Abstract
In order to define a metric on jet space, linear scale space is considered from a statistical standpoint. Given a scale σ, the scale space solution can be interpreted as maximizing a certain Gaussian posterior probability, related to a particular Tikhonov regularization. The Gaussian prior, which governs this solution, in fact induces a Mahalanobis distance on the space of functions. This metric on the function space gives, in a rather precise way, rise to a metric on n-jets. The latter, in turn, can be employed to define a norm on jet space, as the metric is translation invariant and homogeneous. Recently, [1] derived a metric on jet space and our results reinforce his findings, while providing a totally different approach to defining a scale space jet metric.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Griffin, L.D.: The 2nd order local-image-structure solid. IEEE Transactions on Pattern Analysis and Machine Intelligence (in press)
Cao, G., Chen, J., Jiang, J.: A novel local invariant descriptor adapted to mobile robot vision. In: Proceedings of the 2004 American Control Conference, vol. 3, pp. 2196–2201 (2004)
Gevers, T., Smeulders, A.W.M.: Image retrieval by multi-scale illumination invariant indexing. In: Ip, H.H.-S., Smeulders, A.W.M. (eds.) MINAR 1998. LNCS, vol. 1464, pp. 96–108. Springer, Heidelberg (1998)
van Ginneken, B., Stegmann, M.B., Loog, M.: Segmentation of anatomical structures in chest radiographs using supervised methods: a comparative study on a public database. Medical Image Analysis 10, 19–40 (2006)
Kanters, F., et al.: Content based image retrieval using multiscale top points. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 33–43. Springer, Heidelberg (2003)
Lillholm, M., Pedersen, K.S.: Jet based feature classification. In: Proceedings of the 17th International Conference on Pattern Recognition, vol. 2 (2004)
Pham, T.V., Smeulders, A.W.M.: Sparse representation for coarse and fine object recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 28, 555–567 (2006)
Ravela, S., Manmatha, R.: Retrieving images by appearance. In: Proceedings of the 6th International Conference on Computer Vision, pp. 608–613 (1998)
Schmid, C., Mohr, R.: Local grayvalue invariants for image retrieval. IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 530–535 (1997)
Eberly, D.: A differential geometric approach to anisotropic diffusion. In: Geometry-Driven Diffusion in Computer Vision, pp. 371–392. Kluwer Academic Publishers, Dordrecht (1994)
Eberly, D.: Ridges in Image and Data Analysis. Kluwer Academic Publishers, Dordrecht (1996)
Florack, L.M.J.: Deep structure from a geometric point of view. In: Fogh Olsen, O., Florack, L.M.J., Kuijper, A. (eds.) DSSCV 2005. LNCS, vol. 3753, pp. 135–145. Springer, Heidelberg (2005)
Griffin, L.D., Lillholm, M.: Hypotheses for image features, icons and textons. International Journal of Computer Vision 70, 213–230 (2006)
Koenderink, J.J., van Doorn, A.J.: Receptive field assembly specificity. Journal of Visual Communication and Image Representation 3, 1–12 (1992)
Koenderink, J.J., van Doorn, A.J.: Metamerism in complete sets of image operators. In: Advances in Image Understading ’96, pp. 113–129 (1996)
Lillholm, M., Nielsen, M., Griffin, L.D.: Feature-based image analysis. International Journal of Computer Vision 52, 73–95 (2003)
Nielsen, M., Florack, L.M.J., Deriche, R.: Regularization, scale-space, and edge detection filters. Journal of Mathematical Imaging and Vision 7, 291–307 (1997)
Tikhonov, A.N., Arseninn, V.Y.: Solution of Ill-Posed Problems. Winston and Sons, Washington (1977)
Marroquin, J., Mitter, S., Poggio, T.: Probabilistic solution of ill-posed problems in computational vision. Journal of the American Statistical Association 82, 76–89 (1987)
Wahba, G.: Spline models for observational data. Society for Industrial and Applied Mathematics, Philadelphia (1990)
Rasmussen, C.E., Williams, C.K.I.: Gaussian processes for machine learning. MIT Press, Cambridge (2005)
Loog, M.: Support blob machines. the sparsification of linear scale space. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 14–24. Springer, Heidelberg (2004)
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. John Wiley & Sons, Chichester (1984)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Heidelberg (1991)
Baxter, J.: Learning internal representations. In: Proceedings of the eighth annual conference on computational learning theory, New York, USA, pp. 311–320. ACM Press, New York (1995)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Loog, M. (2007). The Jet Metric. In: Sgallari, F., Murli, A., Paragios, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72823-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-72823-8_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72822-1
Online ISBN: 978-3-540-72823-8
eBook Packages: Computer ScienceComputer Science (R0)