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46
Reconstructing convex sets from support line measurements
 ieeePAMI
, 1990
"... This paper proposes algorithms for reconstructing convex sets given noisy support line measurements. We begin by observing that a set of measured support lines may not be consistent with any set in the plane. We then develop a theory of consistent support lines which serves as a basis for reconstruc ..."
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Cited by 19 (6 self)
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This paper proposes algorithms for reconstructing convex sets given noisy support line measurements. We begin by observing that a set of measured support lines may not be consistent with any set in the plane. We then develop a theory of consistent support lines which serves as a basis for reconstruction algorithms that take the form of constrained optimization algorithms. The formal statement of the problem and constraints reveals a rich geometry which allows us to include prior information about object position and boundary smoothness. The algorithms, which use explicit noise models and prior knowledge, are based on ML and MAP estimation principles, and are implemented using efficient linear and quadratic programming codes. Experimental results are presented. This research sets the stage for a more general approach to the incorporation of prior information concerning and the estimation of object shape.
Objectbased 3D reconstruction of arterial trees from magnetic resonance angiograms
 IEEE Transactions on Medical Imaging
, 1991
"... This paper describes an objectbased approach to the problem of reconstructing threedimensional descriptions of arterial trees from a few angiographic projections. The method incorporates aprioriknowledge of the structure of branching arteries into a natural optimality criterion that encompasses ..."
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Cited by 15 (3 self)
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This paper describes an objectbased approach to the problem of reconstructing threedimensional descriptions of arterial trees from a few angiographic projections. The method incorporates aprioriknowledge of the structure of branching arteries into a natural optimality criterion that encompasses the entire arterial tree. This global approach enables reconstruction from a few noisy projection images. We present an efficient optimization algorithm for object estimation, and demonstrate its performance on simulated, phantom, and in vivo magnetic resonance angiograms.
XRay Metrology for Quality Assurance
"... There is considerable currentinterest in deriving accurate dimensional measurements of the internal geometry of complex manufactured parts, particularly castings. This paper describes an approachtothereconstruction of 3D part geometry from multiple digital Xray images. A novel method for radiograp ..."
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Cited by 11 (4 self)
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There is considerable currentinterest in deriving accurate dimensional measurements of the internal geometry of complex manufactured parts, particularly castings. This paper describes an approachtothereconstruction of 3D part geometry from multiple digital Xray images. A novel method for radiographic stereo is described whichtakes into accountthespecial imaging geometry of the digital Xray sensor modeled by a linear moving array, or pushbroom, camera. The 3D reconstruction algorithm employs a nominal geometric model which is perturbed by Xrayimage constraints. Manufacturing applications are discussed and illustrated by experimental results on synthetic phantoms and actual casting images.
A Multiscale Hypothesis Testing Approach to Anomaly Detection and Localization from Noisy Tomograhic Data
 IEEE Transactions on Image Processing
, 1998
"... Abstract—In this paper, we investigate the problems of anomaly detection and localization from noisy tomographic data. These are characteristic of a class of problems that cannot be optimally solved because they involve hypothesis testing over hypothesis spaces with extremely large cardinality. Our ..."
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Cited by 10 (1 self)
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Abstract—In this paper, we investigate the problems of anomaly detection and localization from noisy tomographic data. These are characteristic of a class of problems that cannot be optimally solved because they involve hypothesis testing over hypothesis spaces with extremely large cardinality. Our multiscale hypothesis testing approach addresses the key issues associated with this class of problems. A multiscale hypothesis test is a hierarchical sequence of composite hypothesis tests that discards large portions of the hypothesis space with minimal computational burden and zooms in on the likely true hypothesis. For the anomaly detection and localization problems, hypothesis zooming corresponds to spatial zooming—anomalies are successively localized to finer and finer spatial scales. The key challenges we address include how to hierarchically divide a large hypothesis space and how to process the data at each stage of the hierarchy to decide which parts of the hypothesis space deserve more attention. To answer the former we draw on [1] and [7]–[10]. For the latter, we pose and solve a nonlinear optimization problem for a decision statistic that maximally disambiguates composite hypotheses. With no more computational complexity, our optimized statistic shows substantial improvement over conventional approaches. We provide examples that demonstrate this and quantify how much performance is sacrificed by the use of a suboptimal method as compared to that achievable if the optimal approach were computationally feasible. Index Terms—Anomaly detection, composite hypothesis testing, hypothesis zooming, nonlinear optimization, quadratic programming, tomography. I.
Polygonal and Polyhedral Contour Reconstruction in Computed Tomography
, 2004
"... This paper is about threedimensional (3D) reconstruction of a binary image from its Xray tomographic data. We study the special case of a compact uniform polyhedron totally included in a uniform background and directly perform the polyhedral surface estimation. We formulate this problem as a nonl ..."
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Cited by 9 (3 self)
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This paper is about threedimensional (3D) reconstruction of a binary image from its Xray tomographic data. We study the special case of a compact uniform polyhedron totally included in a uniform background and directly perform the polyhedral surface estimation. We formulate this problem as a nonlinear inverse problem using the Bayesian framework. Vertice estimation is done without using a voxel approximation of the 3D image. It is based on the construction and optimization of a regularized criterion that accounts for surface smoothness. We investigate original deterministic local algorithms, based on the exact computation of the line projections, their update, and their derivatives with respect to the vertice coordinates. Results are first derived in the twodimensional (2D) case, which consists of reconstructing a 2D object of deformable polygonal contour from its tomographic data. Then, we investigate the 3D extension that requires technical adaptations. Simulation results illustrate the performance of polygonal and polyhedral reconstruction algorithms in terms of quality and computation time.
CramérRao Bounds for Parametric Shape Estimation in Inverse Problems
 IEEE Trans. on Image Processing
, 2003
"... We address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems, when the boundaries are parameterized by a finite number of unknown variables. Our model applies to multiple unknown objects, each with its own unknown ..."
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Cited by 9 (2 self)
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We address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems, when the boundaries are parameterized by a finite number of unknown variables. Our model applies to multiple unknown objects, each with its own unknown gray level, or color, and boundary parameterization, on an arbitrary known background. While such fundamental bounds on the performance of shape estimation algorithms can in principle be derived from the CramerRao lower bounds, very few results have been reported due to the di#culty of computing the derivatives of a functional with respect to shape deformation. In this paper, we provide a general formula for computing CramerRao lower bounds in inverse problems where the observations are related to the object by a general linear transform, followed by a possibly nonlinear and noisy measurement system.
Multiscale, statistical anomaly detection analysis and algorithms for linearized inverse scattering problems
 Multidimensional Systems and Signal Processing
, 1997
"... Abstract. In this paper we explore the utility of multiscale and statistical techniques for detecting and characterizing the structure of localized anomalies in a medium based upon observations of scattered energy obtained at the boundaries of the region of interest. Wavelet transform techniques are ..."
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Cited by 9 (5 self)
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Abstract. In this paper we explore the utility of multiscale and statistical techniques for detecting and characterizing the structure of localized anomalies in a medium based upon observations of scattered energy obtained at the boundaries of the region of interest. Wavelet transform techniques are used to provide an efficient and physically meaningful method for modeling the nonanomalous structure of the medium under investigation. We employ decisiontheoretic methods both to analyze a variety of difficulties associated with the anomaly detection problem and as the basis for an algorithm to perform anomaly detection and estimation. These methods allow for a quantitative evaluation of the manner in which the performance of the algorithms is impacted by the amplitudes, spatial sizes, and positions of anomalous areas in the overall region of interest. Given the insight provided by this work, we formulate and analyze an algorithm for determining the number, location, and magnitudes associated with a set of anomaly structures. This approach is based upon the use of a Generalized, Mary Likelihood Ratio Test to successively subdivide the region as a means of localizing anomalous areas in both space and scale. Examples of our multiscale inversion algorithm are presented using the Born approximation of an electrical conductivity problem formulated so as to illustrate many of the features associated with similar detection problems arising in fields such as geophysical prospecting, ultrasonic imaging, and medical imaging. Key Words: 1.
Local tests for consistency of support hyperplane data
 J. Math. Imaging and Vision
, 1995
"... Abstract. Support functions and samples of convex bodies in R ~ are studied with regard to conditions for their validity or consistency. Necessary and sufficient conditions for a function to be a support function are reviewed in a general setting. An apparently little known classical such result for ..."
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Cited by 7 (0 self)
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Abstract. Support functions and samples of convex bodies in R ~ are studied with regard to conditions for their validity or consistency. Necessary and sufficient conditions for a function to be a support function are reviewed in a general setting. An apparently little known classical such result for the planar case due to Rademacher and based on a determinantal inequality is presented and a generalization to, arbitrary dimensions is developed. These conditions are global in the sense that they involve values of the support function at widely separated points. The corresponding discrete problem of determining the validity of a set of samples of a support function is treated. Conditions similar to the continuous inequality results are given for the consistency of a set of discrete support observations. These conditions are in terms of a series of local inequality tests involving only neighboring support samples. Our results serve to generalize existing planar conditions to arbitrary dimensions by providing a generalization of the notion of nearest neighbor for plane vectors which utilizes a simple positive cone condition on the respective support sample normals.
CramérRao Bounds for 2D Target Shape Estimation in Nonlinear Inverse Scattering Problems with Application to Passive Radar
 IEEE Trans. on Antennas and Propagat
, 2001
"... We present new methods for computing fundamental performance limits for twodimensional (2D) parametric shape estimation in nonlinear inverse scattering problems with an application to passive radar imaging. We evaluate CramrRao lower bounds (CRB) on shape estimation accuracy using the domain der ..."
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Cited by 7 (3 self)
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We present new methods for computing fundamental performance limits for twodimensional (2D) parametric shape estimation in nonlinear inverse scattering problems with an application to passive radar imaging. We evaluate CramrRao lower bounds (CRB) on shape estimation accuracy using the domain derivative technique from nonlinear inverse scattering theory. The CRB provides an unbeatable performance limit for any unbiased estimator, and under fairly mild regularity conditions is asymptotically achieved by the maximum likelihood estimator (MLE). The resultant CRBs are used to define an asymptotic global confidence region, centered around the true boundary, in which the boundary estimate lies with a prescribed probability. These global confidence regions conveniently display the uncertainty in various geometric parameters such as shape, size, orientation, and position of the estimated target and facilitate geometric inferences. Numerical simulations are performed using the layer approach and the Nystrm method for computation of domain derivatives and using Fourier descriptors for target shape parameterization. This analysis demonstrates the accuracy and generality of the proposed methods. Index TermsCramrRao bounds, Fourier descriptors, global confidence regions, nonlinear inverse scattering, passive radar imaging, shape estimation. I.
Multiscale Bayesian Methods for Discrete Tomography
, 1999
"... Statistical methods of discrete tomographic reconstruction pose new problems both in stochastic modeling to define an optimal reconstruction, and in optimization to find that reconstruction. Multiscale models have succeeded in improving representation of structure of varying scale in imagery, a ..."
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Cited by 7 (2 self)
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Statistical methods of discrete tomographic reconstruction pose new problems both in stochastic modeling to define an optimal reconstruction, and in optimization to find that reconstruction. Multiscale models have succeeded in improving representation of structure of varying scale in imagery, a chronic problem for common Markov random fields. This chapter shows that associated multiscale methods of optimization also avoid local minima of the log a posteriori probability better than singleresolution techniques. These methods are applied here to both segmentation/reconstruction of the unknown crosssections, and estimation of unknown parameters represented by the discrete levels. 1.1 Introduction The reconstruction of images from projections is important in a variety of problems including tasks in medical imaging and nondestructive testing. Perhaps, the reconstruction technique most frequently used in commercial applications is convolution backprojection (CBP) [1]. While CBP...