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I will describe a sparse image reconstruction method for Poisson-distributed polychromatic X-ray computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. To obtain a parsimonious mean measurement-model parameterization, we first rewrite the measurement equation by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. The mass-attenuation spectrum is then expanded into basis functions using a B-spline basis of order one. We develop a block coordinate-descent algorithm for minimization of a penalized Poisson negative log-likelihood (NLL) cost function, where penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density map image. This algorithm alternates between a Nesterov’s proximal-gradient (NPG) step for estimating the density map image and a limited-memory Broyden–Fletcher–Goldfarb–Shanno with box constraints (L- BFGS-B) step for estimating the incident-spectrum parameters. To accelerate convergence of the density-map NPG steps, we apply a step-size selection scheme that accounts for varying local Lipschitz constants of the objective function. I will discuss the biconvexity of the penalized NLL function and outline preliminary results on convergence of PG-BFGS schemes. Finally, I will present real X-ray CT reconstruction examples that demonstrate the performance of the proposed scheme.

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- 1. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Blind Polychromatic X-ray CT Reconstruction from Poisson MeasurementsŽ Renliang Gu and Aleksandar Dogandžić Electrical and Computer Engineering Iowa State University Ž presented by Aleksandar Dogandžić supported by 1 / 42
- 2. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References References R. G. and A. D., “Blind polychromatic X-ray CT reconstruction from Poisson measurements,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Shanghai, China, Mar. 2016, pp. 898–902. R. G. and A. D., “Blind X-ray CT image reconstruction from polychromatic Poisson measurements,” IEEE Trans. Comput. Imag., vol. 2, no. 2, pp. 150–165, 2016. DOI: 10.1109/TCI.2016.2523431. 2 / 42
- 3. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Terminology and Notation I “ ” is the elementwise version of “ ”; B1 spline means B-spline of order 1, For a vector a D .an/ 2 RN , define nonnegativity indicator function IŒ0;C1/.a/ , ( 0; a 0 C1; o.w. I elementwise logarithm Œlnı an D ln an; 8n: 3 / 42
- 4. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Terminology and Notation II ιL.s/ is the Laplace transform of ι.Ä/: ιL .s/ , Z ι.Ä/e sÄ dÄ; Laplace transform with vector argument: bL ı.s/ D bL ı ˇ2 6 6 6 6 4 s1 s2 ::: sN 3 7 7 7 7 5 D 2 6 6 6 6 4 bL .s1/ bL .s2/ ::: bL .sN / 3 7 7 7 7 5 : 4 / 42
- 5. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References X-ray CT An X-ray computed tomography (CT) scan consists of multiple projections with the beam intensity measured by multiple detectors. Figure 1: Fan-beam CT system. 5 / 42
- 6. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Third Generation X-ray CT (a) medical CT Extremely fast and accurate For con The feat con blad to r By v requ GE R When compared with conventional cone beam CT, highly collimated fa quality CT slice result, due to the reduction of image artifacts caused b Robotic sample manipulation Turbine blade GE Jupiter linear detector array X-ray fan beam Blade rotation X-ray projection imagesAcquisition of slice projections ISOVOLT Titan 450 kV X-ray source In m on tu(b) GE’s system for CT turbine-blade inspection 6 / 42
- 7. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Introduction to CT Imaging (Parallel Beam) A detector array is deployed parallel to the t axis and rotates against the X-ray source collecting projections. Sinogram is the set of collected projections as a function of angle at which they are taken. x-ray source Â t Image Â t sinogram 7 / 42
- 8. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Iterative Reconstruction exactly the inverse of backprojection. It is per- formed by summing up the intensities along the potential ray paths for all projections through the estimated image. The set of projections (or sinogram) generated from the estimated image then is compared with and for this reason have been more slowly adopted in the clinical setting. However, as computer speeds continue to improve, and with a combination of computer acceleration techniques (e.g., parallel processors), and intelligent coding (e.g., exploiting symmetries FIGURE 16-17 Schematic illustration of the steps in iterative reconstruction. An initial image estimate is made and projections that would have been recorded from the initial estimate then are calculated by forward projection. The calculated forward projection proﬁles for the estimated image are compared to the proﬁles actually recorded from the object and the difference is used to modify the estimated image to provide a closer match. The process is repeated until Object ⍺(x,y) CT system Measured projection data sinogram p(r,φ) Image estimate ɑ *(x,y) Calculated projection data p(r,φ) Compare converged? Yes No Update image estimate Forward projection Reconstructed image from (Cherry et al. 2012) 8 / 42
- 9. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Exponential Law of Absorption The fraction dI=I of plane-wave intensity lost in traversing an infinitesimal thickness d` at Carte- sian coordinates .x; y/ is proportional to d`: dI I D .x; y; "/™ attenuation d` D Ä."/˛.x; y/š separable d` where " is photon energy and Ä."/ 0 is the mass attenuation function of the material and ˛.x; y/ 0 is the density map of the inspected object. (κ, α) Iin Iout 9 / 42
- 10. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Exponential Law of Absorption The fraction dI=I of plane-wave intensity lost in traversing an infinitesimal thickness d` at Carte- sian coordinates .x; y/ is proportional to d`: dI I D .x; y; "/™ attenuation d` D Ä."/˛.x; y/š separable d` where " is photon energy and Ä."/ 0 is the mass attenuation function of the material and ˛.x; y/ 0 is the density map of the inspected object. (κ, α) Iin Iout To obtain the intensity decrease along a straight-line path ` D `.x; y/, integrate along ` and over ". The underlying measurement model is nonlinear. 9 / 42
- 11. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Polychromatic X-ray CT Model Incident energy Iin spreads along photon energy " with density Ã."/: Z Ã."/ d" D Iin : Noiseless energy measurement obtained upon traversing a straight line ` D `.x; y/ through an object composed of a single material: Iout D l Ã."/ exp Ä Ä."/ Z ` ˛.x; y/ d` d": (κ, α) Iin Iout 10 / 42
- 12. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Problem with Linear-Model Based Reconstructions (a) (b) Figure 3: (a) FBP and (b) linear-model reconstructions from polychromatic X-ray CT measurements. 11 / 42
- 13. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Problem Formulation and Goal Assume that both o the incident spectrum Ã."/ of X-ray source and o mass attenuation function Ä."/ of the object are unknown. 12 / 42
- 14. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Problem Formulation and Goal Assume that both o the incident spectrum Ã."/ of X-ray source and o mass attenuation function Ä."/ of the object are unknown. Goal: Estimate the density map ˛.x; y/. 12 / 42
- 15. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Polychromatic X-ray CT Model Using Mass-Attenuation Spectrum Mass attenuation Ä."/ and incident spectrum density Ã."/ are both functions of ". κ(ε) ι(ε) 0 0 ε ε 13 / 42
- 16. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Polychromatic X-ray CT Model Using Mass-Attenuation Spectrum Mass attenuation Ä."/ and incident spectrum density Ã."/ are both functions of ". Idea. Write the model as integrals of Ä rather than ": Iin D Z ι.Ä/ dÄ D ιL .0/ Iout D Z ι.Ä/ exp Ä Ä Z ` ˛.x; y/ d` dÄ D ιL ÂZ ` ˛.x; y/ d` Ã : 4 Need to estimate one function, ι.Ä/, rather than two, Ã."/ and Ä."/! κ(ε) ι(ε) ∆κj ∆εj 0 0 ε ε 13 / 42
- 17. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Mass-Attenuation Spectrum κ(ε) κ ι(ε) ι(κ) ∆κj ∆κj ∆εj 0 0 0ε ε Figure 4: Relation between mass attenuation Ä, incident spectrum Ã, photon energy ", and mass attenuation spectrum ι.Ä/. 14 / 42
- 18. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Noiseless Polychromatic X-ray CT Model Using Mass-Attenuation Spectrum ι.Ä/: Summary Iin D ιL .0/ Iout D ιL ÂZ ` ˛.x; y/ d` Ã (κ, α) Iin Iout 15 / 42
- 19. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Mass-Attenuation Spectrum and Linearization Function For s > 0, the function ιL .s/ D Z C1 0 ι.Ä/e sÄ dÄ is an invertible decreasing function of s. 16 / 42
- 20. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Mass-Attenuation Spectrum and Linearization Function For s > 0, the function ιL .s/ D Z C1 0 ι.Ä/e sÄ dÄ is an invertible decreasing function of s. .ιL/ 1 converts the noiseless measurement Iout D ιL ÂZ ` ˛.x; y/ d` Ã into a linear noiseless “measurement” R ` ˛.x; y/ d`. 16 / 42
- 21. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Mass-Attenuation Spectrum and Linearization Function The .ιL/ 1 ı exp. / mapping corresponds to the linearization function in (Herman 1979) and converts ln Iout into a linear noiseless “measurement” R ` ˛.x; y/ d`. 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 Polychromaticprojections Monochromatic projections − ln ιL (·) 17 / 42
- 22. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Basis-function expansion of mass-attenuation spectrum ι.Ä/ D b.Ä/I š.Ä/ Ä š.Ä/ b.Ä/I Figure 5: B1-spline expansion ι.Ä/ D b.Ä/I, where the B1-spline basis is b.Ä/ “ 1 J D b1.Ä/; b2.Ä/; : : : ; bJ .Ä/ . ι.Ä/ 0 implies I 0. 18 / 42
- 23. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Discretization of ˛.x; y/ ˛ 0 is a p 1 vector representing the 2D image that we wish to reconstruct and 0 is a p 1 vector of known weights quantifying how much each element of ˛ contributes to the X-ray attenuation on the straight-line path `. detector array X-ray source ϕi αi R ` ˛.x; y/ d` T ˛. 19 / 42
- 24. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Multiple Measurements and Projection Matrix Denote by N the total number of measurements from all projections collected at the detector array. For the nth measurement, define its discretized line integral as T n ˛. Stacking all N such integrals into a vector yields ˆ˛’ monochromatic projection of ˛ where ˆ projection matrix D 2 6 6 6 6 4 T 1 T 2 ::: T N 3 7 7 7 7 5 N p : 20 / 42
- 25. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Noiseless Measurement Model The N 1 vector of noiseless measurements is Iout .˛; I/ D bL ı.ˆ˛/ ˜ output basis-function matrix I where bL ı.s/ D 2 6 6 6 6 4 bL .s1/ bL .s2/ ::: bL .sN / 3 7 7 7 7 5 and s D ˆ˛ is the monochromatic projection. 21 / 42
- 26. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Noise Model Assume noisy Poisson-distributed measurements E D .En/N nD1, with the negative log-likelihood (NLL) function L.ˆ˛; I/ where L.s; I/ D 1T ŒbL ı.s/I E ET ˚ lnıŒbL ı.s/I lnı E « : The Poisson model is a good approximation for the more precise compound-Poisson distribution (Xu and Tsui 2014; Lasio et al. 2007). 22 / 42
- 27. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References NLL of I Define A D bL ı.ˆ˛/. The NLL of I for known ˛ reduces to the NLL for Poisson generalized linear model (GLM)‡ with identity link and design matrix A: LA.I/ D 1T .AI E/ ET lnı.AI/ lnı E : ‡See (McCullagh and Nelder 1989) for introduction to GLMs. 23 / 42
- 28. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References NLL of ˛ The NLL of ˛ for fixed ι.Ä/ is also a Poisson GLM: Lι.˛/ D 1T ιL ı.ˆ˛/ E ET ˚ lnı ιL ı.ˆ˛/ lnı E « with the link function equal to the inverse of ιL. /. Since ι.Ä/ is known, we do not need its basis-function expansion. 24 / 42
- 29. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References NLL of ˛ The NLL of ˛ for fixed ι.Ä/ is also a Poisson GLM: Lι.˛/ D 1T ιL ı.ˆ˛/ E ET ˚ lnı ιL ı.ˆ˛/ lnı E « with the link function equal to the inverse of ιL. /. Since ι.Ä/ is known, we do not need its basis-function expansion. 4 Lι.˛/ is convex, under conditions that we established in (G. and D. 2016b)! 24 / 42
- 30. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Penalized NLL objective function f .˛; I/ D L.˛; I/ C u k‰H ˛k1 C IR p C .˛/ C IRJ C .I/ 25 / 42
- 31. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Penalized NLL objective function f .˛; I/ D L.˛; I/ C u k‰H ˛k1 C IR p C .˛/ C IRJ C .I/ NLL 25 / 42
- 32. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Penalized NLL objective function f .˛; I/ D L.˛; I/ C u k‰H ˛k1 C IR p C .˛/ C IRJ C .I/ NLL penalty term u > 0 is a scalar tuning constant we select gradient-map sparsifying transform k‰H ˛k1 25 / 42
- 33. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References TV Sparsifying Transform p pixels $ grad. map ˇ ˇŒ‰H ˛i ˇ ˇ # significant coeffs p Total-variation (TV) regularization. 26 / 42
- 34. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Goal and Minimization Approach Goal: Estimate the density-map and mass-attenuation spectrum parameters .˛; I/ by minimizing the penalized NLL f .˛; I/. Approach: A block coordinate-descent that uses Nesterov’s proximal-gradient (NPG) (Nesterov 1983) and limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) (Byrd et al. 1995; Zhu et al. 1997) methods to update estimates of the density map and mass-attenuation spectrum parameters. We refer to this iteration as NPG-BFGS algorithm. 27 / 42
- 35. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Numerical Examples B1-spline constants set to satisfy J D 20; # basis functions qJ D 103 ; span Ä0qd0:5.J C1/e D 1; centering 28 / 42
- 36. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Real X-ray CT Example I 360 equi-spaced fan-beam projections with 1° spacing, X-ray source to rotation center is 3492 detector size, measurement array size of 694 elements, projection matrix ˆ constructed directly on GPU, x y detector array X-ray source D rotate imaginary detector array yielding a nonlinear estimation problem with N D 694 360 measurements and an 512 512 image to reconstruct. Implementation available at github.com/isucsp/imgRecSrc. Real data provided by Joe Gray, CNDE. Thanks! 29 / 42
- 37. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References (a) FBP (b) NPG-BFGS (u D 10 5 ) Figure 6: Real X-ray CT: Full projections. 30 / 42
- 38. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Comments Our reconstruction eliminates the streaking artifacts across the air around the object, the cupping artifacts with high intensity along the border. The regularization constant u has been tuned for good reconstruction performance. 31 / 42
- 39. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Inverse Linearization Function Estimate 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 9 Polychromaticprojections Monochromatic projections NPG-BFGS FBP ﬁtted − ln [ bL (·)I ] Figure 7: The polychromatic measurements as function of the monochromatic projections and its corresponding fitted curve. Observe the biased residual for FBP, the unbiased residual for NPG-BFGS and its increasing variance. 32 / 42
- 40. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Real X-ray CT Example II 360 and 120 equi-spaced fan-beam projections, X-ray source to rotation center is 8696 times of a single detector size, measurement array size of 1380 elements, projection matrix ˆ constructed on GPU with full circular mask. yielding a nonlinear estimation problem with N D 1380 360 measurements and an 1024 1024 image to reconstruct. We employ same convergence constants as in the previous example. 33 / 42
- 41. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References (a) FBP (b) NPG-BFGS (u D 10 5 ) Figure 8: Reconstructions from 360 fan-beam projections with 1° spacing. 34 / 42
- 42. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Figure 9: Estimated ˛ and ln bL . /I from 360 fan-beam projections. 35 / 42 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 4 5 Itr=0 poly.proj.−lnImea mono. proj. ϕT α data ﬁtted − ln ιL (·)
- 43. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Inverse Linearization Function Estimate −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Polychromaticprojections Monochromatic projections NPG-BFGS FBP ﬁtted − ln [ bL (·)I ] Figure 10: The polychromatic measurements as function of the monochromatic projections and its corresponding fitted curve. 36 / 42
- 44. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References (a) FBP (b) NPG-BFGS (u D 10 5 ) Figure 11: Reconstructions from 120 fan-beam projections with 3° spacing. Observe aliasing artifacts in the FBP reconstruction. 37 / 42
- 45. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References (a) 360 projections (b) 120 projections Figure 12: NPG-BFGS (u D 10 5 ) reconstructions from fan-beam projections. The reconstructed density maps are uniform, except the defect region. 38 / 42
- 46. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Simulated X-ray CT Example 10−1 100 101 102 103 40 80 120 160 200 240 280 320 360 RSE/% Number of projections FBP linearized FBP NPG-BFGS0 linearized BPDN NPG-BFGS NPG (known ι(κ)) Figure 13: Average relative square errors (RSEs) as functions of the number of projections. 39 / 42
- 47. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Biconvexity, KL Property, and Convergence Under certain condition, o L.ˆ˛; I/ is biconvex with respect to ˛ and I. o our objective function f .˛; I/ satisfies the Kurdyka-Łojasiewicz (KL) inequality. The above facts can be used to establish the local convergence for alternating proximal minimization methods (Attouch et al. 2010; Xu and Yin 2013) o e.g., PG-BFGS, o not NPG-BFGS. See (G. and D. 2016b; G. and D. 2015). 40 / 42
- 48. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Conclusion Developed a blind method for sparse density-map image reconstruction from polychromatic X-ray CT measurements in Poisson noise. 41 / 42
- 49. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Future Work Generalize our polychromatic signal model to handle multiple materials and develop corresponding reconstruction schemes. 42 / 42
- 50. References I H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, “Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Łojasiewicz inequality,” Math. Oper. Res., vol. 35, no. 2, pp. 438–457, May 2010 (cit. on p. 47). R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput., vol. 16, no. 5, pp. 1190–1208, 1995 (cit. on p. 34). S. R. Cherry, J. A. Sorenson, and M. E. Phelps, 4th ed. Philadelphia, PA: W. B. Saunders, 2012, pp. 493–523 (cit. on p. 8).
- 51. References II R. G. and A. D. (Sep. 2015), Polychromatic X-ray CT image reconstruction and mass-attenuation spectrum estimation, arXiv: 1509.02193 [stat.ME] (cit. on p. 47). R. G. and A. D., “Blind polychromatic X-ray CT reconstruction from Poisson measurements,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Shanghai, China, Mar. 2016, pp. 898–902 (cit. on p. 2). R. G. and A. D., “Blind X-ray CT image reconstruction from polychromatic Poisson measurements,” IEEE Trans. Comput. Imag., vol. 2, no. 2, pp. 150–165, 2016 (cit. on pp. 2, 28, 29, 47). G. T. Herman, “Correction for beam hardening in computed tomography,” Phys. Med. Biol., vol. 24, no. 1, pp. 81–106, 1979 (cit. on p. 21).
- 52. References III G. M. Lasio, B. R. Whiting, and J. F. Williamson, “Statistical reconstruction for X-ray computed tomography using energy-integrating detectors,” Phys. Med. Biol., vol. 52, no. 8, p. 2247, 2007 (cit. on p. 26). P. McCullagh and J. Nelder, Generalized Linear Models, 2nd ed. New York: Chapman & Hall, 1989 (cit. on p. 27). Y. Nesterov, “A method of solving a convex programming problem with convergence rate O.1=k2 /,” Sov. Math. Dokl., vol. 27, no. 2, pp. 372–376, 1983 (cit. on p. 34).
- 53. References IV J. Xu and B. M. W. Tsui, “Quantifying the importance of the statistical assumption in statistical X-ray CT image reconstruction,” IEEE Trans. Med. Imag., vol. 33, no. 1, pp. 61–73, 2014 (cit. on p. 26). Y. Xu and W. Yin, “A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion,” SIAM J. Imag. Sci., vol. 6, no. 3, pp. 1758–1789, 2013 (cit. on p. 47). C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw., vol. 23, no. 4, pp. 550–560, Dec. 1997 (cit. on p. 34).