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Optimization for Computer Vision pp 87–128Cite as

Variational Methods

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Part of the book series: Advances in Computer Vision and Pattern Recognition ((ACVPR))

Abstract

The main goal of many computer vision tasks can in summary be described by finding a function which is optimal according to some criteria. Examples of such functions are the two-dimensional intensity/color function of the image itself in image restoration or deblurring, two-dimensional vector fields like optical flow, or the course of a curve separating the image into multiple areas (which are all presented as examples in this chapter). This is the domain of variational optimization, which aims at estimating those functional relationships. The functional quantifying the quality of a particular solution typically consists of two terms: one for measuring the fidelity of the solution to the observed data and a second term for incorporating prior assumptions about the expected solution, e.g., smoothness constraints. There exist several ways of finding the solution, such as closed-form solutions via the so-called Euler-Lagrange equation, or iterative gradient-based schemes. Despite of the iterative and therefore inherently slow nature of the last-mentioned approach, quite simple iterative update rules can be found for some applications, which allow for a very fast implementation on massively parallel hardware like GPUs. Therefore, variational methods currently are an active area of research in computer vision.

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Notes

  1. 1.

    Actually, (4.32) describes two ROF models: one for \( {{\mathbf{u}}_1} \) (where \( d=1 \)) and one for \( {{\mathbf{u}}_2} \) (where \( d=2 \)). Consequently, we have to perform two (separate) optimizations.

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Authors and Affiliations

  1. ASM Assembly Systems GmbH & Co. KG, Munich, Germany

    Marco Alexander Treiber

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  1. Marco Alexander Treiber
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Treiber, M.A. (2013). Variational Methods. In: Optimization for Computer Vision. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-5283-5_4

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  • DOI: https://doi.org/10.1007/978-1-4471-5283-5_4

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  • Print ISBN: 978-1-4471-5282-8

  • Online ISBN: 978-1-4471-5283-5

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