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Ill-Posed Problems in Imaging and Computer Vision Read chapter Read first chapter

2016 | OriginalPaper | Chapter

6. Frequency-Domain Implementation of Regularization

Authors : Mongi A. Abidi, Andrei V. Gribok, Joonki Paik

Published in: Optimization Techniques in Computer Vision

Publisher: Springer International Publishing

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Abstract

Regularization methods play an important role in solving linear equations of the form
$$ y=Hx, $$
with prior knowledge about the solution. The corresponding regularization results in minimization of
$$ f(x)={\left\Vert y-Hx\kern0.1em \right\Vert}^2+\lambda {\left\Vert \kern0.1em Cx\kern0.1em \right\Vert}^2. $$

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previous chapter Constrained Optimization
next chapter Iterative Methods
Footnotes
1
In order to determine boundary samples of output signal, we assumed circularly symmetric or periodic input with period N.
 
2
Most one-dimensional filters have a causal impulse response because the future input is not available for convolution with the filter. In this case, the filtered output comes with a certain amount of delay. On the other hand, in two-dimensional image processing, noncausal filters are used in order to avoid a shifted output image, caused by the two-dimensional delay.
 
3
The impulse response of a two-dimensional filter is called the point spread function if each coefficient has a nonnegative value.
 
4
A sample procedure to make the periodically extended PSF is illustrated in Fig. 6.2.
 
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Metadata
Title
Frequency-Domain Implementation of Regularization
Authors
Mongi A. Abidi
Andrei V. Gribok
Joonki Paik
Copyright Year
2016
Book
Optimization Techniques in Computer Vision

Print ISBN: 978-3-319-46363-6

Electronic ISBN: 978-3-319-46364-3

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-46364-3_6

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