Uniform Spatial Resolution for Statistical Image Reconstruction for 3D Cone-Beam CT

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1 Uniform Spatial Resolution for Statistical Image Reconstruction for 3D Cone-Beam CT Jang Hwan Cho, Member, IEEE, and Jeffrey A. Fessler, Fellow, IEEE Department of EECS, University of Michigan, Ann Arbor, MI USA. Abstract Statistical image reconstruction methods for X-ray computed tomography (CT) provide improved spatial resolution and noise properties over conventional filtered back-projection (FBP) reconstruction along with other potential advantages such as reduced patient dose and artifacts. Regularized image reconstruction leads to spatially-variant spatial resolution because of the interaction between the system models and the regularization. Previous regularization design methods aiming to solve such issue mostly rely on the circulant approximation of the Fisher information matrix, which fails for the undersampled image locations. This paper extends the spatially-invariant regularization method of Fessler et al. [1] to 3D cone-beam CT by introducing a hypothetical scanning geometry that deals with the undersampling problem. The proposed regularization design were compared with the original method in [1] with both phantom simulation and clinical reconstructions in 3D axial X-ray CT. The proposed regularization method yields improved uniform resolution characteristics in statistical image reconstruction for axial cone-beam CT. I. INTRODUCTION STATISTICAL image reconstruction methods for X-ray computed tomography (CT) use realistic models that incorporate the statistical properties of the noise and the physics of the data acquisition system [2]. Potential advantages of statistical image reconstruction methods over FBP reconstruction have been demonstrated in terms of noise, resolution, and artifacts [3] [5]. However, there are still many factors need to be addressed to ensure the success of statistical methods in clinical applications. Diagnostic readability of the reconstructed images depends on various characteristics such as texture, resolution, noise, and artifacts. Among such characteristics, uniformity of the resolution throughout the reconstructed image is desirable. This paper proposes new modified space-variant regularization design that yields images with improved uniformity of resolution. Regularization is necessary to control noise since unregularized image reconstruction leads to excessively noisy images. However, the interaction between the regularization, system models, and statistical weighting causes the reconstructed images to have object-dependent nonuniform and anisotropic spatial resolution and noise properties, even for idealized shiftinvariant imaging systems [1]. Nonuniformity becomes severe for a short scan in cone-beam CT (CBCT), which has angular span of π +2γ where γ is the fan angle of the projection, compared to a full scan, and also for undersampled voxels 1 in This work is supported in part by GE Healthcare and CPU donations from Intel.. 3D axial or helical scanning geometries. In [1], a regularizer based on the aggregated certainty was developed for 2D PET to yield images with approximately uniform spatial resolution, and that regularizer has been used widely for other geometries and modalities [8] [12]. However, the aggregated certainty (AC) regularizer does not provide uniform resolution when applied to other modalities such as 2D fan beam CT or 3D cone beam CT because of the inherent differences in scanning geometry, such as cone angle and short scan orbit. In [9] and [11], the original AC regularizer was modified with a diagonal scaling factor for 3D PET. Recently, it was also extended to both static and multi-frame reconstruction in 3D PET by considering spatially variant and frame-dependent sensitivity [13]. This paper extends [1] by proposing a modified aggregated certainty function for 3D CT that yields uniform resolution throughout the reconstructed image including undersampled voxel locations. II. SPATIAL RESOLUTION OF STATISTICAL IMAGE RECONSTRUCTION The CT image reconstruction may be formulated as a minimization problem with a penalized weighted least squares (PWLS) objective function of the form Ψ(x) =L- (x)+r(x), L- (x) = 1 2 y Ax 2 W (1) ˆx =argminψ(x), (2) x where A is the system matrix, x = (x 1,, x N ) is the discrete vector of the imaged object, y = (y 1,,y nd ) is the noisy sinogram measurements, R(x) is a regularizer that controls the noise characteristics in the reconstructed image typically by penalizing local differences between voxels, and W =diag{w i } is a statistical weighting matrix. Aregularizationfunctionhasthefollowingtypicalform: R(x) = N j=1 κ j jl β l ψ l ((c l x)[n, m, z]), (3) κ j N l l=1 where N l is the size of the neighborhood, j l denotes the offset of the jth voxel s lth neighbor, β l is the regularization 1 In this study, full sampling do not mean that they satisfy the complete sampling conditions derived in [6], [7], but rather mean that the voxel is seen in every projection view. Thus, undersampling indicates the voxel is seen from only part of the projection views.

2 parameter that balances between the data-fitting term and the regularizer, κ j is the user-defined weight for controlling spatial resolution in the reconstructed image [1], ψ l is the potential function, we define a first-order differencing function that penalizes lth neighbor as c l [n, m, z] = δ(n, m, z) δ(n n l,m m l,z z l ) n 2 l + m 2 l +, z2 l and n l,m l,z l denote the offset of the lth neighbor. The goal of this paper is to refine the regularizer R(x), by designing κ j so that the reconstructed image has more uniform resolution. We want to improve upon the design proposed in [1] which was κ j = nd i=1 a2 ij w i nd. (4) i=1 a2 ij The local impulse response describes the spatial resolution properties. The following definition of local impulse response at jth voxel was used [14]: l j ˆx (ȳ (x true + ϵδ j )) ˆx (ȳ (x true )) lim ϵ ϵ (5) = y ˆx (y) y=ȳ(x true ) x ȳ (x) x=x trueδ j, (6) where δ j is a Kronecker impulse at jth voxel, and the gradient operations are matrices with the following elements: [ y ˆx (y)] ji = y i ˆx j (y), [ x ȳ (x)] ij = x i ȳ i (x). For simplicity we focus on quadratic regularization, for which, from (6), the local impulse response of the PWLS estimator (2) is expressed as l j =[A WA+ R] 1 A WAδ j, (7) where R is the Hessian of the regularizer R(x) [1]. In this paper, we use the constant recovery coefficient (CRC) [8] of the local impulse response to quantify spatial resolution. III. NEW REGULARIZATION DESIGNS A. Ideal Factorization for The System Matrix The aggregated certainty in [1] was developed for shift invariant systems like 2D PET. For shift-variant systems like CBCT, the formulation in [1] must be modified. The Fisher information matrix A WA is shift variant for both emission and transmission tomography, causing nonuniform properties of the reconstructed image. In [1], the system matrix A was factored into three elements as follows A = D[c i ]GD[s j ]. (8) where {c i } denote ray-dependent factors, {s j } denote voxeldependent factors, and G = {g ij } represents objectindependent geometric portion of the tomographic system response. Ideally we would like to choose {c i } and {s j } and G such that G G is approximately shift invariant. Since this representation is not unique, we can design each factors to make G G very shift-invariant. The original design presented in [1] assumed uniform voxel-dependent factors, i.e., s j =1, j, andtheray-dependentpartincludedonlynongeometric aspects such as detector efficiency and dead time. In 2D PET, this leads to geometric factors g ij for which G G is nearly shift invariant. However, this conventional choice of G leads to G G that is highly shift variant for 3D cone beam CT and even for 2D fan beam CT when half scan geometry is considered. Here, we present a new factorization of the form (8) that works in various geometries including 3D cone beam CT. First, we consider the geometric sampling properties of A and consider what rays are missing that cause A A to be shift variant. We then define G to be the N g n d system matrix corresponding to the ideal, fully sampled geometry, for which G G is approximately shift invariant. The matrix G has the same number of columns as A but has more rows (N g >n d ); the rows of A are a subset of the rows of G. Second, we replace the usual diagonal matrix D[c i ] in (8) with a n d N g matrix P [p i ] that picks the rows of the hypothetical geometry G corresponding to those of the actual geometry A. EachrowofP is entirely zero except for a single element that is unity. By ordering the rows of G appropriately, we can use P =[I nd nd (N g n d )]. Animportantproperty of the row picking matrix P is that P WP is a N g N g diagonal matrix D[ω i ] where each element ω i corresponds to a w i value for actual rays and is zero for the hypothetical rays. With above designs, we can rewrite the Fisher information matrix of the data fitting term as follows: A WA = D[s j ]G PWPGD[s j ] = D[s j ]G D[p 2 i w i ]GD[s j ]. (9) Since the Fisher information matrix is fairly concentrated near its diagonal elements [1], we can approximate (9) as where ζ j = A WA D[s j ]D[ζ j ]G GD[ζ j ]D[s j ] = D[η j ]G GD[η j ]=ΛG GΛ, (1) p2 i w i η j = s j ζ j = s j p2 i w i =, (11) i=1 a2 ij w i nd,(12) Λ D[η j ]. (13) B. Regularization with Uniform Resolution Property Substituting (1) into (7), we obtain the following approximation for the local impulse response at jth voxel: l j (ΛG GΛ+R) 1 ΛG GΛδ j = Λ 1 ( G G +Λ 1 RΛ 1) 1 G GΛδ j. (14) Typically, the local impulse response l j is concentrated about voxel j and we have Λδ j = η j δ j.soweapproximate(14)as

3 the following final expression for the local impulse response: ( l j G G + 1 η 2 j R) 1 G Gδ j. (15) Now that we have analyzed the local impulse response, we focus on designing the coefficients κ j in the regularizer (3); these coefficients affect the Hessian R in (15) and thus control the spatial resolution. Our goal here is to choose {κ j } to provide approximately uniform spatial resolution by matching the local impulse response at jth voxel, l j,toatargetlocal impulse response, l ref, i.e., l j l ref.usingtheapproximation (15), we write the target local impulse response at a reference point, such as the isocenter, as follows: l ref (G G + R ) 1 G Gδ ref, (16) where R is the Hessian of a regularizer that provides desirable spatial resolution property at the reference point. Our design for G leads to locally circulant G Gδ j,andweassume Rδ j is approximately locally shift invariant [9]. Taking Fourier transform of (15) yields the following expression for the local frequency response L j F (G Gδ j ) F (G Gδ j )+ 1 F (Rδ ηj 2 j ), (17) where F ( ) denotes the 3-D DFT. We match the local frequency response of jth voxel to the target frequency response, i.e., L j F (G Gδ j ) F (G Gδ j )+ 1 F (Rδ ηj 2 j ) F (G Gδ ref ) F (G Gδ ref )+F (R δ ref ) Lref. Cross multiplying and simplifying yields (18) 1 F (Rδ j ) F (G Gδ ref ) F (R δ ref ) F (G Gδ j ). (19) η 2 j We design κ j by minimizing the least squares difference between both sides of (19) ˆκ j =argmin κ j F (Rδ j ) F (G Gδ ref ) η 2 j F (R δ ref ) F (G Gδ j ) 2. (2) For quadratic potential function, the Fourier transform of the Hessian of the regularization can be written as [14] F (Rδ j )=κ 2 j N l l=1 β l C l (ω 1,ω 2,ω 3 ) 2 κ 2 jc ω, (21) where we assumed that κ j κ l for l within the neighborhood of j. Withoutlossofgenerality,wecanchooseβ l such that the following expression holds N l F (R δ ref )= β l C l (ω 1,ω 2,ω 3 ) 2 = C ω. (22) l=1 Substituting (21) and (22) into (2) yields the following simplified expression: ˆκ j =argmin κ j κ 2 jc ω F (G Gδ ref ) η 2 j C ω F (G Gδ j ) 2. Solving (23), we obtain (23) Cω F (G Gδ ref ),C ω F (G Gδ j ) ˆκ j = η j C ω F (G, (24) Gδ ref ) where, denotes the inner product for 3D DFT space. Since G G is approximately shift-invariant, we have F (G Gδ j ) F (G Gδ ref ), (25) then (24) simplifies to κ j = η j. Our new regularizer for uniform resolution properties in the reconstructed image (hereafter USR-REG) is given by (3) with ˆκ j = nd i=1 a2 ij w i nd i=1 a ijw i i=1 g. (26) ij New aggregated certainty function (26) has very similar form to that of the original certainty (4) proposed in [1], but with adifferentdenominator.thisnewdenominatortakeseffect when the voxel j is at an undersampled location. When it is fully-sampled, the new certainty is exactly the same as the original certainty since i g2 ij = i a2 ij for such locations. For undersampled region, this new denominator results in decreaseing regularization strength at those locations, which will lead to sharper and possibly noisier reconstructed image. Even though the new regularizer design was derived for aquadraticregularization,itcanbealsoappliedtoregularizations with non-quadratic potential functions, following the spirit of [15]. A. Phantom Simulation IV. RESULTS An anthropomorphic phantom simulation was used to demonstrate the improved spatial resolution uniformity induced by the regularization with the modified aggregated certainty (26). We used XCAT phantom [16] with voxel size x = y =.9766 mm and z =.625 mm as our true image, x true (Fig. 1). We used the separable footprint projector [17] to simulate a monoenergetic, noiseless sinogram for a 3rd-generation axial cone-beam CT system having N s =888channels and N t = 64 detector rows with spacings s =1.239 mm and t = mm. We assumed a short scan protocol that covers an angular range of degree with N a =622evenly spaced views. The measurements were obtained with monoenergetic assumption, and no noise was added. The statistical weights were w i = b exp( [Ax] i ) where the x-ray intensity b = 1 6. To obtain local impulse responses at various locations, we added impulses with the magnitude ϵ = mm 1, which corresponds to approximately 25 HU, to 6 different locations in each of the selected 9 slices (see Fig. 1 for impulse

4 isocenter isocenter (a) Regularization with original aggregated certainty (b) Regularization with proposed certainty Fig. 2. Comparison of xy plane through the center of each impulses responses at selected location (see Fig. 1 for the index of locations). Edge-preserving potential function was used. Top row is from a center slice, middle row is from 1st slice of ROI, and bottom row is from outside ROI. (a) Regularization with original aggregated certainty (4) (AC-REG) (b) Regularization with proposed certainty (26) (USR-REG) (a) (b) (c) (d) (e) (f) Fig. 3. Comparison of s through the center of each impulse response in Fig. 2. Left column is from a center slice, middle column is from 1st slice of ROI, and right column is from outside of ROI. Top and bottom rows represent the regularization with original aggregated certainty (4) and that with the proposed certainty (26), respectively. (a) AC-REG, center slice (b) AC-REG, 1st slice of ROI (c) AC-REG, outside ROI (d) USR-REG, center slice (e) USR-REG, 1st slice of ROI (f) USR-REG, outside ROI. locations in xy plane). We used (6) to evaluate the local impulse response at each location for regularizer designs with both the original aggregated certainty (4) and the modified certainty function (26). Edge-preserving regularization was investigated to show that the proposed regularizer design (USR-REG) is applicable to both cases. Image reconstruction was done using the ordered-subsets with double surrogates (OSDS) method [18] on a grid with voxel size x = y =.9766 mm and z =.625 mm. First, we present the results for regularization with a quadratic potential function. Fig. 2 illustrates that the proposed regularizer (26) leads to local impulse response functions having more uniform CRC values than the conventional aggregated certainty design (4), particularly for off-center slices. CRC values of local impulse responses at different locations become nonuniform when we use the regularization with original aggregated certainty (4). This nonuniformity becomes severe as we move away from center slices. When the proposed regularization (26) is used, the CRC values, hence the spatial resolution, become much more uniform regardless of the location or the amount of sampling. Note that the proposed regularization (26) is only correcting the nonuniformity of peak values of local impulse responses. Anisotropy in the shape of the impulse response could be

5 XCAT phantom 12 AC REG USR REG isocenter Fig. 4. Center slice of the reconstructed image of a patient. Proposed regularization USR-REG was used. Display range is [8 12] (HU). 95 Uniform (QUAD) AC REG (QUAD) USR REG (QUAD) Uniform (EDGE) AC REG (EDGE) USR REG (EDGE) Fig. 1. XCAT phantom used in the simulation. Middle 3 planes (xy, xz, and yz planes through the isocenter) are shown. Red and blue dots indicate locations of the added impulses and the isocenter, respectively. Red lines indicate the 1st and last slices of 35 mm coverage in longitudinal direction, which is our ROI. Blue line displays the location of the center slice. Finally, green lines show the 1st and last slices of 4 mm coverage. improved by designing directional coefficients [19]. Fig. 3 shows s through the center of all local impulse responses to compare CRC values more closely. Using the aggregated certainty (4) leads to resolution nonuniformity even in the center slice, primarily due to short scan geometry. Nonuniformity in resolution becomes most severe for locations 3to5thathavemuchlesssamplingcomparedtotheisocenter due to both axial cone-beam geometry and short scan orbit. Even though proposed designs were obtained from approximations, such as (15), we demonstrated that the local impulse response calculated by (6) yields the targeted results. For both quadratic and edge-preserving regularizers, the proposed designs provide nearly uniform CRC values. B. Real Clinical Data We reconstructed a clinical cardiac CT scan as a image with 7 cm field-of-view (FOV). Measurements were obtained from a 64 row axial CT scanner with a short scan protocol and 48 mas tube current. The sinogram dimension was [N s,n t,n a ]=[888, 64, 642]. We used ICD with spatially non-homogeneous updates [2] for reconstruction. We show results from both quadratic and edgepreserving regularization, for which we used the q-generalized Gaussian potential function with p =2, q =1.2, andc =1 [3] Fig. 5 compares the reconstructed images with both quadratic and edge-preserving potential functions and the following different regularizers: Uniform, AC-REG, and USR- REG. When uniform regularization is used, i.e. κ j =1, j, the reconstructed image becomes over-smoothed, illustrating the importance of the certainty function in the regularization. Fig. 5. Comparison of left and right sides of reconstructed images (separated by blue dash lines) at the last slice of ROI obtained from AC-REG and USR-REG (separated by red line). Top and bottom rows are from quadratic and edge-preserving regularizations, respectively. Display range is [8 12] (HU). Fig. 4 shows that reconstructed images from both the original aggregated certainty (4) and the proposed certainty (26) have similar image quality in terms of spatial resolution in the center slice. However, in the end slices of the ROI, the resolution nonuniformity becomes more apparent. The aggregated certainty (4) fails to provide resolution uniformity at undersampled locations, leading to visible differences in smoothness between left and right side of the reconstructed images. On the other hand, the reconstucted image with the proposed regularization (26) has more uniform resolution properties even in these under-sampled region. V. DISCUSSION We have proposed a new regularization method by modifying the aggregated certainty presented in [1] using the hypothetical geometry concept. Proposed regularization (26) improved the spatial resolution uniformity in the reconstructed image for both quadratic and edge-preserving regularizations compared to uniform and aggregated certainty regularization methods. Future work will investigate the proposed design method more thoroughly. ACKNOWLEDGEMENT The authors would like to thank J. B. Thibault and Debashish Pal for providing the scan data and valuable comments.

6 REFERENCES [1] J. A. Fessler and W. L. Rogers, Spatial resolution properties of penalized-likelihood image reconstruction methods: Space-invariant tomographs, IEEE Trans. Im. Proc., vol.5,no.9,pp ,Sep [2] J. A. Fessler, Statistical image reconstruction methods for transmission tomography, in Handbook of Medical Imaging, Volume 2. Medical Image Processing and Analysis, M. Sonka and J. M. Fitzpatrick, Eds. Bellingham: SPIE, 2, pp [3] J.-B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, A three-dimensional statistical approach to improved image quality for multi-slice helical CT, Med. Phys., vol.34,no.11,pp ,Nov.27. [4] J. Qi and R. H. Huesman, Theoretical study of lesion detectability of MAP reconstruction using computer observers, IEEE Trans. Med. Imag., vol.2,no.8,pp ,Aug.21. [5] A. Chatziioannou, J. Qi, A. Annala, A. Moore, R. M. Leahy, and S. R. Cherry, Comparison of 3D MAP and FBP algorithms for image reconstruction in micropet, IEEE Trans. Med. Imag., vol. 19, no. 5, pp , May 2. [6] S. S. Orlov, Theory of three dimensional reconstruction. I. Conditions for a complete set of projections, Sov. Phys. Crystallography, vol. 2, pp , [7] H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., vol.43,no.3,pp ,Jun [8] J. Qi and R. M. Leahy, A theoretical study of the contrast recovery and variance of MAP reconstructions from PET data, IEEE Trans. Med. Imag., vol.18,no.4,pp ,Apr [9], Resolution and noise properties of MAP reconstruction for fully 3D PET, IEEE Trans. Med. Imag., vol. 19, no. 5, pp , May 2. [1] J. Nuyts and J. A. Fessler, A penalized-likelihood image reconstruction method for emission tomography, compared to post-smoothed maximum-likelihood with matched spatial resolution, IEEE Trans. Med. Imag., vol.22,no.9,pp ,Sep.23. [11] S. Mustafovic and K. Thielemans, Object dependency of resolution in reconstruction algorithms with inter-iteration filtering applied to PET data, IEEE Trans. Med. Imag., vol. 23, no. 4, pp , Apr. 24. [12] S. Ramani and J. A. Fessler, A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction, IEEE Trans. Med. Imag., vol.31,no.3,pp ,Mar.212. [13] Y. T. Jian and R. E. Carson, Feasible uniform-resolution penalized likelihood reconstruction for static- and multi-frame 3D PET, in Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., 213. [14] H. R. Shi and J. A. Fessler, Quadratic regularization design for 2D CT, IEEE Trans. Med. Imag., vol.28,no.5,pp ,May29. [15] S. Ahn and R. M. Leahy, Analysis of resolution and noise properties of nonquadratically regularized image reconstruction methods for PET, IEEE Trans. Med. Imag., vol.27,no.3,pp ,Mar.28. [16] W. P. Segars, M. Mahesh, T. J. Beck, E. C. Frey, and B. M. W. Tsui, Realistic CT simulation using the 4D XCAT phantom, Med. Phys., vol. 35, no. 8, pp. 38 8, Aug. 28. [17] Y. Long, J. A. Fessler, and J. M. Balter, 3D forward and back-projection for X-ray CT using separable footprints, IEEE Trans. Med. Imag., vol. 29, no. 11, pp , Nov. 21. [18] J. H. Cho and J. A. Fessler, Accelerating ordered-subsets image reconstruction for X-ray CT using double surrogates, in Proc. SPIE Medical Imaging 212: Phys. Med. Im., 212,p.83131X. [19], Quadratic regularization design for 3D axial CT, in Proc. Intl. Mtg. on Fully 3D Image Recon. in Rad. and Nuc. Med, 213,pp [2] Z. Yu, J.-B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, Fast model-based X-ray CT reconstruction using spatially non-homogeneous ICD optimization, IEEE Trans. Im. Proc., vol. 2, no. 1, pp , Jan. 211.