International Journal on Magnetic Particle Imaging IJMPI
Vol. 6 No. 2 (2020): Int J Mag Part Imag
https://doi.org/10.18416/IJMPI.2020.2012001
L1 data fitting for robust reconstruction in magnetic particle imaging: quantitative evaluation on Open MPI dataset
Main Article Content
Copyright (c) 2020 Tobias Kluth, Bangti Jin
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
Magnetic particle imaging is an emerging quantitative imaging modality, exploiting the unique nonlinear magnetization phenomenon of superparamagnetic iron oxide nanoparticles for recovering the concentration. Traditionally the reconstruction is formulated into a penalized least-squares problem with nonnegativity constraint, and then solved using a variant of Kaczmarz method which is often stopped early after a small number of iterations. Besides the phantom signal, measurements additionally include a background signal and a noise signal. In order to obtain good reconstructions, a preprocessing step of frequency selection to remove the deleterious influences of the noise is often adopted. In this work, we propose a complementary pure variational approach to noise treatment, by viewing highly noisy measurements as outliers, and employing the l1 data fitting, one popular approach from robust statistics. When compared with the standard approach, it is easy to implement with a comparable computational complexity. Experiments with a public domain dataset, i.e., Open MPI dataset, show that it can give accurate reconstructions, and is less prone to noisy measurements, which is illustrated by quantitative (PSNR / SSIM) and qualitative comparisons with the Kaczmarz method. We also investigate the performance of the Kaczmarz method for small iteration numbers quantitatively.
Int. J. Mag. Part. Imag. 6(2), 2020, Article ID: 2012001, DOI: 10.18416/IJMPI.2020.2012001
Article Details
References
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[3] J.Weizenecker, B. Gleich, and J. Borgert.Magnetic particle imaging using a field free line. Journal of Physics D: Applied Physics, 41(10):105009, 2008, doi:10.1088/0022-3727/41/10/105009.
[4] T. Knopp and T. M. Buzug,Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation. Berlin, Heidelberg: Springer, 2012, doi:10.1007/978-3-642-04199-0.
[5] T. Knopp, N. Gdaniec, and M. Möddel. Magnetic particle imaging: from proof of principle to preclinical applications. Physics in Medicine & Biology, 62(14):R124–R178, 2017, doi:10.1088/1361-6560/aa6c99.
[6] T. Kluth. Mathematical models for magnetic particle imaging. Inverse Problems, 34(8):083001, 2018, doi:10.1088/1361-6420/aac535.
[7] J. Weizenecker, B. Gleich, J. Rahmer, H. Dahnke, and J. Borgert. Three-dimensional real-time in vivo magnetic particle imaging. Physics in Medicine and Biology, 54(5):L1–L10, 2009, doi:10.1088/0031-9155/54/5/L01.
[8] A. P. Khandhar, P. Keselman, S. J. Kemp, R. M. Ferguson, P. W. Goodwill, S. M. Conolly, and K. M. Krishnan. Evaluation of PEGcoated iron oxide nanoparticles as blood pool tracers for preclinical magnetic particle imaging. Nanoscale, 9(3):1299–1306, 2017, doi:10.1039/C6NR08468K.
[9] J. Franke, R. Lacroix, H. Lehr, M. Heidenreich, U. Heinen, and V. Schulz. MPI Flow Analysis Toolbox exploiting pulsed tracer information - an aneurysm phantom proof. International Journal on Magnetic Particle Imaging, 3(1), 2017, doi:10.18416/IJMPI.2017.1703020.
[10] J. Haegele, J. Rahmer, B. Gleich, J. Borgert, H. Wojtczyk, N. Panagiotopoulos, T. M. Buzug, J. Barkhausen, and F. M. Vogt. Magnetic Particle Imaging: Visualization of Instruments for Cardiovascular Intervention. Radiology, 265(3):933–938, 2012, doi:10.1148/radiol.12120424.
[11] J. Salamon, M. Hofmann, C. Jung, M. G. Kaul, F. Werner, K. Them, R. Reimer, P. Nielsen, A. vom Scheidt, G. Adam, T. Knopp, and H. Ittrich. Magnetic Particle / Magnetic Resonance Imaging: In-Vitro MPI-Guided Real Time Catheter Tracking and 4D Angioplasty Using a Road Map and Blood Pool Tracer Approach. PLOS ONE, 11(6):e0156899M. Yamamoto, Ed., 2016, doi:10.1371/journal.pone.0156899.
[12] E. Y. Yu, M. Bishop, B. Zheng, R. M. Ferguson, A. P. Khandhar, S. J. Kemp, K. M. Krishnan, P. W. Goodwill, and S. M. Conolly.Magnetic Particle Imaging: A Novel in Vivo Imaging Platform for Cancer Detection. Nano Letters, 17(3):1648–1654, 2017, doi:10.1021/acs.nanolett.6b04865.
[13] K. Murase, M. Aoki, N. Banura, K. Nishimoto, A. Mimura, T. Kuboyabu, and I. Yabata. Usefulness ofMagnetic Particle Imaging for Predicting the Therapeutic Effect of Magnetic Hyperthermia. Open Journal of Medical Imaging, 05(02):85–99, 2015, doi:10.4236/ojmi.2015.52013.
[14] J. Rahmer, J.Weizenecker, B. Gleich, and J. Borgert. Signal encoding in magnetic particle imaging: properties of the system function. BMC Medical Imaging, 9:4, 2009, doi:10.1186/1471-2342-9-4.
[15] T. Knopp, J. Rahmer, T. F. Sattel, S. Biederer, J.Weizenecker, B. Gleich, J. Borgert, and T. M. Buzug.Weighted iterative reconstruction for magnetic particle imaging. Physics in Medicine and Biology, 55(6):1577–1589, 2010, doi:10.1088/0031-9155/55/6/003.
[16] P.W. Goodwill and S. M. Conolly.Multidimensional X-Space Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 30(9):1581–1590, 2011, doi:10.1109/TMI.2011.2125982.
[17] P. W. Goodwill, E. U. Saritas, L. R. Croft, T. N. Kim, K. M. Krishnan, D. V. Schaffer, and S. M. Conolly. X-Space MPI:Magnetic Nanoparticles for Safe Medical Imaging. Advanced Materials, 24(28):3870–3877, 2012, doi:10.1002/adma.201200221.
[18] J. Rahmer, A. Halkola, B. Gleich, I. Schmale, and J. Borgert. First experimental evidence of the feasibility of multi-color magnetic particle imaging. Physics inMedicine and Biology, 60(5):1775–91, 2015, doi:10.1088/0031-9155/60/5/1775.
[19] T. Knopp and M. Hofmann. Online reconstruction of 3D magnetic particle imaging data. Physics in Medicine and Biology, 61(11):N257–N267, 2016, doi:10.1088/0031-9155/61/11/N257.
[20] M. Storath, C. Brandt, M. Hofmann, T. Knopp, J. Salamon, A. Weber, and A. Weinmann. Edge Preserving and Noise Reducing Reconstruction for Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 36(1):74–85, 2017, doi:10.1109/TMI.2016.2593954.
[21] T. Kluth and P. Maass. Model uncertainty in magnetic particle imaging:Nonlinear problem formulation and model-based sparse reconstruction. International Journal onMagnetic Particle Imaging, 3(2), 2017, doi:10.18416/IJMPI.2017.1707004.
[22] C. Brandt and A. Seppänen. Recovery from Errors Due to Domain Truncation in Magnetic Particle Imaging: Approximation Error Modeling Approach. Journal ofMathematical Imaging and Vision, 60(8):1196–1208, 2018, doi:10.1007/s10851-018-0807-z.
[23] T. Kluth and B. Jin. Enhanced reconstruction in magnetic particle imaging by whitening and randomized SVD approximation. Physics in Medicine & Biology, 64(12):125026, 2019, doi:10.1088/1361-6560/ab1a4f.
[24] S. Dittmer, T. Kluth, P.Maass, and D. Otero Baguer. Regularization by Architecture: A Deep Prior Approach for Inverse Problems. Journal ofMathematical Imaging and Vision, 62(3):456–470, 2020, doi:10.1007/s10851-019-00923-x.
[25] T. Kluth, B. Jin, and G. Li. On the degree of ill-posedness of multi-dimensional magnetic particle imaging. Inverse Problems, 34(9):095006, 2018, doi:10.1088/1361-6420/aad015.
[26] K. Them, M. G. Kaul, C. Jung, M.Hofmann, T.Mummert, F.Werner, and T. Knopp. Sensitivity Enhancement inMagnetic Particle Imaging by Background Subtraction. IEEE Transactions on Medical Imaging, 35(3):893–900, 2016, doi:10.1109/TMI.2015.2501462.
[27] T. Knopp, N. Gdaniec, R. Rehr, M. Graeser, and T. Gerkmann. Correction of linear system drifts in magnetic particle imaging. Physics in Medicine & Biology, 64(12):125013, 2019, doi:10.1088/1361-6560/ab2480.
[28] M. Straub and V. Schulz. Joint Reconstruction of Tracer Distribution and Background in Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 37(5):1192–1203, 2018, doi:10.1109/TMI.2017.2777878.
[29] J. Franke, U. Heinen, H. Lehr, A. Weber, F. Jaspard, W. Ruhm, M. Heidenreich, and V. Schulz. System Characterization of a Highly Integrated Preclinical Hybrid MPI-MRI Scanner. IEEE Transactions on Medical Imaging, 35(9):1993–2004, 2016, doi:10.1109/TMI.2016.2542041.
[30] S. Kaczmarz. Angenäherte Auflösung von Systemen linearer Gleichungen. Bulletin International de l’Académie Polonaise des Sciences et des Lettres, (35)pp. 355–357, 1937.
[31] T. Kluth, P. Szwargulski, and T. Knopp. Towards accurate modeling of the multidimensional magnetic particle imaging physics. New Journal of Physics, 21(10):103032, 2019, doi:10.1088/1367-2630/ab4938.
[32] J. J.Konkle, P.W.Goodwill,D.W.Hensley, R.D. Orendorff, M. Lustig, and S. M. Conolly. A Convex Formulation for Magnetic Particle Imaging X-Space Reconstruction. PLOS ONE, 10(10):e0140137J. Najbauer, Ed., 2015, doi:10.1371/journal.pone.0140137.
[33] C. Bathke, T. Kluth, and P.Maaß, MPI reconstruction using structural prior information and sparsity, in International Workshop on Magnetic Particle Imaging, 129–130, 2018.
[34] S. Alliney and S. Ruzinsky. An algorithm for the minimization of mixed l1 and l2 norms with application to Bayesian estimation. IEEE Transactions on Signal Processing, 42(3):618–627, 1994, doi:10.1109/78.277854.
[35] C. Clason, B. Jin, and K. Kunisch. A Semismooth Newton Method forL1 Data Fitting with Automatic Choice of Regularization Parameters and Noise Calibration. SIAM Journal on Imaging Sciences, 3(2):199–231, 2010, doi:10.1137/090758003.
[36] A. Gelman, J. B. Carlin, H. S. Stern, andD. B. Rubin., Bayesian Data Analysis. CRC Press, 2004,
[37] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific Computing, 16(5):1190–1208, 1995, doi:10.1137/0916069.
[38] C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal. Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization. ACM Transactions on Mathematical Software, 23(4):550–560, 1997, doi:10.1145/279232.279236.
[39] D. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, 2006, doi:10.1109/TIT.2006.871582.
[40] S. Ilbey, C. B. Top, A. Gungor, T. Cukur, E. U. Saritas, and H. E. Guven. Fast System Calibration with Coded Calibration Scenes for Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, pp. 1–1, 2019, doi:10.1109/TMI.2019.2896289.
[41] K. Ito and B. Jin, Inverse Problems. World Scientific, 2014, 22. doi:10.1142/9120.
[42] J.Weizenecker, J. Borgert, and B. Gleich. A simulation study on the resolution and sensitivity of magnetic particle imaging. Physics in Medicine and Biology, 52(21):6363–6374, 2007, doi:10.1088/0031-9155/52/21/001.
[43] I. Schmale, J. Borgert, B. Gleich, and J.Weizenecker, Noise within magnetic particle imaging, inMagnetic Nanoparticles, 154–161, World Scientific, 2010. doi:10.1142/9789814324687_0022.
[44] A. C. Bovik, Handbook of Image and Video Processing. Elsevier, 2005, doi:10.1016/B978-0-12-119792-6.X5062-1.
[45] P. J. Huber, Robust Statistics, ser. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: JohnWiley & Sons, Inc., 1981, doi:10.1002/0471725250.
[46] H. Hwang and R.Haddad. Adaptive median filters: new algorithms and results. IEEE Transactions on Image Processing, 4(4):499–502, 1995, doi:10.1109/83.370679.
[47] A. Buades, B. Coll, and J.-M. Morel. Nonlocal Image and Movie Denoising. International Journal of Computer Vision, 76(2):123–139, 2008, doi:10.1007/s11263-007-0052-1.
[48] S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, ser. Springer Series in Operations Research and Financial Engineering. New York, NY: Springer New York, 2013, doi:10.1007/978-1-4614-7101-1.
[49] P. Rodriguez and B. Wohlberg. Efficient Minimization Method for a Generalized Total Variation Functional. IEEE Transactions on Image Processing, 18(2):322–332, 2009, doi:10.1109/TIP.2008.2008420.
[50] J. Yang and Y. Zhang. Alternating Direction Algorithms for `1-Problems in Compressive Sensing. SIAM Journal on Scientific Computing, 33(1):250–278, 2011, doi:10.1137/090777761.
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