A Box Spline Calculus for Computed Tomography

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 extended 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 T is usually achieved by selecting image-domain basis func- expressed in a rotated coordinate system as x = tθ + Pθ⊥ y PT × ( − 1) tions and by mathematically simulating the acquisition pro- where θ⊥ is a d d matrix specifying an orthogonal basis of the hyperplane perpendicular to θ. The correspond- This work was partially supported by the Swiss National Science Foun- y =( )=P x dation under Grant 200020-109415, the Center for Biomedical Imaging ing hyperplane coordinates are y1,...,yd−1 θ⊥ ; (CIBM) and the foundations Leenaards and Louis-Jeantet. they are obtained by projecting x onto the basis vectors per-

978-1-4244-4126-6/10/$25.00 ©2010 IEEE 600 ISBI 2010 ) 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 d−1 d if θ ∈ S (e.g., θ ∈ R with θ =1), then θ T θ P{f}(y; θ)= f(tθ + Pθ⊥ y)dt, (1) R 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 Pθ⊥ = [cos θ sin θ] projection of a pixel (tensor-product first-order B-spline) on so that y = x1 cos θ + x2 sin θ. various angles. The X-ray transform is a bounded linear operator that has the convenient property of pseudo-commuting with the convolution 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: d Pθ{f(·−x0)}(y)=Pθ{f}(y − Pθ⊥ x0). (3) 1 |det Ξ| x = n=1 tnξn for some 0 ≤ tn ≤ 1 MΞ(x)= . 0 These properties are essential to the present work where we otherwise 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 0 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 Pθf(y)= ckPθ{ϕ}(y − Pθ⊥ k). (5) each other, box splines amount to tensor-product B-splines. k∈Zd The continuity and approximation power of the spline space 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 = P (y ) trix that relates the line integrals pm,n θm f n 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 (x)= ∗···∗ (x) MΞ Mξ1 MξN (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 d dimensional space, R (N ≥ d). A box spline is deﬁned for a elementary box splines are in direct geometric correspon- d set of N vectors ξ1, ξ2,...,ξN in R . 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 origin. The matrix of directions Ξ := [ξ1 ξ2 ... ξN ] com- d (x)=box( ) ( ··· ) pletely speciﬁes the box spline in R . Note that the vectors in Me1 x1 δ x2, ,xd

601 where δ(x2, ··· ,xd) is the (d − 1)-dimensional Dirac distri- bution and 10≤ x ≤ 1 box(x)= . ξ1 0 otherwise ξ2 Moreover, they integrate to 1 which is a property that is shared 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. (x) x := y The center of the support of MΞ is given by Ξ,0 1 N Ξ = [cos(θ), sin(θ)] 2 n=1 ξn. The Fourier transform of the box spline is therefore given by: Fig. 2. The X-ray transform of a box spline is a box spline N ω ξ ˆ −jω,xΞ,0 , n whose directions are projections of the directions of the origi- MΞ(ω)=e sinc , 2 nal box spline onto the projected plane. On the right: a trivari- n=1 π ate box spline (a tensor-product B-spline) projected to 2-D. where ω := (ω1,...,ωd) is the multivariate frequency vector.

T Since P ⊥ ω , ξ = ω , Pθ⊥ ξ, we immediately deduce 4. X-RAY PROJECTION OF BOX SPLINES θ Pθ{ ξ}(y)= P ξ(y) that M M θ⊥ , which proves the theorem =1 Ξ =[P ξ ···P ξ ]= We now turn to our main objective, which is the derivation for N . By noting that θ⊥ 1 θ⊥ N Pθ⊥ Ξ, we are then able to transfer the result to the general of an explicit formula for Pθ{MΞ}(y) where MΞ(x) is a case using convolution properties (2) and (7). given box spline generator specified by N direction vectors ξn ∈ Ξ. In the following discussion, ω = ω ,...,ω 1 d−1 The theorem is illustrated in Figure 2. The box spline denotes the (d − 1)-variate frequency vector corresponding on the right is a trivariate tensor-product B-spline (first order) to the projection-domain spatial coordinate vector y ∈ Rd−1, whose direction vectors are (1, 0, 0), (0, 1, 0) and (0, 0, 1). while the projection geometry is the specified in Section 2. When projected to the plane orthogonal to θ, it yields a bi- Theorem 1. The X-ray transform of a d-variate box spline variate, three-direction, box spline that is a hat function with specified by the direction set, Ξ,isa(d−1)-variate box spline hexagonal support. Likewise, the X-ray transform of the tri- whose direction set, Ξ, is the geometric projection of the for- linear B-spline (second order) is again a three-direction box mer. Specifically, spline, but with multiplicity of 2. The concept carries over to higher-order tensor-product B-splines which are transformed Pθ{ Ξ}(y)= P Ξ(y) M M θ⊥ . into three-direction box splines with repeated directions, the main point being that these can be evaluated efficiently. where Pθ⊥ is the transformation matrix that geometrically projects the x-coordinate system onto the y-coordinate system perpendicular to θ. 5. EXPLICIT FORMULAE IN 2-D

Proof. We start with the derivation of the X-ray transform of For d =2, we will now show that the X-ray transforms of the elementary (Dirac-type) box spline Mξ(x) whose distri- box splines are polynomial splines of degree N − 1.The butional Fourier transform is geometric conﬁguration is the one shown in Figure 1 with

the x-to-y projection matrix given by Pθ⊥ = [cos θ sin θ]. ˆ −j ω,ξ ω, ξ Mξ(ω)=e 2 sinc . The application of Theorem 1 together with the convolution 2π formula (7) yields We can proceed geometrically by determining the “shadow” Pθ Ξ( )= ξ ∗ ξ ∗···∗ ξ ( ) of the direction vector ξ since the latter speciﬁes the support M y M 1 M 2 M N y (8) of the elementary box spline as a line segment in Rd. The al- = P ξ =[ξ ] cos +[ξ ] sin ternative is to apply the central slice theorem which states that with ξn θ⊥ n n 1 θ n 2 θ, and the Fourier transform of Pθϕ(y) corresponds to the restric- 1 y tion of ϕˆ(ω) to the hyperplane perpendicular to θ. Speciﬁ- Mξ (y)= box , n | | cally, we have that ξn ξn

=0 P (ω)= ˆ (ω) = ˆ (ω) which is a rectangular box of width ξn when ξn .Note θMξ Mξ MP ⊥ ξ . ω=PT ω θ that the convolution factors with ξn =0may be eliminated θ⊥

602 from (8) since M0(y)=δ(y). To evaluate the above convo- to evaluate. We should note, however, that the present box ( ) lution product, we write Mξn y as spline framework includes other non-separable basis functions with increased isotropy and same approximation order, ( )=Δ ( ) Mξn y ξn u y (9) but lower polynomial degree and smaller support than their tensor-product counterparts. This directly relates to the effi- f(y)−f(y−h) where Δhf(y)= |h| is the finite-difference opera- ciency issue, which deserves further investigation. tor with step h, and where u(y) is the unit-step (or Heaviside) Another concept for consideration is the geometry of the function. By substituting (9) in (8), we find that reconstruction lattice. While the common solution is to use a Cartesian grid, the 2-D hexagonal lattice and the 3-D Body Pθ Ξ( )=(Δ ∗···∗Δ )( ) M y ξ1 u ξN u y Centered Cubic (BCC) and Face Centered Cubic (FCC) lat- N−1 tices have been demonstrated to outperform the sampling ef- Δξ ···Δξ y+ = 1 N (10) ficiency of the Cartesian lattice due to their densest spectral (N − 1)! packing properties. The present box spline formalism can per- where we have used the fact that the (n +1)-fold convolu- fectly handle those lattices as well. A possible option would n y+ n n be to discretize the tomographic reconstruction problem using tion of a step function is with y+ =max(y,0) . Finally, n! the BCC/FCC box splines investigated in [10] since these are we may expand the finite-difference operators which yields a quite favorable from the sampling and computational aspects. linear expansion of PθMΞ(y) in terms of some shifted ver- N−1 sions of y+ . The result therefore implies that PθMΞ(y) is a non-uniform polynomial spline of degree N − 1,orlessif References =0 some ξn . We can also infer that this box spline function N is bell-shaped and that its support is n=1 ξn. [1] F. Natterer, The mathematics of computerized tomogra- A case of special interest is when the 2-D basis func- phy, Society for Industrial and Applied Mathematics, tion (or generator) is a tensor-product B-spline of degree Philadelphia, PA, USA, 2001. n n n: ϕ(x)=β (x1)β (x2) [11]. In the present formal- [2] E. Meijering, “Quantitative evaluation of convolution- ism, this corresponds to a box spline with directions vectors based methods for medical image interpolation,” Med. ξ1 =(1, 0) and ξ2 =(1, 0), each having a multiplicity Img. Ana., vol. 5, no. 2, pp. 111–126, June 2001. (n +1)so that N =2n +2. The specialization of (10) [3] P. Thevenaz,´ T. Blu, and M. Unser, “Interpolation revis- for these particular values yields an explicit formula for the ited,” IEEE Transactions on Medical Imaging, vol. 19, Radon transform of a separable B-spline of degree n: no. 7, pp. 739–758, July 2000. [4] M. Lautsch, “A spline inversion formula for the Radon Δn+1 Δn+1 2n+1 n n cos θ sin θ y+ transform,” SIAM J. Numer. Anal., vol. 26, no. 2, pp. Pθ{β (x1)β (x2)}(y)= (2n +1)! 456–467, 1989. [5] S. Horbelt, M. Liebling, and M. Unser, “Discretization which corresponds to the spline bikernel identified by Horbelt of the radon transform and of its inverse by spline con- et al. in [5]. The more general result (10), which is valid for volutions,” IEEE Transactions on Medical Imaging,vol. any 2-D box spline, including the Zwart-Powell element [6], 21, no. 4, pp. 363–376, 2002. is new to the best of our knowledge. Similar explicit expan- [6] M. Richter, “Use of box splines in computer tomogra- sions can be derived for 3-D to 2-D box splines [9, 10]. phy,” Computing, vol. 61, no. 2, pp. 133–150, 1998. [7] C. de Boor, K. Hollig,¨ and S. Riemenschneider, Box 6. DISCUSSION Splines,vol.98ofApplied Mathematical Sciences, Springer-Verlag, New York, 1993. The choice of the box spline generator in (4) should be dic- [8] L. Kobbelt, “Stable Evaluation of Box Splines,” Numer- tated by computational and approximation theoretic consid- ical Algorithms, vol. 14, no. 4, pp. 377–382, 1997. erations. Basis functions with larger support and smoothness [9] L. Condat and D. Van De Ville, “Three-directional usually offer better approximation quality, but they also re- box-splines: Characterization and efficient evaluation,” quire more computations. This suggests the possibility of IEEE Sig. Proc. Let., vol. 13, no. 7, pp. 417–420, 2006. a tradeoff between approximation order and the density of [10] A. Entezari, D. Van De Ville, and T. Moller,¨ “Practi- the reconstruction grid. In particular, it has been demon- cal box splines for reconstruction on the body centered strated that it is computationally advantageous asymptotically cubic lattice,” IEEE Trans. Visualization and Comp. to switch to a higher-order basis function than to increase the Graph., vol. 14, no. 2, pp. 313–328, March-April 2008. sampling rate [5]. [11] M. Unser, A. Aldroubi, and M. Eden, “B-Spline signal Tensor-product B-splines constitute a preferred set of ba- processing: Part I—Theory,” IEEE Transactions on Sig- sis functions because they are made up from univariate B- nal Processing, vol. 41, no. 2, pp. 821–833, Feb. 1993. splines building blocks which are widely studied and efficient

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