...eloped in [90]–[92]. The basic idea behind this approach is quite simple. Given a sample image, we form an MR pyramid by performing an MR decomposition of the image—the specific decomposition used in =-=[91]-=- is an overcomplete steerable pyramid [298]. Thus, at each node , on a quadtree, we have a vector of coefficients, denoted by , sensitive to variations in different directions at the location and scal... |

72 |
Simple parallel hierarchical and relaxation algorithms for segmenting textured images based on noncasual Markovian random field models
- Cohen, Cooper, et al.
- 1984
(Show Context)
Citation Context ...so extensive literature on the use of MRF models32 together with either full multigrid computational algorithms or purely coarse-to-fine algorithmic structures. Examples of the former can be found in =-=[70]-=-, [109], [319], and [356], where the treatment in [319] represents what to the author’s knowledge is the first thorough examination of the application of multigrid methods to image pro32 In some of th... |

70 | Linear inverses and ill-posed problems, in: - Bertero - 1989 |

63 | Multiscale Bayesian segmentation using a trainable context model.
- Cheng, Bouman
- 2001
(Show Context)
Citation Context ... [272], [281], [302], in which the hidden variables may represent higher level descriptors which we wish to estimate, as in speech analysis, image segmentation, and higher level vision problems [42], =-=[53]-=-, [59], [175], [179], [180], [183], [199], [283], [323]. Whatever the nature of the variables defined on such a tree, there is one critical property that they must satisfy, namely, that collectively t... |

57 | Texture recognition using a nonparametric multi-scale statistical model. In: - Debonet, Viola - 1998 |

53 | Segmentation of textured images using a multiresolution Gaussian autoregressive model,”
- Comer, Delp
- 1999
(Show Context)
Citation Context ...ly some advantages to this approach that result from the usual multigrid/coarse-to-fine philosophy of using coarser grids to guide solutions at finer ones. We also refer the reader to other MR models =-=[74]-=-, [125], [127], [212], [213], [289], that involve structures other than trees, together with algorithmic structures that are reminiscent of multigrid and coarse-to-fine procedures. One final point to ... |

52 |
A wavelet-based method for multifractal image analysis: from theoretical concepts to experimental applications
- Arneodo, Decoster, et al.
- 2003
(Show Context)
Citation Context ...ge analysis and fusion [77], [119], [160], [185], [309]; geographic systems [93], [189]; medical image analysis [290]; models of neural responses in human vision [274]; and mathematical physics [15], =-=[94]-=-, [136]. In this section, we introduce several of these applications which serve to provide context, motivation, and illustrations for the development that follows, as well as to indicate the breadth ... |

51 |
Spatially adaptive wavelet-based multiscale image restoration
- Banham, Katsaggelos
- 1996
(Show Context)
Citation Context ...ribe in this paper have been employed in a wide variety of applications, including: low-level computer vision and image processing problems (image denoising [59], [67], [80], [261], [281], deblurring =-=[19]-=-, edge detection [292], optical flow estimation [10], [223], surface reconstruction [111], texture classification [225], WILLSKY: MR MARKOV MODELS FOR SIGNAL AND IMAGE PROCESSING 1399and image segmen... |

50 |
Bayes smoothing algorithms for segmentation of binary images modeled by Markov random fields:
- Derin, Elliott, et al.
- 1984
(Show Context)
Citation Context ...earlier in the scan order. Another example is the class so-called “Markov mesh” models (for both Gaussian and discrete-state processes), which impose a partial order on pixels in 2-D [1], [78], [88], =-=[98]-=-, [213]. However, in many problems, imposing such total or partial orders is clearly artificial. Moreover, often very high-order models of these types are needed to capture accurately the statistics o... |

46 |
Stochastic theory of minimal realization
- Akaike
- 1974
(Show Context)
Citation Context ...e is associated with a discussion in Section VI-B1. nomenon under study and/or capturing more global quantities whose estimation is desired. For example, in analogy with stochastic realization theory =-=[8]-=-, [9], [214] and the concept of state for dynamic systems, such variables may simply play the role of capturing the intrinsic memory in the signals that are observed or of primary interest. The models... |

42 | A wavelet-based method for multiscale tomographic reconstruction.
- Bhatia, WC, et al.
- 1996
(Show Context)
Citation Context ...6], [280] approximate the effects of distant parts of a random field with coarser aggregate values, providing substantial computational gains for many problems. Similarly, wavelet-based methods [37], =-=[38]-=-, [89], [95], [215], [228], [247], [264], [276], [286], [329], [335], [361] provide potentially significant speed-ups for a variety of computationally intensive problems. (a) B. Our Starting Point A k... |

42 |
Multiscale systems, Kalman filters, and Riccati equations.
- Chou, Willsky, et al.
- 1994
(Show Context)
Citation Context ... variables. To be sure, more efficient methods can be devised that exploit the structure of particular graphs, but it is for trees that we obtain especially simple and scalable algorithms [23], [60], =-=[61]-=-, [225], [226] for such computations. In particular, consider the linear model (6), where is a white noise process, with covariance , independent of the state at the root node whose covariance we deno... |

35 |
Multiresolution tomographic reconstruction using wavelets
- Delaney, Bresler
- 1995
(Show Context)
Citation Context ...proximate the effects of distant parts of a random field with coarser aggregate values, providing substantial computational gains for many problems. Similarly, wavelet-based methods [37], [38], [89], =-=[95]-=-, [215], [228], [247], [264], [276], [286], [329], [335], [361] provide potentially significant speed-ups for a variety of computationally intensive problems. (a) B. Our Starting Point A key character... |

34 |
Linear estimation of boundary value stochastic processes, Part II: 1-D smoothing problems
- Adams, B, et al.
- 1984
(Show Context)
Citation Context ...ssentially corresponds to solving (a discretized version of) an elliptic PDE [208] for which extremely fast algorithms (conjugate gradient, multipole, etc. [138], [256], [280]) exist. However, 28 See =-=[4]-=-, [5], [141], [210], [258], and [260] for a methodology applied to time series, [101] for analogous results for MR trees, and [337] for results for discrete-state graphical models on single loops. 141... |

33 | Long-lead prediction of Pacific SST’s via Bayesian dynamic modeling,
- Berliner, Wikle, et al.
- 2000
(Show Context)
Citation Context ...les or resolutions. For example, many physical processes—e.g., geophysical fields such as atmospheric or oceanographic phenomena—possess behavior over vast ranges of spatial or spatio–temporal scales =-=[31]-=-, [112], [198], [219], [220], [352], [354]. Studies of large classes of natural imagery also show characteristic variability at multiple scales [46], [47], [140], [157], [218], [243], [250], [261], [2... |

30 |
Balanced approximation of stochastic systems.
- Arun, Kung
- 1990
(Show Context)
Citation Context ...ization: In this section, we describe a more general and formal construction of linear MR models [82], [85], [122], [161]. The approach makes use of concepts adapted from state space theory [8], [9], =-=[16]-=-, [214]; however, the adaptation to trees uncovers some important differences with the temporal case. First, in contrast to the usual temporal state space framework—and, for that matter, to the framew... |

27 |
Random cascades on wavelet dyadic trees,”
- Arneodo, Bacry, et al.
- 1998
(Show Context)
Citation Context ...ar image analysis and fusion [77], [119], [160], [185], [309]; geographic systems [93], [189]; medical image analysis [290]; models of neural responses in human vision [274]; and mathematical physics =-=[15]-=-, [94], [136]. In this section, we introduce several of these applications which serve to provide context, motivation, and illustrations for the development that follows, as well as to indicate the br... |

27 |
Image analysis with partially ordered Markov models.
- Cressie, Davidson
- 1998
(Show Context)
Citation Context ...e that come earlier in the scan order. Another example is the class so-called “Markov mesh” models (for both Gaussian and discrete-state processes), which impose a partial order on pixels in 2-D [1], =-=[78]-=-, [88], [98], [213]. However, in many problems, imposing such total or partial orders is clearly artificial. Moreover, often very high-order models of these types are needed to capture accurately the ... |

27 | A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing - Daniel, Willsky - 1996 |

26 | Multiscale autoregressive models and wavelets
- Daoudi, Frakt, et al.
- 1999
(Show Context)
Citation Context ...[62], [100], [151]. Second, one can do better than this in both modeling and estimation by taking any residual correlation into account. Indeed, several authors have considered methods for doing this =-=[85]-=-, [146], [275], [358], and (44) suggests a very simple method of this type, similar to an approach described in Section VI-B. In particular, suppose that the objective is to construct a model as in (4... |

25 | Sequential Filtering for Multi-frame Visual Reconstruction,” - Chin, Karl, et al. - 1992 |

24 |
Willsky "Multiscale Autoregressive Processes, Part II: Lattice Structures for Whitening and Modeling", submitted to
- Basseville, Benveniste, et al.
(Show Context)
Citation Context ...e significant differences—with state space modeling for time series. For example, we refer the reader to [29] for the development of a state space theory for deterministic MR dynamics on trees and to =-=[24]-=-, [25], [65], and [66] in which MR counterparts to autoregressive modeling and efficient algorithms analogous to Levinson’s algorithm for time series are developed for the class of isotropic processes... |

22 |
Wavelet representations of stochastic processes and multiresolution stochastic models
- Dijkerman, Mazumdar
- 1994
(Show Context)
Citation Context ... is raised. Much of the reason for this stems either from analyses that demonstrate that wavelet transforms provide substantial decorrelation of important classes of processes such as fBm [69], [83], =-=[100]-=-, [102], [114], [117], [137], [146], [154], [176], [191], [235], [273], [293], [320], [346], [360], or from constructions of processes using wavelet synthesis [62], [80], [83], [100], [114], [151], [2... |

22 |
Convergence of a reconstruction method for the inverse conductivity problem
- Dobson
- 1992
(Show Context)
Citation Context ...truction in order to develop deterministic inversion algorithms that are either very efficient or that allow fast high-resolution reconstructions of localized regions. Also, a number of authors [39], =-=[103]-=-, [107], [246]–[248], [305], [335], [362] have used both the decorrelation and sparsification properties of orthogonal wavelet decompositions in order to develop statistical reconstruction algorithms ... |

19 | Probabilistic and sequential computation of optical ow using temporal coherence - Chin, Karl, et al. - 1994 |

19 |
On the correlation structure of the wavelet coefficients of fractionally Brownian motion.
- Dijkerman, Mazumdar
- 1994
(Show Context)
Citation Context ...al processes [288] such as fractional Brownian motion (fBm) [30], [83], [116], [232], [313], motivating examinations of the properties of the wavelet transforms of such signals and images [69], [83], =-=[102]-=-, [114], [117], [154], [176], [191], [235], [273], [293], [320], [346]–[350], [359]. Second, whether the phenomenon displays MR behavior or not, it may be the case that the available data are at multi... |

18 | Multiscale System Theory
- Benveniste, Nikoukhah, et al.
- 1990
(Show Context)
Citation Context ...ith the graph structure fixed the problem of MR modeling bears a number of similarities—and some significant differences—with state space modeling for time series. For example, we refer the reader to =-=[29]-=- for the development of a state space theory for deterministic MR dynamics on trees and to [24], [25], [65], and [66] in which MR counterparts to autoregressive modeling and efficient algorithms analo... |

17 | The modeling and estimation of statistically self-similar processes in a multiresolution framework - Daniel, Willsky - 1999 |

16 | Multiscale classification using complex wavelets and hidden Markov tree models
- Romberg, Choi, et al.
(Show Context)
Citation Context ..., [281], [302], in which the hidden variables may represent higher level descriptors which we wish to estimate, as in speech analysis, image segmentation, and higher level vision problems [42], [53], =-=[59]-=-, [175], [179], [180], [183], [199], [283], [323]. Whatever the nature of the variables defined on such a tree, there is one critical property that they must satisfy, namely, that collectively they de... |

16 |
Estimating Gaussian Markov random field parameters in a non-stationary framework
- Descombes, Sigelle, et al.
- 1998
(Show Context)
Citation Context ...00], and [269], in all but a special set of circumstances, coarser scale fields resulting from other coarsening procedures such as subsampling [195], [200] and so-called renormalization group methods =-=[99]-=-, [131], [135], [257] do not have such simple exact descriptions—and indeed may correspond to graphical models with fully connected graphs. In such cases, what are generally used at coarser scales are... |

15 |
A stochastic realization approach to the smoothing problem
- Badawi, Lindquist, et al.
- 1979
(Show Context)
Citation Context ... in this and subsequent sections), this is a major benefit of the use of MR tree models. The second implication of the tree structure of , which directly generalizes known results for temporal models =-=[17]-=-, [27], [28], [226], is that this implies that the error process is an MR process on the same tree, with parameters (i.e., matrices analogous to and for the original process) that are automatically av... |

15 |
Texture synthesis and pattern recognition for partially ordered Markov models.
- Davidson, Cressie, et al.
- 1999
(Show Context)
Citation Context ... come earlier in the scan order. Another example is the class so-called “Markov mesh” models (for both Gaussian and discrete-state processes), which impose a partial order on pixels in 2-D [1], [78], =-=[88]-=-, [98], [213]. However, in many problems, imposing such total or partial orders is clearly artificial. Moreover, often very high-order models of these types are needed to capture accurately the statis... |

14 | Image data compression with the Laplacian pyramid. In
- Adelson, Burt
- 1981
(Show Context)
Citation Context ...er scale variables in such a case might simply represent decompositions of the finest scale variables into coarser scale components, e.g., as in the use of wavelet decompositions or Laplacian pyramid =-=[6]-=-, [47] representations of images. In other problems, some of these coarser scale variables may be measured directly, as occurs in problems in which we wish to fuse data sets collected at differing res... |

13 | On the approximation of free discontinuity problems. Bollettino Della Unione Matematica Italiana - Ambrosio, Tortorelli - 1992 |

13 | Image segmentation in pyramids,” - Antonisse - 1982 |

13 | Multiresolution Stochastic Models, Data Fusion, and Wavelet Transforms
- Chou, Golden, et al.
- 1993
(Show Context)
Citation Context ...f processes such as fBm [69], [83], [100], [102], [114], [117], [137], [146], [154], [176], [191], [235], [273], [293], [320], [346], [360], or from constructions of processes using wavelet synthesis =-=[62]-=-, [80], [83], [100], [114], [151], [275], [346]–[350], [358]. In this section we take a brief look at some of the relationships between wavelets and MR models on trees, a subject we have divided into ... |

11 |
Determinantal formulae for matrix completions with chordal graphs. Linear Algebra and Its Applications
- Barrett, Johnson, et al.
- 1989
(Show Context)
Citation Context ...lar, if is an matrix, consider the undirected graph with nodes labeled , where we include the edge between distinct nodes and if the th element of has been specified. Then, the following results hold =-=[21]-=-, [75], [142]. 1) Given a particular graph of this type, extensions and completions exist for any valid partially specified covariance with this graph structure if and only if the graph is chordal. 57... |

10 |
Special issue on wavelet transforms and multiresolution signal analysis
- Daubechies, Mallat, et al.
- 1992
(Show Context)
Citation Context ...ds for the statistical analysis of phenomena and data have been and remain topics of tremendous interest in a wide variety of disciplines (see, for example, two special issues devoted to this subject =-=[87]-=- and [194] as well as the book [304]). The reasons for the intensity of activity and the dizzying variety of methods that have been developed are myriad, and it is not the intent of this paper to put ... |

10 |
Optimal filterbanks for signal reconstruction from noisy subband components
- Delopoulos, Kollias
- 1996
(Show Context)
Citation Context ...investigation of methods for space–time problems, either in which time is treated in an MR fashion as well (e.g., as is found in so-called multirate Kalman filtering and estimation theory [51], [79], =-=[96]-=-, [150], [151], ) or, as in the methods discussed in Section VII-A, in which time is treated as a sequential variable but space is treated in an MR graphical manner. The results we have presented (and... |

9 | Tomographic reconstruction and estimation based on multi-scale natural pixel bases - Bhatia, Karl, et al. - 1997 |

9 |
Optimal signal reconstruction in noisy filterbanks: Multirate kalman synthesis filtering approach
- Chen, Lin, et al.
- 1995
(Show Context)
Citation Context ...the further investigation of methods for space–time problems, either in which time is treated in an MR fashion as well (e.g., as is found in so-called multirate Kalman filtering and estimation theory =-=[51]-=-, [79], [96], [150], [151], ) or, as in the methods discussed in Section VII-A, in which time is treated as a sequential variable but space is treated in an MR graphical manner. The results we have pr... |

9 |
multi-resolution, recursive Kalman Filtering. Signal Processing,
- Cristi, Multi-rate
- 2000
(Show Context)
Citation Context ...rther investigation of methods for space–time problems, either in which time is treated in an MR fashion as well (e.g., as is found in so-called multirate Kalman filtering and estimation theory [51], =-=[79]-=-, [96], [150], [151], ) or, as in the methods discussed in Section VII-A, in which time is treated as a sequential variable but space is treated in an MR graphical manner. The results we have presente... |

7 |
Least Squares Image Estimations on a Multiresolution Pyramid
- Clippingdale, Wilson
(Show Context)
Citation Context ...G OF APPLICATIONS The methods we describe in this paper have been employed in a wide variety of applications, including: low-level computer vision and image processing problems (image denoising [59], =-=[67]-=-, [80], [261], [281], deblurring [19], edge detection [292], optical flow estimation [10], [223], surface reconstruction [111], texture classification [225], WILLSKY: MR MARKOV MODELS FOR SIGNAL AND I... |

6 |
Critical Graphs for the Positive Definite Completion Problem
- Barrett, Johnson, et al.
- 1998
(Show Context)
Citation Context ...f the structure that is exploited in multipole algorithms [256], [280] for the solution of PDEs. We refer the reader to [115] for a first attempt to adapt multipole ideas to MR estimation and also to =-=[22]-=-, [167], and [325] for some results on MR modeling and covariance extension on graphs with cycles. Other directions for further research can be found in virtually every corner of this paper. One is th... |

6 |
Construction and applications of discretetime smoothing error models
- Bello, Willsky, et al.
- 1989
(Show Context)
Citation Context ...is and subsequent sections), this is a major benefit of the use of MR tree models. The second implication of the tree structure of , which directly generalizes known results for temporal models [17], =-=[27]-=-, [28], [226], is that this implies that the error process is an MR process on the same tree, with parameters (i.e., matrices analogous to and for the original process) that are automatically availabl... |

6 |
Analyse multirésolution des signaux aléatoires
- Cohen, Froment, et al.
- 1991
(Show Context)
Citation Context ...lar or fractal processes [288] such as fractional Brownian motion (fBm) [30], [83], [116], [232], [313], motivating examinations of the properties of the wavelet transforms of such signals and images =-=[69]-=-, [83], [102], [114], [117], [154], [176], [191], [235], [273], [293], [320], [346]–[350], [359]. Second, whether the phenomenon displays MR behavior or not, it may be the case that the available data... |

5 |
Multiscale signal processing: Isotropic random fields on homogeneous trees
- Claus, Chartier
- 1994
(Show Context)
Citation Context ...ces—with state space modeling for time series. For example, we refer the reader to [29] for the development of a state space theory for deterministic MR dynamics on trees and to [24], [25], [65], and =-=[66]-=- in which MR counterparts to autoregressive modeling and efficient algorithms analogous to Levinson’s algorithm for time series are developed for the class of isotropic processes on trees (i.e., proce... |

5 | Zirilli F. The fusion of different resolution SAR images [A - Costantiti, Farina - 1997 |

5 |
Simulating and mapping spatial complexity using multi-scale techniques
- DeCola
- 1994
(Show Context)
Citation Context ...usion for hydrology applications [84], [139], [198]; process control [18], [196], [306], [327]; synthetic aperture radar image analysis and fusion [77], [119], [160], [185], [309]; geographic systems =-=[93]-=-, [189]; medical image analysis [290]; models of neural responses in human vision [274]; and mathematical physics [15], [94], [136]. In this section, we introduce several of these applications which s... |

4 |
Castañon, “Smoothing error dynamics and their use in the solution of smoothing and mapping problems
- Bello, Willsky, et al.
- 1986
(Show Context)
Citation Context ... subsequent sections), this is a major benefit of the use of MR tree models. The second implication of the tree structure of , which directly generalizes known results for temporal models [17], [27], =-=[28]-=-, [226], is that this implies that the error process is an MR process on the same tree, with parameters (i.e., matrices analogous to and for the original process) that are automatically available as a... |

4 |
Space-time interpolation of oceanic fronts
- Chin, Mariano
- 1997
(Show Context)
Citation Context ...ls remains an open topic whose resolution would provide a way in which to relate local approximations at each node in a tree with the impact on global model accuracy. We also refer the reader to [54]–=-=[56]-=- for related research in space–time estimation in which the spatial phenomenon can be 2-D or 3-D, and for which the objective is to propagate an MRF model (e.g., a first-order or higher order MRF on a... |

4 | A Multi-resolution, Probabilistic Approach to Twodimensional Inverse Conductivity Problems
- Chou, Willsky
- 1989
(Show Context)
Citation Context ...ed as a random field but rather is simply viewed as an unknown, which is represented in an MR fashion to allow coarse-to-fine algorithms for ML estimation. One example of such an approach is given in =-=[63]-=- in the context of an inverse conductivity estimation problem, in which the unknown conductivity field is modeled as piecewise constant at a sequence of resolutions from coarse-to-fine (corresponding ... |

2 |
Classification of binary patterns
- Abend, Harley, et al.
- 1965
(Show Context)
Citation Context ...e line that come earlier in the scan order. Another example is the class so-called “Markov mesh” models (for both Gaussian and discrete-state processes), which impose a partial order on pixels in 2-D =-=[1]-=-, [78], [88], [98], [213]. However, in many problems, imposing such total or partial orders is clearly artificial. Moreover, often very high-order models of these types are needed to capture accuratel... |

2 | A multiscale approach for estimating solute transport travel time distributions
- Daniel, Willsky, et al.
(Show Context)
Citation Context ...ples can be found in a variety of other problems involving remotely sensed or probed data including the fusion of synthetic aperture radar (SAR) imagery [77] and geophysical inversion and data fusion =-=[84]-=-, [139], [198], [247]. In addition, advances in biomedical sensing [317] require the development of new methods for fusing data sets with very different characteristics (e.g., positron emission tomogr... |

2 |
Reciprocal processes on a tree-modeling and estimation issues
- Dijkerman, Mazumdar, et al.
- 1995
(Show Context)
Citation Context ...red from many of the graphical model references given at the start of this section. Other discussions of this can be found in [156] and in the discussion of so-called reciprocal processes on trees in =-=[101]-=-. 1408 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 8, AUGUST 2002graphical structure connecting each time point to its successor. Specifically, if we take the point as the root node of the (acyclic) graph ... |

1 | Multiscale approaches to moving target detection in image sequences
- Allen, Luettgen, et al.
- 1994
(Show Context)
Citation Context ...ety of applications, including: low-level computer vision and image processing problems (image denoising [59], [67], [80], [261], [281], deblurring [19], edge detection [292], optical flow estimation =-=[10]-=-, [223], surface reconstruction [111], texture classification [225], WILLSKY: MR MARKOV MODELS FOR SIGNAL AND IMAGE PROCESSING 1399and image segmentation [42], [58], [199], [212], [324], to name a fe... |

1 |
Analysis of flow in gas-liquid bubble-columns using multiresolution methods
- Bakshi, Zhong, et al.
- 1995
(Show Context)
Citation Context ...ion, and data fusion [112], [113], [158], [184], [242], [255], [326]; speech [42], [162], [175], [241], [249], [331]; multisensor fusion for hydrology applications [84], [139], [198]; process control =-=[18]-=-, [196], [306], [327]; synthetic aperture radar image analysis and fusion [77], [119], [160], [185], [309]; geographic systems [93], [189]; medical image analysis [290]; models of neural responses in ... |

1 |
Multiscale statistical signal processing identification of a multiscale AR process from a sample of an ordinary signal
- Claus
- 1993
(Show Context)
Citation Context ...t differences—with state space modeling for time series. For example, we refer the reader to [29] for the development of a state space theory for deterministic MR dynamics on trees and to [24], [25], =-=[65]-=-, and [66] in which MR counterparts to autoregressive modeling and efficient algorithms analogous to Levinson’s algorithm for time series are developed for the class of isotropic processes on trees (i... |

1 |
Efficient implementations of twodimensional noncausal IIR filters
- Daniel, Willsky
- 1997
(Show Context)
Citation Context ...cribed has close relationships both to well-known methods for the numerical solution of PDEs [106], [138] and to algorithms and ideas for space–time processes and DBNs. In particular, as described in =-=[81]-=- (see also [181]), suppose that, instead of beginning with the middle red row in Fig. 14, we begin either with the top row or the leftmost column and then “march” either downward row by row or from le... |