Reply to “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’”

doi:10.1364/BOE.2.001265

Biomedical Optics Express

Editor: Joseph A. Izatt
Vol. 2, Iss. 5 — May. 1, 2011
pp: 1265–1267

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Reply to “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’”

Haiou Shen and Ge Wang »View Author Affiliations

Biomedical Optics Express, Vol. 2, Issue 5, pp. 1265-1267 (2011)
http://dx.doi.org/10.1364/BOE.2.001265

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Abstract

We compare the accuracy of TIM-OS and MMCM in response to the recent analysis made by Fang [Biomed. Opt. Express 2, 1258 (2011)]. Our results show that the tetrahedron-based energy deposition algorithm used in TIM-OS is more accurate than the node-based energy deposition algorithm used in MMCM.

Reply

Simulation speed

In [2

H. Shen and G. Wang, “A study on tetrahedron-based inhomogeneous Monte Carlo optical simulation,” Biomed. Opt. Express 2(1), 44–57 (2011). [CrossRef] [PubMed]

], we compared the latest versions of several optical Monte Carlo (MC) simulation packages with our recently developed TIM-OS [1

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55(4), 947–962 (2010). [CrossRef] [PubMed]

]. Particularly, MMCM was downloaded on September 29, 2010 from its website (http://mcx.sourceforgo.net/mmc) and compiled with the best setting in the package. As shown in Dr. Fang’s comment [5

Q. Fang, “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’,” Biomed. Opt. Express 2, 1258–1264 (2011).

], he recently updated the MMCM package that now takes advantage of the SSE instructions and the Intel compiler, yielding a substantial performance gain. However, the latest MMCM still does not take the thread racing condition into account. As pointed out by Alerstam [4

E. Alerstam, W. C. Yip Lo, T. D. Han, J. Rose, S. Andersson-Engels, and L. Lilge, “Next-generation acceleration and code optimization for light transport in turbid media using GPUs,” Biomed. Opt. Express 1(2), 658–675 (2010). [CrossRef] [PubMed]

], thread racing may compromise data integrity. We also observed this problem in the MMCM results.

It is underlined that TIM-OS photon-tetrahedron intersection style has a less computational complexity than the Plücker-coordinate scheme used in MMCM [2

H. Shen and G. Wang, “A study on tetrahedron-based inhomogeneous Monte Carlo optical simulation,” Biomed. Opt. Express 2(1), 44–57 (2011). [CrossRef] [PubMed]

,5

Q. Fang, “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’,” Biomed. Opt. Express 2, 1258–1264 (2011).

]. When we do photon-tetrahedron intersection tests, a photon is actually inside a tetrahedron. Such a tight restriction on the position of the photon greatly reduces the computational complexity. As a result, while the Plücker-coordinate algorithm utilizes all the equations in [3

J. Havel and A. Herout, “Yet faster ray-triangle intersection (using SSE4),” IEEE Trans. Vis. Comput. Graph. 16(3), 434–438 (2010). [CrossRef] [PubMed]

], the original TIM-OS algorithm only uses the popular ray-plane intersection equation.

Simulation accuracy

Figure 1 illustrates the problem in [5

Q. Fang, “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’,” Biomed. Opt. Express 2, 1258–1264 (2011).

]. While the solid curve shows the true value

y_{t r u t h}

,

y_{m m c} (i)

and

y_{t i m o s} (i)

are the values used in [5

Q. Fang, “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’,” Biomed. Opt. Express 2, 1258–1264 (2011).

] to compare MMCM and TIM-OS. However, each

y_{t i m o s} (i)

datum he used had two parts:

y_{t i m o s} (i) = (\int_{(i - 1) Δ x}^{i Δ x} f (x) d x + \int_{i Δ x}^{(i + 1) Δ x} f (x) d x) / 2

, where

\int_{(i - 1) Δ x}^{i Δ x} f (x) d x

and

\int_{i Δ x}^{(i + 1) Δ x} f (x) d x

were the values TIM-OS estimated at the positions

(i - 1 / 2) Δ x

and

(i + 1 / 2) Δ x

, respectively. Hence,

y_{t i m o s} (i)

actually was a linear interpolation of two TIM-OS results. It is not fair to compare a linearly interpolated TIM-OS result to a directly computed MMCM result.

Fig. 1 Illustration of the problem in Dr. Fang’s Comment.

To address this discrepancy for the problem shown in Fig. 1, we compared the results of MMCM and TIM-OS to the true value

1 / (i Δ x)

at an arbitrarily selected point

i Δ x

. In this case, by the meshing requirements of the two simulators, the integral range for MMCM was from

(i - 1) Δ x

to

(i + 1) Δ x

and the range for TIM-OS was from

(i - 1 / 2) Δ x

to

(i + 1 / 2) Δ x

. We have

\begin{array}{l} y_{t r u t h} = f (x) = 1 / x \\ y_{m m c} = (\int_{(i - 1) Δ x}^{(i + 1) Δ x} f (x) φ_{i} (x) d x) / Δ x = ((i + 1) \ln (i + 1) + (i - 1) \ln (i - 1) - 2 i \ln (i)) / Δ x \\ y_{t i m o s} = (\int_{(i - 1 / 2) Δ x}^{(i + 1 / 2) Δ x} f (x) d x) / Δ x = (\ln (i + 1 / 2) - \ln (i - 1 / 2)) / Δ x \end{array}

Then, the relative errors for MMCM and TIM-OS were derived as

\begin{array}{l} e r r o r_{m m c} = (y_{m m c} - 1 / i Δ x) i Δ x = i (i + 1) \ln ((i + 1) / i) - i (i - 1) \ln (i / (i - 1)) - 1 \\ e r r o r_{t i m o s} = (y_{t i m o s} - 1 / i Δ x) i Δ x = i (\ln (i + 1 / 2) - \ln (i - 1 / 2)) - 1 \end{array}

Therefore

\lim_{i - > \infty} e r r o r_{m m c} / e r r o r_{t i m o s} = 2

. Figure 2 plots

e r r o r_{m m c} / e r r o r_{t i m o s}

for

2 \leq i \leq 20

.

Fig. 2 Comparison of MMCM and TIM-OS in terms of the relative error.

Furthermore, we considered a more realistic example in which a pencil beam passed through an absorbing-only media, and the intensity of the light beam would obey Beer’s law along the light path. We got similar result:

\lim_{Δ x - > 0} e r r o r_{m m c} / e r r o r_{t i m o s} = 2

and

e r r o r_{m m c} / e r r o r_{t i m o s} > 1

for

Δ x > 0

. We also set up a mesh to test MMCM and TIM-OS under the above condition. Our experimental results are in an excellent agreement with the analytical prediction. We prepared a package containing all the files for the reader to repeat the experiments, which can be downloaded from http://imaging.sbes.vt.edu/software/tim-os.

Acknowledgment

The work is partially supported by NIHR01HL098912.

References and links

1.	H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55(4), 947–962 (2010). [CrossRef] [PubMed]
2.	H. Shen and G. Wang, “A study on tetrahedron-based inhomogeneous Monte Carlo optical simulation,” Biomed. Opt. Express 2(1), 44–57 (2011). [CrossRef] [PubMed]
3.	J. Havel and A. Herout, “Yet faster ray-triangle intersection (using SSE4),” IEEE Trans. Vis. Comput. Graph. 16(3), 434–438 (2010). [CrossRef] [PubMed]
4.	E. Alerstam, W. C. Yip Lo, T. D. Han, J. Rose, S. Andersson-Engels, and L. Lilge, “Next-generation acceleration and code optimization for light transport in turbid media using GPUs,” Biomed. Opt. Express 1(2), 658–675 (2010). [CrossRef] [PubMed]
5.	Q. Fang, “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’,” Biomed. Opt. Express 2, 1258–1264 (2011).

OCIS Codes
(170.3660) Medical optics and biotechnology : Light propagation in tissues

ToC Category:
Optics of Tissue and Turbid Media

History
Original Manuscript: April 11, 2011
Manuscript Accepted: April 13, 2011
Published: April 19, 2011

Citation
Haiou Shen and Ge Wang, "Reply to “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’”," Biomed. Opt. Express 2, 1265-1267 (2011)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-5-1265

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References

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55(4), 947–962 (2010). [CrossRef] [PubMed]
H. Shen and G. Wang, “A study on tetrahedron-based inhomogeneous Monte Carlo optical simulation,” Biomed. Opt. Express 2(1), 44–57 (2011). [CrossRef] [PubMed]
J. Havel and A. Herout, “Yet faster ray-triangle intersection (using SSE4),” IEEE Trans. Vis. Comput. Graph. 16(3), 434–438 (2010). [CrossRef] [PubMed]
E. Alerstam, W. C. Yip Lo, T. D. Han, J. Rose, S. Andersson-Engels, and L. Lilge, “Next-generation acceleration and code optimization for light transport in turbid media using GPUs,” Biomed. Opt. Express 1(2), 658–675 (2010). [CrossRef] [PubMed]
Q. Fang, “Comment on ‘A study on tetrahedron-based inhomogeneous Monte-Carlo optical simulation’,” Biomed. Opt. Express 2, 1258–1264 (2011).

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