Recursive and iterative estimation algorithms for multiresolution stochastic processes

Chou, K.C.; Willsky, Alan S.; Benveniste, Albert; Basseville, Michèle

doi:10.1109/cdc.1989.70321

Cited by 43 publications

(65 citation statements)

References 18 publications

(11 reference statements)

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“…Our objective has been to develop probabilistic counterparts to this deterministic theory that can then be used as the basis for signal modeling, optimal estimation, and related algorithms. During the past year we have built on our earlier work to establish the foundation of a theory for multiscale stochastic processes and their estimation [32,33,41,42,49,[53][54][55][56][58][59][60].…”

Section: Multiresolution Modeling and Signal Processingmentioning

confidence: 99%

“…Our work has focused on developing and exploiting stochastic models for the coefficients x(m,n) as the scale, m, evolves. As we show in [32,33,41,42], there is a natural structure that arises in examining wavelet transforms from this dynamic synthesis perspective. In particular the (m, n) index set should be thought of as a weighted lattice, where each level of the lattice corresponds to a particular scale and the lattice structure and weights from scale to scale are determined by the wavelet transforms structure.…”

Section: Multiresolution Modeling and Signal Processingmentioning

confidence: 99%

“…The results here are far more complex than their time series counterparts because of the tree structure, but we have been able to develop Levinson-like efficient algorithms for constructing such models. In another part of our work [32,42,44,56,59] we have studied a class of "state" processes evolving from coarse-to-fine scales. Much of our work in this area has focused on the analysis of processes on trees, but we have also obtained some results on the more general class of weighted lattices.…”

Section: Multiresolution Modeling and Signal Processingmentioning

confidence: 99%

See 2 more Smart Citations

Analysis, Estimation, and Control for Perturbed and Singular Systems and for Systems Subject to Discrete Events

Willsky¹,

Verghese²

1988

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Section: Multiresolution Modeling and Signal Processingmentioning

confidence: 99%

Section: Multiresolution Modeling and Signal Processingmentioning

confidence: 99%

Section: Multiresolution Modeling and Signal Processingmentioning

confidence: 99%

See 1 more Smart Citation

Analysis, Estimation, and Control for Perturbed and Singular Systems and for Systems Subject to Discrete Events

Willsky¹,

Verghese²

1988

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“…One of the most important consequences of this alternate smoothness constraint is that it allows us to make use of the extremely efficient scale-recursive optimal estimation algorithm that this statistical model admits [11,9,10]. In particular, the resulting algorithm is not iterative and in fact requires a fixed number of floating point operations per pixel independent of image size.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we do just that as we introduce an approach based on substituting a fractal-like class of prior models recently introduced in [11,9,10,13] for the smoothness constraint prior. The key idea behind this approach is that instead of the Brownian motion fractal prior that describes the optical flow field as one that has independent increments in space, we use a statistical model for optical flow that has independent increments in scale.…”

Section: Introductionmentioning

confidence: 99%

Efficient Multiscale Regularization with Applications to the Computation of Optical Flow

Luettgen¹,

Karl²,

Willsky³

1993

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A new aplproach to regularization methods for image processing is iltro(duced and developed using as a. vehicle the problem of coml)uting dleilse optical flow fields in an image sequence. Standard formulations of this problemi re(luire the computationally intensive solution of an ellil)tic pa.rtial differential equation which arises froin the often used "smootlhness constraint' type regularization. We utilize the illterl)reta.tion of' .he smoot.hness constraint as a. "fractal prior" to motivate regularization based on a. recently introduced class of multiscale stochastic models. The solution of the new l)robleln formulat.ion is computed with an efficient multiscale algorithm. Experiiments on several ima.ge sequences (demonstrate the sutl)sta.nftial compult.a.tional savings tlia.t. canll I)e achieved due to the fact tha.t the algorit.hm is non-iterative and in fact has a. per pixel computational complexity which is independent. of imiage size. Tlie new a.pp)l)roach also has a. numbler of other important. advantages. Specifically, nitltiresolution flow field estimates are availabl)le, allowing grea.t. flexibility in dealing wvithl the t.radeoff b.)et.ween resolution a.nd(l accuracy. NIultiscale error covariance information is also availabLle. which is of considerable use inll assessing the accuracy of the estimates. Ini particular, these error statistics calln be used as the b)asis for a rational procedure for dletermlining the spatially-varying optimal reconstruction resolution. Furthermore, if there are conipelling reasons to insist upon a standard smoothness constraint, our algorithm provides an excellent initialization for the iterative algoritlhms associated with the smoot. ness constraint l)roblemi formulation. Finally, the usefulness of otir appI)roach should extend t-o a. wide variety of ill-posed inverse problems inll which variational techniques seeking a. "smooth" solution are generally usedl. EDICS category 1.11. Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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Multiscale Statistical Signal Processing: Stochastic Processes Indexed by Trees

Basseville¹,

Benveniste²,

Chou³

et al. 1990

Realization and Modelling in System Theory

Self Cite

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Motivated by the recently-developed theory of multiscale signal models and wavelet transforms, we introduce stochastic dynamic models evolving on homogeneous trees. In particular we introduce and investigate both AR and state models on trees. Our analysis yields generalizations of Levinson and Schur recursions and of Kalman filters, Riccati equations, and Rauch-Tung-Striebel smoothing. MULTISCALE REPRESENTATIONS AND HOMOGENEOUS TREESThe recently-introduced theory of multiscale representations and wavelet transforms [4] provides a sequence of approximations of signals at finer and finer scales. In 1-D a signal f(x) is represented at the mth scale by a sequence f(m, n) which provides the amplitudes of time-scaled pulses located at the points n2 -m . The progression from one scale to the next thus introduces twice as many points and indeed provides a tree structure with the pair (2 -m , n) at one scale associated with ( 2 -( m+ l ), 2n) and (2-(m+l),2n + 1) at the next. This provides the motivation for the development of a system and stochastic process theory when the index set is taken to be a homogeneous dyadic tree. In this paper we outline some of the basic ideas behind our work.Let T denote the index set of the tree and we use the single symbol t for nodes on the tree. The scale associated with t is denoted by m(t), and we write s -< t (s --t) if m(s) < m(t)(m(s) < m(t)).We also let d (s, t) denote the distance between s and t, and s A t the common "parent" node of s and t (e.g. ( 2 -m, n) is the parent of (2-(m+1),2n) and (2-(m+1),2n + 1). In analogy with the shift operator z-' used as the basis for describing discrete-time dynamics we also define several shift operators on the tree: 0, the identity operator (no move); 7 -l, the fine-to-coarse shift (e.g.from (2-(m+l),2n or 2n + 1) to (2-m, n)); a, the left coarse-to-fine shift ((2-m,n) to

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Recursive and iterative estimation algorithms for multiresolution stochastic processes

Cited by 43 publications

References 18 publications

Analysis, Estimation, and Control for Perturbed and Singular Systems and for Systems Subject to Discrete Events

Analysis, Estimation, and Control for Perturbed and Singular Systems and for Systems Subject to Discrete Events

Efficient Multiscale Regularization with Applications to the Computation of Optical Flow

Multiscale Statistical Signal Processing: Stochastic Processes Indexed by Trees

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