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94
Multiresolution markov models for signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coheren ..."
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Cited by 122 (18 self)
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This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for selfsimilar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden
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 ..."
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Cited by 98 (33 self)
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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.
Dynamic Model of Visual Recognition Predicts Neural Response Properties in the Visual Cortex
 Neural Computation
, 1995
"... this paper, we describe a hierarchical network model of visual recognition that explains these experimental observations by using a form of the extended Kalman filter as given by the Minimum Description Length (MDL) principle. The model dynamically combines inputdriven bottomup signals with expec ..."
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Cited by 86 (21 self)
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this paper, we describe a hierarchical network model of visual recognition that explains these experimental observations by using a form of the extended Kalman filter as given by the Minimum Description Length (MDL) principle. The model dynamically combines inputdriven bottomup signals with expectationdriven topdown signals to predict current recognition state. Synaptic weights in the model are adapted in a Hebbian manner according to a learning rule also derived from the MDL principle. The resulting prediction/learning scheme can be viewed as implementing a form of the ExpectationMaximization (EM) algorithm. The architecture of the model posits an active computational role for the reciprocal connections between adjoining visual cortical areas in determining neural response properties. In particular, the model demonstrates the possible role of feedback from higher cortical areas in mediating neurophysiological effects due to stimuli from beyond the classical receptive field. Si
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 ..."
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Cited by 85 (27 self)
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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.
Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination
 IEEE Transactions on Image Processing
, 1995
"... Abstruct A class of multiscale stochastic models based on scalerecursive dynamics on trees has recently been introduced. Theoretical and experimental results have shown that these models provide an extremely rich framework for representing both processes which are intrinsically multiscale, e.g., l ..."
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Cited by 58 (20 self)
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Abstruct A class of multiscale stochastic models based on scalerecursive dynamics on trees has recently been introduced. Theoretical and experimental results have shown that these models provide an extremely rich framework for representing both processes which are intrinsically multiscale, e.g., llf processes, as well as 1D Markov processes and 2D Markov random fields. Moreover, efficient optimal estimation algorithms have been developed for these models by exploiting their scalerecursive structure. In this paper, we exploit this structure in order to develop a computationally efficient and parallelizable algorithm for likelihood calculation. We illustrate one possible application to texture discrimination and demonstrate that likelihoodbased methods using our algorithm achieve performance comparable to that of Gaussian Markov random field based techniques, which in general are prohibitively complex computationally. I.
An optimal estimation approach to visual perception and learning
 VISION RESEARCH
, 1999
"... How does the visual system learn an internal model of the external environment? How is this internal model used during visual perception? How are occlusions and background clutter so effortlessly discounted for when recognizing a familiar object? How is a particular object of interest attended to an ..."
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Cited by 39 (8 self)
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How does the visual system learn an internal model of the external environment? How is this internal model used during visual perception? How are occlusions and background clutter so effortlessly discounted for when recognizing a familiar object? How is a particular object of interest attended to and recognized in the presence of other objects in the field of view? In this paper, we attempt to address these questions from the perspective of Bayesian optimal estimation theory. Using the concept of generative models and the statistical theory of Kalman filtering, we show how static and dynamic events occurring in the visual environment may be learned and recognized given only the input images. We also describe an extension of the Kalman filter model that can handle multiple objects in the field of view. The resulting robust Kalman filter model demonstrates how certain forms of attention can be viewed as an emergent property of the interaction between top–down expectations and bottom–up signals. Experimental results are provided to help demonstrate the ability of such a model to perform robust segmentation and recognition of objects and image sequences in the presence of occlusions and clutter.
A Categorization of Multiscaledecompositionbased Image Fusion Schemes with a Performance Study for a Digital Camera Application
 Proceedings of the IEEE
, 1999
"... The objective of image fusion is to combine information from multiple images of the same scene. The result of image fusion is a single image which is more suitable for human and machine perception or further image processing tasks. In this paper, a generic image fusion framework based on multiscale ..."
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Cited by 39 (2 self)
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The objective of image fusion is to combine information from multiple images of the same scene. The result of image fusion is a single image which is more suitable for human and machine perception or further image processing tasks. In this paper, a generic image fusion framework based on multiscale decomposition is studied. This framework provides freedom to choose different multiscale decomposition methods and different fusion rules. The framework includes all of the existing multiscaledecompositionbased fusion approaches we found in the literature which did not assume a statistical model for the source images. Different image fusion approaches are investigated based on this framework. Some evaluation measures are suggested and applied to compare the performance of these fusion schemes for a digital camera application. The comparisons indicate that our framework includes some new approaches which outperform the existing approaches for the cases we consider. 1 Introduction There ha...
An overlapping tree approach to multiscale stochastic modeling and estimation
 IEEE Trans. on Image Processing
, 1997
"... Abstract — Recently, a class of multiscale stochastic models has been introduced in which random processes and fields are described by scalerecursive dynamic trees. A major advantage of this framework is that it leads to an extremely efficient, statistically optimal algorithm for leastsquares esti ..."
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Cited by 37 (11 self)
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Abstract — Recently, a class of multiscale stochastic models has been introduced in which random processes and fields are described by scalerecursive dynamic trees. A major advantage of this framework is that it leads to an extremely efficient, statistically optimal algorithm for leastsquares estimation. In certain applications, however, estimates based on the types of multiscale models previously proposed may not be adequate, as they have tended to exhibit a visually distracting blockiness. In this paper, we eliminate this blockiness by discarding the standard assumption that distinct nodes on a given level of the multiscale process correspond to disjoint portions of the image domain; instead, we allow a correspondence to overlapping portions of the image domain. We use these socalled overlappingtree models for both modeling and estimation. In particular, we develop an efficient multiscale algorithm for generating sample paths of a random field whose secondorder statistics match a prespecified covariance structure, to any desired degree of fidelity. Furthermore, we demonstrate that under easily satisfied conditions, we can “lift ” a random field estimation problem to one defined on an overlapped tree, resulting in an estimation algorithm that is computationally efficient, directly produces estimation error covariances, and eliminates blockiness in the reconstructed imagery without any sacrifice in the resolution of finescale detail. Index Terms—Least squares estimation, multiscale, quadtrees, stochastic modeling.
Embedded Trees: Estimation of Gaussian Processes on Graphs with Cycles
 IEEE Transactions on Signal Processing
, 2002
"... Graphical models provide a powerful general framework for encoding the structure of largescale estimation problems. However, the graphs describing typical realworld phenomena contain many cycles, making direct estimation procedures prohibitively costly. In this paper, we develop an iterative infer ..."
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Cited by 36 (13 self)
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Graphical models provide a powerful general framework for encoding the structure of largescale estimation problems. However, the graphs describing typical realworld phenomena contain many cycles, making direct estimation procedures prohibitively costly. In this paper, we develop an iterative inference algorithm for general Gaussian graphical models. It operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph. When these subproblems are suitably chosen, the algorithm converges to the correct conditional means. Moreover, and in contrast to many other iterative methods, the treebased procedures we propose can also be used to calculate exact error variances. Although the conditional mean iteration is effective for quite densely connected graphical models, the error variance computation is most efficient for sparser graphs. In this context, we present a modeling example which suggests that very sparsely connected graphs with cycles may provide significant advantages relative to their treestructured counterparts, thanks both to the expressive power of these models and to the efficient inference algorithms developed herein.
A WaveletBased Method For Multiscale Tomographic Reconstruction
, 1995
"... We represent the standard ramp filter operator of the filtered backprojection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp filter matrix operator is approximately diagonal. The accuracy of t ..."
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Cited by 32 (4 self)
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We represent the standard ramp filter operator of the filtered backprojection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp filter matrix operator is approximately diagonal. The accuracy of this diagonal approximation becomes better as wavelets with larger number of vanishing moments are used. This waveletbased representation enables us to formulate a multiscale tomographic reconstruction technique wherein the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. Our multiscale reconstruction technique has the same computational complexity as the FBP reconstruction method. It differs from other multiscale reconstruction techniques in that 1) the object is defined through a multiscale transformation of the projection domain, and 2) we explicitly account for noise in the projection da...