Abstract
Inner products of Sobolev type are extremely useful for image reconstruction of images from a sparse set of α-scale space features. The common (non)-linear reconstruction frameworks, follow an Euler Lagrange minimization. If the Lagrangian (prior) is a norm induced by an inner product of a Hilbert space, this Euler Lagrange minimization boils down to a simple orthogonal projection within the corresponding Hilbert space. This basic observation has been overlooked in image analysis for the cases where the Lagrangian equals a norm of Sobolev type, resulting in iterative (non-linear) numerical methods, where already an exact solution with non-iterative linear algorithm is at hand. Therefore we provide a general theory on linear image reconstructions and metameric classes of images. By applying this theory we obtain visually more attractive reconstructions than the previously proposed linear methods and we find connected curves in the metameric class of images, determined by a fixed set of linear features, with a monotonic increase of smoothness. Although the theory can be applied to any linear feature reconstruction or principle component analysis, we mainly focus on reconstructions from so-called topological features (such as top-points and grey-value flux) in scale space, obtained from geometrical observations in the deep structure of a scale space.
The Netherlands Organization for Scientific Research is gratefully acknowledged for financial support.
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
Duits, R., Florack, L., de Graaf, J., ter Haar Romeny, B.: On the axioms of scale space theory. Journal of Mathematical Imaging and Vision 20, 267–298 (2004)
Duits, R., Duits, M., van Almsick, M.: Invertible orientation scores as an application of generalized wavelet theory. Technical report, TUE, Eindhoven, Technical Report 04-04, Biomedical Image and Analysis, Department of Biomedical Engineering, Eindhoven University of Technology (2004)
Loog, M., Duistermaat, J.J., Florack, L.M.J.: On the behavior of spatial critical points under Gaussian blurring. a folklore theorem and scale-space constraints [14], 183–192
Kuijper, A., Florack, L.M.J.: Hierarchical pre-segmentation without prior knowledge. In: Proceedings of the 8th International Conference on Computer Vision, Vancouver, Canada, July 9-12, pp. 487–493. IEEE Computer Society Press, Los Alamitos (2001)
Balmachnova, E., Florack, L., Platel, B., Kanters, F., ter Haar Romeny, B.: Stability of top-points in scale space. In: Proceedings 5th Scale Space conference 2005, pp. 62–72 (2005)
Felsberg, M., Duits, R., Florack, L.: The monogenic scale space on a bounded domain and its applications. In: Proceedings Scale Space Conference, Isle of Skye, UK, pp. 209–224 (2003)
Nielsen, M., Lillholm, M.: What do features tell about images? [14], 39–50
Lillholm, M., Nielsen, M., Griffin, L.D.: Feature-based image analysis. International Journal of Computer Vision 52, 73–95 (2003)
Kanters, B., Platel, L.M.J., ter Haar Romeny, B.M.: Image reconstruction from multiscale critical points. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 464–478. Springer, Heidelberg (2003)
Yosida, K.: Functional Analysis. Springer, New York (1980)
Robinson, D.: Elliptic Operators and Lie groups. Clarendon Press, Oxford (1991)
Janssen, B., Kanters, F., Duits, R., Florack, L.M.J., ter Haar Romeny, B.M.: A linear image reconstruction framework based on sobolev type inner products. To Appear in Proc. of the Scale Space Conference (2005)
Kanters, F., Lillholm, M., Duits, R., Janssen, B., Platel, B., Florack, L.M.J.: Image reconstruction from multiscale top points. To appear at Proceedings 5th Scale Space Conference (2005)
Kerckhove, M. (ed.): Scale-Space 2001. LNCS, vol. 2106. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duits, R., Janssen, B., Kanters, F., Florack, L. (2005). Linear Image Reconstruction from a Sparse Set of α-Scale Space Features by Means of Inner Products of Sobolev Type. In: Fogh Olsen, O., Florack, L., Kuijper, A. (eds) Deep Structure, Singularities, and Computer Vision. DSSCV 2005. Lecture Notes in Computer Science, vol 3753. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11577812_9
Download citation
DOI: https://doi.org/10.1007/11577812_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29836-6
Online ISBN: 978-3-540-32097-5
eBook Packages: Computer ScienceComputer Science (R0)