Estimating view parameters from random projections for Tomography using spherical MDS

Abstract

BackgroundDuring the past decade, the computed tomography has been successfully applied to various fields especially in medicine. The estimation of view angles for projections is necessary in some special applications of tomography, for example, the structuring of viruses using electron microscopy and the compensation of the patient's motion over long scanning period.MethodsThis work introduces a novel approach, based on the spherical multidimensional scaling (sMDS), which transforms the problem of the angle estimation to a sphere constrained embedding problem. The proposed approach views each projection as a high dimensional vector with dimensionality equal to the number of sampling points on the projection. By using SMDS, then each projection vector is embedded onto a 1D sphere which parameterizes the projection with respect to view angles in a globally consistent manner. The parameterized projections are used for the final reconstruction of the image through the inverse radon transform. The entire reconstruction process is non-iterative and computationally efficient.ResultsThe effectiveness of the sMDS is verified with various experiments, including the evaluation of the reconstruction quality from different number of projections and resistance to different noise levels. The experimental results demonstrate the efficiency of the proposed method.ConclusionOur study provides an effective technique for the solution of 2D tomography with unknown acquisition view angles. The proposed method will be extended to three dimensional reconstructions in our future work. All materials, including source code and demos, are available on https://engineering.purdue.edu/PRECISE/SMDS.

Moment-Based Angular Difference Estimation Between Two Tomographic Projections in 2D and 3D

“…In Proposition 4, we give bounds on the error due not only to the finite expansion but also to the unknown extremum moments that intervene in Eq. (6). The first result of this section is Proposition 3 that gives an upper bound on the difference between the theoretical extremum moments μ max 2 , μ min 2 and the empirical ones μ max 2 ,μ min 2 .…”

“…(6), Salzman uses odd-order moments to disambiguate the angle values. The angular difference between two projections can be estimated by subtracting the two arcsines calculated from (6). However, due to the high slope of the function x → arcsin( √ x) near the abscissae 0 and 1, the calculation of d ang (P(θ i ), P(θ j )) derived from (6) is not robust to noise when the moment of one of the projections is close to μ min 2 or μ max 2 , especially as the exact values of μ min 2 and μ max 2 cannot be known precisely.…”

“…Thus, the projection directions need to be estimated in the tomographic reconstruction process of these cases. During this process, the Euclidean distance between two projections is often used for projection set clustering or refinement [4,6,7,25]. Instead of using the Euclidean distance for projection refinement, the cross-correlation coefficient [26] can be used to measure the distances between two projections, but the projection refinements obtained using the Euclidean distance or the cross-correlation coefficient are not different as shown in [9].…”

A New Iterative Method for CT Reconstruction with Uncertain View Angles

“…Typically such methods are based on reference objects or reference instruments for the calibration. Recently, in the case of uncertain or unknown view angles a few reconstruction methods that only use the measured sinogram without reference objects or instruments have been proposed, see [7][8][9]. These methods aim to estimate the view angles θ in addition to the CT image x, and can be categorized into two groups: (1) Estimating view angles directly from projection data and then estimating the CT image and (2) simultaneously estimating view angles and CT image.…”

“…According to this result, we can estimate angles directly from complete measurements. However, if the object is partly symmetrical or the measurements are not sufficient, we cannot obtain an accurate angle estimation, see [8]. Then, due to error propagation, an inaccurate angle estimation would lead to an unsatisfactory reconstruction.…”