International Journal of Computer Vision

, Volume 101, Issue 2, pp 254–269 | Cite as

A Linear Optimal Transportation Framework for Quantifying and Visualizing Variations in Sets of Images

  • Wei Wang
  • Dejan Slepčev
  • Saurav Basu
  • John A. Ozolek
  • Gustavo K. Rohde
Article

Abstract

Transportation-based metrics for comparing images have long been applied to analyze images, especially where one can interpret the pixel intensities (or derived quantities) as a distribution of ‘mass’ that can be transported without strict geometric constraints. Here we describe a new transportation-based framework for analyzing sets of images. More specifically, we describe a new transportation-related distance between pairs of images, which we denote as linear optimal transportation (LOT). The LOT can be used directly on pixel intensities, and is based on a linearized version of the Kantorovich-Wasserstein metric (an optimal transportation distance, as is the earth mover’s distance). The new framework is especially well suited for computing all pairwise distances for a large database of images efficiently, and thus it can be used for pattern recognition in sets of images. In addition, the new LOT framework also allows for an isometric linear embedding, greatly facilitating the ability to visualize discriminant information in different classes of images. We demonstrate the application of the framework to several tasks such as discriminating nuclear chromatin patterns in cancer cells, decoding differences in facial expressions, galaxy morphologies, as well as sub cellular protein distributions.

Keywords

Optimal transportation Linear embedding 

Notes

Acknowledgements

The authors wish to thank the anonymous reviewers for helping significantly improve this paper. W. Wang, S. Basu, and G.K. Rohde acknowledge support from NIH grants GM088816 and GM090033 (PI GKR) for supporting portions of this work. D. Slepčev was also supported by NIH grant GM088816, as well as NSF grant DMS-0908415. He is also grateful to the Center for Nonlinear Analysis (NSF grant DMS-0635983 and NSF PIRE grant OISE-0967140) for its support.

References

  1. Ambrosio, L., Gigli, N., & Savaré, G. (2008). Lectures in mathematics ETH zürich. Gradient flows in metric spaces and in the space of probability measures (2nd ed.). Basel: Birkhäuser. MATHGoogle Scholar
  2. Angenent, S., Haker, S., & Tannenbaum, A. (2003). Minimizing flows for the Monge-Kantorovich problem. SIAM Journal on Mathematical Analysis, 35(1), 61–97. (electronic). MathSciNetMATHCrossRefGoogle Scholar
  3. Barrett, J. W., & Prigozhin, L. (2009). Partial L 1 Monge-Kantorovich problem: variational formulation and numerical approximation. Interfaces and Free Boundaries, 11(2), 201–238. MathSciNetMATHCrossRefGoogle Scholar
  4. Beg, M., Miller, M., Trouvé, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61(2), 139–157. CrossRefGoogle Scholar
  5. Benamou, J. D., & Brenier, Y. (2000). A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84(3), 375–393. MathSciNetMATHCrossRefGoogle Scholar
  6. Bengtsson, E. (1999). Fifty years of attempts to automate screening for cervical cancer. Medical Imaging Technology, 17, 203–210. Google Scholar
  7. Bishop, C. M. (2006). Pattern recognition and machine learning (information science and statistics). Berlin: Springer. Google Scholar
  8. Blum, H., et al. (1967). A transformation for extracting new descriptors of shape. In Models for the perception of speech and visual form (Vol. 19, pp. 362–380). Google Scholar
  9. Boland, M. V., & Murphy, R. F. (2001). A neural network classifier capable of recognizing the patterns of all major subcellular structures in fluorescence microscope images of hela cells. Bioinformatics, 17(12), 1213–1223. CrossRefGoogle Scholar
  10. do Carmo, M. P. (1992). Riemannian geometry. Mathematics: theory & applications. Boston: Birkhäuser Boston. Translated from the second Portuguese edition by Francis Flaherty. Google Scholar
  11. Chefd’hotel, C., & Bousquet, G. (2007). Intensity-based image registration using earth mover’s distance. Proceedings of SPIE, vol. 6512, p. 65122B. CrossRefGoogle Scholar
  12. Delzanno, G. L., & Finn, J. M. (2010). Generalized Monge-Kantorovich optimization for grid generation and adaptation in L p. SIAM Journal on Scientific Computing, 32(6), 3524–3547. MathSciNetMATHCrossRefGoogle Scholar
  13. Dialynas, G. K., Vitalini, M. W., & Wallrath, L. L. (2008). Linking heterochromatin protein 1 (hp1) to cancer progression. Mutation Research, 647(1–2), 13–20. CrossRefGoogle Scholar
  14. Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7, 179–188. CrossRefGoogle Scholar
  15. Gardner, M., Sprague, B., Pearson, C., Cosgrove, B., Bicek, A., Bloom, K., Salmon, E., & Odde, D. (2010). Model convolution: a computational approach to digital image interpretation. Cellular and Molecular Bioengineering , 3(2), 163–170. CrossRefGoogle Scholar
  16. Grauman, K., & Darrell, T. (2004). Fast contour matching using approximate earth mover’s distance. In Proceedings of the 2004 IEEE computer society conference on computer vision and pattern recognition, CVPR 2004. Google Scholar
  17. Haber, E., Rehman, T., & Tannenbaum, A. (2010). An efficient numerical method for the solution of the L 2 optimal mass transfer problem. SIAM Journal on Scientific Computing, 32(1), 197–211. MathSciNetMATHCrossRefGoogle Scholar
  18. Haker, S., Zhu, L., Tennenbaum, A., & Angenent, S. (2004). Optimal mass transport for registration and warping. International Journal of Computer Vision, 60(3), 225–240. CrossRefGoogle Scholar
  19. Kong, J., Sertel, O., Shimada, H., BOyer, K. L., Saltz, J. H., & Gurcan, M. N. (2009). Computer-aided evaluation of neuroblastoma on whole slide histology images: classifying grade of neuroblastic differentiation. Pattern Recognition, 42, 1080–1092. CrossRefGoogle Scholar
  20. Ling, H., & Okada, K. (2007). An efficient earth mover’s distance algorithm for robust histogram comparison. In IEEE transactions on pattern analysis and machine intelligence (pp. 840–853). Google Scholar
  21. Lloyd, S. P. (1982). Least squares quantization in pcm. IEEE Transactions on Information Theory, 28(2), 129–137. MathSciNetMATHCrossRefGoogle Scholar
  22. Loo, L., Wu, L., & Altschuler, S. (2007). Image-based multivariate profiling of drug responses from single cells. Nature Methods, 4(5), 445–454. Google Scholar
  23. Methora, S. (1992). On the implementation of a primal-dual interior point method. SIAM Journal on Scientific and Statistical Computing, 2, 575–601. Google Scholar
  24. Miller, M. I., Priebe, C. E., Qiu, A., Fischl, B., Kolasny, A., Brown, T., Park, Y., Ratnanather, J. T., Busa, E., Jovicich, J., Yu, P., Dickerson, B. C., & Buckner, R. L. (2009). Collaborative computational anatomy: an mri morphometry study of the human brain via diffeomorphic metric mapping. Human Brain Mapping, 30(7), 2132–2141. CrossRefGoogle Scholar
  25. Moss, T. J., & Wallrath, L. L. (2007). Connections between epigenetic gene silencing and human disease. Mutation Research, 618(1–2), 163–174. CrossRefGoogle Scholar
  26. Orlin, J. B. (1993). A faster strongly polynomial minimum cost flow algorithm. Operations Research, 41(2), 338–350. MathSciNetMATHCrossRefGoogle Scholar
  27. Pele, O., & Werman, M. (2008). A linear time histogram metric for improved sift matching. In ECCV. Google Scholar
  28. Pele, O., & Werman, M. (2009). Fast and robust earth mover’s distances. In Computer vision, 2009 IEEE 12th international conference on (pp. 460–467). New York: IEEE Press. CrossRefGoogle Scholar
  29. Pincus, Z., & Theriot, J. A. (2007). Comparison of quantitative methods for cell-shape analysis. Journal of Microscopy , 227(2), 140–156. MathSciNetCrossRefGoogle Scholar
  30. Rohde, G. K., Ribeiro, A. J. S., Dahl, K. N., & Murphy, R. F. (2008). Deformation-based nuclear morphometry: capturing nuclear shape variation in hela cells. Cytometry, 73(4), 341–350. CrossRefGoogle Scholar
  31. Roweis, S. T., & Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500), 2323–2326. doi: 10.1126/science.290.5500.2323. CrossRefGoogle Scholar
  32. Rubner, Y., Tomassi, C., & Guibas, L. J. (2000). The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision, 40(2), 99–121. MATHCrossRefGoogle Scholar
  33. Rueckert, D., Frangi, A. F., & Schnabel, J. A. (2003). Automatic construction of 3-d statistical deformation models of the brain using nonrigid registration. IEEE Transactions on Medical Imaging, 22(8), 1014–1025. CrossRefGoogle Scholar
  34. Shamir, L. (2009). Automatic morphological classification of galaxy images. Monthly Notices of the Royal Astronomical Society. Google Scholar
  35. Shirdhonkar, S., & Jacobs, D. (2008). Approximate earth mover’s distance in linear time. In Proceedings of the 2008 IEEE computer society conference on computer vision and pattern recognition, CVPR 2008. Google Scholar
  36. Stegmann, M., Ersboll, B., & Larsen, R. (2003). Fame—a flexible appearance modeling environment. IEEE Transactions on Medical Imaging, 22(10), 1319–1331. CrossRefGoogle Scholar
  37. Tenenbaum, J. B., de Silva, V., & Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500), 2319–2323. doi: 10.1126/science.290.5500.2319. CrossRefGoogle Scholar
  38. Vaillant, M., Miller, M., Younes, L., & Trouvé, A. (2004). Statistics on diffeomorphisms via tangent space representations. NeuroImage, 23, S161–S169. CrossRefGoogle Scholar
  39. Villani, C. (2003). Graduate studies in mathematics: Vol. 58. Topics in optimal transportation. Providence: Am. Math. Soc.. MATHGoogle Scholar
  40. Villani, C. (2009). In Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences]: Vol. 338. Optimal transport. Berlin: Springer. doi: 10.1007/978-3-540-71050-9. MATHCrossRefGoogle Scholar
  41. Wang, W., Mo, Y., Ozolek, J. A., & Rohde, G. K. (2011). Penalized fisher discriminant analysis and its application to image-based morphometry. Pattern Recognition Letters, 32(15), 2128–2135. CrossRefGoogle Scholar
  42. Wang, W., Ozolek, J., & Rohde, G. (2010). Detection and classification of thyroid follicular lesions based on nuclear structure from histopathology images. Cytometry Part A, 77(5), 485–494. Google Scholar
  43. Wang, W., Ozolek, J., Slepčev, D., Lee, A., Chen, C., & Rohde, G. (2011). An optimal transportation approach for nuclear structure-based pathology. IEEE Transactions on Medical Imaging, 30(3), 621–631. CrossRefGoogle Scholar
  44. Yang, L., Chen, W., Meer, P., Salaru, G., Goodell, L., Berstis, V., & Foran, D. (2009). Virtual microscopy and grid-enabled decision support for large scale analysis of imaged pathology specimens. IEEE Transactions on Information Technology in Biomedicine, 13(4), 636–644. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Wei Wang
    • 1
  • Dejan Slepčev
    • 3
  • Saurav Basu
    • 1
  • John A. Ozolek
    • 4
  • Gustavo K. Rohde
    • 2
  1. 1.Center for Bioimage Informatics, Department of Biomedical EngineeringCarnegie Mellon UniversityPittsburghUSA
  2. 2.Center for Bioimage Informatics, Department of Biomedical Engineering, Department of Electrical and Computer Engineering, Lane Center for Computational BiologyCarnegie Mellon UniversityPittsburghUSA
  3. 3.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  4. 4.Department of PathologyChildren’s Hospital of PittsburghPittsburghUSA

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