Abstract
In this paper, we present a method to compute an embedding matrix which maximises the dependence of the embedding space upon the graph-vertex coordinates and the incidence mapping of the graph. This treatment leads to a convex cost function which, by construction, attains its maximum at the leading singular value of a matrix whose columns are given by the incidence mapping and the embedded vertex coordinates. This, in turn, maximises the correlation between the spaces in which the embedding and the graph vertex coordinates are defined. It also maximises the dependence between the embedding and the incidence mapping of the graph. We illustrate the utility of the method for purposes of approximating the colour sensitivity functions of a set of over 20 commercially available digital cameras using a library of spectral reflectance measurements.
Chapter PDF
References
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. Neural Information Processing Systems 14, 634–640 (2002)
Christmas, W.J., Kittler, J., Petrou, M.: Structural matching in computer vision using probabilistic relaxation. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(8), 749–764 (1995)
Gold, S., Rangarajan, A.: A graduated assignment algorithm for graph matching. PAMI 18(4), 377–388 (1996)
Chung, F.: Spectral Graph Theory. American Mathematical Society (1997)
Umeyama, S.: An eigen decomposition approach to weighted graph matching problems. PAMI 10(5), 695–703 (1988)
Scott, G., Longuet-Higgins, H.: An algorithm for associating the features of two images. Proceedings of the Royal Society of London 244(B), 21–26 (1991)
Shapiro, L., Brady, J.M.: Feature-based correspondance - an eigenvector approach. Image and Vision Computing 10, 283–288 (1992)
Caelli, T., Kosinov, S.: An eigenspace projection clustering method for inexact graph matching. IEEE Trans. on Pattern Analysis and Machine Intelligence 26(4), 515–519 (2004)
Sebastian, T.B., Klein, P.N., Kimia, B.B.: Shock-based indexing into large shape databases. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002, Part III. LNCS, vol. 2352, pp. 731–746. Springer, Heidelberg (2002)
Wong, A.K.C., You, M.: Entropy and distance of random graphs with application to structural pattern recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 7, 599–609 (1985)
Wilson, R., Hancock, E.R.: Structural matching by discrete relaxation. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(6), 634–648 (1997)
Caetano, T., Cheng, L., Le, Q., Smola, A.: Learning graph matching. In: Proceedings of the 11th International Conference on Computer Vision, pp. 14–21 (2007)
Biggs, N.L.: Algebraic Graph Theory. Cambridge University Press (1993)
Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938)
Björck, A.: Numerical methods for least squares problems. SIAM (1996)
Golub, G.H., Loan, C.F.V.: Matrix Computations. The Johns Hopkins Press (1996)
Torgerson, W.S.: Multidimensional scaling I: Theory and method. Psychometrika 17, 401–419 (1952)
Varga, R.S.: Matrix Iterative Analysis, 2nd edn. Springer (2000)
Chung, F.: Discrete Isoperimetric Inequalities. Surveys in Differential Geometry IX (2004)
Wandell, B.A.: The synthesis and analysis of color images. IEEE Transactions on Pattern Analysis and Machine Intelligence 9(1), 2–13 (1987)
Brainard, D.H., Stockman, A.: Colorimetry. McGraw-Hill (1995)
Finlayson, G.D., Drew, M.S.: The maximum ignorance assumption with positivity. In: Proceedings of the IS&T/SID 4th Color Imaging Conference, pp. 202–204 (1996)
Longere, P., Brainard, D.H.: Simulation of digital camera images from hyperspectral input. In: van den Branden Lambrecht, C. (ed.) Vision Models and Applications to Image and Video Processing, pp. 123–150. Kluwer (2001)
Ejaz, T., Horiuchi, T., Ohashi, G., Shimodaira, Y.: Development of a camera system for the acquisition of high-fidelity colors. IEICE Transactions on Electronics E89–C(10), 1441–1447 (2006)
Kimmel, R., Elad, M., Shaked, D., Keshet, R., Sobel, I.: A variational framework for retinex. International Journal of Computer Vision 52(1), 7–23 (2003)
Finlayson, G.D., Schaefer, G.: Solving for colour constancy using a constrained dichromatic reflection model. International Journal of Computer Vision 42(3), 127–144 (2001)
Jiang, J., Liu, D., Gu, J., Süsstrunk, S.: What is the space of spectral sensitivity functions for digital color cameras? In: Workshop on Applications of Computer Vision, pp. 168–179 (2013)
Wyszecki, G., Stiles, W.: Color Science: Concepts and Methods, Quantitative Data and Formulae. Wiley (2000)
Judd, D.B., Macadam, D.L., Wyszecki, G., Budde, H.W., Condit, H.R., Henderson, S.T., Simonds, J.L.: Spectral distribution of typical daylight as a function of correlated color temperature. Journal of the Optical Society of America 54(8), 1031–1036 (1964)
Robles-Kelly, A., Huynh, C.P.: Imaging Spectroscopy for Scene Analysis. Springer (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Robles-Kelly, A. (2014). Max-Correlation Embedding Computation. In: Fränti, P., Brown, G., Loog, M., Escolano, F., Pelillo, M. (eds) Structural, Syntactic, and Statistical Pattern Recognition. S+SSPR 2014. Lecture Notes in Computer Science, vol 8621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44415-3_12
Download citation
DOI: https://doi.org/10.1007/978-3-662-44415-3_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44414-6
Online ISBN: 978-3-662-44415-3
eBook Packages: Computer ScienceComputer Science (R0)