The fiber orientation distribution (FOD) function is an advanced model for high angular resolution diffusion MRI, capable of representing complex crossing or fanning fiber geometries. However, the intricate mathematical structures of FOD functions pose significant challenges for data processing and analysis. Current frameworks often fail to consider fiber bundle rotation information among FOD peaks, leading to improper data processing, such as inaccurate FOD interpolation and, consequently, anatomically incorrect fiber tracking. This paper presents a novel Wasserstein-space-based framework for processing and analyzing FOD functions that systematically considers fiber-bundle-specific geometry. Our approach begins with a spherical deconvolution method to accurately detect and decompose FOD functions into single-peak lobes. These single-peak lobes are then embedded into the Wasserstein-space, where a new metric for FOD functions is defined, capable of handling rotations among peak lobes. We introduce a geometry-aware clustering method to regroup the single-peak lobes for further bundle-specific FOD processing. The proposed framework is applied to the essential task of FOD interpolation, computed as the Barycenter of the new metric, with a fast approximation method for efficient computation. Experiments conducted on synthetic data, as well as datasets from the Human Connectome Project (HCP) and the Alzheimer's Disease Neuroimaging Initiative (ADNI), demonstrate that our framework effectively handles complex fiber geometries, provides anatomically meaningful FOD interpolations, and significantly enhances the performance of FOD-based tractography.

Susceptibility-induced distortions in diffusion MRI (dMRI) data significantly affect the study of human brain fiber pathways. Most correction methods using B0 images in the opposite phase encodings (PEs) struggle to correct the distortion in certain regions due to the low intensity distribution in B0 images. Recently, some methods use complementary information from Fiber Orientation Distribution (FOD) images to further correct the residual distortion. However, these methods follow a two-step correction pipeline and require dMRI data acquisition with both PEs to estimate FOD images, limiting their applicability. To accommodate different acquisition protocols, we propose a novel distortion correction framework MAC. Specifically, by mapping the B0 image onto the intensity distribution of the FOD image, our proposed MAC extracts complementary features from multiple modalities and progressively estimates the distortion field through cross-modal attention embedding. Extensive experiments in multiple datasets demonstrated the superiority and robustness of our method in all modal combinations.

The short association fibers or U-fibers travel in the superficial white matter (SWM) beneath the cortical layer. While the U-fibers play a crucial role in various brain disorders, there is a lack of effective tools to reconstruct their highly curved trajectory from diffusion MRI (dMRI). In this work, we propose a novel surface-based framework for the probabilistic tracking of fibers on the triangular mesh representation of the SWM. By deriving a closed-form solution to transform the spherical harmonics (SPHARM) coefficients of 3D fiber orientation distributions (FODs) to local coordinate systems on each triangle, we develop a novel approach to project the FODs onto the tangent space of the SWM. After that, we utilize parallel transport to realize the intrinsic propagation of streamlines on SWM following probabilistically sampled fiber directions. Our intrinsic and surface-based method eliminates the need to perform the necessary but challenging sharp turns in 3D compared with conventional volume-based tractography methods. Using data from the Human Connectome Project (HCP), we performed quantitative comparisons to demonstrate the proposed algorithm can more effectively reconstruct the U-fibers connecting the precentral and postcentral gyrus than previous methods. Quantitative validations were then performed on post-mortem MRIs to show the reconstructed U-fibers from our method more faithfully follow the SWM than volume-based tractography. Finally, we applied our algorithm to study the parietal U-fiber connectivity changes in autosomal dominant Alzheimer's disease (ADAD) patients and successfully detected significant associations between U-fiber connectivity and disease severity.

Susceptibility induced distortion is a major artifact that affects the diffusion MRI (dMRI) data analysis. In the Human Connectome Project (HCP), the state-of-the-art method adopted to correct this kind of distortion is to exploit the displacement field from the B0 image in the reversed phase encoding images. However, both the traditional and learning-based approaches have limitations in achieving high correction accuracy in certain brain regions, such as brainstem. By utilizing the fiber orientation distribution (FOD) computed from the dMRI, we propose a novel deep learning framework named DistoRtion Correction Net (DrC-Net), which consists of the U-Net to capture the latent information from the 4D FOD images and the spatial transformer network to propagate the displacement field and back propagate the losses between the deformed FOD images. The experiments are performed on two datasets acquired with different phase encoding (PE) directions including the HCP and the Human Connectome Low Vision (HCLV) dataset. Compared to two traditional methods topup and FODReg and two deep learning methods S-Net and flow-net, the proposed method achieves significant improvements in terms of the mean squared difference (MSD) of fractional anisotropy (FA) images and minimum angular difference between two PEs in white matter and also brainstem regions. In the meantime, the proposed DrC-Net takes only several seconds to predict a displacement field, which is much faster than the FODReg method.

Tractography is an important technique that allows the in vivo reconstruction of structural connections in the brain using diffusion MRI. Although tracking algorithms have improved during the last two decades, results of validation studies and international challenges warn about the reliability of tractography and point out the need for improved algorithms. In propagationbased tracking, connections have traditionally been modeled as piece-wise linear segments. In this work, we propose a novel propagation-based tracker that is capable of generating geometrically smooth (C1) curves using parallel transport frames. Notably, our approach does not increase the complexity of the propagation problem that remains two-dimensional. Moreover, our tracker has a novel mechanism to reduce noise related propagation errors by incorporating topographic regularity of connections, a neuroanatomic property of many brain pathways. We ran extensive experiments and compared our approach against deterministic and other probabilistic algorithms. Our experiments on FiberCup and ISMRM 2015 challenge datasets as well as on 56 subjects of the Human Connectome Project show highly promising results both visually and quantitatively. Open-source implementations of the algorithm are shared publicly.publicly.

Optical Coherence Tomography Angiography (OCTA) is a novel, non-invasive imaging modality of retinal capillaries at micron resolution. Recent studies have correlated macular OCTA vascular measures with retinal disease severity and supported their use as a diagnostic tool. However, these measurements mostly rely on a few summary statistics in retinal layers or regions of interest in the two-dimensional (2D) en face projection images. To enable 3D and localized comparisons of retinal vasculature between longitudinal scans and across populations, we develop a novel approach for mapping retinal vessel density from OCTA images. We first obtain a high-quality 3D representation of OCTA-based vessel networks via curvelet-based denoising and optimally oriented flux (OOF). Then, an effective 3D retinal vessel density mapping method is proposed. In this framework, a vessel density image (VDI) is constructed by diffusing the vessel mask derived from OOF-based analysis to the entire image volume. Subsequently, we utilize a non-linear, 3D OCT image registration method to provide localized comparisons of retinal vasculature across subjects. In our experimental results, we demonstrate an application of our method for longitudinal qualitative analysis of two pathological subjects with edema during the course of clinical care. Additionally, we quantitatively validate our method on synthetic data with simulated capillary dropout, a dataset obtained from a normal control (NC) population divided into two age groups and a dataset obtained from patients with diabetic retinopathy (DR). Our results show that we can successfully detect localized vascular changes caused by simulated capillary loss, normal aging, and DR pathology even in presence of edema. These results demonstrate the potential of the proposed framework in localized detection of microvascular changes and monitoring retinal disease progression.

Diabetic retinopathy (DR) is a significant microvascular complication of diabetes mellitus and a leading cause of vision impairment in working age adults. Optical coherence tomography (OCT) is a routinely used clinical tool to observe retinal structural and thickness alterations in DR. Pathological changes that alter the normal anatomy of the retina, such as intraretinal edema, pose great challenges for conventional layer-based analysis of OCT images. We present an alternative approach for the automated analysis of OCT volumes in DR research based on nonlinear registration. In this paper, we first obtain an anatomically consistent volume of interest (VOI) in different OCT images via carefully designed masking and affine registration. After that, efficient B-spline transformations are computed using stochastic gradient descent optimization. Using the OCT volumes of normal controls, for which layer-based segmentation works well, we demonstrate the accuracy of our registration-based analysis in aligning layer boundaries. By nonlinearly registering the OCT volumes of DR subjects to an atlas constructed from normal controls and measuring the Jacobian determinant of the deformation, we can simultaneously visualize tissue contraction and expansion due to DR pathology. Tensor-based morphometry (TBM) can also be performed for quantitative analysis of local structural changes. In our experimental results, we apply our method to a dataset of 105 subjects and demonstrate that volumetric OCT registration and TBM analysis can successfully detect local retinal structural alterations due to DR.

3D optical coherence tomography angiography (OCT-A) is a novel and non-invasive imaging modality for analyzing retinal diseases. The studies of microvasculature in 2D en face projection images have been widely implemented, but comprehensive 3D analysis of OCT-A images with rich depth-resolved microvascular information is rarely considered. In this paper, we propose a robust, effective, and automatic 3D shape modeling framework to provide a high-quality 3D vessel representation and to preserve valuable 3D geometric and topological information for vessel analysis. Effective vessel enhancement and extraction steps by means of curvelet denoising and optimally oriented flux (OOF) filtering are first designed to produce 3D microvascular networks. Afterwards, a novel 3D data representation of OCT-A microvasculature is reconstructed via advanced mesh reconstruction techniques. Based on the 3D surfaces, shape analysis is established to extract novel shape-based microvascular area distortion via the Laplace-Beltrami eigen-projection. The extracted feature is integrated into a graph-cut segmentation system to categorize large vessels and small capillaries for more precise shape analysis. The proposed framework is validated on a dedicated repeated scan dataset including 260 volume images and shows high repeatability. Statistical analysis using the surface area biomarker is performed on small capillaries to avoid the effect of tailing artifact from large vessels. It shows significant differences ( ) between DR stages on 100 subjects in a OCTA-DR dataset. The proposed shape modeling and analysis framework opens the possibility for further investigating OCT-A microvasculature in a new perspective.

Precise characterization and analysis of corneal nerve fiber tortuosity are of great importance in facilitating examination and diagnosis of many eye-related diseases. In this paper we propose a fully automated method for image-level tortuosity estimation, comprising image enhancement, exponential curvature estimation, and tortuosity level classification. The image enhancement component is based on an extended Retinex model, which not only corrects imbalanced illumination and improves image contrast in an image, but also models noise explicitly to aid removal of imaging noise. Afterwards, we take advantage of exponential curvature estimation in the 3D space of positions and orientations to directly measure curvature based on the enhanced images, rather than relying on the explicit segmentation and skeletonization steps in a conventional pipeline usually with accumulated pre-processing errors. The proposed method has been applied over two corneal nerve microscopy datasets for the estimation of a tortuosity level for each image. The experimental results show that it performs better than several selected state-of-the-art methods. Furthermore, we have performed manual gradings at tortuosity level of four hundred and three corneal nerve microscopic images, and this dataset has been released for public access to facilitate other researchers in the community in carrying out further research on the same and related topics.

In the above article[1], there were two errors in the printed article that the authors want to correct.

Multiple testing for statistical maps remains a critical and challenging problem in brain mapping. Since the false discovery rate (FDR) criterion was introduced to the neuroimaging community a decade ago, many variations have been proposed, mainly to enhance detection power. However, a fundamental geometrical property known as transformation invariance has not been adequately addressed, especially for the voxel-wise FDR. Correction of multiple testing applied after spatial transformation is not necessarily equivalent to transformation applied after correction in the original space. Without the invariance property, assigning different testing spaces will yield different results. We find that normalized residuals of linear models with Gaussian noises are uniformly distributed on a unit high-dimensional sphere, independent of t-statistics and F-statistics. By defining volumetric measure in the hyper-spherical space mapped by normalized residuals, instead of the image's Euclidean space, we can achieve invariant control of the FDR under diffeomorphic transformation. This hyper-spherical measure also reflects intrinsic "volume of randomness" in signals. Experiments with synthetic, semi-synthetic and real images demonstrate that our method significantly reduces FDR inconsistency introduced by the choice of testing spaces.

The measurement of cell viability is an important and challenging topic in the cellular property researches. In this paper, we have presented a novel framework to measure the cell viability from both contour deformation and cytoplasm streaming for image sequences. The framework is distinguished by two aspects: First, we construct the appearance change field on the basis of the displacement field in order to more completely evaluate the streaming of protoplasm. Second, Temporal Bag of Words (TBoW) is introduced to capture the variation of cell viability across the temporal dimension. Experiments show that the proposed method is effective to quantize the cell viability.

Diffusion MRI offers the unique opportunity of assessing the structural connections of human brains in vivo. With the advance of diffusion MRI technology, multi-shell imaging methods are becoming increasingly practical for large scale studies and clinical application. In this work, we propose a novel method for the analysis of multi-shell diffusion imaging data by incorporating compartment models into a spherical deconvolution framework for fiber orientation distribution (FOD) reconstruction. For numerical implementation, we develop an adaptively constrained energy minimization approach to efficiently compute the solution. On simulated and real data from Human Connectome Project (HCP), we show that our method not only reconstructs sharp and clean FODs for the modeling of fiber crossings, but also generates reliable estimation of compartment parameters with great potential for clinical research of neurological diseases. In comparisons with publicly available DSI-Studio and BEDPOSTX of FSL, we demonstrate that our method reconstructs sharper FODs with more precise estimation of fiber directions. By applying probabilistic tractography to the FODs computed by our method, we show that more complete reconstruction of the corpus callosum bundle can be achieved. On a clinical, two-shell diffusion imaging data, we also demonstrate the feasibility of our method in analyzing white matter lesions.

Diffusion tensor imaging is widely used in brain connectivity research. As more and more studies recruit large numbers of subjects, it is important to design registration methods which are not only theoretically rigorous, but also computationally efficient. However, the requirement of reorienting diffusion tensors complicates and considerably slows down registration procedures, due to the correlated impacts of registration forces at adjacent voxel locations. Based on the diffeomorphic Demons algorithm (Vercauteren , 2009), we propose a fast local trust region algorithm for handling inseparable registration forces for quadratic energy functions. The method guarantees that, at any time and at any voxel location, the velocity is always within its local trust region. This local regularization allows efficient calculation of the transformation update with numeric integration instead of completely solving a large linear system at every iteration. It is able to incorporate exact reorientation and regularization into the velocity optimization, and preserve the linear complexity of the diffeomorphic Demons algorithm. In an experiment with 84 diffusion tensor images involving both pair-wise and group-wise registrations, the proposed algorithm achieves better registration in comparison with other methods solving large linear systems (Yeo , 2009). At the same time, this algorithm reduces the computation time and memory demand tenfold.

In this paper, we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research. Using the Laplace-Beltrami eigen-system, we represent each surface with an isometry invariant embedding in a high dimensional space. The key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding. By minimizing a distance measure in the embedding space with metric optimization, our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features. Besides pairwise surface maps, we also extend the metric optimization approach for group-wise atlas construction and multi-atlas cortical label fusion. In experimental results, we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies. For cortical labeling, our method achieves excellent performance in a cross-validation experiment with 40 manually labeled surfaces, and successfully models localized brain development in a pediatric study of 80 subjects. For hippocampal mapping, our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects.

In this paper we present a novel system for the automated reconstruction of cortical surfaces from T1-weighted magnetic resonance images. At the core of our system is a unified Reeb analysis framework for the detection and removal of geometric and topological outliers on tissue boundaries. Using intrinsic Reeb analysis, our system can pinpoint the location of spurious branches and topological outliers, and correct them with localized filtering using information from both image intensity distributions and geometric regularity. In this system, we have also developed enhanced tissue classification with Hessian features for improved robustness to image inhomogeneity, and adaptive interpolation to achieve sub-voxel accuracy in reconstructed surfaces. By integrating these novel developments, we have a system that can automatically reconstruct cortical surfaces with improved quality and dramatically reduced computational cost as compared with the popular FreeSurfer software. In our experiments, we demonstrate on 40 simulated MR images and the MR images of 200 subjects from two databases: the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and International Consortium of Brain Mapping (ICBM), the robustness of our method in large scale studies. In comparisons with FreeSurfer, we show that our system is able to generate surfaces that better represent cortical anatomy and produce thickness features with higher statistical power in population studies.

Corpus callosum (CC) is an important structure in human brain anatomy. In this work, we propose a fully automated and robust approach to extract corpus callosum from T1-weighted structural MR images. The novelty of our method is composed of two key steps. In the first step, we find an initial guess for the curve representation of CC by using the zero level set of the first nontrivial Laplace-Beltrami (LB) eigenfunction on the white matter surface. In the second step, the initial curve is deformed toward the final solution with a geodesic curvature flow on the white matter surface. For numerical solution of the geodesic curvature flow on surfaces, we represent the contour implicitly on a triangular mesh and develop efficient numerical schemes based on finite element method. Because our method depends only on the intrinsic geometry of the white matter surface, it is robust to orientation differences of the brain across population. In our experiments, we validate the proposed algorithm on 32 brains from a clinical study of multiple sclerosis disease and demonstrate that the accuracy of our results.

For various 3D shape analysis tasks, the Laplace-Beltrami(LB) embedding has become increasingly popular as it enables the efficient comparison of shapes based on intrinsic geometry. One fundamental difficulty in using the LB embedding, however, is the ambiguity in the eigen-system, and it is conventionally only handled in a heuristic way. In this work, we propose a novel and intrinsic metric, the spectral l2-distance, to overcome this difficulty. We prove mathematically that this new distance satisfies the conditions of a rigorous metric. Using the resulting optimal embedding determined by the spectral l2-distance, we can perform both local and global shape analysis intrinsically in the embedding space. We demonstrate this by developing a template matching approach in the optimal embedding space to solve the challenging problem of identifying major sulci on vervet cortical surfaces. In our experiments, we validate the robustness of our method by the successful identification of major sulcal lines on a large data set of 698 cortical surfaces and illustrate its potential in brain mapping studies.

We develop a computational model of shape that extends existing Riemannian models of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a formulation that accounts for regional differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals or groups, and modeling and simulation of shape dynamics. We give multiple examples of geodesic interpolations and illustrations of the use of the models in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data.

In medical shape analysis, a critical problem is reconstructing a smooth surface of correct topology from a binary mask that typically has spurious features due to segmentation artifacts. The challenge is the robust removal of these outliers without affecting the accuracy of other parts of the boundary. In this paper, we propose a novel approach for this problem based on the Laplace-Beltrami (LB) eigen-projection and properly designed boundary deformations. Using the metric distortion during the LB eigen-projection, our method automatically detects the location of outliers and feeds this information to a well-composed and topology-preserving deformation. By iterating between these two steps of outlier detection and boundary deformation, we can robustly filter out the outliers without moving the smooth part of the boundary. The final surface is the eigen-projection of the filtered mask boundary that has the correct topology, desired accuracy and smoothness. In our experiments, we illustrate the robustness of our method on different input masks of the same structure, and compare with the popular SPHARM tool and the topology preserving level set method to show that our method can reconstruct accurate surface representations without introducing artificial oscillations. We also successfully validate our method on a large data set of more than 900 hippocampal masks and demonstrate that the reconstructed surfaces retain volume information accurately.

In this paper, we propose an automated approach for the joint detection of major sulci on cortical surfaces. By representing sulci as nodes in a graphical model, we incorporate Markovian relations between sulci and formulate their detection as a maximum a posteriori (MAP) estimation problem over the joint space of major sulci. To make the inference tractable, a sample space with a finite number of candidate curves is automatically generated at each node based on the Hamilton-Jacobi skeleton of sulcal regions. Using the AdaBoost algorithm, we learn both individual and pairwise shape priors of sulcal curves from training data, which are then used to define potential functions in the graphical model based on the connection between AdaBoost and logistic regression. Finally belief propagation is used to perform the MAP inference and select the joint detection results from the sample spaces of candidate curves. In our experiments, we quantitatively validate our algorithm with manually traced curves and demonstrate the automatically detected curves can capture the main body of sulci very accurately. A comparison with independently detected results is also conducted to illustrate the advantage of the joint detection approach.

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.

In this paper, we propose a new method to construct graphical representations of cortical folding patterns by computing skeletons on triangulated cortical surfaces. In our approach, a cortical surface is first partitioned into sulcal and gyral regions via the solution of a variational problem using graph cuts, which can guarantee global optimality. After that, we extend the method of Hamilton-Jacobi skeleton [1] to subsets of triangulated surfaces, together with a geometrically intuitive pruning process that can trade off between skeleton complexity and the completeness of representing folding patterns. Compared with previous work that uses skeletons of 3-D volumes to represent sulcal patterns, the skeletons on cortical surfaces can be easily decomposed into branches and provide a simpler way to construct graphical representations of cortical morphometry. In our experiments, we demonstrate our method on two different cortical surface models, its ability of capturing major sulcal patterns and its application to compute skeletons of gyral regions.

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.

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 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 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.