Spline-based Sparse Tomographic Reconstruction with Besov Priors

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1 Spline-based Sparse Tomographic Reconstruction with Besov Priors Elham Sakhaee a and Alireza Entezari a a Department of CISE, University of Florida, Gainesville, FL ABSTRACT Tomographic reconstruction from limited X-ray data is an ill-posed inverse problem. A common Bayesian approach is to search for the maximum a posteriori (MAP) estimate of the unknowns that integrates the prior knowledge, about the nature of biomedical images, into the reconstruction process. Recent results on the Bayesian inversion have shown the advantages of Besov priors for the convergence of the estimates as the discretization of the image is refined. We present a spline framework for sparse tomographic reconstruction that leverages higher-order basis functions for image discretization while incorporating Besov space priors to obtain the MAP estimate. Our method leverages tensor-product B-splines and box splines, as higher order basis functions for image discretization, that are shown to improve accuracy compared to the standard, first-order, pixel-basis. Our experiments show that the synergy produced from higher order B-splines for image discretization together with the discretization-invariant Besov priors leads to significant improvements in tomographic reconstruction. The advantages of the proposed Bayesian inversion framework are examined for image reconstruction from limited number of projections in a few-view setting. Keywords: Sparse approximation, limited-data computed tomography, tomographic reconstruction, splines, Bayesian inversion, Besov priors. 1. INTRODUCTION Tomographic reconstruction of 2-D and 3-D images from projection data is a classical problem in biomedical imaging. 1 While X-ray computed tomography (CT) is widely used in clinical setting, accurate image reconstruction from limited X-ray data continues to attract research in modeling the physics of the acquisition process 2 as well as new mathematical models for the inversion of X-ray transform. 3 In few-view CT, the number of measurements is fewer than image unknowns, due to the limited number of projection views, making the image reconstruction an ill-posed problem. Optimization-based approaches for image reconstruction in X-ray imaging as well as other image modalities (e.g., PET and SPECT) have been shown to outperform analytical methods such as conventional filtered back projection (FBP). 3 A necessary step in optimization-based algorithms is the discretization of the attenuation image which is represented by a basis expansion. Iterative methods are, then, used to solve for the unknowns in the expansion. Most current algorithms recover the discretized attenuation coefficients where the image is represented in pixel basis (i.e., indicator function of pixel). Lewitt 4 proposed spherically symmetric blob functions (e.g., Kaiser-Bessel window) as an alternative to pixel basis. However, Thevenaz et. al. 5 demonstrated that from the approximationtheoretic view point, pixel and blob function bases only have a first order of approximation. This implies that as the reconstruction grid (or the sampling step) gets finer, the approximation error decays slowly (at best order of 1) for pixel-basis and blob functions. Recently a spline-theoretic framework for discretization in tomographic reconstruction problems was introduced 6 that allows for employing basis functions with higher orders of approximation in the context of CT. 7 The proposed approach leverages generic box splines 8 (and tensor-product B-splines as a special case) as suitable E. Sakhaee and A. Entezari are currently with the Department of Computer & Information Science & Engineering, University of Florida, esakhaee@cise.ufl.edu, entezari@cise.ufl.edu Send correspondence to esakhaee@cise.ufl.edu This research was supported in part by the NSF grant CCF/CIF and the ONR grant N

2 alternative basis functions for tomographic reconstruction. The fact that box spline bases of any order have analytical Radon (or X-ray) transforms, enables us to analytically construct the forward and backward projection models in a higher-order basis representation. On the other hand, developments in compressed sensing and sparse approximation demonstrated the possibility of accurate image reconstruction from limited data by exploiting the prior knowledge in the form of sparsity in some transform domain. 3,9 11 For example, motivated by sparsity of the image gradient field, Pan et al. 3 studied total variation (TV) minimization, subject to image positivity and consistency of given X-ray measurements with the projection of the recovered image. However, TV-based approaches are favorable for piece-wise constant images. Siltanen et al. 10 integrated the TV norm into a Bayesian framework for limited-data tomographic reconstruction, where the goal is to recover an image with minimal variations as maximum a posteriori (MAP) estimate of the image. Xu et al. 11 employed dictionary learning to enforce prior knowledge of patch-based sparsity in an statistical iterative reconstruction (SIR) framework with global and adaptive dictionaries, where the update stage is performed through surrogate functionals. While the majority of existing approaches sparsely represent the attenuation map in pixel-basis, higher order representations can also be utilized in the context of sparse reconstruction (e.g., using wavelets 12 or adaptive dictionaries 13 ). The ill-posed problem of image reconstruction based on higher-order representations can also be formulated in a Bayesian framework. In addition to finding the point estimate (MAP) of the spline coefficients, such Bayesian formalism has the potential to provide a full posterior density function to be used for evaluating the confidence in approximation of the unknowns. 14 Saksman et. al. 15 demonstrated that not every choice of priors is discretization invariant in a Bayesian framework. They demonstrated that Gaussian smoothness and Besov space priors are discretization invariant in the sense that as the reconstruction grid gets finer, the MAP estimate in Bayesian inversion problem, converges. Given the measurements p, the Bayesian solution of an inverse problem is to find the posterior estimate,p(c p), of the image unknowns c, given the likelihood P(p c), and a prior knowledge P(c). The importance of discretization invariant priors lies in three concepts: 15 The posterior estimates c n (where n is the number of coefficients representing the image) must converge to a limit estimate as the reconstruction grid gets finer (n ) and hence closer to the underlying continuous object. The posterior estimates must converge to a limit estimate as the number of measurements tends to infinity. Otherwise increasing number of angles or obtaining more X-ray data leads to less accurate recovery. Ensures that the same a priori knowledge is used for all n. Otherwise a priori information will be incompatible with some discretization level. Rantala et. al. 16 employed Besov space priors, that have a close connection with Bounded Variation (BV) functions, for limited-angle CT reconstruction. Again, their reconstruction is based on pixel basis representation which, as discussed above, has a first order of approximation. We propose to exploit Besov space priors in a higher-order box spline representation for the image reconstruction problem in a Bayesian framework. This approach satisfies the discretization invariance property while benefiting from the higher order of approximation. Our experimental results show that recovering higher order box spline coefficients, in this Bayesian framework, results in a more accurate image reconstruction which in turn can be exploited for reducing the number of projection angles. 2. SPLINE-BASED BAYESIAN INVERSION In this section, we describe our spline-based Bayesian framework for limited-data tomographic reconstruction. We briefly explain Radon transform in box spline representation and why higher order B-splines or more generally box splines are suitable alternatives for pixel representation, particularly in the context of tomographic reconstruction. A brief description of Besov spaces and their connection with wavelets are also provided in this section.

3 2.1 Tomography in box spline basis The discretized image can be represented as a linear combination of the translates of a basis function: 7 f(x) = N c n ϕ(x k n ) (1) n=1 where x R 2 is the image domain coordinate, k n is the nth grid point and c := [c n ] is the vector of coefficients of the image represented in basis ϕ. For tomographic applications often the basis function is the indicator function of pixel (aka pixel basis) which coincides with the first-order box spline in our setting. Let h denote the sampling step, which implies the coefficients, c n, describing the discretized image f, are h units apart. When the reconstruction grid gets finer, i.e. h gets smaller, more details of the underlying continuous image are recovered. Based on the theory of approximation, it is shown in 5 that for a B-spline of order L, the approximation error decreases like h L, therefore, the discretized image converges rapidly to the underlying continuous function when higher order basis functions are used. In the limit case, if h approaches zero (h 0)equivalentlyn. AsdiscussedinSection1, choiceofbesovspacepriors, asthe a priori knowledge, is favorable for higher order methods, since these priors are also discretization invariant and guarantee convergence of reconstructed image to the continuous image as n. Let P θ be the projection matrix that projects f onto Radon coordinate system (parametrized as y(l)) along a direction orthogonal to [cos(θ),sin(θ)] T as shown in Fig. 1. The image domain coordinate x can be then parametrized as x = tθ+lp θ where t is a parameter along direction θ. The Radon transform of the bivariate function f(x) along the integration direction θ, can be now expressed as: P θ f(l) = f(tθ +lp θ )dt, (2) R Figure 1. Radon transform (projection) of the image f(x), represented in box spline basis ϕ(.), along direction θ Exploiting convolution and translation properties of Radon transform, 7 projection of f(x) along the direction θ can be written as: P θ f(l) = N c n P θ (ϕ)(l k n P θ ) (3) n=1 This intuitively means the projection of f (i.e., P θ f(l)) is the sum of non-uniform translations of the projected basis function P θ (ϕ) weighted by the image domain coefficients. The geometry of projection of an image represented in box spline bases ϕ is shown in Fig. 1. Entezari et. al. 6,7 derived the analytic projection of a wide range of basis functions ϕ such as the pixel basis, higher order B-splines and non-separable box splines.

4 Assuming the measurements consist of m projection angles, each discretized to k samples, the system matrix His formed suchthat itrelates eachsinogram data(i.e. P θm f(l k )) tolinear combination of boxspline coefficients c n. In a limited angle CT the number of sinogram data is less than number of coefficients to be recovered, hence the linear system of equations: Hc = p (4) with p := [P θm f(l k )], is an under-determined system and makes the image reconstruction an ill-posed inversion problem which can be solved in a Bayesian framework with assumption of a prior knowledge. 2.2 Besov space priors Motivated by the discretization invariance and sparsity promoting properties of Besov priors, 15 we assume that the image representation lies in Besov space B s p,q and hence satisfies smoothness properties of that space. Here, s denotes the smoothness index, p and q are integrability components. Besov priors promote further sparsity in wavelet domain as they enforce rapid decay of L p norm of wavelet coefficients across the scales. We are particularly interested in the case of p = 1, q = 1 and s = 1 since the norm in B 1 1,1 corresponds to total variation norm. More specifically, B 1 1,1 BV 17 and bounded variation functions are suitable for accurately estimating discontinuities in the biomedical images. 3 MAP estimate of the image coefficients is the maximum likelihood (ML) estimator (that minimizes the discrepancy with given measurements) and simultaneously satisfies desired smoothness in Besov space. The regularization parameter λ controls the level of smoothness: minimize c Hc p 2 2 +λ c B 1 1,1 (5) In other words, the optimization problem in (5) seeks image coefficients, c, that minimize the error in sinogram domain while the Besov norm is as small as possible. Müller and Siltanen discussed in 9 that B 1 1,1 norm, in its finite form, can be written as sum of weighted wavelet coefficients at different scales c B 1 1,1 = w 0 + N 1 i=0 2 i 1 j=0 2 i/2 w i,j (6) where w i,j represent the wavelet coefficient corresponding to the wavelet basis at scale i translated by j. In the following section we investigate and establish the effectiveness of higher-order basis functions when the prior knowledge is smoothness in Besov space. 3. EXPERIMENTS AND RESULTS Existing optimization-based reconstruction algorithms rely on the pixel-basis for image representation. Since the first-order box spline coincides with the pixel-basis, we compare our higher order reconstructions with the first-order box spline whose reconstruction is equivalent to that of the existing methods (e.g., 3,15 ). Moreover, we compare our Bayesian framework against solving (4) with least-squares approach as well as total variation (TV) minimization as in Experimental Setup We investigate effectiveness of linear and cubic tensor-product B-splines as well as Zwart-Powell function which is a non-separable box spline 7 in the proposed framework. We choose Haar wavelet for Besov norm implementation, as Kolehmainen, et. al. 18 demonstrated that choice of Haar wavelet leads to edge-preserving reconstruction. λ is empirically found and set to 0.4. Our ground truth images consist of an analytical brain phantom, 19 a real image of the human ankle [ and MATLAB Shepp-Logan phantom, all of size , as shown in Fig. 2(a), Fig. 5(a) and Fig. 6(a). In order for the simulated sinogram data to be close to the projection of a continuous model, we project a high resolution image and resample the sinogram at a lower sampling rate, corresponding to the discretization of the reconstruction grid. In the current implementation, (5) is solved using the l 1 solver provided as L1-LS package [

5 (a) ground truth (b) SNR:11.92 db (c) SNR:14.64 db (d) SNR:15.15 db (e) SNR:16.54 db (f) SNR:16.67 db Figure 2. Reconstructing brain phantom from only 45 projection angles (12.5% of full-angle tomography) using (b) FBP, (c) pixel-basis (first-order), (d) linear (second-order), (e) Zwart-Powell and (f) cubic (fourth-order) box spline. 3.2 Results and Discussion In the first set of experiments, we reconstruct brain phantom from only 45 projection views (12.5% of full-angle tomography). Fig. 2 provides reconstruction results for FBP, pixel-basis and higher order box splines in the proposed approach. We notice that when number of measurements is not enough for analytical approaches, i.e., FBP, the resultant image exhibits streaking artifacts and strong noise. Fig. 2(c) shows direct recovery of attenuation map as pixel values in the Bayesian framework with Besov priors. The experiments suggest that, while recovering image unknowns represented in pixel-basis improves reconstruction quality compared to FBP, higher order methods are more successful in recovering accurate images, with higher SNR values and less noise and artifacts (Fig. 2(d-f)). (a) SNR:17.85 db (b) SNR:19.07 db Figure 3. Reconstruction based on cubic box spline coefficients from 60 projection angles with (a) least-squares solution of (4) without prior knowledge and (b) integrating Besov space priors.

6 In the next experiment we compare the impact of incorporating Besov a priori knowledge into the system of equations in (4). Fig. 3 compares the result of least square solution of (4) without Besov priors (the left image) and incorporating sparsity in Besov space (the right image). We can clearly see that integrating prior knowledge of Besov space sparsity into tomographic reconstruction problem, while higher order box spline representations are used, results in more accurate reconstruction SNR (db) pixel basis 14 Linear Zwart Powell Cubic number of projection angles (a) SNR SSIM pixel basis Linear 0.8 Zwart Powell Cubic number of projection angles (b) SSIM Figure 4. Employing higher order splines in a Bayesian framework results in a more accurate reconstruction even with fewer number of angles. The graphs in Fig. 4 demonstrate effectiveness of higher order box splines in the Bayesian framework with Besov priors. The left graph presents reconstruction accuracy (SNR) of the brain phantom when the number of projection views are reduced from 90 views down to 30 views. The plot on the right demonstrates SSIM for the same experiment. We observe that higher-order methods (such as cubic box spline or Zwart-Powell function) achieve higher accuracy even with fewer number of views. For example SNR of reconstruction with 45 views is db for cubic box spline, while pixel-basis achieves lower SNR of db with 15 (25%) more projection angles (60 views). (a) ground truth (b) SNR:15.50 db (c) SNR:19.94 db Figure 5. The ground truth ankle dataset (a), and resultant images from 120 projection views for TV minimization approach (b) and proposed method with cubic box spline (c). We also examined the effectiveness of the proposed approach compared to commonly used TV minimization. In TV-based reconstruction, the image is represented in pixel-basis and the prior knowledge is sparsity in gradient domain, essentially sparsity of the image gradient magnitude (TV norm). 3,9 In 10 TV norm is integrated into a Bayesian framework for limited-data X-ray tomography, therefore, we also compare our higher-order method with this TV-based approach. The results in Fig. 5 demonstrates that Besov priors integrated to box spline representation, outperforms TV-based method for real data which contain many details and do not necessarily consist of a set of homogenous regions.

7 (a) ground truth (b) SNR:15.92 db (c) SNR:15.27 db (d) SNR:14.67 db Figure 6. Reconstruction of Shepp-Logan (a) using cubic box spline with Besov space priors, from 90 views (b), 60 views (c) and 45 views (d). The proposed method is resilient to reduction of projection views. The next experiment demonstrates the resiliency of the reconstruction using higher-order box spline (cubic) with Besov space priors, against reduction of projection views. Experimental results for Shepp-Logan image, provided in Fig. 6 demonstrates that when the projection angles are reduced from 90 views to 45 views, the reconstruction results from the proposed approach still present acceptable results and fine features are still recovered. 4. CONCLUSION We studied the effect of incorporating Besov priors into a spline tomography framework. Experimental results suggest that integrating prior knowledge of sparsity in Besov space, results in higher accuracy in a few-view tomographic reconstruction problem. Since the forward and backward system matrices are pre-computed based on the formula in (3), the computational cost of recovering higher order box spline coefficients is similar to recovering attenuation coefficients in the commonly used pixel-basis, which means we can achieve higher accuracy without sacrificing reconstruction time. Although in this research we found the box spline coefficients as the MAP estimate of a Bayesian formulation, such framework can be utilized for estimating the full posterior density function, which is more informative about the degree of confidence in the recovered coefficients and hence the resultant images. We leave this study as the future work. REFERENCES [1] F. Natterer. The mathematics of computerized tomography. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, [2] I. A Elbakri and J. Fessler. Statistical image reconstruction for polyenergetic x-ray computed tomography. IEEE T Med Img, 21(2):89 99, [3] X. Pan, E. Y Sidky, and M. Vannier. Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? Inverse Problems, 25(12):123009, [4] R. M. Lewitt. Alternatives to voxels for image representation in iterative reconstruction algorithms. Physics in Medicine and Biology, 37(3):705, [5] P. Thévenaz, T. Blu, and M. Unser. Interpolation revisited. IEEE Transactions on Medical Imaging, 19(7): , July [6] A. Entezari and M. Unser. A box spline calculus for computed tomography. In IEEE International Symposium on Biomedical Imaging, pages , [7] A. Entezari, M. Nilchian, and M. Unser. A box spline calculus for the discretization of computed tomography reconstruction problems. Medical Imaging, IEEE Transactions on, 31(8): , 2012.

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