American Institute of Mathematical Sciences

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April  2020, 14(2): 291-316. doi: 10.3934/ipi.2020013

Robust and stable region-of-interest tomographic reconstruction using a robust width prior

Bart Goossens 1, , Demetrio Labate 2, and  Bernhard G Bodmann 2,

1. 

imec-IPI-Ghent University, B9000 Gent, Belgium
2. 

Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA

Received  May 2019 Revised  September 2019 Published  February 2020

Fund Project: The second author is supported by NSF grants DMS 1720487 and 1720452 and the third author is supported by NSF grant DMS 1715735.

Region-of-interest computed tomography (ROI CT) aims at reconstructing a region within the field of view by using only ROI-focused projections. The solution of this inverse problem is challenging and methods of tomographic reconstruction that are designed to work with full projection data may perform poorly or fail when applied to this setting. In this work, we study the ROI CT problem in the presence of measurement noise and formulate the reconstruction problem by relaxing data fidelity and consistency requirements. Under the assumption of a robust width prior that provides a form of stability for data satisfying appropriate sparsity-inducing norms, we derive reconstruction performance guarantees and controllable error bounds. Based on this theoretical setting, we introduce a novel iterative reconstruction algorithm from ROI-focused projection data that is guaranteed to converge with controllable error while satisfying predetermined fidelity and consistency tolerances. Numerical tests on experimental data show that our algorithm for ROI CT is competitive with state-of-the-art methods especially when the ROI radius is small.

Keywords: Region-of-interest tomography, robust width, sparsity, iterative reconstruction, convex optimization.
Mathematics Subject Classification: Primary: 44A12, 65R32; Secondary: 92C55.
Citation: Bart Goossens, Demetrio Labate, Bernhard G Bodmann. Robust and stable region-of-interest tomographic reconstruction using a robust width prior. Inverse Problems and Imaging, 2020, 14 (2) : 291-316. doi: 10.3934/ipi.2020013
References:
[1]

T. A. Bubba, G. Kutyniok, M. Lassas, M. Maerz, W. Samek, S. Siltanen and V. Srinivasan, Learning the invisible: A hybrid deep learning-shearlet framework for limited angle computed tomography, Inverse Problems, 35 (2019), 064002, 38pp. doi: 10.1088/1361-6420/ab10ca.

[2]

J. Cahill and D. G. Mixon, Robust width: A characterization of uniformly stable and robust compressed sensing, arXiv: 1408.4409.

[3]

J. Cahill, X. Chen and R. Wang, The gap between the null space property and the restricted isometry property, Linear Algebra and its Applications, 501 (2016), 363-375.  doi: 10.1016/j.laa.2016.03.022.

[4]

E. Candès and D. Donoho, Ridgelets: A key to higher-dimensional intermittency?, Phil. Trans. R. Soc. Lond. A., 357 (1999), 2495-2509.  doi: 10.1098/rsta.1999.0444.

[5]

E. Candès and D. Donoho, Recovering edges in ill-posed inverse problems: Optimality of curvelet frames, Annals of Statistics, 30 (2002), 784-842.  doi: 10.1214/aos/1028674842.

[6]

R. Clackdoyle and M. Defrise, Tomographic Reconstruction in the 21st century. Region-of-interest reconstruction from incomplete data, IEEE Signal Processing, 60 (2010), 60-80. 

[7]

R. Clackdoyle, F. Noo, J. Guo and J. Roberts, Quantitative reconstruction from truncated projections in classical tomography, IEEE Trans Nuclear Science, 51 (2004), 2570-2578.  doi: 10.1109/TNS.2004.835781.

[8]

F. Colonna, G. Easley, K. Guo and D. Labate, Radon transform inversion using the shearlet representation, Appl. Comput. Harmon. Anal., 29 (2000), 232-250.  doi: 10.1016/j.acha.2009.10.005.

[9]

G. Easley, D. Labate and W. Lim, Sparse directional image representations using the discrete shearlet transform, Applied and Computational Harmonic Analysis, 25 (2008), 25-46.  doi: 10.1016/j.acha.2007.09.003.

[10]

M. Felsberg, A novel two-step method for ct reconstruction, in Bildverarbeitung für die Medizin, Springer, 2008,303–307. doi: 10.1007/978-3-540-78640-5_61.

[11]

S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Birkhäuser Basel, 2013. doi: 10.1007/978-0-8176-4948-7.

[12]

B. Goossens, Dataflow management, dynamic load balancing, and concurrent processing for real-time embedded vision applications using quasar, International Journal of Circuit Theory and Applications, 46 (2018), 1733-1755.  doi: 10.1002/cta.2494.

[13]

Y. Han, J. Gu and J. C. Ye, Deep learning interior tomography for region-of-interest reconstruction, arXiv preprint, arXiv: 1712.10248.

[14]

Y. Han and J. C. Ye, Framing u-net via deep convolutional framelets: Application to sparse-view ct, IEEE Transactions on Medical Imaging, 37 (2018), 1418-1429. 

[15]

G. T. Herman and A. Lent, Iterative reconstruction algorithms, Computers in Biology and Medicine, 6 (1976), 273-294.  doi: 10.1016/0010-4825(76)90066-4.

[16]

M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409-436.  doi: 10.6028/jres.049.044.

[17]

X. Jin, A. Katsevich, H. Yu, G. Wang, L. Li and Z. Chen, Interior tomography with continuous singular value decomposition, IEEE Transactions on Medical Imaging, 31 (2012), 2108-2119.  doi: 10.1109/TMI.2012.2213304.

[18]

S. Kawata and O. Nalcioglu, Constrained iterative reconstruction by the conjugate gradient method, IEEE Transactions on Medical Imaging, 4 (1985), 65-71.  doi: 10.1109/TMI.1985.4307698.

[19]

E. Klann, E. Quinto and R. Ramlau, Wavelet methods for a weighted sparsity penalty for region of interest tomography, Inverse Problems, 31 (2015), 025001, 22pp. doi: 10.1088/0266-5611/31/2/025001.

[20]

H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Phys. Med. Biol., 53 (2008), 2207-3923. 

[21]

H. Kudo, T. Suzuki and E. A. Rashed, Image reconstruction for sparse-view CT and interior CT - introduction to compressed sensing and differentiated backprojection, Quantitative Imaging in Medicine and Surgery, 3 (2013), 147.

[22]

C. Metzler, A. Maleki and R. Baraniuk, From denoising to compressed sensing, IEEE Transactions on Information Theory, 62 (2016), 5117-5144.  doi: 10.1109/TIT.2016.2556683.

[23]

F. Natterer, The Mathematics of Computerized Tomography, SIAM: Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898719284.

[24]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM: Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898718324.

[25]

F. Noo, R. Clackdoyle and J. Pack, A two-step Hilbert transform method for 2D image reconstruction, Phys. Med. Biol., 49 (2004), 3903-3923.  doi: 10.1088/0031-9155/49/17/006.

[26]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An interative regularization method for total variation-based image restoration, SIAM Multiscale Modeling and Simulation, 4 (2005), 460-489.  doi: 10.1137/040605412.

[27]

L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography., IEEE Trans Med Imaging, 1 (1982), 113-122. 

[28]

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pižurica, S. Vandenberghe and S. Staelens, Iterative CT reconstruction using shearlet-based regularization, IEEE Trans. Nuclear Science, 60 (2013), 3305-3317. 

[29]

T. Würfl, F. C. Ghesu, V. Christlein and A. Maier, Deep learning computed tomography, in International conference on medical image computing and computer-assisted intervention, Springer, 2016,432–440.

[30]

Q. Xu, X. Mou, G. Wang, J. Sieren, E. Hoffman and H. Yu, Statistical interior tomography, IEEE Transactions on Medical Imaging, 30 (2011), 1116-1128.  doi: 10.1109/TMI.2011.2106161.

[31]

W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for l1-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences, 1 (2008), 143-168.  doi: 10.1137/070703983.

[32]

D. F. Yu, J. A. Fessler and E. P. Ficaro, Maximum-likelihood transmission image reconstruction for overlapping transmission beams, IEEE Transactions on Medical Imaging, 19 (2000), 1094-1105.  doi: 10.1109/42.896785.

[33]

H. Yu, Y. Ye and G. Wang, Interior reconstruction using the truncated Hilbert transform via singular value decomposition, J. Xray Sci. Technol., 16 (2008), 243-251. 

[34]

G. Zeng and G. Gullberg, Exact iterative reconstruction for the interior problem, Physics of Medical Biology, 54 (2009), 5805-5814. 

[35]

B. Zhang and G. L. Zeng, Two-dimensional iterative region-of-interest (ROI) reconstruction from truncated projection data, Med. Phys., 34 (2007), 935-944.  doi: 10.1118/1.2436969.

show all references

References:
[1]

T. A. Bubba, G. Kutyniok, M. Lassas, M. Maerz, W. Samek, S. Siltanen and V. Srinivasan, Learning the invisible: A hybrid deep learning-shearlet framework for limited angle computed tomography, Inverse Problems, 35 (2019), 064002, 38pp. doi: 10.1088/1361-6420/ab10ca.

[2]

J. Cahill and D. G. Mixon, Robust width: A characterization of uniformly stable and robust compressed sensing, arXiv: 1408.4409.

[3]

J. Cahill, X. Chen and R. Wang, The gap between the null space property and the restricted isometry property, Linear Algebra and its Applications, 501 (2016), 363-375.  doi: 10.1016/j.laa.2016.03.022.

[4]

E. Candès and D. Donoho, Ridgelets: A key to higher-dimensional intermittency?, Phil. Trans. R. Soc. Lond. A., 357 (1999), 2495-2509.  doi: 10.1098/rsta.1999.0444.

[5]

E. Candès and D. Donoho, Recovering edges in ill-posed inverse problems: Optimality of curvelet frames, Annals of Statistics, 30 (2002), 784-842.  doi: 10.1214/aos/1028674842.

[6]

R. Clackdoyle and M. Defrise, Tomographic Reconstruction in the 21st century. Region-of-interest reconstruction from incomplete data, IEEE Signal Processing, 60 (2010), 60-80. 

[7]

R. Clackdoyle, F. Noo, J. Guo and J. Roberts, Quantitative reconstruction from truncated projections in classical tomography, IEEE Trans Nuclear Science, 51 (2004), 2570-2578.  doi: 10.1109/TNS.2004.835781.

[8]

F. Colonna, G. Easley, K. Guo and D. Labate, Radon transform inversion using the shearlet representation, Appl. Comput. Harmon. Anal., 29 (2000), 232-250.  doi: 10.1016/j.acha.2009.10.005.

[9]

G. Easley, D. Labate and W. Lim, Sparse directional image representations using the discrete shearlet transform, Applied and Computational Harmonic Analysis, 25 (2008), 25-46.  doi: 10.1016/j.acha.2007.09.003.

[10]

M. Felsberg, A novel two-step method for ct reconstruction, in Bildverarbeitung für die Medizin, Springer, 2008,303–307. doi: 10.1007/978-3-540-78640-5_61.

[11]

S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Birkhäuser Basel, 2013. doi: 10.1007/978-0-8176-4948-7.

[12]

B. Goossens, Dataflow management, dynamic load balancing, and concurrent processing for real-time embedded vision applications using quasar, International Journal of Circuit Theory and Applications, 46 (2018), 1733-1755.  doi: 10.1002/cta.2494.

[13]

Y. Han, J. Gu and J. C. Ye, Deep learning interior tomography for region-of-interest reconstruction, arXiv preprint, arXiv: 1712.10248.

[14]

Y. Han and J. C. Ye, Framing u-net via deep convolutional framelets: Application to sparse-view ct, IEEE Transactions on Medical Imaging, 37 (2018), 1418-1429. 

[15]

G. T. Herman and A. Lent, Iterative reconstruction algorithms, Computers in Biology and Medicine, 6 (1976), 273-294.  doi: 10.1016/0010-4825(76)90066-4.

[16]

M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409-436.  doi: 10.6028/jres.049.044.

[17]

X. Jin, A. Katsevich, H. Yu, G. Wang, L. Li and Z. Chen, Interior tomography with continuous singular value decomposition, IEEE Transactions on Medical Imaging, 31 (2012), 2108-2119.  doi: 10.1109/TMI.2012.2213304.

[18]

S. Kawata and O. Nalcioglu, Constrained iterative reconstruction by the conjugate gradient method, IEEE Transactions on Medical Imaging, 4 (1985), 65-71.  doi: 10.1109/TMI.1985.4307698.

[19]

E. Klann, E. Quinto and R. Ramlau, Wavelet methods for a weighted sparsity penalty for region of interest tomography, Inverse Problems, 31 (2015), 025001, 22pp. doi: 10.1088/0266-5611/31/2/025001.

[20]

H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Phys. Med. Biol., 53 (2008), 2207-3923. 

[21]

H. Kudo, T. Suzuki and E. A. Rashed, Image reconstruction for sparse-view CT and interior CT - introduction to compressed sensing and differentiated backprojection, Quantitative Imaging in Medicine and Surgery, 3 (2013), 147.

[22]

C. Metzler, A. Maleki and R. Baraniuk, From denoising to compressed sensing, IEEE Transactions on Information Theory, 62 (2016), 5117-5144.  doi: 10.1109/TIT.2016.2556683.

[23]

F. Natterer, The Mathematics of Computerized Tomography, SIAM: Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898719284.

[24]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM: Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898718324.

[25]

F. Noo, R. Clackdoyle and J. Pack, A two-step Hilbert transform method for 2D image reconstruction, Phys. Med. Biol., 49 (2004), 3903-3923.  doi: 10.1088/0031-9155/49/17/006.

[26]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An interative regularization method for total variation-based image restoration, SIAM Multiscale Modeling and Simulation, 4 (2005), 460-489.  doi: 10.1137/040605412.

[27]

L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography., IEEE Trans Med Imaging, 1 (1982), 113-122. 

[28]

B. Vandeghinste, B. Goossens, R. Van Holen, C. Vanhove, A. Pižurica, S. Vandenberghe and S. Staelens, Iterative CT reconstruction using shearlet-based regularization, IEEE Trans. Nuclear Science, 60 (2013), 3305-3317. 

[29]

T. Würfl, F. C. Ghesu, V. Christlein and A. Maier, Deep learning computed tomography, in International conference on medical image computing and computer-assisted intervention, Springer, 2016,432–440.

[30]

Q. Xu, X. Mou, G. Wang, J. Sieren, E. Hoffman and H. Yu, Statistical interior tomography, IEEE Transactions on Medical Imaging, 30 (2011), 1116-1128.  doi: 10.1109/TMI.2011.2106161.

[31]

W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for l1-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences, 1 (2008), 143-168.  doi: 10.1137/070703983.

[32]

D. F. Yu, J. A. Fessler and E. P. Ficaro, Maximum-likelihood transmission image reconstruction for overlapping transmission beams, IEEE Transactions on Medical Imaging, 19 (2000), 1094-1105.  doi: 10.1109/42.896785.

[33]

H. Yu, Y. Ye and G. Wang, Interior reconstruction using the truncated Hilbert transform via singular value decomposition, J. Xray Sci. Technol., 16 (2008), 243-251. 

[34]

G. Zeng and G. Gullberg, Exact iterative reconstruction for the interior problem, Physics of Medical Biology, 54 (2009), 5805-5814. 

[35]

B. Zhang and G. L. Zeng, Two-dimensional iterative region-of-interest (ROI) reconstruction from truncated projection data, Med. Phys., 34 (2007), 935-944.  doi: 10.1118/1.2436969.

Figure 1.  Illustrations of recoverable regions for truncated projection data: (a) initial DBP methods [25,7,35] require at least one projection view in which the complete object is covered, (b) interior reconstruction is possible given a known subregion [20,34,30,17] and (c) no assumptions are made other than that the shape of the ROI is convex and approximate sparsity within a ridgelet domain (this paper). The gray dashed line indicates the measured area on the detector array for one particular source angle
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Figure 2.  Illustration of ROI tomographic reconstruction problem. (a) In ROI tomography, only projections intersecting the Region-of-Interest $ S $ are known. (b) In the Radon-transform domain, the region $ P(S) $ corresponding to $ S $ is a strip-like domain
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Figure 3.  Schematic illustration of the solution space for $ y $, given a current estimate $ f $, as intersection of the balls $ \left\Vert My-y_{0}\right\Vert ^{2}\leq \alpha $ and $ \left\Vert y-Wf\right\Vert ^{2}\leq \beta $, as well as solutions favored by the sparsity prior $ \left\Vert {y}\right\Vert _{\sharp} $ (see Sec. 3). Data consistent solutions may have a non-zero data fidelity, while data fidelity solutions are in general not consistent. We control the reconstruction error by combining fidelity and consistency constraints with the additional sparsity assumption
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Figure 4.  Commutative diagram of the measurement operator $ \Phi $ and the restricted measurement operator $ \Phi ' $ (see Theorem 3.5). The following relationship holds: $ P_{\tilde{\mathcal{H}}'}\Phi = \Phi 'P_{\mathcal{H}'} $.
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Figure 5.  Full LSCG reconstruction of the fan-beam data sets: (a) Preclinical - lungs, (b) Preclinical - abdomen. Images are cropped for visualization purposes
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Figure 6.  PSNR results for 2D fan-beam ROI reconstruction with increasing radius using (a) lungs and (b) adbomen. The PSNR is calculated inside the ROI
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Figure 7.  ROI reconstruction results for a fixed radius of $ 64 $ pixels (or 3.2 mm). (a), (b): SIRA with different values of $ \alpha $, $ \beta $ and $ u = \Vert (I-\tilde{W}\tilde{W}^{+})y_{0}\Vert _{2}^{2} $; (c): LSCG; (d): CS-TV, (e): CS-ridgelet, (f) Full view LSCG reconstruction (used for computing PSNR)
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Figure 8.  (a)-(c): Reconstruction results for different ROI radii using SIRA with fixed parameters $ \alpha = 0 $, $ \beta = 0.25\left\Vert \left(I-\tilde{W}\tilde{W}^{+}\right)y_{0}\right\Vert _{2}^{2}. $ (d)-(f): Reconstruction results using LSCG for same ROI radii
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Table 1.  CT fan-beam acquisition geometries (X-O CT system from Gamma Medica-Ideas) used in the experiments for this paper
Data set
Geometry parameter Preclinical - lungs Preclinical - abdomen
Distance source-detector 146.09 mm 145.60 mm
Distance source-object 41.70 mm 57.92 mm
Detector offset -15.00 mm 12.14 mm
Detector elements 592 592
Projection angles 512 640
Pixel pitch 0.20 mm 0.20 mm
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Data set
Geometry parameter Preclinical - lungs Preclinical - abdomen
Distance source-detector 146.09 mm 145.60 mm
Distance source-object 41.70 mm 57.92 mm
Detector offset -15.00 mm 12.14 mm
Detector elements 592 592
Projection angles 512 640
Pixel pitch 0.20 mm 0.20 mm
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