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  • Technomathematik

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Mathematische Methoden der Bild- und Signalanalyse

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Weitere Veröffentlichungen

Publikationen in mathematischer Physik, angewandter Mathematik, Bild- und Signalverarbeitung.

Lehrbuch:
Wavelets and Signal Processing. An Application-Based Introduction
Springer-Verlag, 2005
ISBN: 3-540-23433-0, DOI: http://dx.doi.org/10.1007/3-540-27481-2

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Image atoms

Any digital image, i.e., rectangular array of pixels of a given size, may be decomposed in "atomic images". Using mathematical terminology these atoms constitute a "basis system", or - more generally - a frame.

These atoms are fixed a priori and each given image is characterized uniquely by numbers measuring the contributions of the respective atoms to this image. In this sense every possible image having a certain pixel resolution is made up from the same atoms and its individuality is characterized exclusively by the contribution of the respective atoms.
The basic idea of image compression is: For a given image, find out the "important" atoms and neglect the other ones. Importance is measured by the absolute value of the numbers quoted above.

There are several kinds of atoms. The most popular set of atoms may be traced back to Joseph Fourier. Each member of this set may be visualized as a periodic horizontal, vertical or checkerboard-like structure covering the whole image region. In this sense the atoms are global. The various atoms differ by frequency and intensity of these structures.

Another set of atoms is built from wavelets. Here the atoms show an oscillatory behavior as well, but the structures are well localized.

The videos below show, how images gradually emerge (right), when they are composed from the actual atoms running left. The atoms are selected in decreasing order of their importance (in the sense defined above). Thus, if the emerging image improves rapidly, this indicates that the atom set captures the essential features of this image with relatively few atoms and therefore is useful for compression purposes.

Demos

Projects

Acronym/name Funded by Duration URL
IWAZA BMBF 2000-2002
Robidsteel StMWFK 2008-2011
Intelligent Sensor Systems StMWFK 2008-2011
UnLocX EU/FP7 2010-2013 URL
STRAT-AB BMBF 2015-2019
MPI2: Modellbasierte Parameteridentifikation in Magnetic Particle Imaging BMBF 2017-2019 URL