## On the Rate of Structural Change in Scale Spaces

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

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**On the Rate of Structural Change in Scale Spaces.** / Gustafsson, David Karl John; Pedersen, Kim Steenstrup; Lauze, Francois Bernard; Nielsen, Mads.

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

#### Harvard

*Proceedings of Scale Space and Variational Methods in Computer Vision (SSVM) 09.*vol. 5567, Springer, Lecture notes in computer science, vol. 5567/209, pp. 832-843, Scale Space and Variational Methods in Computer Vision (SSVM) 09, Voss, Norway, 01/06/2009. https://doi.org/10.1007/978-3-642-02256-2

#### APA

*Proceedings of Scale Space and Variational Methods in Computer Vision (SSVM) 09*(Vol. 5567, pp. 832-843). Springer. Lecture notes in computer science Vol. 5567/209 https://doi.org/10.1007/978-3-642-02256-2

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#### Bibtex

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#### RIS

TY - GEN

T1 - On the Rate of Structural Change in Scale Spaces

AU - Gustafsson, David Karl John

AU - Pedersen, Kim Steenstrup

AU - Lauze, Francois Bernard

AU - Nielsen, Mads

N1 - Conference code: 2

PY - 2009

Y1 - 2009

N2 - We analyze the rate in which image details are suppressed as a functionof the regularization parameter, using first order Tikhonov regularization,Linear Gaussian Scale Space and Total Variation image decomposition. Thesquared L2-norm of the regularized solution and the residual are studied as afunction of the regularization parameter. For first order Tikhonov regularizationit is shown that the norm of the regularized solution is a convex function, whilethe norm of the residual is not a concave function. The same result holds forGaussian Scale Space when the parameter is the variance of the Gaussian, butmay fail when the parameter is the standard deviation. Essentially this implythat the norm of regularized solution can not be used for global scale selectionbecause it does not contain enough information. An empirical study basedon synthetic images as well as a database of natural images confirms that thesquared residual norms contain important scale information.

AB - We analyze the rate in which image details are suppressed as a functionof the regularization parameter, using first order Tikhonov regularization,Linear Gaussian Scale Space and Total Variation image decomposition. Thesquared L2-norm of the regularized solution and the residual are studied as afunction of the regularization parameter. For first order Tikhonov regularizationit is shown that the norm of the regularized solution is a convex function, whilethe norm of the residual is not a concave function. The same result holds forGaussian Scale Space when the parameter is the variance of the Gaussian, butmay fail when the parameter is the standard deviation. Essentially this implythat the norm of regularized solution can not be used for global scale selectionbecause it does not contain enough information. An empirical study basedon synthetic images as well as a database of natural images confirms that thesquared residual norms contain important scale information.

U2 - 10.1007/978-3-642-02256-2

DO - 10.1007/978-3-642-02256-2

M3 - Article in proceedings

SN - 978-3-642-02255-5

VL - 5567

T3 - Lecture notes in computer science

SP - 832

EP - 843

BT - Proceedings of Scale Space and Variational Methods in Computer Vision (SSVM) 09

PB - Springer

Y2 - 1 June 2009 through 5 June 2009

ER -

ID: 11574833