Top

International Journal of Computer Vision

Published in:

17-02-2017

Iterative Multiplicative Filters for Data Labeling

Authors: Ronny Bergmann, Jan Henrik Fitschen, Johannes Persch, Gabriele Steidl

Published in: International Journal of Computer Vision | Issue 3/2017

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Based on an idea in Åström et al. (J Math ImagingVis, doi:10.1007/s10851-016-0702-4, 2017) we propose a new iterative multiplicative filtering algorithm for label assignment matrices which can be used for the supervised partitioning of data. Starting with a row-normalized matrix containing the averaged distances between prior features and observed ones, the method assigns in a very efficient way labels to the data. We interpret the algorithm as a gradient ascent method with respect to a certain function on the product manifold of positive numbers followed by a reprojection onto a subset of the probability simplex consisting of vectors whose components are bounded away from zero by a small constant. While such boundedness away from zero is necessary to avoid an arithmetic underflow, our convergence results imply that they are also necessary for theoretical reasons. Numerical examples show that the proposed simple and fast algorithm leads to very good results. In particular we apply the method for the partitioning of manifold-valued images.

previous article Directed Acyclic Graph Continuous Max-Flow Image Segmentation for Unconstrained Label Orderings

next article Lie-X: Depth Image Based Articulated Object Pose Estimation, Tracking, and Action Recognition on Lie Groups

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

See http://camino.cs.ucl.ac.uk.

Adams, B. L., Wright, S. I., & Kunze, K. (1993). Orientation imaging: The emergence of a new microscopy. Journal Metallurgical and Materials Transactions A, 24, 819–831.CrossRef

Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2006). Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine, 56(2), 411–421.CrossRef

Åström, F., Petra, S., Schmitzer, B., & Schnörr, C. (2016a). A geometric approach to image labeling. In Proceedings of the ECCV.

Åström, F., Petra, S., Schmitzer, B., & Schnörr, C. (2016b). The Assignment Manifold: A smooth model for image labeling. In Proceedings of the 2nd international workshop on differential geometry in computer vision and machine learning.

Åström, F., Petra, S., Schmitzer, B., & Schnörr, C. (2017). Image labeling by assignment. Journal of Mathematical Imaging and Vision. doi:10.1007/s10851-016-0702-4.

Bachmann, F., & Hielscher, R. (2005–2016). MTEX—MATLAB toolbox for quantitative texture analysis. http://mtex-toolbox.github.io/.

Bachmann, F., Hielscher, R., & Schaeben, H. (2011). Grain detection from 2d and 3d EBSD data-specification of the MTEX algorithm. Ultramicroscopy, 111(12), 1720–1733.CrossRef

Bae, E., Yuan, J., & Tai, X.-C. (2011). Global minimization for continuous multiphase partitioning problems using a dual approach. International Journal of Computer Vision, 92(1), 112–129.MathSciNetCrossRefMATH

Belkin, M., & Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15, 1373–1396.CrossRefMATH

Buades, A., Coll, B., & Morel, J.-M. (2006). Neighborhood filters and PDEs. Numerische Mathematik, 105, 1–34.MathSciNetCrossRefMATH

Buades, A., Coll, B., & Morel, J.-M. (2010). Image denoising methods. A new nonlocal principle. SIAM Review, 52(1), 113–147.MathSciNetCrossRefMATH

Burger, M., Sawatzky, A., & Steidl, G. (2016). First order algorithms in variational image processing. In R. Glowinski, S. Osher, & W. Yin (Eds.), Operator splittings and alternating direction methods. Berlin: Springer.

Cai, X., Chan, R., & Zeng, T. (2013). A two-stage image segmentation method using a convex variant of the Mumford–Shah model and thresholding. SIAM Journal on Imaging Sciences, 6(1), 368–390.MathSciNetCrossRefMATH

Chambolle, A., Cremers, D., & Pock, T. (2012). A convex approach to minimal partitions. SIAM Journal on Imaging Sciences, 5(4), 1113–1158.MathSciNetCrossRefMATH

Chambolle, A., & Pock, T. (2011). A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1), 120–145.MathSciNetCrossRefMATH

Chan, T. F., Esedoglu, S., & Nikolova, M. (2006). Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal on Applied Mathematics, 66(5), 1632–1648.MathSciNetCrossRefMATH

Chaux, C., Jezierska, A., Pesquet, J.-C., & Talbot, H. (2011). A spatial regularization approach for vector quantization. Journal of Mathematical Imaging and Vision, 41(1–2), 23–38.MathSciNetCrossRefMATH

Cook, P. A., Bai, Y., Nedjati-Gilani, S., Seunarine, K. K., Hall, M. G., Parker, G. J., & Alexander, D. C. (2006)(2006) Camino: Open-source diffusion-MRI reconstruction and processing. In 14th Scientific meeting of the international society for magnetic resonance in medicine (p. 2759). Seattle, WA.

Deledalle, C. A., Denis, L., & Tupin, F. (2009). Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Transactions on Image Processing, 18(12), 2661–2672.MathSciNetCrossRef

Elad, M., & Aharon, M. (2006). Image denoising via sparse and redundant representations over learned dictionaries. IEEE Transactions on Image Processing, 15(12), 3736–3745.MathSciNetCrossRef

Folland, G. B. (1999). Real analysis. New York: Wiley.MATH

Frobenius, G. F. (1912). Über Matrizen aus nichtnegativen Elementen. Königliche Akademie der Wissenschaften.

Gilboa, G., & Osher, S. (2007). Nonlocal linear image regularization and supervised segmentation. SIAM Journal on Multiscale Modeling and Simulation, 6(2), 595–630.MathSciNetCrossRefMATH

Gräf, M. (2012). A unified approach to scattered data approximation on \(\mathbb{S}^{3}\) and \(\operatorname{SO}(3)\). Advances in Computational Mathematics, 37, 379–392.MathSciNetCrossRefMATH

Häuser, S., & Steidl, G. (2013). Convex multiclass segmentation with shearlet regularization. International Journal of Computer Mathematics, 90(1), 62–81.MathSciNetCrossRefMATH

Herault, L., & Horaud, R. (1993). Figure-ground discrimination: A combinatorial optimization approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(9), 899–914.CrossRef

Hofmann, T., & Buhmann, J. M. (1997). Pairwise data clustering by deterministic annealing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(1), 1–14.CrossRef

Horn, R. A., & Johnson, C. R. (2013). Matrix analysis (2nd ed.). Cambridge, MA: Cambridge University Press.MATH

Hummel, R. A., & Zucker, S. W. (1983). On the foundation of relaxation labeling processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI, 5(3), 267–287.CrossRefMATH

Kappes, J. H., Andres, B., Hamprecht, F. A., Schnörr, C., Nowozin, S., Batra, D., et al. (2015). A comparative study of modern inference techniques for structured discrete energy minimization problems. International Journal of Computer Vision, 115(2), 155–184.MathSciNetCrossRef

Kolmogorov, V. (2006). Convergent tree-reweighted message passing for energy minimization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(10), 1568–1583.CrossRef

Kolmogorov, V., & Zabih, R. (2004). What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(2), 147–159.CrossRefMATH

Kunze, K., Wright, S. I., Adams, B. L., & Dingley, D. J. (1993). Advances in automatic EBSP single orientation measurements. Textures and Microstructures, 20, 41–54.CrossRef

Laus, F., Persch, J., & Steidl, G. (2016). A nonlocal denoising algorithm for manifold-valued images using second order statistics. SIAM Journal on Imaging Sciences. ArXiv, Preprint arXiv:1607.08481.

Lellmann, J., Lenzen, F., & Schnörr, C. (2013). Optimality bounds for a variational relaxation of the image partitioning problem. Journal of Mathematical Imaging and Vision, 47(3), 239–257.MathSciNetCrossRefMATH

Lellmann, J., & Schnörr, C. (2011). Continuous multiclass labeling approaches and algorithms. SIAM Journal on Imaging Sciences, 4(4), 1049–1096.MathSciNetCrossRefMATH

Moakher, M., & Batchelor, P. G. (2006). Symmetric positive-definite matrices: From geometry to applications and visualization. In Visualization and processing of tensor fields (pp. 285–298). Berlin: Springer.

Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42(5), 577–685.MathSciNetCrossRefMATH

Nolze, G., & Hielscher, R. (2016). IPF coloring of crystal orientation data. Preprint Technische Universität Chemnitz.

Orland, H. (1985). Mean-field theory for optimization problems. Journal de Physique Lettres, 46(17), 763–770.CrossRef

Pelillo, M. (1997). The dynamics of nonlinear relaxation labeling processes. Journal of Mathematical Imaging and Vision, 7, 309–323.MathSciNetCrossRef

Pennec, X., Fillard, P., & Ayache, N. (2006). A Riemannian framework for tensor computing. International Journal of Computer Vision, 66, 41–66.CrossRefMATH

Perron, O. (1907). Zur Theorie der Matrizen. Mathematische Annalen, 64(2), 248–263.MathSciNetCrossRefMATH

Peyré, G. (2015). Entropic Wasserstein gradient flows. ArXiv Preprint arXiv:1502.06216v3.

Rosenfeld, A., Hummel, R. A., & Zucker, S. W. (1976). Scene labeling by relaxation operations. IEEE Transactions on Systems, Man and Cybernetics, SMC, 6(6), 420–433.MathSciNetCrossRefMATH

Rudin, L. I., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D, 60(1), 259–268.MathSciNetCrossRefMATH

Szeliski, R., Zabih, R., Scharstein, D., Veksler, O., Kolmogorov, V., Agarwala, A., et al. (2008). A comparative study of energy minimizing methods for Markov random fields with smoothness-based priors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(6), 1068–1080.CrossRef

Tomasi, C., & Manduchi, R. (1998). Bilateral filtering for gray and color images. In Proceedings of the sixth international conference on computer vision (pp. 839–846). Bombay. Narosa Publishing House.

Tuzel, O., Porikli, F., & Meer, P. (2006). Region covariance: A fast descriptor for detection and classification. In European conference on computer vision (pp. 589–600). Berlin: Springer.

Wainwright, M. J., & Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 1(1–2), 1–305.MATH

Weickert, J. (1998). Anisotropic diffusion in image processing. Stuttgart: Teubner.MATH

Wielandt, H. (1950). Unzerlegbare, nicht negative Matrizen. Mathematische Zeitschrift, 52(1), 642–648.MathSciNetCrossRefMATH

Yedidia, J. S., Freeman, W. T., & Weiss, Y. (2005). Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Transactions on Information Theory, 51(7), 2282–2312.MathSciNetCrossRefMATH

Title: Iterative Multiplicative Filters for Data Labeling
Authors: Ronny Bergmann
Jan Henrik Fitschen
Johannes Persch
Gabriele Steidl
Publication date: 17-02-2017
Publisher: Springer US
Published in: International Journal of Computer Vision / Issue 3/2017
Print ISSN: 0920-5691
Electronic ISSN: 1573-1405
DOI: https://doi.org/10.1007/s11263-017-0995-9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 3/2017

Empowering Simple Binary Classifiers for Image Set Based Face Recognition

Transductive Zero-Shot Action Recognition by Word-Vector Embedding

Exemplar-Guided Similarity Learning on Polynomial Kernel Feature Map for Person Re-identification

Max-Margin Heterogeneous Information Machine for RGB-D Action Recognition

Visual Object Detection Using Cascades of Binary and One-Class Classifiers

A Branch-and-Bound Framework for Unsupervised Common Event Discovery

Premium Partner