American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  Google Scholar

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">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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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}'} $.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Full LSCG reconstruction of the fan-beam data sets: (a) Preclinical - lungs, (b) Preclinical - abdomen. Images are cropped for visualization purposes
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

Dong liu, Ville Kolehmainen, Samuli Siltanen, Anne-maria Laukkanen, Aku Seppänen. Estimation of conductivity changes in a region of interest with electrical impedance tomography. Inverse Problems & Imaging, 2015, 9 (1) : 211-229. doi: 10.3934/ipi.2015.9.211
[2]

Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171
[3]

Henrik Garde, Kim Knudsen. 3D reconstruction for partial data electrical impedance tomography using a sparsity prior. Conference Publications, 2015, 2015 (special) : 495-504. doi: 10.3934/proc.2015.0495
[4]

Yan Liu, Wuwei Ren, Habib Ammari. Robust reconstruction of fluorescence molecular tomography with an optimized illumination pattern. Inverse Problems & Imaging, 2020, 14 (3) : 535-568. doi: 10.3934/ipi.2020025
[5]

Jutamas Kerdkaew, Rabian Wangkeeree. Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2651-2673. doi: 10.3934/jimo.2019074
[6]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100
[7]

Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems & Imaging, 2014, 8 (1) : 173-197. doi: 10.3934/ipi.2014.8.173
[8]

Nithirat Sisarat, Rabian Wangkeeree, Gue Myung Lee. Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 469-493. doi: 10.3934/jimo.2018163
[9]

Jian Lu, Lixin Shen, Chen Xu, Yuesheng Xu. Multiplicative noise removal with a sparsity-aware optimization model. Inverse Problems & Imaging, 2017, 11 (6) : 949-974. doi: 10.3934/ipi.2017044
[10]

Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial & Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189
[11]

Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems & Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111
[12]

Li Shen, Eric Todd Quinto, Shiqiang Wang, Ming Jiang. Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography. Inverse Problems & Imaging, 2018, 12 (6) : 1343-1364. doi: 10.3934/ipi.2018056
[13]

Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919
[14]

Lijuan Zhao, Wenyu Sun. Nonmonotone retrospective conic trust region method for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 309-325. doi: 10.3934/naco.2013.3.309
[15]

Bülent Karasözen. Survey of trust-region derivative free optimization methods. Journal of Industrial & Management Optimization, 2007, 3 (2) : 321-334. doi: 10.3934/jimo.2007.3.321
[16]

Xin Zhang, Jie Wen, Qin Ni. Subspace trust-region algorithm with conic model for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 223-234. doi: 10.3934/naco.2013.3.223
[17]

Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 79-89. doi: 10.3934/naco.2015.5.79
[18]

Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357
[19]

Tianyu Yang, Yang Yang. A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021038
[20]

Rasmus Backholm, Tatiana A. Bubba, Camille Bélanger-Champagne, Tapio Helin, Peter Dendooven, Samuli Siltanen. Simultaneous reconstruction of emission and attenuation in passive gamma emission tomography of spent nuclear fuel. Inverse Problems & Imaging, 2020, 14 (2) : 317-337. doi: 10.3934/ipi.2020014

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (151)
  • HTML views (170)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences