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A Box Spline Calculus for the Discretization of Computed 1532 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 31, NO. 8, AUGUST 2012 A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems Alireza Entezari*, Member, IEEE, Masih Nilchian, Student Member, IEEE, and Michael Unser, Fellow, IEEE Abstract—B-splines are attractive basis functions for the contin- gebraic reconstruction (ART) or the statistical OSEM method, uous-domain representation of biomedical images and volumes. which offer greater flexibility. In this paper, we prove that the extended family of box splines are A crucial step in the design of iterative reconstruction algo- closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known rithms is the discretization of the forward model. This is usu- brands of compactly-supported box splines (tensor-product ally achieved by selecting image-domain basis functions and B-splines, separable or not) in any number of dimensions. The by mathematically simulating the acquisition process on them. proposed box spline approach extends to non-Cartesian lattices For a parallel projection geometry, the suitable model is the used for discretizing the image space. In particular, we prove Radon transform (or the X-ray transform in 3-D). Current re- that the 2-D Radon transform of an -direction box spline is generally a (nonuniform) polynomial spline of degree . construction algorithms typically use the natural (square) pixel The proposed framework allows for a proper discretization of a basis or some blob (or Kaiser–Bessel) functions which have the variety of tomographic reconstruction problems in a box spline advantage of being circularly symmetric. While these particular basis. It is of relevance for imaging modalities such as X-ray choices simplify the derivation of the forward model, they are computed tomography and cryo-electron microscopy. We provide not as favorable from an approximation theoretic point of view. experimental results that demonstrate the practical advantages of the box spline formulation for improving the quality and efficiency Both bases have—at best—a first order of approximation which of tomographic reconstruction algorithms. implies that the rate of decay of the discretization error as the re- construction grid gets finer is relatively slow [17], [23]. In prin- Index Terms—B-splines, box splines, computed tomography, Radon transform. ciple, choosing higher order B-spline basis functions would be more advantageous since these are optimal in the sense of their support being minimal for a given order of approximation [23]. I. INTRODUCTION Splines have been found to be useful for improving the perfor- mance of FBP reconstruction [14], [12], [22] but have hardly IOMEDICAL imaging heavily relies on tomographic been deployed in the context of iterative algorithms, probably B algorithms for the reconstruction of 2-D and 3-D im- due to the increased complexity of the corresponding forward ages from projection data [19]. Prominent medical imaging model. examples are X-ray computed tomography, emission tomog- Recently, splines have been used for determining the Hilbert raphy [positron emission tomography (PET) and single-photon transform of the sinogram signal, instead of the traditional emission computed tomography (SPECT)], and portal imaging Fourier-based method [6], [7] which is widely used for FBP. for radio therapy. Tomographic reconstruction is also relevant The problem of reconstructing a multivariate polynomial for biology, for instance, for small animal imaging using from its Radon projections is studied in [1]. This problem microscanners, optical projection tomography, and molecular offers an approximation-theoretic view for the analysis of structure determination from 3-D cryo-electron tomography. X-ray tomography algorithms. Moreover, using the orthogonal When the projection angles are evenly distributed, the tomo- expansion of multivariate polynomials (e.g., Chebyshev expan- gram is usually reconstructed by filtered back-projection (FBP) sions), it is possible to provide partial sum approximations to [20], [19]. When the acquisition conditions are less ideal (e.g., the tomography problem along with closed-form error analysis noisy and/or missing data, noneven angular distribution of the [26]. Multivariate polynomials can be expanded in box spline projections), it is better to apply iterative techniques such as al- basis (via Marsden’s identity), as spline spaces spanned by the compactly supported box splines contain multivariate poly- nomials of certain degrees (up to their approximation order). Manuscript received December 24, 2011; revised March 08, 2012; accepted March 08, 2012. Date of publication March 20, 2012; date of current version Hence, our box spline approach offers a direct method for local July 27, 2012. This work was supported in part by the U.S. National Science reconstruction of multivariate polynomials from their Radon or Foundation under Grant CCF/CIF-1018149, in part by the Swiss National Sci- X-ray projections. ence Foundation under Grant 200020-109415, and in part by the Center for Biomedical Imaging of the Geneva-Lausanne Universities and EPFL. Asterisk The contribution of this paper is to provide a general char- indicates corresponding author. acterization of the Radon (or X-ray) transform for the extended *A. Entezari is with the Computer and Information Science and Engineering family of box splines.1 While bringing in this level of gener- Department, University of Florida, Gainesville, FL 32611 USA (e-mail: [email protected]fl.edu). ality may look like overkill at first sight, we find that it actu- M. Nilchian and M. Unser are with the Biomedical Imaging Group, Ecole allysimplifies the analysis because the family happens to be Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland (e-mail: masih.nilchian@epfl.ch; michael.unser@epfl.ch). 1Apreliminary version of this approach with 2-D B-splines was presented at Digital Object Identifier 10.1109/TMI.2012.2191417 the ISBI conference. [4] 0278-0062/$31.00 © 2012 IEEE Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on July 14,2020 at 16:18:49 UTC from IEEE Xplore. Restrictions apply. ENTEZARI et al.: BOX SPLINE CALCULUS FOR THE DISCRETIZATION OF CT RECONSTRUCTION PROBLEMS 1533 transform of any spline or box spline function definedonareg- ular grid. III. DISCRETIZATION OF RADON TRANSFORM To formulate a spline-based forward model for computed to- mography, we consider an input function that is expressed in an integer-shift invariant basis (5) where the generator is a tensor-product B-spline or, more gen- erally, a box spline. This is not a restriction because there is usu- Fig. 1. X-ray projection geometry for . On the right: projection of a pixel (tensor-product first-order B-spline) on various angles. ally a one-to-one linear correspondence between the samples of a function (or pixels) and the B-spline coefficients [23]. The direct application of (4) then yields closed under the Radon/X-ray transform. Since all commonly (6) used brands of B-splines are special instances of box splines [2], it therefore makes sense to investigate these functions in detail to obtain a complete analytical picture. This means that we can discretize the X-ray reconstruction problem exactly, provided that we have an explicit formula II. PROJECTION GEOMETRY for , the X-ray (or Radon) transform of the B-spline The X-ray transform relates a function to its projections along generator. It is then possible to use (6) to form the system some directions . The projections (or shadows) are obtained by matrix that relates the line integrals to integrating the function along a set of parallel rays. In two di- the B-spline coefficients of the signal . The corresponding mensions, this is equivalent to performing the Radon transform. adjoint operator (back-projection process) is represented by To specify the geometry in the general -dimensional setting, , the transpose of the system matrix. we introduce the unit vector that points along the direc- We also note that the approach is applicable to nonparallel ge- tion of integration. The spatial coordinates of the input function ometries (e.g., fan beam or cone beam), as long as the measure- are denoted by .Let be a ments corresponds to pure line integrals. For example, one can matrix specifying an orthogonal basis of the hyperplane perpen- define a geometric transformation where each ray in a fan-beam dicular to the direction of integration . The corresponding hy- setup is transformed to a single ray in a parallel geometry. This perplane coordinates are ;theyare transformation allows us to compute the ray contribution using obtained by projecting onto the basis vectors perpendicular to in the corresponding parallel geometry. These are also expressed in a rotated coordinate system as In practice, one may rely on the analytical forms derived in this paper to precompute the B-spline projections with (1) an adequate sampling along the angular and spatial dimensions. An effective way to reduce the algorithmic cost of (6) is by The ( -dimensional) X-ray transform maps a function storing an oversampled version of these functions in a table. It is into the set of its line integrals [19]. Specifically, then possible to implement an efficient version of a spline-based if (e.g., with ), then forward model (as well as its adjoint backprojection) using the fact that the influence of a basis function is very local (sparse (2) system matrix). Specifically, one should simply scan through the projection lines hitting the detector, while forming an ap- with . In the sequel, we will also use the shorthand propriate linear combination of the B-spline coefficients in (5) notation for the previous map. The projection geometry with nonzero contributions, that is, the ones for which the sup- for the 2-D case is illustrated in Fig. 1; the direction of inte- port of the B-spline intersects the projection line .The gration is and the projection matrix onto required weights (nonzero system matrix entries) the Radon coordinate system is so that are retrieved by simple table lookup (nearest neighbor interpo- .
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