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We study the limited data problem of the spherical Radon transform in two and three-dimensional spaces with general acquisition surfaces. In such situations, it is known that the application of filtered-backprojection reconstruction formulas might generate added artifacts and degrade the quality of reconstructions. In this article, we explicitly analyze a family of such inversion formulas, depending on a smoothing function that vanishes to order k on the boundary of the acquisition surfaces. We show that the artifacts are k orders smoother than their generating singularity. Moreover, in two-dimensional space, if the generating singularity is conormal satisfying a generic condition then the artifacts are even k+1/2 orders smoother than the generating singularity. Our analysis for three-dimensional space contains an important idea of lifting up space. We also explore the theoretical findings in a series of numerical experiments. Our experiments show that a good choice of the smoothing function leads to a significant improvement of reconstruction quality.
This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray computed tomography (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 regionof-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.
This article provides a mathematical classification of artifacts from arbitrary incom-plete X-ray tomography data when using the classical filtered backprojection algorithm. Usingmicrolocal analysis, we prove that all artifacts arise from points at the boundary of the data set.Our results show that, depending on the geometry of the data set boundary, two types of artifactscan arise: object-dependent and object-independent artifacts. The object-dependent artifacts aregenerated by singularities of the object being scanned and these artifacts can extend all along lines.This is a generalization of the streak artifacts observed in limited angle CT. The article also char-acterizes two new phenomena: the object-independent artifacts are caused only by the geometryof the data set boundary; they occur along lines if the boundary of the data set is not smooth andalong curves if the boundary of the data set is smooth. In addition to the geometric descriptionof artifacts, the article also provides characterizations of their strength in Sobolev scale in certaincases. Moreover, numerical reconstructions from simulated and real data are presented illustratingour theorems.This work is motivated by a reconstruction we present from a synchrotron data set in whichartifacts along lines appeared that were independent of the object.The results of this article apply to a wide range of well-known incomplete data problems, in-cluding limited angle CT and region of interest tomography, as well as to unconventional x-ray CTimaging setups. Some of those problems are explicitly addressed in this article, theoretically and numerically.
Reduction of variable-truncation artifacts from beam occlusion during in situ x-ray tomography
(2017)
Many in situ x-ray tomography studies require experimental rigs which may partially occlude the beam and cause parts of the projection data to be missing. In a study of fluid flow in porous chalk using a percolation cell with four metal bars drastic streak artifacts arise in the filtered backprojection (FBP) reconstruction at certain orientations. Projections with non-trivial variable truncation caused by the metal bars are the source of these variable-truncation artifacts. To understand the artifacts a mathematical model of variable-truncation data as a function of metal bar radius and distance to sample is derived and verified numerically and with experimental data. The model accurately describes the arising variable-truncation artifacts across simulated variations of the experimental setup. Three variable-truncation artifact-reduction methods are proposed, all aimed at addressing sinogram discontinuities that are shown to be the source of the streaks. The 'reduction to limited angle' (RLA) method simply keeps only non-truncated projections; the 'detector-directed smoothing' (DDS) method smooths the discontinuities; while the 'reflexive boundary condition' (RBC) method enforces a zero derivative at the discontinuities. Experimental results using both simulated and real data show that the proposed methods effectively reduce variable- truncation artifacts. The RBC method is found to provide the best artifact reduction and preservation of image features using both visual and quantitative assessment. The analysis and artifact-reduction methods are designed in context of FBP reconstruction motivated by computational efficiency practical for large, real synchrotron data. While a specific variable- truncation case is considered, the proposed methods can be applied to general data cut-offs arising in different in situ x-ray tomography experiments.
We study samples with full and partial occlusion causing streak artifacts, and propose two mod-ifications of filtered backprojection for artifact removal. Data is obtained by the SPring-8 synchrotron using a monochromatic parallel-beam scan [1]. Thresholding in the sinogram segments the metal, resulting in edges on which we apply 1) a smooth transition, or 2) a Dirichlet boundary condition.
We derive a new 3D model for magnetic particle imaging (MPI) that is able to incorporate realistic magnetic fields in the reconstruction process. In real MPI scanners, the generated magnetic fields have distortions that lead to deformed magnetic low-field volumes with the shapes of ellipsoids or bananas instead of ideal field-free points (FFP) or lines (FFL), respectively. Most of the common model-based reconstruction schemes in MPI use however the idealized assumption of an ideal FFP or FFL topology and, thus, generate artifacts in the reconstruction. Our model-based approach is able to deal with these distortions and can generally be applied to dynamic magnetic fields that are approximately parallel to their velocity field. We show how this new 3D model can be discretized and inverted algebraically in order to recover the magnetic particle concentration. To model and describe the magnetic fields, we use decompositions of the fields in spherical harmonics. We complement the description of the new model with several simulations and experiments, exploring the effects of magnetic fields distortion and reconstruction parameters on the reconstruction.
Inverse problems are at the heart of many practical problems such as image reconstruction or nondestructive testing. A characteristic feature is their instability with respect to data perturbations. To stabilize the inversion process, regularization methods must be developed and applied. In this paper, we introduce the concept of filtered diagonal frame decomposition, which extends the classical filtered SVD to the case of frames. The use of frames as generalized singular systems allows a better match to a given class of potential solutions and is also beneficial for problems where the SVD is not analytically available. We show that filtered diagonal frame decompositions yield convergent regularization methods, derive convergence rates under source conditions and prove order optimality. Our analysis applies to bounded and unbounded forward operators. As a practical application of our tools, we study filtered diagonal frame decompositions for inverting the Radon transform as an unbounded operator on L2(R2).
The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover, we derive convergence rates under source type conditions and prove order optimality under the assumption that the considered frame is a Riesz-basis.
Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging
(2018)
Magnetic particle imaging (MPI) is a promising new in vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator.