Discrete tomography problems require the generation of an underlying attenuation field over a combinatorial set that renders the direct application of conventional tomographic reconstruction techniques obsolete. In this work, we propose a solution approach based on a new variable splitting coupled with the alternating direction method of multipliers (ADMM). The new approach creates subproblems that can be solved using existing and fast techniques, such as graph-cut methods, and results in overall solutions of excellent quality. In particular, compared to state-of-the art techniques, we obtain reliable representations of the underlying fields with substantially lower angular coverage. We develop the method under both Gaussian and Poisson noise models and show results with both binary and multilabel phantoms.

In recent years, baggage screening at airports has included the use of dual-energy X-ray computed tomography (DECT), an advanced technology for nondestructive evaluation. The main challenge remains to reliably find and identify threat objects in the bag from DECT data. This task is particularly hard due to the wide variety of objects, the high clutter, and the presence of metal, which causes streaks and shading in the scanner images. Image noise and artifacts are generally much more severe than in medical CT and can lead to splitting of objects and inaccurate object labeling. The conventional approach performs object segmentation and material identification in two decoupled processes. Dual-energy information is typically not used for the segmentation, and object localization is not explicitly used to stabilize the material parameter estimates. We propose a novel learning-based framework for joint segmentation and identification of objects directly from volumetric DECT images, which is robust to streaks, noise and variability due to clutter. We focus on segmenting and identifying a small set of objects of interest with characteristics that are learned from training images, and consider everything else as background. We include data weighting to mitigate metal artifacts and incorporate an object boundary field to reduce object splitting. The overall formulation is posed as a multilabel discrete optimization problem and solved using an efficient graph-cut algorithm. We test the method on real data and show its potential for producing accurate labels of the objects of interest without splits in the presence of metal and clutter.

Efficient graph-cut methods have been used with great success for labeling and denoising problems occurring in computer vision. Unfortunately, the presence of linear image mappings has prevented the use of these techniques in most discrete-amplitude image reconstruction problems. In this paper, we develop a graph-cut based framework for the direct solution of discrete amplitude linear image reconstruction problems cast as regularized energy function minimizations. We first analyze the structure of discrete linear inverse problem cost functions to show that the obstacle to the application of graph-cut methods to their solution is the variable mixing caused by the presence of the linear sensing operator. We then propose to use a surrogate energy functional that overcomes the challenges imposed by the sensing operator yet can be utilized efficiently in existing graph-cut frameworks. We use this surrogate energy functional to devise a monotonic iterative algorithm for the solution of discrete valued inverse problems. We first provide experiments using local convolutional operators and show the robustness of the proposed technique to noise and stability to changes in regularization parameter. Then we focus on nonlocal, tomographic examples where we consider limited-angle data problems. We compare our technique with state-of-the-art discrete and continuous image reconstruction techniques. Experiments show that the proposed method outperforms state-of-the-art techniques in challenging scenarios involving discrete valued unknowns.

We propose a novel approach for designing kernels for support vector machines (SVMs) when the class label is linked to the observation through a latent state and the likelihood function of the observation given the state (the sensing model) is available. We show that the Bayes-optimum decision boundary is a hyperplane under a mapping defined by the likelihood function. Combining this with the maximum margin principle yields kernels for SVMs that leverage knowledge of the sensing model in an optimal way. We derive the optimum kernel for the bag-of-words (BoWs) sensing model and demonstrate its superior performance over other kernels in document and image classification tasks. These results indicate that such optimum sensing-aware kernel SVMs can match the performance of rather sophisticated state-of-the-art approaches.

We propose a new structure-preserving dual-energy (SPDE) CT inversion technique for luggage screening, which can mitigate metal artifacts and provide precise object localization. Such artifact reduction can increase material identification accuracy in security applications. Our main objective is formation of enhanced photoelectric and Compton pixel property images from dual-energy X-ray tomographic data. We achieve this aim by incorporating three important elements in a single unified framework. First, we generate our images as the solution of a joint optimization problem, which explicitly models the projection process. Second, we include metal aware data weighting to reduce streaks and metal artifacts. Third, we estimate a regularized joint boundary field and apply it to both the photoelectric and Compton images in order to improve object localization as well as smoothing inside the objects. We evaluate the performance of the method using real dual-energy data. We demonstrate a significant reduction in noise and metal artifacts.

The rapid decrease in the dimensions of integrated circuits has necessitated corresponding higher-resolution methods for defect imaging. Current state of the art, defect imaging systems are reaching the limits of their resolution. In this work, we are proposing a new overcomplete dictionary based sparse signal imaging framework to improve the resolution and localization in confocal microscopy systems for backside optical integrated circuit defect imaging. The domain of integrated circuit imaging is particularly suitable for the application of overcomplete dictionaries in an image reconstruction framework because the images are highly structured, containing predictable building blocks derivable from the corresponding computer aided design layouts. This structure provides a strong and natural a-priori dictionary for scene reconstruction. This dictionary prior is coupled with a physically-based observation model to create enhanced scene reconstructions. The approach is described and results on simulated data are provided.

We study high-dimensional asymptotic performance limits of binary supervised classification problems where the class conditional densities are Gaussian with unknown means and covariances and the number of signal dimensions scales faster than the number of labeled training samples. We show that the Bayes error, namely the minimum attainable error probability with complete distributional knowledge and equally likely classes, can be arbitrarily close to zero and yet the limiting minimax error probability of every supervised learning algorithm is no better than a random coin toss. In contrast to related studies where the classification difficulty (Bayes error) is made to vanish, we hold it constant when taking high-dimensional limits. In contrast to VC-dimension based minimax lower bounds that consider the worst case error probability over all distributions that have a fixed Bayes error, our worst case is over the family of Gaussian distributions with constant Bayes error. We also show that a nontrivial asymptotic minimax error probability can only be attained for parametric subsets of zero measure (in a suitable measure space). These results expose the fundamental importance of prior knowledge and suggest that unless we impose strong structural constraints, such as sparsity, on the parametric space, supervised learning may be ineffective in high dimensional small sample settings.

In this letter, we study the impact of compressed data collections from a synthetic aperture radar (SAR) sensor on the reconstruction quality of a scene of interest. Different monostatic and multistatic SAR measurement configurations produce different Fourier sampling patterns. These patterns reflect different spectral and spatial diversity tradeoffs that must be made during task planning. Compressed sensing theory argues that the mutual coherence of the measurement probes is related to the reconstruction performance of sparse domains. With this motivation, we propose a closely related t%-average mutual coherence parameter as a sensing configuration quality parameter and examine its relationship to the reconstruction behavior of various monostatic and ultranarrow-band multistatic configurations. We investigate how this easily computed metric is related to SAR reconstruction quality.

In this paper we consider the task of classifying materials into explosives and non-explosives according to features obtainable from Multi-Energy X-ray Computed Tomography (MECT) measurements. The discriminative ability of MECT derives from its sensitivity to the attenuation versus energy curves of materials. Thus we focus on the fundamental information available in these curves and features extracted from them. We study the dimensionality and span of these curves for a set of explosive and non-explosive compounds and show that their space is larger than two-dimensional, as is typically assumed. In addition, we build support vector machine classifiers with different feature sets and find superior classification performance when using more than two features and when using features different than the standard photoelectric and Compton coefficients. These results suggest the potential for improved detection performance relative to conventional dual-energy X-ray systems.

In many applications decisions must be made about the state of an object based on indirect noisy observation of high-dimensional data. An example is the determination of the presence or absence of stroke from tomographic projections. Conventionally, the sensing process is inverted and a classifier is built in the reconstructed domain, which requires complete knowledge of the sensing mechanism. Alternatively, a direct data domain classifier might be constructed, but the constraints imposed by the sensing process are then lost. In this work we study the behavior of a third path we term “sensing-aware classification.” Our aim is to contribute to the development of a rigorous theory for such challenging problems. To this end, we consider an abstracted binary classification problem with very high dimensional observations, a restricting sensing configuration, and unknown statistical models of noise and object which must be learned from constrained training data. We analyze the impact of different levels of prior knowledge concerning the sensing mechanism for various classification strategies. In particular we prove that the strategies based on the naive estimation of all model elements results in a classification performance asymptotically no better than guessing whereas sensing-aware, projection-based classification rules attain Bayes-optimal risk. Simulation results are also provided.

In this paper we propose a graph cut method for solving a multi-label linear inverse problem with an arbitrary system matrix. Graph cuts are efficient methods for solving pixel-labeling and early vision problems. Energy function minimization problems that occur in image denoising are easily solved by graph cut techniques. However, applying graph cuts to inverse problems which have a non-diagonal system matrix becomes challenging, as a data cost of one pixel depends on intensities of other pixels. Such cost functions are not graph representable. In this paper, we propose an iterative method for minimization of energy functions occurring in inverse problems, where a graph-representable Taylor approximation of the original cost function is rapidly solved via a graph cut method at each iteration.

Perfusion imaging is a useful adjunct to anatomic imaging in numerous diagnostic and therapy-monitoring settings. One approach to perfusion imaging is to assume a convolution relationship between a local arterial input function and the tissue enhancement profile of the region of interest via a "residue function" and subsequently solve for this residue function. This ill-posed problem is generally solved using singular-value decomposition based approaches, and the hemodynamic parameters are solved for each voxel independently. In this paper, we present a formulation which incorporates both spatial and temporal correlations, and show through simulations that this new formulation yields higher accuracy and greater robustness with respect to image noise. We also show using rectal cancer tumor images that this new formulation results in better segregation of normal and cancerous voxels.

In this paper, we develop a new unified approach for laser radar range anomaly suppression, range profiling, and segmentation. This approach combines an object-based hybrid scene model for representing the range distribution of the field and a statistical mixture model for the range data measurement noise. The image segmentation problem is formulated as a minimization problem which jointly estimates the target boundary together with the target region range variation and background range variation directly from the noisy and anomaly-filled range data. This formulation allows direct incorporation of prior information concerning the target boundary, target ranges, and background ranges into an optimal reconstruction process. Curve evolution techniques and a generalized expectation-maximization algorithm are jointly employed as an efficient solver for minimizing the objective energy, resulting in a coupled pair of object and intensity optimization tasks. The method directly and optimally extracts the target boundary, avoiding a suboptimal two-step process involving image smoothing followed by boundary extraction. Experiments are presented demonstrating that the proposed approach is robust to anomalous pixels (missing data) and capable of producing accurate estimation of the target boundary and range values from noisy data.

In this paper, we present a complete and practical algorithm for the approximation of level-set-based curve evolution suitable for real-time implementation. In particular, we propose a two-cycle algorithm to approximate level-set-based curve evolution without the need of solving partial differential equations (PDEs). Our algorithm is applicable to a broad class of evolution speeds that can be viewed as composed of a data-dependent term and a curve smoothness regularization term. We achieve curve evolution corresponding to such evolution speeds by separating the evolution process into two different cycles: one cycle for the data-dependent term and a second cycle for the smoothness regularization. The smoothing term is derived from a Gaussian filtering process. In both cycles, the evolution is realized through a simple element switching mechanism between two linked lists, that implicitly represents the curve using an integer valued level-set function. By careful construction, all the key evolution steps require only integer operations. A consequence is that we obtain significant computation speedups compared to exact PDE-based approaches while obtaining excellent agreement with these methods for problems of practical engineering interest. In particular, the resulting algorithm is fast enough for use in real-time video processing applications, which we demonstrate through several image segmentation and video tracking experiments.

Recent development of Multi-detector Computerized Tomography (MDCT) holds the promise of a non-invasive diagnostic method to evaluate coronary artery disease as an alternative to standard angiography. A major impediment preventing its utilization in routine clinical practice is the presence of image "blooming" artifacts due to vascular calcium. For high-risk patients with serious coronary stenosis who often have significant calcified plaque, this problem becomes very acute. In this paper, we examine an approach to ameliorate the degradations due to the presence of calcium and thus increase the applicability of MDCT. We assume the blooming effect is well modeled as a linear convolutional blur. We then use this model as the basis to perform image restoration, thus deblurring the original image. The aim is to produce CT images with reduced calcium artifacts. We evaluate our method on simulated, ex vivo and clinical data and show that this method has the potential to improve coronary CT image and thus improve stenosis diagnosis.

We propose an image formation algorithm for ultrasound imaging based on sparsity-driven regularization functionals. We consider data collected by synthetic transducer arrays, with the primary motivating application being nondestructive evaluation. Our framework involves the use of a physical optics-based forward model of the observation process; the formulation of an optimization problem for image formation; and the solution of that problem through efficient numerical algorithms. Our sparsity-driven, model-based approach achieves the preservation of physical features while suppressing spurious artifacts. It also provides robust reconstructions in the case of sparse observation apertures. We demonstrate the effectiveness of our imaging strategy on real ultrasound data

The problem of determining the location and orientation of straight lines in images is of great importance in the fields of computer vision and image processing. Traditionally the Hough transform, (a special case of the Radon transform) has been widely used to solve this problem for binary images. In this paper, we pose the problem of detecting straight lines in gray-scale images as an inverse problem. Our formulation is based on use of the inverse Radon operator, which relates the parameters determining the location and orientation of the lines in the image to the noisy input image. The advantage of this formulation is that we can then approach the problem of line detection within a regularization framework and enhance the performance of the Hough-based line detector through the incorporation of prior information in the form of regularization. We discuss the type of regularizers that are useful for this problem and derive efficient computational schemes to solve the resulting optimization problems enabling their use in large applications. Finally, we show how our new approach can be alternatively viewed as one of finding an optimal representation of the noisy image in terms of elements chosen from a dictionary of lines. This interpretation relates the problem of Hough-based line finding to the body of work on adaptive signal representation.

In this paper we propose a novel implementation of the level set method that achieves real-time level-set-based video tracking. In our fast algorithm, the evolution of the curve is realized by simple operations such as switching elements between two linked lists and there is no need to solve any partial differential equations. Furthermore, a novel procedure based on Gaussian filtering is introduced to incorporate boundary smoothness regularization. By replacing the standard curve length penalty with this new smoothing procedure, further speedups are obtained. Another advantage of our fast algorithm is that the topology of the curves can be controlled easily. For the tracking of multiple objects, we extend our fast algorithm to maintain the desired topology for multiple object boundaries based on ideas from discrete topology. With our fast algorithm, a real-time system has been implemented on a standard PC and only a small fraction of the CPU power is used for tracking. Results from standard test sequences and our realtime system are presented.

In this paper, we propose a novel and fast level set method without the need for solving PDEs (partial differential equations) while preserving the advantages of level set methods, such as the automatic handling of topological changes. The foundation of our method is the direct use of an optimality condition for the final curve location based on the speed field. By testing this condition, only simple operations like insertion and deletion on two lists of boundary points are needed to evolve the curve. Our method is suitable for a set of general evolution speeds that are composed of two parts: an external speed derived from the image data and a speed term imposing boundary smoothness or regularization. In our experiments, we demonstrate that our algorithm is approximately two orders of magnitude faster than previous optimized narrow band algorithms for image segmentation tasks.

We propose a novel framework for constructing and using a shape prior in a curve evolution framework. The prior shape information is captured through shape distributions, which are histograms of features derived from the shape boundary. The resulting prior captures perceptual shape similarity, is robust to small sample size, and is flexible. We further derive a curve evolution force that corresponds to this prior. This enables us to use this prior to perform tasks such as mean shape calculation and image segmentation within a curve evolution framework.

The sensor network for monitoring distributed spatial phenomena has emerged as an area of significant practical interest. In this paper we investigate fundamental issues in detection of spatially distributed phenomena under communication constraints. The novelty of the paper is in providing a tradeoff between global performance and costs involved in communication. In particular we focus our attention on boundary estimation and develop a framework to optimize communication costs subject to worst-case misclassification guarantees. It is shown that the communication cost is primarily a function of two parameters: (1) length of the boundary; (2) overall misclassification error - which leads us to the conclusion that wireless sensor network performance is comparable to that obtained with a wired network of sensors.

We propose a new method for shape reconstruction from noisy and unorganized point data. We represent a shape through its signed distance function and formulate shape reconstruction as a constrained energy minimization problem directly based on the observed point set. The associated energy function includes both the likelihood of the observed data points and a smoothness prior on the reconstructed shape. To solve this optimization problem, an efficient data-driven level set method is developed. Our method is robust to local minima, clutter, and noise. It is also applicable to situations where the data are sparse. The topological nature of the underlying shape is handled automatically through the level set formalism.

In this paper, we propose a new variational formulation for simultaneous multiple motion segmentation and occlusion detection in an image sequence. For the representation of segmented regions, we use the multiphase level set method proposed by Vese and Chan. This method allows an efficient representation of up to 2^L regions with L level-set functions. Moreover, by construction, it enforces a domain partition with no gaps and overlaps. This is unlike previous variational approaches to multiple motion segmentation, where additional constraints were needed. The variational framework we propose can incorporate an arbitrary number of motion transformations as well as occlusion areas. In order to minimize the resulting energy, we developed a two-step algorithm. In the first step, we use a feature-based method to estimate the motions present in the image sequence. In the second step, based on the extracted motion information, we iteratively evolve all level set functions in the gradient descent direction to find the final segmentation. We have tested the above algorithm on both synthetic- and natural-motion data with very promising results. We show here segmentation results for two real video sequences.

In this paper, we develop a new approach to tomographic reconstruction problems based on geometric curve evolution techniques. We use a small set of texture coefficients to represent the object and background inhomogeneities and a contour to represent the boundary of multiple connected or unconnected objects. Instead of reconstructing pixel values on a fixed rectangular grid, we then find a reconstruction by jointly estimating these unknown contours and texture coefficients of the object and background. By designing a new "tomographic flow", the resulting problem is recast into a curve evolution problem and an efficient algorithm based on level set techniques is developed. The performance of the curve evolution method is demonstrated using examples with noisy limited-view Radon transformed data and noisy ground-penetrating radar data. The reconstruction results and computational cost are compared with those of conventional, pixel-based regularization methods. The results indicate that the curve evolution methods achieve improved shape reconstruction and have potential computation and memory advantages over conventional regularized inversion methods.

In the domain of shape modeling researchers have investigated various shape models in the abstract. In particular, Zhu and co-workers have proposed a maximum-entropy based model of object shape derived from data. In this paper we investigate the possibility of using such a shape model for image segmentation. To our knowledge, this is the first attempt at using such models to solve practical problems. One of the challenges of these models is their computational cost. We develop an efficient Monte Carlo Markov Chain (MCMC) method to train the model. A Bayesian framework is then used to perform segmentation based on the identified prior shape model and observed data. Based on limited evaluation we can demonstrate superior segmentation results relative to segmentations performed with no prior information and segmentations performed using a curve evolution approach with a generic boundary length penalizing prior.

In this paper, we propose a curve evolution method for dynamic tomography. We jointly estimate both object boundaries and intensity dynamics using all the data. The image intensity is assumed piecewise constant and the activity in each region is modeled as a parameterized dynamic process. The boundaries of objects are modeled as a series of curves. The boundaries and dynamics are estimated jointly through a curve evolution process. This curve evolution process is implemented with level set methods, which can handle topology changes easily.

We propose a method for edge-preserving regularized reconstruction in coherent imaging systems. In our framework, image formation from measured data is achieved through the minimization of a cost function, which includes nonquadratic regularizing constraints for suppressing noise artifacts, while preserving the object boundaries in the reconstruction. The cost function we use effectively deals with the complex-valued and random-phase nature of the scattered field, which is inherent in many coherent systems. We solve the challenging optimization problems posed in our framework by a novel extension of half-quadratic regularization methods. We present experimental results from three coherent imaging applications: digital holography, synthetic aperture radar, and medical ultrasound. The proposed technique produces images where coherent speckle artifacts are effectively suppressed, and boundaries between different regions in the scene are preserved.

Most multiresolution methods of segmenting images have hitherto resolved an image into successively coarser scales and discarded high-pass edge information. We present a robust and computationally tractable method for the automated segmentation of images using spatially varying kernels derived from multiscale edge information of the image. We include examples of segmentation of synthetic and real images that demonstrate the performance of the algorithm in preserving fine detail and edge information in the segmentation maps while being robust to heavy noise.

We develop a variational method for dynamic tomography using curve evolution with spatial-temporal regularization. We take into account both the dynamics of the object boundaries as well as the dynamics of region intensities. These quantities are jointly estimated through minimization of an appropriately defined energy function. The temporal correlation of the object boundary curves are included through a weighted area difference distance measure between curves. The intensity dynamics are captured through a simple autoregressive model. Curve evolution methods are used to solve for the optimal reconstruction, which are implemented with efficient level set techniques. The strength of this approach is demonstrated through the successful reconstruction of an object sequence based on observations of only a single projection angle over time.

Our work focuses on applying boundary-preserving smoothing techniques to the fusion of multiple image modalities in an effort to improve image-based classification of atherosclerotic lesions. No single imaging modality has yet demonstrated the ability to reliably detect "vulnerable" lesions. We present an approach for estimating multi-modality biomedical imagery when tissue classes are sharply delimited by boundaries. Our approach is based on the Mumford-Shah framework for edge-preserving smoothing. We exploit this framework to fuse heterogeneous sensing modalities that image unrelated physicochemical parameters of a piecewise-homogeneous tissue field. We demonstrate this approach by fusing boundary field estimates from MR and CT atherosclerotic lesion imagery into a single estimated underlying tissue boundary field, while simultaneously estimating the original imagery to better estimate tissue characteristics and structure.

Characterizing the response of the brain to a stimulus based on functional magnetic resonance imaging data is a major challenge due to the fact that the response time delay of the brain may be different from one stimulus phase to the next and from pixel to pixel. To enhance detectability, this work introduces the use of a curve evolution approach that provides separate estimates of the response time shifts at each phase of the stimulus on a pixel-by-pixel basis. The approach relies on a parsimonious but simple model that is nonlinear in the time shifts of the response relative to the stimulus and linear in the gains. To effectively use the response time shift estimates in a subspace detection framework, we implement a robust hypothesis test based on a Laplacian noise model. The algorithm provides a pixel-by-pixel functional characterization of the brain's response. The results based on experimental data show that response time shift estimates, when properly implemented, enhance detectability without sacrificing robustness.

In this paper, we develop a new active contour model for image segmentation using adaptive flows. This active contour model can be derived from minimizing a limiting form of the Mumford-Shah functional, where the segmented image is assumed to consist of piecewise constant regions. This paper is an extension of an active contour model developed by Chan-Vese. The segmentation method proposed in this paper adaptively estimates mean intensities for each separated region and uses a single curve to capture multiple regions with different intensities. The class of imagery that our new active model can handle is greater than the bimodal images. In particular, our method segments images with an arbitrary number of intensity levels and separated regions while avoiding the complexity of solving a full Mumford-Shah problem. The adaptive flow developed in this paper is easily formulated and solved using level set methods. We illustrate the performance of our segmentation methods on images generated by different modalities.

We develop a method for the formation of spotlight-mode synthetic aperture radar (SAR) images with enhanced features. The approach is based on a regularized reconstruction of the scattering field which combines a tomographic model of the SAR observation process with prior information regarding the nature of the features of interest. Compared to conventional SAR techniques, the method we propose produces images with increased resolution, reduced sidelobes, reduced speckle and easier-to-segment regions. Our technique effectively deals with the complex-valued, random-phase nature of the underlying SAR reflectivities. An efficient and robust numerical solution is achieved through extensions of half-quadratic regularization methods to the complex-valued SAR problem. We demonstrate the performance of the method on synthetic and real SAR scenes.

Motivated by work in the area of dynamic magnetic resonance imaging (MRI), we develop a new approach to the problem of reduced-order MRI acquisition. Efforts in this field have concentrated on the use of Fourier and singular value decomposition (SVD) methods to obtain low-order representations of an entire image plane. We augment this work to the case of imaging an arbitrarily-shaped region of interest (ROI) embedded within the full image. After developing a natural error metric for this problem, we show that determining the minimal order required to meet a prescribed error level is in general intractable, but can be solved under certain assumptions. We then develop an optimization approach to the related problem of minimizing the error for a given order. Finally, we demonstrate the utility of this approach and its advantages over existing Fourier and SVD methods on a number of MRI images.

In this paper, we develop a new approach to tomographic reconstruction problems based on geometric curve evolution techniques. We use a low order parametric model to describe the shape and texture of the object support as well as the background. This model uses a set of texture coefficients to represent the object and background inhomogeneities and a contour to represent the boundary of multiple connected or unconnected objects. The problem of determining the unknown contour and texture coefficients of the object and background medium is then formulated as a non-linear estimation problem. By designing a new, "tomographic flow", the resulting problem is recast into a curve evolution problem and an efficient algorithm based on level set techniques is developed. The performance of the curve evolution method is demonstrated using examples with noisy Radon transformed data and noisy ground penetrating radar data. The reconstruction results and computational cost are compared with those of conventional regularization methods. The results indicate that our curve evolution methods achieve improved shape reconstruction with reduced computation requirements.

We propose an enhanced image reconstruction method for spotlight-mode synthetic aperture radar (SAR). Our approach involves extension of feature preserving regularization techniques developed in other applications to the complex-valued SAR imaging problem. Compared to conventional image formation schemes, our approach offers increased resolvability of point-scatterers, enhancement of object shapes, reduced sidelobes and reduced speckle. We present the effectiveness of the proposed method on synthetic and real SAR scenes.

The problem of determining the location and orientation of straight lines in images is often encountered in the fields of computer vision and image processing. Traditionally the Hough transform has been widely used to solve this problem for binary images, due to it's simplicity and effectiveness. In this paper we pose the problem of detecting straight lines in gray-scale images as an inverse problem. We treat the input image as noisy observations, which are related to the underlying transform domain image through the inverse Hough transform operator. We then regularize this inverse problem using constraints that accentuate peaks in the Hough parameter space. We present four different forms of such constraints and demonstrate their effectiveness. Finally we show how our scheme can be alternatively viewed as one of finding an optimal representation of the image in terms of elements chosen from a redundant dictionary of lines, and thus is a form of adaptive signal representation.

We propose a regularization-based method for the complex-valued synthetic aperture radar (SAR) image formation problem. The method can produce images with higher resolution than that supported by the measured data, as well as images with reduced variability of reflectivity magnitudes within homogeneous regions and preserved region boundaries. This is achieved by the inclusion of prior information in the regularized data inversion process through nonquadratic potential functions. For this task, we demonstrate the use of a variety of potential functions in our framework. The technique effectively deals with the random-phase nature of the underlying SAR reflectivities. For an efficient numerical solution, we extend half-quadratic regularization methods to this complex-valued problem. We demonstrate the performance of the method on real SAR data.

The estimation of local arterial structure and vascular deformation is hampered by regions of poor contrast and by the presence of ultrasound imaging artifacts such as speckle noise and specular reflections. Conventional methods for boundary extraction and velocity estimation can exhibit poor performance under these conditions. We demonstrate a new multi-stage processing scheme which uses a directionally sensitive variational energy segmentation to suppress noise and preserve image boundaries for robust extraction of the vessel contour. This contour information is used to generate a motion region-of-interest (mROI) for a modified optical flow scheme which estimates local boundary motion based only on image data lying within the mROI.

We develop a statistical approach for imaging weak-contrast objects buried under a rough soil/air interface using data from a ground penetrating radar array. The imaging system uses short-pulse electromagnetic plane waves as illumination signals. We develop an approximate physical model based on a flat soil/air interface and weak-scattering to relate the collected data to the buried objects. This model uses ray optics to represent the soil/air interface, and a Born approximation to model the weak contrast backscattering from the buried objects. In order to form images of objects buried under rough soil/air interfaces, we embed our model in a statistical framework to account for modeling errors and ill-posedness. The statistical framework leads to optimal reconstruction algorithms for buried objects. We evaluate the performance of the reconstruction algorithms for different soil/air interface realizations using data generated from detailed nonlinear electromagnetic simulations of ground penetrating radar.

We develop a new geometric curve evolution approach to tomographic reconstruction problems including multiple objects having different levels of intensity, and background with varying intensity. We use two contours to represent the boundaries of distinct object regions with different intensity levels and a set of basis functions and expansion coefficients to describe the background. By designing coupled "tomographic flows", the problem of estimating the two contours and intensity coefficients is recast into a curve evolution problem. A new "repelling" force between the two coupled curves is proposed for the reconstruction algorithms to accelerate convergence. The performance of the coupled curve evolution method is demonstrated via a Radon inversion example. The results show that our curve evolution method achieves improved shape reconstruction over comparable regularization techniques.

This paper presents a systematic and computable method for choosing the regularization parameter appearing in Tikhonov-type regularization based on non-quadratic regularizers. First, we extend the notion of the L-curve, originally defined for quadratically regularized problems, to the case of non-quadratic functions. We then associate the optimal value of the regularization parameter for these non-quadratic problems with the corner of the resulting generalized L-curve. We identify the corner of this L-curve as the point of tangency between a straight line of arbitrary slope and the L-curve. This definition results in a corresponding algebraic equation which the optimal regularization parameter must satisfy. This algebraic equation naturally leads to an iterative algorithm for the optimal value of the regularization parameter. The convergence of this iterative algorithm is established. Simulation results confirm that the proposed method yields values of the regularization parameters that result in good reconstructions for non-quadratic problems.

We examine the problem of edge-preserving smoothing from the perspective of explicit regions rather than the explicit edge approach of Shah (1996), Ambrosio and Tortorelli (1992), and others. We reformulate Mumford and Shah's (1985) original edge-preserving smoothing functional in terms of regions rather than their boundaries. We then introduce an energy functional that is similar to Ambrosio and Tortorelli's continuous approximation of Mumford and Shah's functional, but uses a region label function rather than an edge-strength function. We also show that the spatially discretized version of our energy functional results from a particular continuation method as applied to Mumford and Shah's energy functional. We base this continuation method on Hummel and Zucker's (1983) relaxation labeling approach to solving discrete Markov random field labeling problems. We demonstrate this approach for a test case.

This paper addresses the problem of both segmenting and reconstructing a noisy signal or image. The work is motivated by large problems arising in certain scientific applications, such as medical imaging. Two objectives for a segmentation and denoising algorithm are laid out: it should be computationally efficient and capable of generating statistics for the errors in the reconstruction and estimates of the boundary locations. The starting point for the development of a suitable algorithm is a variational approach to segmentation (Shah 1992). This paper then develops a precise statistical interpretation of a one dimensional (1-D) version of this variational approach to segmentation. The 1-D algorithm that arises as a result of this analysis is computationally efficient and capable of generating error statistics. A straightforward extension of this algorithm to two dimensions would incorporate recursive procedures for computing estimates of inhomogeneous Gaussian Markov random fields. Such procedures require an unacceptably large number of operations. To meet the objective of developing a computationally efficient algorithm, the use of previously developed multiscale statistical methods is investigated. This results in the development of an algorithm for segmenting and denoising which is not only computationally efficient but also capable of generating error statistics, as desired.

Arterial diameter estimation from X-ray ciné angiograms is important for quantifying coronary artery disease (CAD) and for evaluating therapy. However, diameter measurement in vessel cross sections < or =1.0 mm is associated with large measurement errors. We present a novel diameter estimator which reduces both magnitude and variability of measurement error. We use a parametric nonlinear imaging model for X-ray ciné angiography and estimate unknown model parameters directly from the image data. Our technique allows us to exploit additional diameter information contained within the intensity profile amplitude, a feature which is overlooked by existing methods. This method uses a two-step procedure: the first step estimates the imaging model parameters directly from the angiographic frame and the second step uses these measurements to estimate the diameter of vessels in the same image. In Monte-Carlo simulation over a range of imaging conditions, our approach consistently produced lower estimation error and variability than conventional methods. With actual X-ray images, our estimator is also better than existing methods for the diameters examined (0.4-4.0 mm). These improvements are most significant in the range of narrow vessel widths associated with severe coronary artery disease.

We present here a novel multiresolution-based image segmentation algorithm. The proposed method extends and improves the Gaussian mixture model (GMM) paradigm by incorporating a multiscale correlation model of pixel dependence into the standard approach. In particular, the standard GMM is modified by introducing a multiscale neighborhood clique that incorporates the correlation between pixels in space and scale. We modify the log likelihood function of the image field by a penalization term that is derived from a multiscale neighborhood clique. Maximum likelihood (ML) estimation via the expectation-maximization (EM) algorithm is used to estimate the parameters of the new model. Then, utilizing the parameter estimates, the image field is segmented with a MAP classifier. It is demonstrated that the proposed algorithm provides superior segmentations of synthetic images, yet is computationally efficient.

We propose a method for discrimination of underlying textural structures from spotlight-mode synthetic aperture radar (SAR) returns by using a tomographic data acquisition model as the basis for statistical reasoning. We model the hypothesized textures by statistically self-similar processes and formulate the problem in a hypothesis testing framework in the SAR range profile domain without any image formation. We achieve a near-optimal, computationally efficient evaluation of the likelihood test by transforming the data into the multiscale domain.

Modeling the electrical properties of neurons (e.g. with a distributed transmission line) requires extensive knowledge of 3D morphological structure, such as dendritic trajectories from the soma. Confocal microscopy provides sharp contrast for acquiring such 3D information, but these microscopes are expensive and not widely accessible, which prompts many researchers to resort to the more accessible optical sectioning microscopy (OSM) techniques using camera lucida to recover 3D structure. Unfortunately, this method is slower and subjective because it requires extensive human intervention. We present and validate a new elegantly simple nonparametric method for semi-automated 3D spatial reconstruction of dendritic trajectories within an OSM slice. We exploit the thin structure of the dendrites and apply depth from defocus techniques for automatic range-finding in the microscope slide without specific knowledge of the optical transfer function of the microscope. Specifically, we interpolate the depth of focus which maximizes the radial high frequency energy in an image patch. In this way, we accurately and efficiently estimate the 3D structure of the dendritic arborization.

Many large multidimensional space-time signal processing and data inversion applications (e.g. deconvolution) require some form of regularization to extract meaningful information. A popular approach to regularizing such problems in a statistical context is via a Gauss-Markov random field (GMRF) prior model in a maximum a posteriori (MAP) estimation framework. While providing good reconstructions, the high dimensionality of these problems can lead to prohibitive computational constraints which limit their practical applicability, particularly in real or near-real time applications. It has been shown that GMRF models possess a particular recursive structure. Conversely, complementary work in suboptimal filtering has been based on reduced order GMRF modeling. In, this work, we combine these two results to present a suboptimal filter design which repeatedly takes advantage of this recursive GMRF structure to subdivide a large problem into a series of smaller, more tractable problems. In this way we present a method for approximate, model-based, recursive solution to such high dimensional problems based on their inherent recursive structure.

Variational methods have been employed with considerable success in computer vision, particularly for surface reconstruction problems. Formulations of this type require the solution of computationally complex Euler–Lagrange partial differential equations (PDEs) to obtain the desired reconstructions. Further, the calculation of reconstruction error covariances for such approaches are usually neglected. In this paper we describe a computationally efficient multiscale approach to surface reconstruction which differs fundamentally from other multiresolution methods that are used to solve the Euler–Lagrange PDEs. Instead, we interpret the variational problem as a statistical estimation problem in order to define a nearby, but slightly different , multiscale estimation problem that admits efficient solutions for both surface reconstruction and the calculation of error statistics. In particular, the membrane and thin-plate variational models for surfaces are interpreted as 1/ f 2 prior statistical models for the surface and its gradients, respectively. Such 1/ f 2 behavior is then achieved using a recently introduced class of multiresolution models that admits algorithms with constant per-pixel computational complexity.

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. 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.

We formulate the spotlight-mode synthetic aperture radar (SAR) imaging problem as a maximum a posteriori (MAP) estimation problem. Our statistical perspective offers an alternative to standard SAR processing techniques by incorporating prior knowledge about either the data quality or the underlying reflectivity function. In our formulation we consider exploiting the similarity between computer assisted tomography (CAT) and spotlight-mode SAR, which was first exposed by Munson et al. (1983). We present the effectiveness of the proposed approach by imagery formed from simulated spotlight-mode SAR data.

The problem of estimating the diameter of narrow coronary arteries with low measurement error and variability remains unresolved. Current methods for improving measurements increase accuracy at the expense of reproducibility and vice-versa. To address this problem, we have constructed a parametric model for X-ray cine-angiography that accurately captures the non-linearity of the process. In addition, we have derived a practical method for the estimation of the image model parameters directly from the angiographic data. These developments make it possible to use absolute image intensity values to improve the diameter estimation quality. We have characterized the performance and sensitivity of this technique by using Monte Carlo simulation and by using actual X-ray image data. Our results show that this estimator is robust to the expected range of encountered imaging conditions and produces lower estimation error and variability than existing methods. This quality enhancement is greatest for narrow vessel widths associated with severe coronary disease.

Magnetic resonance imaging (MRI) has become a widely used research and clinical tool in the study of the human brain. The ability to accurately segment the MRI data set into homogeneous regions such as gray matter, white matter, and cerebro spinal fluid aids in morphological quantification of brain features. The large amount of data associated with typical MRI brain scans makes completely manual segmentation prohibitive on a large scale. We develop an estimation-theoretic interpretation of the segmentation problem which leads to a computationally efficient, statistically based recursive technique for its solution. Being statistically based, the method also provides associated measures of uncertainty of the resulting estimates, which are useful both for evaluation of the estimates as well as their combination with other sources of information.

In this paper, we demonstrate that a window-based Gaussian mixture model can be applied in the development of a robust nonlinear filter for image restoration. Via the EM algorithm, we utilize ML estimation of the spatially-varying model parameters to achieve the desired noise suppression and detail preservation. We demonstrate that this approach is a powerful tool which gives us information about the local statistics of noisy images. We demonstrate that the estimated local statistics can be efficiently utilized for outlier detection and edge detection. The advantage of our algorithm is that it can simultaneously suppress additive Gaussian and impulsive noise, while preserving fine details and edges.

Magnetic resonance imaging (MRI) has become a widely used research and clinical tool in the study of the human brain. The ability to robustly and accurately quantify repeatable morphological measures from such data is aided by the ability to accurately segment the MRI data set into homogeneous regions such as gray matter, white matter, and cerebro spinal fluid. The large amount of data associated with typical MRI scans makes completely manual segmentation prohibitive on a large scale. In this paper an efficient approach to the segmentation of such MR imagery is presented. The approach uses an estimation-theoretic interpretation of the segmentation problem to develop a computationally efficient, statistically-based recursive technique for its solution. Being statistically based, the method also provides associated measures of uncertainty of the resulting estimates, which are extremely important both for evaluation of the estimates as well as their combination with other sources of information.

We use a natural pixel-type representation of an object, originally developed for incomplete data tomography problems, to construct nearly orthonormal multiscale basis functions. The nearly orthonormal behavior of the multiscale basis functions results in a system matrix, relating the input (the object coefficients) and the output (the projection data), which is extremely sparse. In addition, the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system. We exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well-conditioned and sparse matrix. This enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions. In case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability (MAP) multiscale reconstruction estimates based on a certain self-similar prior on the multiscale object coefficients. The framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete, irregular, and noisy, and in object feature recognition directly from projection data.

We present efficient multiscale approaches to the segmentation of natural clutter, specifically grass and forest, and to the enhancement of anomalies in synthetic aperture radar (SAR) imagery. The methods we propose exploit the coherent nature of SAR sensors. In particular, they take advantage of the characteristic statistical differences in imagery of different terrain types, as a function of scale, due to radar speckle. We employ a class of multiscale stochastic processes that provide a powerful framework for describing random processes and fields that evolve in scale. We build models representative of each category of terrain of interest (i.e., grass and forest) and employ them in directing decisions on pixel classification, segmentation, and anomalous behaviour. The scale-autoregressive nature of our models allows extremely efficient calculation of likelihoods for different terrain classifications over windows of SAR imagery. We subsequently use these likelihoods as the basis for both image pixel classification and grass-forest boundary estimation. In addition, anomaly enhancement is possible with minimal additional computation. Specifically, the residuals produced by our models in predicting SAR imagery from coarser scale images are theoretically uncorrelated. As a result, potentially anomalous pixels and regions are enhanced and pinpointed by noting regions whose residuals display a high level of correlation throughout scale. We evaluate the performance of our techniques through testing on 0.3-m resolution SAR data gathered with Lincoln Laboratory's millimeter-wave SAR.

This work presents a method for segmenting images based on gradients in the intensity function. Past approaches have centered on formulating the problem in the context of variational calculus as the minimization of a functional involving the image intensity and edge functions. Computational methods for finding the minima of such variational problems are prone to two shortfalls: they are often computationally intensive and almost always incapable of computing error statistics associated with the segmentation. Using a particular variational formulation as a starting point, this paper presents a derivation of an associated statistical formulation using multiscale models. The result is an algorithm which is fast and capable of computing error statistics.

We present an efficient multiscale approach to the segmentation of natural clutter, specifically grass and forest, in synthetic aperture imagery (SAR) and to the enhancement of anomalous image regions therein. The methods we propose exploit the coherent nature of SAR sensors. In particular, they characterize the scale-to-scale statistical differences in imagery of various terrain categories due to radar speckle. To achieve this, we employ a recently introduced class of multiscale stochastic processes that provide a powerful framework for describing random processes and fields that evolve in scale. We build models representative of each relevant category of terrain and use them to direct subsequent decisions on pixel classification, segmentation, and anomaly presence.

Anomaly characterization from tomographic measurements is of interest in a wide range of fields. In this paper we address the problems of single anomaly detection and localization which we formulate as hypothesis testing problems. While the optimal hypothesis test is easy to formulate, it is computationally infeasible due to the overwhelming number of hypotheses which must be considered. We propose the multiscale hypothesis test (MSHT) as an efficient suboptimal alternative. We show how to find decision statistics to achieve maximal composite hypothesis distinguishability for the composite hypothesis tests which comprise the MSHT.

In this work, we develop statistically based algorithms to reconstruct binary polygonal objects from sparse and noisy tomographic-based observation data. Traditional approaches to the reconstruction of geometric objects from projection data often lead to highly nonlinear estimation problems. To avoid the difficulties associated with such nonlinear problems, we first examine the problem of reconstruction of an object based on knot location measurements, i.e., measurements of the locations of abrupt change in the projections. The ties between this problem and that of multitarget radar tracking enable us to develop a sequential hypothesis-testing algorithm requiring only the solution of a series of linear estimation problems. In particular, data association hypotheses are generated, under each of which the inversion is linear. The complexity of the association possibilities are kept in check through the use of constraints on the reconstruction imposed by the tomography problem. The solution of this first problem is then used as an initialization to a more complete reconstruction which, while utilizing all the projection data, is nonlinear. We demonstrate that the estimates provided by the first, efficient algorithm are of good quality on their own, and, when combined with a fully nonlinear inversion, produce excellent object estimates.

We describe a variational framework for the tomographic reconstruction of an image from the maximum likelihood (ML) estimates of its orthogonal moments. We show how these estimated moments and their (correlated) error statistics can be computed directly, and in a linear fashion from given noisy and possibly sparse projection data. Moreover, thanks to the consistency properties of the Radon transform, this two-step approach (moment estimation followed by image reconstruction) can be viewed as a statistically optimal procedure. Furthermore, by focusing on the important role played by the moments of projection data, we immediately see the close connection between tomographic reconstruction of nonnegative valued images and the problem of nonparametric estimation of probability densities given estimates of their moments. Taking advantage of this connection, our proposed variational algorithm is based on the minimization of a cost functional composed of a term measuring the divergence between a given prior estimate of the image and the current estimate of the image and a second quadratic term based on the error incurred in the estimation of the moments of the underlying image from the noisy projection data. We show that an iterative refinement of this algorithm leads to a practical algorithm for the solution of the highly complex equality constrained divergence minimization problem. We show that this iterative refinement results in superior reconstructions of images from very noisy data as compared with the classical filtered back-projection (FBP) algorithm.

The authors represent the standard ramp filter operator of the filtered-back-projection (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 numbers of vanishing moments are used. This wavelet-based representation enables the authors to formulate a multiscale tomographic reconstruction technique in which the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. The authors' 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 one-dimensional multiscale transformation of the projection domain, and (2) the authors explicitly account for noise in the projection data by calculating maximum a posteriori probability (MAP) multiscale reconstruction estimates based on a chosen fractal prior on the multiscale object coefficients. The computational complexity of this maximum a posteriori probability (MAP) solution is also the same as that of the FBP reconstruction. This result is in contrast to commonly used methods of statistical regularization, which result in computationally intensive optimization algorithms.

A significant problem in oceanographic remote sensing is the dense gridding or smoothing of sparsely sampled altimetric data. The smoothing of altimetric measurements has an application much broader than just the regular production of elevation maps for oceanographers, however. In particular, the ability to estimate ocean circulation patterns from altimetric data can serve as an important measure for the verification of ocean acoustic tomographic results. The authors present a multiscale technique capable of extremely efficient interpolation of altimetric data: about 256000 estimates and estimation error variances are computed in one minute on a Sun Sparc-10. They also demonstrate how similar techniques may be used to directly estimate the surface gradient and biases in the geoid-model error.

In this work, a novel method of multiscale geometric feature extraction and object recognition is developed. In particular, the new representation should have the following characteristics. First, the coarse scale features should have a geometric interpretation so that the overall geometry of the object is discernible from just these features. Second, the presence of fine scale detail should not change the coarse scale representation. These two goals are not achieved by current techniques which are based on error as measured by the L/sup 2/ norm. Two methods to accomplish these goals are presented. In the first, morphological filtering and wavelet networks are used. In the second, the correlation criteria of the matching pursuit algorithm of Mallat and Zhang (1993) is modified to obtain a variable, high resolution matching pursuit.

A recently developed multiresolution estimation framework offers the possibility of highly efficient statistical analysis, interpolation, and smoothing of extremely large data sets in a multiscale fashion. This framework enjoys a number of advantages not shared by other statistically-based methods. In particular, the algorithms resulting from this framework have complexity that scales only linearly with problem size, yielding constant complexity load per grid point independent of problem size. Furthermore these algorithms directly provide interpolated estimates at multiple resolutions, accompanying error variance statistics of use in assessing resolutionlaccuracy tradeoffs and in detecting statistically significant anomalies, and maximum likelihood estimates of parameters such as spectral power law coefficients. Moreover, the efficiency of these algorithms is completely insensitive to irregularities in the sampling or spatial distribution of measurements and to heterogeneities in measurement errors or model parameters. For these reasons this approach has the potential of being an effective tool in a variety of remote sensing problems. In this paper, we demonstrate a realization of this potential by applying the multiresolution framework to a problem of considerable current interest-the interpolation and statistical analysis of ocean surface data from the TOPEXPOSEIDON altimeter

The estimation of dynamically evolving ellipsoids from noisy lower-dimensional projections is examined. In particular, this work describes a model-based approach using geometric reconstruction and recursive estimation techniques to obtain a dynamic estimate of left-ventricular ejection fraction from a gated set of planar myocardial perfusion images. The proposed approach differs from current ejection fraction estimation techniques both in the imaging modality used and in the subsequent processing which yields a dynamic ejection fraction estimate. For this work, the left ventricle is modeled as a dynamically evolving three-dimensional (3-D) ellipsoid. The left-ventricular outline observed in the myocardial perfusion images is then modeled as a dynamic, two-dimensional (2-D) ellipsoid, obtained as the projection of the former 3-D ellipsoid. This data is processed in two ways: first, as a 3-D dynamic ellipsoid reconstruction problem; second, each view is considered as a 2-D dynamic ellipse estimation problem and then the 3-D ejection fraction is obtained by combining the effective 2-D ejection fractions of each view. The approximating ellipsoids are reconstructed using a Rauch-Tung-Striebel smoothing filter, which produces an ejection fraction estimate that is more robust to noise since it is based on the entire data set; in contrast, traditional ejection fraction estimates are based only on true frames of data. Further, numerical studies of the sensitivity of this approach to unknown dynamics and projection geometry are presented, providing a rational basis for specifying system parameters. This investigation includes estimation of ejection fraction from both simulated and real data.

Proposes a technique for discrimination of fractal fields with different fractal dimensions, directly from the noisy and sparse tomographic projection data. This application is motivated from the medical field, where a change in fractal dimension is used to differentiate between normal and abnormal conditions in many different contexts, including diagnosis of liver abnormalities. The conventional method for discrimination of fractal fields from tomographic data is based on the calculation of the slope of the power spectra of the corresponding projections. This method, derived from the Radon transform results, breaks down in case the projection data are sparse and/or noisy. In order to avoid any restrictions on the duality and quantity of the projection data, we formulate our discrimination problem in a discrete hypothesis testing framework, the solution to which is given by the maximum-log-likelihood discrimination rule. The problem of discriminating fractal fields through likelihood calculations is, however, complicated by the fact that inverses and determinants of large, full, and generally ill conditioned fractal-field data covariance matrices are required. We show that these complications in the likelihood calculations can be removed by a transformation to the multiscale framework. The multiscale data covariance matrices are sparse and in addition, are naturally partitioned into ill conditioned coarsest scale approximation blocks and relatively well conditioned multiscale detail blocks. We simplify our likelihood calculations by using the class of multiscale stochastic models defined on trees to realize accurate approximations of the detail block of the data covariance matrices.>

Large data assimilation problems present a number of challenges, many of which the authors' multiscale estimation framework is capable of addressing. They demonstrate the application of their estimation framework to an oceanographic data assimilation problem of considerable interest: the smoothing of Topex/Poseidon ocean altimetric data. they produce surface estimates and error statistics on a 512/spl times/512 grid in one minute on a Sparc-10. Using a generic surface reconstruction problem they demonstrate alternative uses of the multiscale framework: higher order state models, combining estimates from an ensemble of trees, and overlapping tree models.>

In this paper, we discuss two interesting instantiations of the moment problem in image processing. The first involves the estimation of moments of an image indirectly from projections, and the reconstruction of the image from these moments. The second relates the reconstruction of binary polygons from moments to well-known algorithm in array signal processing. Through these examples, we place the moment problem into a geometric perspective and illustrate how this perspective leads to a number of interesting practical applications in image processing and other fields.>

In the computation of dense optical flow fields, spatial coherence constraints are commonly used to regularize otherwise ill-posed problem formulations, providing spatial integration of data. We present a temporal, multiframe extension of the dense optical flow estimation formulation proposed by Horn and Schunck (1981) in which we use a temporal coherence constraint to yield the optimal fusing of data from multiple frames of measurements. Conceptually, standard Kalman filtering algorithms are applicable to the resulting multiframe optical flow estimation problem, providing a solution that is sequential and recursive in time. Experiments are presented to demonstrate that the resulting multiframe estimates are more robust to noise than those provided by the original, single-frame formulation. In addition, we demonstrate cases where the aperture problem of motion vision cannot be resolved satisfactorily without the temporal integration of data enabled by the proposed formulation. Practically, the large matrix dimensions involved in the problem prohibit exact implementation of the optimal Kalman filter. To overcome this limitation, we present a computationally efficient, yet near-optimal approximation of the exact filtering algorithm. This approximation has a precise interpretation as the sequential estimation of a reduced-order spatial model for the optical flow estimation error process at each time step and arises from an estimation-theoretic treatment of the filtering problem. Experiments also demonstrate the efficacy of this near-optimal filter.

A 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. 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 spatially-varying optimal reconstruction resolution. Furthermore, if there are compelling reasons to insist upon a standard smoothness constraint, the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation. Finally, the usefulness of the approach should extend to a wide variety of ill-posed inverse problems in which variational techniques seeking a "smooth" solution are generally used.