International Journal of Computer Vision

, Volume 106, Issue 1, pp 93–112 | Cite as

Sampling Minimal Subsets with Large Spans for Robust Estimation

  • Quoc Huy Tran
  • Tat-Jun Chin
  • Wojciech Chojnacki
  • David Suter
Article

Abstract

When sampling minimal subsets for robust parameter estimation, it is commonly known that obtaining an all-inlier minimal subset is not sufficient; the points therein should also have a large spatial extent. This paper investigates a theoretical basis behind this principle, based on a little known result which expresses the least squares regression as a weighted linear combination of all possible minimal subset estimates. It turns out that the weight of a minimal subset estimate is directly related to the span of the associated points. We then derive an analogous result for total least squares which, unlike ordinary least squares, corrects for errors in both dependent and independent variables. We establish the relevance of our result to computer vision by relating total least squares to geometric estimation techniques. As practical contributions, we elaborate why naive distance-based sampling fails as a strategy to maximise the span of all-inlier minimal subsets produced. In addition we propose a novel method which, unlike previous methods, can consciously target all-inlier minimal subsets with large spans.

Keywords

Least squares Total least squares  Minimal subsets Robust fitting Hypothesis sampling 

References

  1. Chin, T. J., Yu, J., & Suter, D. (2010). Accelerated hypothesis generation for multi-structure robust fitting. In European Conference on Computer Vision (ECCV).Google Scholar
  2. Chin, T. J., Yu, J., & Suter, D. (2012). Accelerated hypothesis generation for multi-structure data via preference analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(4), 625–638.CrossRefGoogle Scholar
  3. Chum, O., & Matas, J. (2005). Matching with PROSAC—Progressive sample consensus. In Computer Vision and Pattern Recognition (CVPR).Google Scholar
  4. Chum, O., & Matas, J. (2010). Planar affine rectification from change of scale. In Asian Conference on Computer Vision (ACCV).Google Scholar
  5. Chum, O., Matas, J., & Kittler, J. (2003). Locally optimized RANSAC. In Deutsche Arbeitsgemeinschaft für Mustererkennung (DAGM).Google Scholar
  6. Chum, O., Matas, J., & Obdrzakek, S. (2004). Enhancing RANSAC by generalized model optimization. In Asian Conference on Computer Vision (ACCV) Google Scholar
  7. Chum, O., Werner, T., & Matas, J. (2005). Two-view geometry estimation unaffected by a dominant plane. In Computer Vision and Pattern Recognition (CVPR).Google Scholar
  8. de Groen, P. (1996). An introduction to total least squares. Nieuw Archief voor Wiskunde, 4(14), 237–253.Google Scholar
  9. Fischler, M. A., & Bolles, R. C. (1981). Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, 24, 381–395.CrossRefMathSciNetGoogle Scholar
  10. Frahm, J. M., & Pollefeys, M. (2006). RANSAC for (quasi-)degenerate data (QDEGSAC).Google Scholar
  11. Golub, G. H., Hoffman, A., & Stewart, G. W. (1987). A generalization of the Eckart-Young-Mirksy matrix approximation theorem. Linear Algebra and its Applications, 88–89, 317–327.CrossRefMathSciNetGoogle Scholar
  12. Golub, G. H., & van Loan, C. F. (1980). An analysis of the total least squares problem. Numerical Analysis, 17, 883–893.CrossRefMATHGoogle Scholar
  13. Goshen, L., & Shimshoni, I. (2008). Balanced exploration and exploitation model search for efficient epipolar geometry estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence Google Scholar
  14. Harker, M., & O’Leary, P. (2006). Direct estimation of homogeneous vectors: An ill-solved problem in computer vision. In Indian Conference on Computer Vision, Graphics and Image Processing.Google Scholar
  15. Hartley, R. (1997). In defense of the eight-point algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(6), 580–593.Google Scholar
  16. Hartley, R., & Zisserman, A. (2004). Multiple View Geometry (2nd ed.). Cambridge: Cambridge University Press.CrossRefMATHGoogle Scholar
  17. Hoerl, A. E., & Kennard, R. W. (1980). A note on least squares estimates. Communications in Statistics: Simulation and Computation, 9(3), 315–317.Google Scholar
  18. Jacobi, C. G. J. (1841). De formatione et proprietatibus determinantium. Journal fur die reine und angewandte Mathematik, 9, 315–317.Google Scholar
  19. Kahl, F., & Hartley, R. (2008). Multiple-view geometry under the \(l_\infty \)-norm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(9), 1603–1617.CrossRefGoogle Scholar
  20. Kahl, F., & Henrion, D. (2005). Globally optimal estimates for geometric reconstruction problems. In International Conference on Computer Vision (ICCV).Google Scholar
  21. Kanazawa, Y., & Kawakami, H. (2004). Detection of planar regions with uncalibrated stereo using distributions of feature points. In British Machine Vision Conference (BMVC).Google Scholar
  22. Kemp, C., & Drummond, T. (2005). Dynamic measurement clustering to aid real time tracking. In International Conference on Computer Vision (ICCV).Google Scholar
  23. Kukush, A., Markovsky, I., & Huffel, S. V. (2002). Consistent fundamental matrix estimation in a quadratic measurement error model arising in motion analysis. Computational Statistics and Data Analysis, 3(18), 3–18.CrossRefGoogle Scholar
  24. Lowe, D. G. (2004). Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60(2), 91–110.CrossRefGoogle Scholar
  25. Meer, P. (2004). Robust techniques for computer vision. In G. Medioni & S. B. Kang (Eds.), Emerging topics in computer vision. Prentice Hall.Google Scholar
  26. Mikolajczyk, K., & Schmid, C. (2004). A performance evaluation of local descriptors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(10), 1615–1630.CrossRefGoogle Scholar
  27. Mikolajczyk, K., Tuytelaars, T., Schmid, C., Zisserman, A., Matas, J., Schaffalitzky, F., et al. (2005). A comparison of affine region detectors. International Journal of Computer Vision, 65(1), 43–72.CrossRefGoogle Scholar
  28. Mühlich, M., & Mester, R. (1998). The role of total least squares in motion analysis. In: European Conference on Computer Vision (ECCV).Google Scholar
  29. Myatt, D. R., Torr, P. H. S., Nasuto, S. J., Bishop, J. M., & Craddock, R. (2002). NAPSAC: high noise, high dimensional robust estimation—It’s in the bag. In British Machine Vision Conference (BMVC).Google Scholar
  30. Olsson, C., Eriksson, A., & Hartley, R. (2010). Outlier removal using duality. In Computer Vision and Pattern Recognition (CVPR).Google Scholar
  31. Pham, T. T., Chin, T. J., Yu, J., & Suter, D. (2012). The random cluster model for robust geometric fitting. In Computer Vision and Pattern Recognition (CVPR).Google Scholar
  32. Rousseeuw, P. J., & Leroy, A. M. (1987). Robust regression and outlier detection. New York: Wiley.CrossRefMATHGoogle Scholar
  33. Scherer-Negenborn, N., & Schaefer, R. (2010). Model fitting with sufficient random sample coverage. International Journal of Computer Vision, 89, 120–128.CrossRefGoogle Scholar
  34. Stigler, S. M. (2000). The history of statistics: The measurement of uncertainty before 1900 (8th edn., Chap. 1). The Belknap Press of Harvard University Press.Google Scholar
  35. Subrahmanyam, M. (1972). A property of simple least squares estimates. Sankhya B, 34, 3.MathSciNetGoogle Scholar
  36. Tordoff, B. J., & Murray, D. W. (2005). Guided-MLESAC: Faster image transform estimation by using matching priors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(10), 1523–1535.Google Scholar
  37. van Huffel, S., & Wandewalle, J. (1989). Algebraic connections between the least squares and total least squares problem. Numerical Mathematics, 55, 431–449. Google Scholar
  38. van Huffel, S., & Wandewalle, J. (1991). The total least squares problem: Computational aspects and analysis. Philadelphia, PA: SIAM Publications.Google Scholar
  39. Vedaldi, A., & Fulkerson, B. (2008). VLFeat: An open and portable library of computer vision algorithms. http://www.vlfeat.org/
  40. Wong, H. S., Chin, T. J., Yu, J., & Suter, D. (2011). Dynamic and hierarchical nulti-structure geometric model fitting. In: International Conference on Computer Vision (ICCV).Google Scholar
  41. Zhang, Z. (1997). Parameter estimation techniques: A tutorial with application to conic fitting. Image and Vision Computing, 15(1), 59–76.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Quoc Huy Tran
    • 1
  • Tat-Jun Chin
    • 1
  • Wojciech Chojnacki
    • 1
  • David Suter
    • 1
  1. 1.The Australian Centre for Visual Technologies, and School of Computer ScienceThe University of AdelaideNorth TerraceAustralia

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