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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.
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 ..."
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Cited by 29 (3 self)
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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...
On Variable Brightness Optical Flow For Tagged MRI
 IN INFORMATION PROCESSING IN MEDICAL IMAGING
, 1995
"... The problem of computing velocity fields from tagged MR images is considered, with particular attention paid to the image brightness changes caused by tag pattern fading with time. Shortcomings of existing optical flow methods for this application are pointed out and Gennert and Negahdaripour's opt ..."
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Cited by 27 (4 self)
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The problem of computing velocity fields from tagged MR images is considered, with particular attention paid to the image brightness changes caused by tag pattern fading with time. Shortcomings of existing optical flow methods for this application are pointed out and Gennert and Negahdaripour's optical flow method is discussed. This method involves estimation of a local brightness transformation field in addition to the velocity field. Approximations for these transformations are developed, leading to a faster algorithm which does not require careful selection of the regularization coefficients. This method is validated by tests on both simulated and actual MR data, and is demonstrated to be computationally robust to inaccurate knowledge of MR parameters.
An estimationbased approach to the reconstruction ofoptical flow
, 1987
"... The problem of reconstructing the apparent velocity field (optical flow) in a sequence of images is formulated as a linear estimation problem. Estimationbased interpretations are provided for wellknown formulations and methods, allowing us to use the machinery of recursive estimation theory to con ..."
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Cited by 10 (4 self)
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The problem of reconstructing the apparent velocity field (optical flow) in a sequence of images is formulated as a linear estimation problem. Estimationbased interpretations are provided for wellknown formulations and methods, allowing us to use the machinery of recursive estimation theory to construct both new and efficient algorithms for these problems and a flexible framework for the development of algorithms for modified or related problems. The first problem we address is the estimation of the velocity field along a moving contour given a stochastic model of this field and measurements of the component of velocity normal to the contour. The methods of 1D linear smoothing theory provide recursive algorithms, in contrast to the iterative method of Hildreth for the same problem. We then consider the problem of estimating the optical flow inside a bounded domain, given an estimate on the boundary and observations inside the domain, which we formulate as an estimation problem for a 2D boundary value stochastic process. The resulting estimator is then obtained as the solution of the same system of elliptic partial differential equations derived in a very different way by Horn and Schunck. We then develop an efficient implementation of this estimator using a recently developed local relaxation method.
Optimal Brightness Functions For Optical Flow Estimation Of Deformable Motion
, 1997
"... Estimation accuracy of Horn and Schunck's classical optical flow algorithm depends on many factors including the brightness pattern of the measured images. Since some applications can select brightness functions with which to "paint" the object, it is desirable to know what patterns will lead to the ..."
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Cited by 5 (0 self)
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Estimation accuracy of Horn and Schunck's classical optical flow algorithm depends on many factors including the brightness pattern of the measured images. Since some applications can select brightness functions with which to "paint" the object, it is desirable to know what patterns will lead to the best motion estimates. In this paper we present a method for determining this pattern a priori using mild assumptions about the velocity field and imaging process. Our method is based on formulating Horn and Schunck's algorithm as a linear smoother and rigorously deriving an expression for the corresponding error covariance function. We then specify a scalar performance measure and develop an approach to select an optimal brightness function which minimizes this performance measure from within a parametrized class. Conditions for existence of an optimal brightness function are also given. The resulting optimal performance is demonstrated using simulations, and a discussion of these results ...
A geometric projectionspace reconstruction algorithm
 Linear Algebra and Its Applications, 130:151191
, 1990
"... We present a method to reconstruct images from finite sets of noisy projections that may be available only over limited or sparse angles. The algorithm calculates the maximum a posteriori (MAP) estimate of the full sinogram (which is an image of the 2D Radon transform of the object) from the availa ..."
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Cited by 2 (1 self)
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We present a method to reconstruct images from finite sets of noisy projections that may be available only over limited or sparse angles. The algorithm calculates the maximum a posteriori (MAP) estimate of the full sinogram (which is an image of the 2D Radon transform of the object) from the available data. It is implemented using a primaldual constrained optimization procedure that solves a partial differential equation in the primal phase with an efficient local relaxation algorithm and uses a simple Lagrangemultiplier update in the dual phase. The sinogram prior probability is given by a Markov random field (MRF) that includes information about the mass, center of mass, and convex hul,l of the object, and about the smoothness, fundamental constraints, and periodicity of the 2D Radon transform. The object is reconstructed using convolution backprojection applied to the estimated sinogram. We show several reconstructed objects which are obtained from simulated limited and sparseangle data using the described algorithm, and compare these results with images obtained using convolution backprojection directly. 1.
Stochastic Estimation of Deformable Motion from Magnetic Resonance Tagged Cardiac Images
, 1994
"... The estimation of heart motion from image sequence data has received a great deal of attention in recent years because cardiac motion is complex and analysis of cardiac motion can be used to diagnose damage to the heart muscle caused by a heart attack. Magnetic resonance imaging has shown great prom ..."
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The estimation of heart motion from image sequence data has received a great deal of attention in recent years because cardiac motion is complex and analysis of cardiac motion can be used to diagnose damage to the heart muscle caused by a heart attack. Magnetic resonance imaging has shown great promise for imaging cardiac motion because of a technique called tagging. Tagged images appear with a spatially encoded pattern that moves with the heart tissue as it moves through the heart cycle. In this dissertation we develop two new approaches to cardiac motion estimation that exploit MR tagging techniques. Both approaches are developed within a stochastic estimation framework, which serves as an important unifying theme. In the first approach, we use Horn and Schunck's optical flow (HSOF) algorithm to estimate motion and design a continuously varying tag pattern that is optimal for the HSOF algorithm. An optimal tag pattern is determined by formulating HSOF as a stochastic linear smoother ...
LECTURE 9: 12/7/93 Iterative methods
, 1993
"... is implemented on a MCC MPP with several unknowns per processor. ffl Jacobi and GS are not effective. They require O(N) iterations for convergence compared to O(N 1=2 ) for SOR. Under suitable orderings all methods make use of local information on a meshconnected processor. ffl Except for mode ..."
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is implemented on a MCC MPP with several unknowns per processor. ffl Jacobi and GS are not effective. They require O(N) iterations for convergence compared to O(N 1=2 ) for SOR. Under suitable orderings all methods make use of local information on a meshconnected processor. ffl Except for model problems, SOR requires parameter estimation procedures [HY81]. These require global information in each iteration, implying O(N 1=2 ) communication per step. Hence the total SOR complexity becomes O(N), losing the advantage. A local relaxation scheme was proposed by Ehrlich in [Ehr81]. CS454: December 7, 1993 [ 5 ] Local relaxation Red points (i + j) even: u<F14.
Optical Flow Estimation From Tagged Cardiac MR Images
, 1996
"... INTRODUCTION Cardiovascular disease is the number one cause of death in the United States. Analysis of cardiac motion is an important area of research since it aids in screening and monitoring damage in the heart wall muscle. Characterizing the behavior of normal and abnormal heart can help in earl ..."
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INTRODUCTION Cardiovascular disease is the number one cause of death in the United States. Analysis of cardiac motion is an important area of research since it aids in screening and monitoring damage in the heart wall muscle. Characterizing the behavior of normal and abnormal heart can help in early diagnosis of ischemia (area of tissue resulting from obstruction of blood circulation). It can also aid in patient monitoring after a cardiac arrest and in evaluating the effectiveness of treatment. Currently, contractile behavior of the myocardium is mainly studied by a wallthickening analysis [1]. This does not accurately capture the complicated motion of the heart wall muscle, and produces inaccurate estimates of the amount of myocardial infarction. Conventional MR imaging and computed tomography also do not show motion within the heart wall muscle. Use of implanted radiopaque markers such as beads or ultrasound transducers permits tracking of actual fixed points on the heart w
StateSpace Optical Flow
"... Optical flow estimators of motion in image sequences are sometimes found using variational frameworks, which produce partial differential equations as the estimators. In this paper, we show that certain statespace formulations, solved by linear smoothing theory, lead to these same solutions. The ad ..."
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Optical flow estimators of motion in image sequences are sometimes found using variational frameworks, which produce partial differential equations as the estimators. In this paper, we show that certain statespace formulations, solved by linear smoothing theory, lead to these same solutions. The advantages of viewing these problems in this way include explicit stochastic motion models, a priori error measures, and alternate ways to choose regularization parameters. In this paper, we cast the algorithm of Horn and Schunk and that of Gennert and Negahdaripour and the newer DIVCURL spline formulations in this fashion, with care taken to form wellposed models. Finally, we demonstrate one important use of these models, which is the simulation of the motion fields for which the algorithms are optimal.