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Diffusion, Pre-smoothing and Gradient Descent

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Scale Space and Variational Methods in Computer Vision (SSVM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12679))

Abstract

Nonlinear diffusion of images, both isotropic and anisotropic, has become a well-established and well-understood denoising tool during the last three decades. Moreover, it is a component of partial differential equation methods for various further tasks in image analysis. For the analysis of such methods, their understanding as gradient descents of energy functionals often plays an important role. Often the diffusivity or diffusion tensor field for nonlinear diffusion is computed from pre-smoothed image gradients. What was not clear so far was whether nonlinear diffusion with this pre-smoothing step still is the gradient descent for some energy functional. This question is answered to the negative in the present paper. Triggered by this result, possible modifications of the pre-smoothing step to retain the gradient descent property of diffusion are discussed.

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Authors and Affiliations

  1. Institute of Biomedical Image Analysis, UMIT TIROL – Private University of Health Sciences, Medical Informatics and Technology, Eduard-Wallnöfer-Zentrum 1, 6060, Hall, Tyrol, Austria

    Martin Welk

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  1. Martin Welk
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Correspondence to Martin Welk .

Editor information

Editors and Affiliations

  1. UNICAEN, GREYC – Normandy University, Caen, France

    Abderrahim Elmoataz

  2. ENSICAEN, GREYC – Normandy University, Caen, France

    Jalal Fadili

  3. CNRS, GREYC – Normandy University, Caen, France

    Yvain Quéau

  4. UNICAEN, GREYC – Normandy University, Caen, France

    Julien Rabin

  5. ENSICAEN, GREYC – Normandy University, Caen, France

    Loïc Simon

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Welk, M. (2021). Diffusion, Pre-smoothing and Gradient Descent. In: Elmoataz, A., Fadili, J., Quéau, Y., Rabin, J., Simon, L. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2021. Lecture Notes in Computer Science(), vol 12679. Springer, Cham. https://doi.org/10.1007/978-3-030-75549-2_7

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  • DOI: https://doi.org/10.1007/978-3-030-75549-2_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-75548-5

  • Online ISBN: 978-3-030-75549-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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