Optimized filter functions for filtered back projection reconstructions

Matthias Beckmann; Judith Nickel

doi:10.3934/ipi.2025003

Article Contents

2025, Volume 19, Issue 5: 903-939. Doi: 10.3934/ipi.2025003 open access

This issue Previous Article A variational image segmentation model with intensity correction in the presence of high level multiplicative noise Next Article Examples of non-scattering inhomogeneities

Optimized filter functions for filtered back projection reconstructions

Matthias Beckmann^{1, 2,} and
Judith Nickel^1, ,

1.
Center for Industrial Mathematics, University of Bremen, Germany
2.
Department of Electrical and Electronic Engineering, Imperial College London, UK

^*Corresponding author: Judith Nickel
^*Corresponding author: Judith Nickel

Received: August 2024

Revised: December 2024

Early access: January 20, 2025

Published online: January 20, 2025

The authors are listed in alphabetical order. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) - project numbers 530863002 and 281474342/GRK2224/2 - as well as the Bundesministerium für Bildung und Forschung (BMBF) - funding code 03TR07W11A.

Abstract / Introduction Full Text(HTML) Figure(13) / Table(2) Related Papers Cited by

Abstract

The method of filtered back projection (FBP) is a widely used reconstruction technique in X-ray computerized tomography (CT), which is particularly important in clinical diagnostics. To reduce scanning times and radiation doses in medical CT settings, enhancing the reconstruction quality of the FBP method is essential. To this end, this paper focuses on analytically optimizing the applied filter function. We derive a formula for the filter that minimizes the expected squared $ {\mathrm L}^2 $-norm of the difference between the FBP reconstruction, given infinite noisy measurement data and the true target function. Additionally, we adapt our derived filter to the case of finitely many measurements. The resulting filter functions have a closed-form representation, do not require a training dataset, and, thus, provide an easy-to-implement, out-of-the-box solution. Our theoretical findings are supported by numerical experiments based on both simulated and real CT data.

Keywords:

Mathematics Subject Classification: 44A12, 65R32, 4A12.

Citation:

FullText(HTML)

Figure 1. Selection of classical low-pass filters used in our numerical experiments

Download: Full-size image PowerPoint slide

Figure 2. Phantoms used in our numerical experiments along with their sinograms (Radon data)

Download: Full-size image PowerPoint slide

Figure 3. Noisy sinograms of the Shepp-Logan phantom with different noise levels

Download: Full-size image PowerPoint slide

Figure 4. Plots of the MSE and SSIM of FBP reconstructions for the Shepp-Logan phantom

Download: Full-size image PowerPoint slide

Figure 5. Plots of the optimized filter functions $ A_{L,D}^\ast $, $ A_{L,D}^\varepsilon $ and $ \hat{A}_{L,D}^\varepsilon $ for the noisy Shepp-Logan sinogram with $ N_\varphi \in \{90,180,360\} $ and $ p_{\mathrm{noise}} \in \{0.05, \, 0.1, \, 0.15\} $

Download: Full-size image PowerPoint slide

Figure 6. Reconstructions of Shepp-Logan phantom from noisy Radon data ($ p_\mathrm{noise} = 0.1 $, $ N_\varphi = 360 $)

Download: Full-size image PowerPoint slide

Figure 7. Plots of the MSE and SSIM of FBP reconstructions for the modified Shepp-Logan phantom

Download: Full-size image PowerPoint slide

Figure 8. Reconstructions of modified Shepp-Logan phantom from noisy Radon data ($ p_\mathrm{noise} = 0.1 $, $ N_\varphi = 360 $)

Download: Full-size image PowerPoint slide

Figure 9. 2DeteCT slice 1942 (mode 1) sinogram with target reconstruction

Download: Full-size image PowerPoint slide

Figure 10. Reconstructions of 2DeteCT slice 1942 (mode 1)

Download: Full-size image PowerPoint slide

Figure 11. Zoomed-in reconstructions of 2DeteCT slice 1942 (mode 1)

Download: Full-size image PowerPoint slide

Figure 12. Plots of the MSE and SSIM of FBP reconstructions with the three best filter functions for the Shepp-Logan phantom

Download: Full-size image PowerPoint slide

Figure 13. Plots of the MSE and SSIM of FBP reconstructions with the three best filter functions for the modified Shepp-Logan phantom

Download: Full-size image PowerPoint slide

Table 1. Window functions of classical low-pass filters, where $ W(\sigma) = 0 $ for all $ |\sigma| > 1 $ in all cases

Name	$ W(\sigma) $ for $ \|\sigma\|\leq 1 $	Parameter
Ram-Lak	1	-
Shepp-Logan	$ {\rm{sinc}}(\tfrac{\pi \sigma}{2}) $	-
Cosine	$ \cos(\tfrac{\pi \sigma}{2}) $	-
Hamming	$ \beta + (1-\beta) \cos(\pi \sigma) $	$ \beta \in [\tfrac{1}{2},1] $

| Show Table

DownLoad: CSV

Table 2. MSE of FBP reconstructions for the 2DeteCT slice 1942 (mode 1)

Filter	MSE	Filter	MSE	Filter	MSE
Ram-Lak	$ 1.0703 \cdot 10^{-5} $	MR-FBP	$ 1.2231 \cdot 10^{-5} $	$ \bar{A}_{L,D}^\ast $	$ 8.7733 \cdot 10^{-6} $
Shepp-Logan	$ 9.1803 \cdot 10^{-6} $	MR-FBP$ _\mathrm{GM} $	$ 9.8905 \cdot 10^{-6} $	$ \hat{A}_{L,D}^\varepsilon $	$ 9.1616 \cdot 10^{-6} $
Cosine	$ 9.5116 \cdot 10^{-6} $	SIRT-FBP	$ 9.2900 \cdot 10^{-6} $	$ A_{L,D}^\varepsilon $	$ 9.0792 \cdot 10^{-6} $

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	S. Aziznejad and J. Fageot, Wavelet analysis of the Besov regularity of Lévy white noise, Electronic Journal of Probability, 25 (2020), Paper No. 158, 38 pp. doi: 10.1214/20-EJP554.
[2]	Z. A. Balogh and B. Janos Kis, Comparison of CT noise reduction performances with deep learning-based, conventional, and combined denoising algorithms, Medical Engineering & Physics, 109 (2022), 103897.
[3]	M. Beckmann and A. Iske, Error estimates for filtered back projection, Proceedings of IEEE International Conference on Sampling Theory and Applications (SampTA), Washington, DC, USA, (2015), 553-557.
[4]	M. Beckmann and A. Iske, Error estimates and convergence rates for filtered back projection, Mathematics of Computation, 88 (2019), 801-835. doi: 10.1090/mcom/3343.
[5]	M. Beckmann and A. Iske, Saturation rates of filtered back projection approximations, Calcolo, 57 (2020), Paper No. 12, 35 pp. doi: 10.1007/s10092-020-00360-y.
[6]	M. Beckmann, P. Maass and J. Nickel, Error analysis for filtered back projection reconstructions in Besov spaces, Inverse Problems, 37 (2021), Paper No. 014002, 35 pp. doi: 10.1088/1361-6420/aba5ee.
[7]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.
[8]	E. Bibbona, G. Panfilo and P. Tavella, The Ornstein–Uhlenbeck process as a model of a low pass filtered white noise, Metrologia, 45 (2008), S117-S126. doi: 10.1088/0026-1394/45/6/S17.
[9]	P. Billingsley, Probability and Measure, Wiley, New-York, 1995.
[10]	T. M. Buzug, Computed Tomography, Springer, Berlin, 2008.
[11]	R. C. Dalang and T. Humeau, Lévy processes and Lévy white noise as tempered distributions, The Annals of Probability, 45 (2017), 4389-4418. doi: 10.1214/16-AOP1168.
[12]	M. Diwakar and M. Kumar, A review on CT image noise and its denoising, Biomedical Signal Processing and Control, 42 (2018), 73-88. doi: 10.1016/j.bspc.2018.01.010.
[13]	J. L. Doob, The Brownian movement and stochastic equations, Annals of Mathematics, 43 (1942), 351-369. doi: 10.2307/1968873.
[14]	C. Epstein, Introduction to the Mathematics of Medical Imaging, SIAM, Philadelphia, 2008.
[15]	J. Fageot, On tempered discrete and Lévy white noises, preprint, 2022. arXiv: 2201.00797.
[16]	J. Fageot, A. Fallah and M. Unser, Multidimensional Lévy white noise in weighted Besov spaces, Stochastic Processes and their Applications, 127 (2017), 1599-1621. doi: 10.1016/j.spa.2016.08.011.
[17]	J. Fageot, M. Unser and J. P. Ward, On the Besov regularity of periodic Lévy noises, Applied and Computational Harmonic Analysis, 42 (2017), 21-36. doi: 10.1016/j.acha.2015.07.001.
[18]	T. Feeman, The Mathematics of Medical Imaging: A Beginner's Guide, Springer, Cham, 2015.
[19]	L. Fu, T.-C. Lee, S. M. Kim, A. M. Alessio, P. E. Kinahan, Z. Chang, K. Sauer, M. K. Kalra and B. De Man, Comparison between pre-log and post-log statistical models in ultra-low-dose CT reconstruction, IEEE Transactions on Medical Imaging, 36 (2017), 707-720. doi: 10.1109/TMI.2016.2627004.
[20]	C. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences, Springer, Berlin, 2009.
[21]	I. M. Gel'fand and N. Y. Vilenkin, Generalized Functions. Volume 4: Applications of Harmonic Analysis, Academic Press, New York, 1964.
[22]	G. Herman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections, Springer, London, 2009.
[23]	S. Horbelt, M. Liebling and M. A. Unser, Filter design for filtered back-projection guided by the interpolation model, Proceedings of SPIE Medical Imaging, San Diego, USA, (2002), 806-813.
[24]	S. Hottovy, A. McDaniel and J. Wehr, A small delay and correlation time limit of stochastic differential delay equations with state-dependent colored noise, Journal of Statistical Physics, 175 (2019), 19-46. doi: 10.1007/s10955-019-02242-2.
[25]	S. Kabri, A. Auras, D. Riccio, H. Bauermeister, M. Benning, M. Moeller and M. Burger, Convergent data-driven regularizations for CT reconstruction, Communications on Applied Mathematics and Computation, 6 (2024), 1342-1368. doi: 10.1007/s42967-023-00333-2.
[26]	A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, SIAM, Philadelphia, 2001.
[27]	H. Kekkonen, M. Lassas and S. Siltanen, Analysis of regularized inversion of data corrupted by white Gaussian noise, Inverse Problems, 30 (2014), 045009, 18 pp. doi: 10.1088/0266-5611/30/4/045009.
[28]	M. B. Kiss, S. B. Coban, K. J. Batenburg, T. van Leeuwen and F. Lucka, 2DeteCT - a large 2D expandable, trainable, experimental Computed Tomography dataset for machine learning, Scientific Data, 10 (2023), 576. doi: 10.1038/s41597-023-02484-6.
[29]	T. Kuhn, H.-G. Leopold, W. Sickel and L. Skrzypczak, Entropy numbers of embeddings of weighted Besov spaces, Constructive Approximation, 23 (2006), 61-77. doi: 10.1007/s00365-005-0594-9.
[30]	M. J. Lagerwerf, A. A. Hendriksen, J.-W. Buurlage and K. J. Batenburg, Noise2Filter: Fast, self-supervised learning and real-time reconstruction for 3D computed tomography, Machine Learning: Science and Technology, 2 (2021), 015012. doi: 10.1088/2632-2153/abbd4d.
[31]	T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh and Z. Liang, Nonlinear sinogram smoothing for low-dose X-ray CT, IEEE Transactions on Nuclear Science, 51 (2004), 2505-2513. doi: 10.1109/TNS.2004.834824.
[32]	Z. Liao, S. Hu, M. Li and W. Chen, Noise estimation for single-slice sinogram of low-dose X-ray computed tomography using homogenous patch, Mathematical Problems in Engineering, 2012 (2012), 696212. doi: 10.1155/2012/696212.
[33]	J. S. Lim, Two-Dimensional Signal and Image Processing, Prentice Hall, Englewood Cliffs, 1990.
[34]	H. Lu, I. T. Hsiao, X. Li and Z. Liang, Noise properties of low-dose CT projections and noise treatment by scale transformations, in IEEE Nuclear Science Symposium Conference Record, (2001), 1662-1666.
[35]	M. Meyries and M. Veraar, Sharp embedding results for spaces of smooth functions with power weights, Studia Mathematica, 208 (2012), 257-293. doi: 10.4064/sm208-3-5.
[36]	F. Natterer, The Mathematics of Computerized Tomography, SIAM, Philadelphia, 2001.
[37]	F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, 2001.
[38]	A. A. Omer, O. I. Hassan, A. I. Ahmed and A. Abdelrahman, Denoising CT images using median based filters: A review, Proceedings of International Conference on Computer, Control, Electrical, and Electronics Engineering (ICCCEEE), Khartoum, Sudan, (2018), 1-6.
[39]	R. Paley, N. Wiener and A. Zygmund, Note on random functions, Mathematische Zeitschrift, 37 (1933), 647-668. doi: 10.1007/BF01474606.
[40]	X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 123009. doi: 10.1088/0266-5611/25/12/123009.
[41]	A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastoc Processes, McGraw-Hill, Boston, 2002.
[42]	D. M. Pelt and K. J. Batenburg, Fast tomographic reconstruction from limited data using artificial neural networks, IEEE Transactions on Image Processing, 22 (2013), 5238-5251. doi: 10.1109/TIP.2013.2283142.
[43]	D. M. Pelt and K. J. Batenburg, Improving filtered backprojection reconstruction by data-dependent filtering, IEEE Transactions on Image Processing, 23 (2014), 4750-4762. doi: 10.1109/TIP.2014.2341971.
[44]	D. M. Pelt and K. J. Batenburg, Accurately approximating algebraic tomographic reconstruction by filtered backprojection, Proceedings of International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, Newport, USA, (2015), 158-161.
[45]	D. M. Pelt and V. De Andrade, Improved tomographic reconstruction of large-scale real-world data by filter optimization, Advanced Structural and Chemical Imaging, 2 (2016), 17. doi: 10.1186/s40679-016-0033-y.
[46]	J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Berichte über die Verhandlungen der Sächsische Akademie der Wissenschaften, 69 (1917), 262-277.
[47]	A. Ravishankar, S. Anusha, H. K. Akshatha, A. Raj, S. Jahnavi and J. Madhura, A survey on noise reduction techniques in medical images, Proceedings of International Conference of Electronics, Communication and Aerospace Technology (ICECA), Coimbatore, India, (2017), 385-389.
[48]	R. T. Sadia, J. Chen and J. Zhang, CT image denoising methods for image quality improvement and radiation dose reduction, Journal of Applied Clinical Medical Physics, 25 (2024), e14270. doi: 10.1002/acm2.14270.
[49]	L. A. Shepp and B. F. Logan, The Fourier reconstruction of a head section, IEEE Transactions on Nuclear Science, 21 (1974), 21-43. doi: 10.1109/TNS.1974.6499235.
[50]	H. Shi and S. Luo, A novel scheme to design the filter for CT reconstruction using FBP algorithm, BioMedical Engineering OnLine, 12 (2013), 50. doi: 10.1186/1475-925X-12-50.
[51]	J. Shtok, M. Elad and M. Zibulevsky, Adaptive filtered-back-projection for computed tomography, Proceedings of IEEE Convention of Electrical and Electronics Engineers in Israel, Eilat, Israel, (2008), 528-532.
[52]	K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, Kluwer Academic Publishers, Dordrecht, 1991.
[53]	C. Syben, B. Stimpel, K. Breininger, T. Würfl, R. Fahrig, A. Dörfler and A. Maier, Precision learning: Reconstruction filter kernel discretization, Proceedings of International Conference on Image Formation in X-ray Computed Tomography, Salt Lake City, USA, (2018), 386-390.
[54]	H. Triebel, Function Spaces and Wavelets on Domains, European Mathematical Society Publishing House, Zürich, 2008.
[55]	M. Tsagris, C. Beneki and H. Hassani, On the folded normal distribution, Mathematics, 2 (2014), 12-28. doi: 10.3390/math2010012.
[56]	G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Physical Review, 36 (1930), 823-841. doi: 10.1103/PhysRev.36.823.
[57]	M. Veraar, Regularity of Gaussian white noise on the d-dimensional torus, Marcinkiewicz Centenary Volume, Banach Center Publ., Polish Academy of Sciences, Institute of Mathematics, Warsaw, 95 (2011), 385-398.
[58]	J. Wang, H. Lu, Z. Liang, D. Eremina, G. Zhang, S. Wang, J. Chen and J. Manzione, An experimental study on the noise properties of X-ray CT sinogram data in Radon space, Physics in Medicine & Biology, 53 (2008), 3327-3341.
[59]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.
[60]	B. R. Whiting, P. Massoumzadeh, O. A. Earl, J. A. O'Sullivan, D. L. Snyder and J. F. Williamson, Properties of preprocessed sinogram data in X-ray computed tomography, Medical Physics, 33 (2006), 3290-3303. doi: 10.1118/1.2230762.
[61]	J. Xu, C. Sun, Y. Huang and X. Huang, Residual neural network for filter kernel design in filtered back-projection for CT image reconstruction, Proceedings of German Workshop on Medical Image Computing, Regensburg, Germany, (2021), 164-169. doi: 10.1007/978-3-658-33198-6_39.