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We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such
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This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray CT with arbitrary incomplete data. We prove that these singular artifacts arise from points at the boundary of the data set. Our results show that, depending on the geometry of this boundary, two types of artifacts can arise: object-dependent and object-independent artifacts. Object-dependent artifacts are generated by singularities of the object being scanned and these artifacts can extend along lines. They generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts, on the other hand, are essentially independent of the object and take one of two forms: streaks on lines if the boundary of the data set is not smooth at a point and curved artifacts if the boundary is smooth locally. We prove that these streak and curve artifacts are the only singular artifacts that can occur for FBP in the continuous case. In addition to the geometric description of artifacts, the article provides characterizations of their strength in Sobolev scale in certain cases. The results of this article apply to the well-known incomplete data problems, including limited-angle and region-of-interest tomography, as well as to unconventional X-ray CT imaging setups that arise in new practical applications. Reconstructions from simulated and real data are analyzed to illustrate our theorems, including the reconstruction that motivated this work---a synchrotron data set in which artifacts appear on lines that have no relation to the object.
We describe our submission to the Extreme Value Analysis 2019 Data Challenge in which teams were asked to predict extremes of sea surface temperature anomaly within spatio-temporal regions of missing data. We present a computational framework which reconstructs missing data using convolutional deep neural networks. Conditioned on incomplete data, we employ autoencoder-like models as multivariate conditional distributions from which possible reconstructions of the complete dataset are sampled using imputed noise. In order to mitigate bias introduced by any one particular model, a prediction ensemble is constructed to create the final distribution of extremal values. Our method does not rely on expert knowledge in order to accurately reproduce dynamic features of a complex oceanographic system with minimal assumptions. The obtained results promise reusability and generalization to other domains.
3D shape reconstruction is essential in the navigation of minimally-invasive and auto robot-guided surgeries whose operating environments are indirect and narrow, and there have been some works that focused on reconstructing the 3D shape of the surgical organ through limited 2D information available. However, the lack and incompleteness of such information caused by intraoperative emergencies (such as bleeding) and risk control conditions have not been considered. In this paper, a novel hierarchical shape-perception network (HSPN) is proposed to reconstruct the 3D point clouds (PCs) of specific brains from one single incomplete image with low latency. A tree-structured predictor and several hierarchical attention pipelines are constructed to generate point clouds that accurately describe the incomplete images and then complete these point clouds with high quality. Meanwhile, attention gate blocks (AGBs) are designed to efficiently aggregate geometric local features of incomplete PCs transmitted by hierarchical attention pipelines and internal features of reconstructing point clouds. With the proposed HSPN, 3D shape perception and completion can be achieved spontaneously. Comprehensive results measured by Chamfer distance and PC-to-PC error demonstrate that the performance of the proposed HSPN outperforms other competitive methods in terms of qualitative displays, quantitative experiment, and classification evaluation.
The problem of estimating missing fragments of curves from a functional sample has been widely considered in the literature. However, a majority of the reconstruction methods rely on estimating the covariance matrix or the components of its eigendecomposition, a task that may be difficult. In particular, the accuracy of the estimation might be affected by the complexity of the covariance function and the poor availability of complete functional data. We introduce a non-parametric alternative based on a novel concept of depth for partially observed functional data. Our simulations point out that the available methods are unbeatable when the covariance function is stationary, and there is a large proportion of complete data. However, our approach was superior when considering non-stationary covariance functions or when the proportion of complete functions is scarce. Moreover, even in the most severe case of having all the functions incomplete, our method performs well meanwhile the competitors are unable. The methodology is illustrated with two real data sets: the Spanish daily temperatures observed in different weather stations and the age-specific mortality by prefectures in Japan.
In this article, we consider the limited data problem for spherical mean transform. We characterize the generation and strength of the artifacts in a reconstruction formula. In contrast to the thirds author work [Ngu15b], the observation surface considered in this article is not flat. Our results are comparable to those obtained in [Ngu15b] for flat observation surface. For the two dimensional problem, we show that the artifacts are $k$ orders smoother than the original singularities, where $k$ is vanishing order of the smoothing function. Moreover, if the original singularity is conormal, then the artifacts are $k+frac{1}{2}$ order smoother than the original singularity. We provide some numerical examples and discuss how the smoothing effects the artifacts visually. For three dimensional case, although the result is similar to that [Ngu15b], the proof is significantly different. We introduce a new idea of lifting the space.
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