Abstract
The image labeling problem can be described as assigning to each pixel a single element from a finite set of predefined labels. Recently, a smooth geometric approach for inferring such label assignments was proposed by following the Riemannian gradient flow of a given objective function on the so-called assignment manifold. Due to the specific Riemannian structure, this results in a coupled replicator dynamic incorporating local spatial geometric averages of lifted data-dependent distances. However, in this framework an approximation of the flow is necessary in order to arrive at explicit formulas. We propose an alternative variational model, where lifting and averaging are decoupled in the objective function so as to stay closer to established approaches and at the same time preserve the main ingredients of the original approach: the overall smooth geometric setting and regularization through geometric local averages. As a consequence the resulting flow is explicitly given, without the need for any approximation. Furthermore, there exists an interesting connection to graphical models.
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References
Absil, P.-A., Mathony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)
Åström, F., Petra, S., Schmitzer, B., Schnörr, C.: Image labeling by assignment. J. Math. Imag. Vis. 58(2), 211–238 (2017)
Bergmann, R., Fitschen, J.H., Persch, J., Steidl, G.: Iterative multiplicative filters for data labeling. Int. J. Comput. Vis. 123(3), 435–453 (2017)
Bergmann, R., Tenbrinck, D.: A graph framework for manifold-valued data. SIAM J. Imag. Sci. 11(1), 325–360 (2018)
Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)
Hühnerbein, R., Savarino, F., Åström, F., Schnörr, C.: Image labeling based on graphical models using wasserstein messages and geometric assignment. SIAM J. Imag. Sci. 11(2), 1317–1362 (2018)
Kappes, J.H., et al.: A comparative study of modern inference techniques for structured discrete energy minimization problems. In: IJCV, vol. 115, no. 2, pp. 155–184 (2015)
Lellmann, J., Schnörr, C.: Continuous multiclass labeling approaches and algorithms. SIAM J. Imag. Sci. 4(4), 1049–1096 (2011)
Savarino, F., Hühnerbein, R., Åström, F., Recknagel, J., Schnörr, C.: Numerical integration of riemannian gradient flows for image labeling. In: Lauze, F., Dong, Y., Dahl, A.B. (eds.) SSVM 2017. LNCS, vol. 10302, pp. 361–372. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58771-4_29
Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn. 1(1–2), 1–305 (2008)
Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imag. Sci. 7(4), 2226–2257 (2014)
Acknowledgments
We gratefully acknowledge support by the German Science Foundation, grant GRK 1653.
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Savarino, F., Schnörr, C. (2019). A Variational Perspective on the Assignment Flow. In: Lellmann, J., Burger, M., Modersitzki, J. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2019. Lecture Notes in Computer Science(), vol 11603. Springer, Cham. https://doi.org/10.1007/978-3-030-22368-7_43
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DOI: https://doi.org/10.1007/978-3-030-22368-7_43
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