## Noise reduction with low dose CT data based on a modified ROF model |

Optics Express, Vol. 20, Issue 16, pp. 17987-18004 (2012)

http://dx.doi.org/10.1364/OE.20.017987

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### Abstract

In order to reduce the radiation exposure caused by Computed Tomography (CT) scanning, low dose CT has gained much interest in research as well as in industry. One fundamental difficulty for low dose CT lies in its heavy noise pollution in the raw data which leads to quality deterioration for reconstructed images. In this paper, we propose a modified ROF model to denoise low dose CT measurement data in light of Poisson noise model. Experimental results indicate that the reconstructed CT images based on measurement data processed by our model are in better quality, compared to the original ROF model or bilateral filtering.

© 2012 OSA

## 1. Introduction

1. A. Berrington de Gonzalez, M. Mahesh, K. Kim, M. Bhargavan, R. Lewis, F. Mettler, and C. Land, “Projected cancer risks from computed tomographic scans performed in the united states in 2007,” Arch. Intern Med. **169**, 2071 (2009). [CrossRef] [PubMed]

^{12}(or 2

^{14}). In this case, while scanning large or high density object, the counts of the raw data would be low.

2. A. Schilham, B. van Ginneken, H. Gietema, and M. Prokop, “Local noise weighted filtering for emphysema scoring of low-dose ct images,” IEEE Trans. Med. Imaging **25**, 451–463 (2006). [CrossRef] [PubMed]

3. M. Tabuchi, N. Yamane, and Y. Morikawa, “Adaptive wiener filter based on gaussian mixture model for denoising chest x-ray ct image,” in *IEEE Proceedings of SICE 2007 Annual Conference* (IEEE, 2007), pp. 682–689. [CrossRef]

5. P. La Rivière and D. Billmire, “Reduction of noise-induced streak artifacts in x-ray computed tomography through spline-based penalized-likelihood sinogram smoothing,” IEEE Trans. Med. Imaging **24**, 105–111 (2005). [CrossRef] [PubMed]

6. P. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose ct,” Med. Phys. **32**, 1676 (2005). [CrossRef] [PubMed]

7. 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 Trans. Nucl. Sci. **51**, 2505–2513 (2004). [CrossRef]

8. J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose x-ray computed tomography,” IEEE Trans. Med. Imaging **25**, 1272–1283 (2006). [CrossRef] [PubMed]

## 2. Noise model and the ROF model

### 2.1. Noise model

*I*

_{0}indicates the number of emitted photons,

*I*the number of photons collected,

^{d}*f*(

*x*) the attenuation distribution of the object under examination and

*L*the X-Ray trajectory. CT image reconstruction is an inverse problem, which is to solve

*f*(

*x*) from a series of measurement data

*p*the projection data. For formula (2), we usually use the Filtered-Back-Projection algorithm (FBP) [16] to compute

_{i}*f*(

*x*). In the previous work [16, 17], it’s shown that the low-dose calibrated data could be approximated as the ideal data polluted by non-stationary gaussian noise

*n*and

*n*has a variance and mean relationship as where

*σ*and

_{i}*μ*denote the variance and mean respectively, while

_{i}*f*and

_{i}*T*are parameters that determined by the property of detectors.

*p*. Instead, we deal with the raw data

_{i}*I*, which is named

^{d}*projection raw data*in this paper. The reason is that we think that

*I*is less “polluted” than

^{d}*p*, since less mathematical computations are involved. We also found out that it’s not obvious how to design a denoising method for the noise model of

_{i}*p*. For real CT systems, there are also two kinds of background noise–electrical thermal noise (i.i.d. gaussian) and round-off errors. Moreover, the monoenergetic source is just an ideal assumption. So the Poisson distribution is just an approximate model. Strictly, we can only say that the noise is Poisson-like. In real CT systems,

_{i}*I*

_{0}in formulas (1) and (2) is usually sampled several hundred times for average. So

*I*

_{0}could be thought of as ideal. What we should do is to find the ideal of

*I*, then compute

_{d}*p*.

_{i}### 2.2. ROF model and Chambolle’s method

9. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D **60**, 259–268 (1992). [CrossRef]

*λ*where (for regular enough of

*u*) In formula (6), Ω is the image domain,

*f*is the input image which is assumed to contain additive white Gaussian noise.

18. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision **20**, 89–97 (2004). [CrossRef]

*u*

^{n+1}denotes the denoised image after the (

*n*+ 1)th iteration,

*div*(·) the divergence operator and

*τ*the pseudo time step, which should be less than 1/4 for 2D images to guarantee convergence.

## 3. The modified ROF model - Poisson-ROF

### 3.1. Deviation of the Poisson-ROF model

**Remark 3.1**Mathematically, the data term in (10) suffers from nonlinearity and degeneration problem at zeros. These two difficulties could be overcome by changing the term

19. T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. **36**, 354–367 (1999). [CrossRef]

*u*(

*x*) = 0 means that the X-Ray can not penetrate the object from some trajectory, and the associated information is lost. Insufficient information results in deterioration for the reconstructed images. In this case, people usually increase the intensity of the X-Ray to avoid its happening. So for the CT projection raw data, one could think that

*u*(

*x*) > 0 holds true all the time.

**Remark 3.2**It’s easy to verify that if

*u*is strictly greater than zero, then the minimization problem (10) is strictly convex. So the problem admits a single solution.

### 3.2. Justification of the Poisson-ROF model – statistical analysis

*w*(

*x*) can be expressed as where [·] denotes the

*round*operation. For real CT systems, the count of raw data mostly lies in the range of [1000, 5000]. We compute the density function for

*u*(

*x*) = 100, 1000, 3000 and 5000, and plot them in Fig. 1. From Fig. 1, we see that in real CT systems, even the distributions corresponding to

*w*(

*x*) are not exact the same (pixel dependent), they are very similar to each other. So we can think that

*w*(

*x*) obeys almost the same distribution for different

*x*. This justifies the data term in (9) and the Poisson-ROF model. For Poisson-like noise, we believe that the above argument about

*w*(

*x*) still holds, and the Poisson-ROF model should be valid even for Poisson-like noise. It should be pointed out that the density function for a very low count case that

*u*(

*x*) = 100 (which is unlikely to happen for real CT scanning) is also plotted in Fig. 1. This is to show that the above observation about

*w*(

*x*) could be valid for a much larger spectrum, e.g. our model should be valid for ”very-low” dose CT.

### 3.3. Algorithm to solve the Poisson-ROF model

*u*. To easy the problem, we do a iterative lineation of the data term. Let’s first suppose that the variance is known as

*f*, e.g. where

_{NF}*f*indicates the noise free data of

_{NF}*f*. Then the Poisson-ROF model becomes where The corresponding E-L Eq. is Obviously,

*f*is not available. So we approximate it by

_{NF}*ū*during the iterations of the proposed algorithm described below, and

*ū*is calculated by bilateral filtering where The Chambolle’s algorithm can not be applied to the Poisson-ROF model directly. So we introduce another variable

*v*to relax the problem, just like in [20]. Then we solve (15) by solving the following two subproblems alternatively

- Minimization with
*u*, for which Chambolle’s method could be used, or one can adopt some more recently proposed fast algorithms such as the Split-Bregman [21];21. T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci.

**2**, 323–343 (2009). [CrossRef]

*θ*is set to 0.1,

*v*is updated for each 5 iterations.

## 4. Experiments

### 4.1. Simulation data

26. H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in *IEEE 19th International Conference on Systems Engineering 2008* (IEEE, 2008), pp. 521–526. [CrossRef]

^{4}photons are transmitted for each image in this way. The ideal CT images are computed by analytic method, e.g. FBP. We then pollute the raw data with simulated Poisson noise. Three different denoising methods including bilateral filtering, ROF model and Poisson-ROF model, are applied to the simulated projection raw data. The results are shown in Fig. 2. In the first column, from (a)–(b), the ideal data and noisy data are shown respectively. From (c)–(e), the data denoised by bilateral filtering, ROF model and Poisson-ROF model are shown separately. In the second column, the corresponding images reconstructed by FBP algorithm are shown. By comparing the reconstructed images, we see that the bilateral filtering can preserve the structures of the phantoms, while suppressing some of the noise. The ROF model could remove most of the noise, but the structures (edges) are also diffused, which is unacceptable in real applications. For the Poisson-ROF model, most of the noise are removed, while the details are also preserved, see Fig. 2(e’). We also give the profiles of each image in Fig. 3.

### 4.2. Real CT data

#### 4.2.1. Volcanic rock at different dose

#### 4.2.2. line-pairs phantom at low dose

## 5. Conclusion

## Acknowledgment

## References and links

1. | A. Berrington de Gonzalez, M. Mahesh, K. Kim, M. Bhargavan, R. Lewis, F. Mettler, and C. Land, “Projected cancer risks from computed tomographic scans performed in the united states in 2007,” Arch. Intern Med. |

2. | A. Schilham, B. van Ginneken, H. Gietema, and M. Prokop, “Local noise weighted filtering for emphysema scoring of low-dose ct images,” IEEE Trans. Med. Imaging |

3. | M. Tabuchi, N. Yamane, and Y. Morikawa, “Adaptive wiener filter based on gaussian mixture model for denoising chest x-ray ct image,” in |

4. | E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol. |

5. | P. La Rivière and D. Billmire, “Reduction of noise-induced streak artifacts in x-ray computed tomography through spline-based penalized-likelihood sinogram smoothing,” IEEE Trans. Med. Imaging |

6. | P. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose ct,” Med. Phys. |

7. | 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 Trans. Nucl. Sci. |

8. | J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose x-ray computed tomography,” IEEE Trans. Med. Imaging |

9. | L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D |

10. | J. Hsieh, |

11. | K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography.” J. Comput. Assist. Tomogr. |

12. | T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision |

13. | L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn. |

14. | F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process. |

15. | B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. |

16. | A. C. Kak and M. Slaney, |

17. | H. Lu, T. Hsiao, X. Li, and Z. Liang, “Noise properties of low-dose ct projections and noise treatment by scale transformations,” in in the |

18. | A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision |

19. | T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal. |

20. | M. Unger, T. Pock, and H. Bischof, “Continuous globally optimal image segmentation with local constraints,” in Computer Vision Winter Workshop at Slovenian Pattern Recognition Society, Ljubljana, Slovenia (2008). |

21. | T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci. |

22. | L. Rudin and S. Osher, “Total variation based image restoration with free local constraints,” in |

23. | G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process. |

24. | B. Wohlberg and Y. Lin, “Upre method for total variation parameter selection,” Tech. Report, Los Alamos National Laboratory (LANL) (2008). |

25. | S. Babacan, R. Molina, and A. Katsaggelos, “Parameter estimation in tv image restoration using variational distribution approximation,” IEEE Trans. Image Process. |

26. | H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in |

**OCIS Codes**

(030.4280) Coherence and statistical optics : Noise in imaging systems

(170.7440) Medical optics and biotechnology : X-ray imaging

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: April 16, 2012

Revised Manuscript: July 6, 2012

Manuscript Accepted: July 12, 2012

Published: July 23, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Yining Zhu, Mengliu Zhao, Yunsong Zhao, Hongwei Li, and Peng Zhang, "Noise reduction with low dose CT data based on a modified ROF model," Opt. Express **20**, 17987-18004 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-16-17987

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### References

- A. Berrington de Gonzalez, M. Mahesh, K. Kim, M. Bhargavan, R. Lewis, F. Mettler, and C. Land, “Projected cancer risks from computed tomographic scans performed in the united states in 2007,” Arch. Intern Med.169, 2071 (2009). [CrossRef] [PubMed]
- A. Schilham, B. van Ginneken, H. Gietema, and M. Prokop, “Local noise weighted filtering for emphysema scoring of low-dose ct images,” IEEE Trans. Med. Imaging25, 451–463 (2006). [CrossRef] [PubMed]
- M. Tabuchi, N. Yamane, and Y. Morikawa, “Adaptive wiener filter based on gaussian mixture model for denoising chest x-ray ct image,” in IEEE Proceedings of SICE 2007 Annual Conference (IEEE, 2007), pp. 682–689. [CrossRef]
- E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam ct,” J. X-Ray Sci. Technol.14, 119–139 (2006).
- P. La Rivière and D. Billmire, “Reduction of noise-induced streak artifacts in x-ray computed tomography through spline-based penalized-likelihood sinogram smoothing,” IEEE Trans. Med. Imaging24, 105–111 (2005). [CrossRef] [PubMed]
- P. La Rivière, “Penalized-likelihood sinogram smoothing for low-dose ct,” Med. Phys.32, 1676 (2005). [CrossRef] [PubMed]
- 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 Trans. Nucl. Sci.51, 2505–2513 (2004). [CrossRef]
- J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose x-ray computed tomography,” IEEE Trans. Med. Imaging25, 1272–1283 (2006). [CrossRef] [PubMed]
- L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D60, 259–268 (1992). [CrossRef]
- J. Hsieh, Computed tomography: principles, design, artifacts, and recent advances (Society of Photo Optical, 2003), Vol. 114.
- K. Lange, R. Carson, and , “Em reconstruction algorithms for emission and transmission tomography.” J. Comput. Assist. Tomogr.8, 306 (1984). [PubMed]
- T. Le, R. Chartrand, and T. Asaki, “A variational approach to reconstructing images corrupted by poisson noise,” J. Math. Imaging Vision27, 257–263 (2007). [CrossRef]
- L. Zhang, L. Zhang, D. Zhang, and H. Zhu, “Computer analysis of images and patterns,” Pattern Recogn.44, 1990–1998 (2011). [CrossRef]
- F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed poisson-gaussian noise,” IEEE Trans. Image Process.20(3), 696–708 (2011). [CrossRef]
- B. Zhang, J. Fadili, and J. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process.17, 1093–1108 (2008). [CrossRef] [PubMed]
- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Service Center, 1988), p. 327.
- H. Lu, T. Hsiao, X. Li, and Z. Liang, “Noise properties of low-dose ct projections and noise treatment by scale transformations,” in in the IEEE Nuclear Science Symposium Conference 2001 Record (IEEE, 2001), Vol. 3, pp. 1662–1666.
- A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004). [CrossRef]
- T. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM J. Numer. Anal.36, 354–367 (1999). [CrossRef]
- M. Unger, T. Pock, and H. Bischof, “Continuous globally optimal image segmentation with local constraints,” in Computer Vision Winter Workshop at Slovenian Pattern Recognition Society, Ljubljana, Slovenia (2008).
- T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imag. Sci.2, 323–343 (2009). [CrossRef]
- L. Rudin and S. Osher, “Total variation based image restoration with free local constraints,” in Proceedings of the IEEE International Conference on Image Processing1994 (IEEE, 1994), vol. 1, pp. 31–35.
- G. Gilboa, N. Sochen, and Y. Zeevi, “Estimation of optimal pde-based denoising in the snr sense,” IEEE Trans. Image Process.15, 2269–2280 (2006). [CrossRef] [PubMed]
- B. Wohlberg and Y. Lin, “Upre method for total variation parameter selection,” Tech. Report, Los Alamos National Laboratory (LANL) (2008).
- S. Babacan, R. Molina, and A. Katsaggelos, “Parameter estimation in tv image restoration using variational distribution approximation,” IEEE Trans. Image Process.17, 326–339 (2008). [CrossRef] [PubMed]
- H. Gach, C. Tanase, and F. Boada, “2d & 3d shepp-logan phantom standards for mri,” in IEEE 19th International Conference on Systems Engineering 2008 (IEEE, 2008), pp. 521–526. [CrossRef]

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