We study unique recovery of cosparse signals from limited-angle tomographic measurements of two- and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality @math with respect to the discrete gradient operator. We relate @math both to the number of measurements and to a nullspace condition with respect to the measurement matrix, so as to achieve unique recovery by linear programming. These results are supported by comprehensive numerical experiments that show a high correlation of performance in practice and theoretical predictions. Despite poor properties of the measurement matrix from the viewpoint of compressed sensing, the class of uniquely recoverable signals basically seems large enough to cover practical applications, like contactless quality inspection of compound solid bodies composed of few materials.
We overcome this difficulty by adopting the recently introduced from @cite_9 , that provides an alternative viewpoint to the classical and is more suitable to the problem class considered in this paper. Our present work applies and extends the results from @cite_9 to the 3D recovery problem from few tomographic projections of three-dimensional images consisting of few homogeneous regions. We give a theoretical relation between the image and sufficient sampling, validate it empirically and conclude that TV-reconstructions of a class of synthetic phantoms exhibit a well-defined recovery curve similar to the study in @cite_25 @cite_21 .
@cite_9 @cite_21 @cite_25
2055686029 2147276092 2082344075 2101092098
We propose a probabilistic formulation of joint silhouette extraction and 3D reconstruction given a series of calibrated 2D images. Instead of segmenting each image separately in order to construct a 3D surface consistent with the estimated silhouettes, we compute the most probable 3D shape that gives rise to the observed color information. The probabilistic framework, based on Bayesian inference, enables robust 3D reconstruction by optimally taking into account the contribution of all views. We solve the arising maximum a posteriori shape inference in a globally optimal manner by convex relaxation techniques in a spatially continuous representation. For an interactively provided user input in the form of scribbles specifying foreground and background regions, we build corresponding color distributions as multivariate Gaussians and find a volume occupancy that best fits to this data in a variational sense. Compared to classical methods for silhouette-based multiview reconstruction, the proposed approach does not depend on initialization and enjoys significant resilience to violations of the model assumptions due to background clutter, specular reflections, and camera sensor perturbations. In experiments on several real-world data sets, we show that exploiting a silhouette coherency criterion in a multiview setting allows for dramatic improvements of silhouette quality over independent 2D segmentations without any significant increase of computational efforts. This results in more accurate visual hull estimation, needed by a multitude of image-based modeling approaches. We made use of recent advances in parallel computing with a GPU implementation of the proposed method generating reconstructions on volume grids of more than 20 million voxels in up to 4.41 seconds. Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper, we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modeled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed lscr2 lscr1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP, we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently. Tomographic image reconstruction, such as the reconstruction of computed tomography projection values, of tomosynthesis data, positron emission tomography or SPECT events, and of magnetic resonance imaging data is computationally very demanding. One of the most time-consuming steps is the backprojection. Recently, a novel general purpose architecture optimized for distributed computing became available: the cell broadband engine (CBE). To maximize image reconstruction speed we modified our parallel-beam backprojection algorithm [two dimensional (2D)] and our perspective backprojection algorithm [three dimensional (3D), cone beam for flat–panel detectors] and optimized the code for the CBE. The algorithms are pixel or voxel driven, run with floating point accuracy and use linear (LI) or nearest neighbor (NN) interpolation between detector elements. For the parallel-beam case, 512 projections per half rotation, 1024 detector channels, and an image of size 512 2 was used. The cone-beam backprojection performance was assessed by backprojecting a full circle scan of 512 projections of size 1024 2 into a volume of size 512 3 voxels. The field of view was chosen to completely lie within the field of measurement and the pixel or voxel size was set to correspond to the detector element size projected to the center of rotation divided by 2 . Both the PC and the CBE were clocked at 3 GHz . For the parallel backprojection of 512 projections into a 512 2 image, a throughput of 11 fps (LI) and 15 fps (NN) was measured on the PC, whereas the CBE achieved 126 fps (LI) and 165 fps (NN), respectively. The cone-beam backprojection of 512 projections into the 512 3 volume took 3.2 min on the PC and is as fast as 13.6 s on the cell. Thereby, the cell greatly outperforms today’s top-notch backprojections based on graphical processing units. Using both CBEs of our dual cell-based blade (Mercury Computer Systems) allows to 2D backproject 330 images s and one can complete the 3D cone-beam backprojection in 6.8 s . We present an approach for 3D reconstruction from multiple video streams taken by static, synchronized and calibrated cameras that is capable of enforcing temporal consistency on the reconstruction of successive frames. Our goal is to improve the quality of the reconstruction by finding corresponding pixels in subsequent frames of the same camera using optical flow, and also to at least maintain the quality of the single time-frame reconstruction when these correspondences are wrong or cannot be found. This allows us to process scenes with fast motion, occlusions and self- occlusions where optical flow fails for large numbers of pixels. To this end, we modify the belief propagation algorithm to operate on a 3D graph that includes both spatial and temporal neighbors and to be able to discard messages from outlying neighbors. We also propose methods for introducing a bias and for suppressing noise typically observed in uniform regions. The bias encapsulates information about the background and aids in achieving a temporally consistent reconstruction and in the mitigation of occlusion related errors. We present results on publicly available real video sequences. We also present quantitative comparisons with results obtained by other researchers.