# ACCELERATION OF THE EXPECTATION-MAXIMIZATION ALGORITHM FOR A TWOFOLD GAUSSIAN MIXTURE MODEL BY USING THE HISTOGRAM OF THE OBSERVATIONS INSTEAD OF THE OBSERVATIONS - Evaluation of its Accuracy by Generated Histograms

### J. Bruijns

#### 2009

#### Abstract

Volume representations of blood vessels acquired by 3D rotational angiography are very suitable for diagnosing a stenosis or an aneurysm. For optimal treatment, physicians need to know the shape of the diseased vessel parts. Binary segmentation by thresholding is the first step in our shape extraction procedure. Assuming a twofold Gaussian mixture model, the model parameters (and thus the threshold for binary segmentation) can be extracted from the observations (i.e. the gray values) by the Expectation-Maximization (EM) algorithm. Since the EM algorithm requires a number of iterations through the observations, and because of the large number of observations, the EM algorithm is very time-consuming. Therefore, we developed a method to apply the EM algorithm to the histogram of the observations, requiring a single pass through the observations and a number of iterations through the much smaller histogram. This variant gives almost the same results as the original EM algorithm, at least for our clinical volumes. We have used this variant for an evaluation of the accuracy of the EM algorithm: the maximum relative error in the mixing coefficients was less than 7%, the maximum relative error in the parameters of the two Gaussian components was less than 2.5%.

#### References

- Bilmes, J. (1997). A gentle tutorial on the EM algorithm and its application to parameter estimation for gaussian mixture and hidden markov models. Technical Report ICSI-TR-97-021, University of Berkeley, Berkeley, CA, USA.
- Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statist. Soc. Series B (Methodological), 39(1):1-38.
- Frederix, G. (2005). Beyond Gaussian Mixture Models: Unsupervised Learning with applications to Image Analysis. PhD thesis, Katholieke Universiteit of Leuven, Belgium.
- Frederix, G. and pauwels, E. (2004). A statistically principled approach to histogram segmentation. Technical Report Report PNA-E0401, CWI, Amsterdam, The Netherlands.
- Gan, R., Chung, A., Wong, W., and Yu, S. (2004a). Vascular segmentation in three-dimensional rotational angiography based on maximum intensity projections. In Proc. IEEE International Symposium on Biomedical Imaging: From Nano to Macro, pages 133-136, Arlington, VA, USA.
- Gan, R., Wong, W., and Chung, A. (2005). Statistical cerebrovascular segmentation in three-dimensional rotational angiography based on maximum intensity projections. Med. Phys., 32(9):3017-3028.
- Gan, R., Wong, W., Chung, A., and Yu, S. (2004b). Statistical cerebrovascular segmentation in threedimensional rotational angiography based on maximum intensity projections. In Proc. CAR, pages 195- 200, Chicago, USA.
- Kemkers, R., de Beek, J. O., Aerts, H., Koppe, R., Klotz, E., Grasse, M., and Moret, J. (1998). 3D-rotational angiography: First clinical application with use of a standard philips C-Arm system. In Proc. CAR, pages 182-187, Tokyo, Japan.
- Kittler, J. and Illingworth, J. (1986). Minimum error thresholding. Pattern Recognition, 19(1):41-47.
- Liao, P., Chen, T., and Chung, P. (2001). A fast algorithm for multilevel thresholding. Journal of Information Science and Engineering, 17(5):713-727.
- Moret, J., Kemkers, R., de Beek, J. O., Koppe, R., Klotz, E., and Grass, M. (1998). 3D rotational angiography: Clinical value in endovascular treatment. Medicamundi, 42(3):8-14.
- Otsu, N. (1979). A threshold selection method from gray level histograms. IEEE Trans. Syst., Man, Cybern., 9(1):62-66.
- Philips-Medical-Systems-Nederland (2001). INTEGRIS 3D-RA. instructions for use. release 2.2. Technical Report 9896 001 32943, Philips Medical Systems Nederland, Best, The Netherlands.
- Wilson, D. and Noble, J. (1999). An adaptive segmentation algorithm for time-of-flight MRA data. IEEE Trans. Med. Imag., 18(10):938-945.

#### Paper Citation

#### in Harvard Style

Bruijns J. (2009). **ACCELERATION OF THE EXPECTATION-MAXIMIZATION ALGORITHM FOR A TWOFOLD GAUSSIAN MIXTURE MODEL BY USING THE HISTOGRAM OF THE OBSERVATIONS INSTEAD OF THE OBSERVATIONS - Evaluation of its Accuracy by Generated Histograms** . In *Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)* ISBN 978-989-8111-69-2, pages 229-236. DOI: 10.5220/0001652902290236

#### in Bibtex Style

@conference{visapp09,

author={J. Bruijns},

title={ACCELERATION OF THE EXPECTATION-MAXIMIZATION ALGORITHM FOR A TWOFOLD GAUSSIAN MIXTURE MODEL BY USING THE HISTOGRAM OF THE OBSERVATIONS INSTEAD OF THE OBSERVATIONS - Evaluation of its Accuracy by Generated Histograms},

booktitle={Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)},

year={2009},

pages={229-236},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001652902290236},

isbn={978-989-8111-69-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)

TI - ACCELERATION OF THE EXPECTATION-MAXIMIZATION ALGORITHM FOR A TWOFOLD GAUSSIAN MIXTURE MODEL BY USING THE HISTOGRAM OF THE OBSERVATIONS INSTEAD OF THE OBSERVATIONS - Evaluation of its Accuracy by Generated Histograms

SN - 978-989-8111-69-2

AU - Bruijns J.

PY - 2009

SP - 229

EP - 236

DO - 10.5220/0001652902290236