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A BOX SPLINE CALCULUS FOR COMPUTED TOMOGRAPHY Alireza Entezari1, Michael Unser2 1: CISE Department, University of Florida, Gainesville, FL 32611-6120, USA 2: Biomedical Imaging Group, EPFL, CH-1015, Lausanne, Switzerland ABSTRACT cess on them. For a parallel projection geometry, the suitable model is the Radon transform (or the X-ray transform in 3- B-splines are attractive basis functions for the continuous- D). Current reconstruction algorithms typically use the natu- domain representation of biomedical images and volumes. In ral (square) pixel basis or some blob (or Kaiser-Bessel) func- this paper, we prove that the extended family of box splines tions which have the advantage of being circularly symmetric. are closed under the Radon transform and derive explicit for- While these particular choices simplify the derivation of the mulae for their transforms. Our results are general; they cover forward model, they are not as favorable from an approxima- all known brands of compactly-supported box splines (tensor- tion theoretic point of view. Both bases have—at best—a first product B-splines, separable or not) in any dimensions. In order of approximation which implies that the rate of decay particular, we prove that the 2-D Radon transform of an N- of the discretization error as the reconstruction grid gets finer direction box spline is generally a (non-uniform) polynomial is relatively slow [2, 3]. In principle, choosing higher-order spline of degree N − 1. The proposed framework allows for a B-spline basis functions would be more advantageous since proper discretization of a variety of 2-D and 3-D tomographic these are optimal in the sense of their support being minimal reconstruction problems in a box spline basis. It is of rele- for a given order of approximation [3]. Splines have been vance for imaging modalities such as X-ray computed tomog- found to be useful for improving the performance of FBP re- raphy and 3-D cryo-electron microscopy. construction [4, 5, 6], but have hardly been deployed in the Index Terms— Radon transform, tomography, box context of iterative algorithms, probably due to the increased splines complexity of the corresponding forward model. The contribution of this work is to provide a general char- 1. INTRODUCTION acterization of the Radon (or X-ray) transform for the ex- tended family of box splines. While bringing in this level Biomedical imaging heavily relies on tomographic algorithms of generality may look like overkill at first sight, we find for the reconstruction of 2-D and 3-D images from projec- that it actually simplifies the analysis because the family hap- tion data [1]. Prominent medical imaging examples are X- pens to be closed under the Radon/X-ray transform. Since all ray computed tomography, emission tomography (PET and commonly-used brands of B-splines are special instances of SPECT), and portal imaging for radio therapy. Tomographic box splines [7], it therefore makes sense to investigate these reconstruction is also relevant for biology; for instance, for functions in detail to obtain a complete analytical picture. small animal imaging using micro scanners and molecular structure determination from 3-D cryo-electon tomography. 2. PROJECTION GEOMETRY When the projection angles are evenly distributed, the to- mogram is usually reconstructed by filtered backprojection The X-ray transform relates a function to its projections along (FBP) [1]. When the acquisition conditions are less ideal some directions θ. The projections (or shadows) are obtained (e.g., noisy and/or missing data, non-even angular distribu- by integrating the function along a set of parallel rays. In 2-D, tion of the projections), it is better to apply iterative tech- this is equivalent to performing the Radon transform. niques such as ART (algebraic reconstruction) or the statis- To specify the geometry in the general d-dimensional set- tical OSEM method, which offer greater flexibility. ting, we introduce the unit vector θ ∈ Rd that points along the A crucial step in the design of iterative reconstruction al- direction of integration. The spatial coordinates of the input gorithms is the discretization of the forward model. This function are denoted by x = (x1,x2,...,xd); these are also is usually achieved by selecting image-domain basis func- PT expressed in a rotated coordinate system as x = tθ + θ⊥ y tions and by mathematically simulating the acquisition pro- PT where θ⊥ is a d × (d − 1) matrix specifying an orthogonal basis of the hyperplane perpendicular to θ. The correspond- This work was partially supported by the Swiss National Science Foun- P dation under Grant 200020-109415, the Center for Biomedical Imaging ing hyperplane coordinates are y = (y1,...,yd−1) = θ⊥ x; (CIBM) and the foundations Leenaards and Louis-Jeantet. they are obtained by projecting x onto the basis vectors per- ) pendicular to θ. (y f y The (d-dimensional) X-ray transform P maps a function P θ f(x), x ∈ Rd into the set of its line integrals [1]. Specifically, x2 if θ ∈ Sd−1 (e.g., θ ∈ Rd with kθk = 1), then θ θ PT P{f}(y; θ) = f(tθ + θ⊥ y)dt, (1) R Z x1 with y ∈ Rd−1. In the sequel, we will also use the short-hand notation Pθf(y) for the above map. The projection geometry for the 2-D case is illustrated in Figure 1; the direction of integration is θ = (− sin θ, cos θ) and the projection matrix Fig. 1. X-ray projection geometry for d = 2. On the right: onto the Radon coordinate system is P ⊥ = [cos θ sin θ] θ projection of a pixel (tensor-product first-order B-spline) on so that y = x cos θ + x sin θ. 1 2 various angles. The X-ray transform is a bounded linear operator that has the convenient property of pseudo-commuting with the con- volution and translation operations: this (multi-) set need not be distinct as they can appear with some multiplicity. When N = d, the box spline is simply the Pθ{f ∗ g}(y) = (Pθf ∗ Pθg)(y), (2) (normalized) indicator function of the parallelepiped formed by d vectors in Rd: P d Pθ{f(· − x0)}(y) = Pθ{f}(y − θ⊥ x0). (3) 1 |det Ξ| x = n=1 tnξn for some 0 ≤ tn ≤ 1 MΞ(x) = . These properties are essential to the present work where we (0 otherwiseP are aiming at obtaining a closed-form expression for the X- For N > d, box splines are defined recursively by a “direc- ray transform of a function that is expressed in an integer-shift tional” convolution which makes them particularly suitable invariant basis for the Radon transform: 1 f(x) = ckϕ(x − k) (4) MΞ∪ξ(x) = MΞ(x − tξ)dt. (6) k∈Zd X Z0 where the generator ϕ is a tensor-product B-spline or, more When the lower dimensional space is R (i.e., d = 1), the box generally, a box spline. The direct application of (3) yields splines coincide with univariate B-splines (basic splines). When the distinct column vectors of Ξ are orthogonal to P each other, box splines amount to tensor-product B-splines. Pθf(y) = ckPθ{ϕ}(y − θ⊥ k). (5) k∈Zd The continuity and approximation power of the spline space X formed by the linear combination of shifts of a box spline This means that we can discretize the X-ray reconstruction are determined based on properties of the matrix Ξ. The problem exactly, provided that we have an explicit formula approximation properties, along with the de Boor-Hollig’s¨ for Pθ{ϕ}(y), the X-ray (or Radon) transform of the B-spline recurrence relations for fast evaluation of box splines, are generator. It is then possible to use (5) to form the system ma- discussed in [7] which is the definitive reference on the sub- H trix that relates the line integrals pm,n = Pθm f(yn) to the ject. There are several algorithms for efficient and stable B-spline coefficients of the signal ck. Note that the approach evaluation of box splines [8, 9, 10]. is also applicable to nonparallel geometries (e.g., fan beam or Another way of constructing box splines, which is prob- cone beam), as long as the measurements correspond to pure ably more transparent to engineers, is by repeated convolu- line integrals. tion of elementary line-segment-like distributions. Specifi- cally, we have 3. BOX SPLINES REVIEW Ξ M (x) = Mξ1 ∗···∗ MξN (x) (7) Geometrically, a box spline is the shadow (i.e., X-ray im- where the elementary box splines, Mξ , are Dirac-like line RN n age) of a hypercube, in , when projected to a lower- distributions supported over x = tξn with t ∈ [0, 1]. These dimensional space, Rd (N ≥ d). A box spline is defined for a elementary box splines are in direct geometric correspon- Rd set of N vectors ξ1, ξ2,..., ξN in . Each of these vectors dence (via a rotation and a proper scaling) with the primary is the shadow of an edge of the N-hypercube adjacent to its box spline Ξ origin. The matrix of directions := [ξ1 ξ2 . ξN ] com- d x pletely specifies the box spline in R . Note that the vectors in M~e1 ( ) = box(x1)δ(x2, · · · ,xd) where δ(x2, · · · ,xd) is the (d − 1)-dimensional Dirac distri- bution and 1 0 ≤ x ≤ 1 box(x) = . ξ1 (0 otherwise 2 Moreover, they integrate to 1 which is a property that is shared ξ by all box splines (and also preserved through convolution). Based on (7), one directly infers that the box splines are ′ ′ θ ξ1 ξ2 positive, compactly-supported functions. Their support is a θ zonotope, which is the Minkowski sum of N vectors in Ξ.

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