The same signal can be expressed in many different ways: either directly in its original area (e.g., its course in time) or in the spectral region using the usual Fourier transform or in the wavelet area, etc. All these representations have one thing in common: the main signal is always expressed as the sum of signals. The signal thus can be expressed as a sum of consecutive multiples of the Dirac pulse (time domain) or as a sum of harmonic functions (frequency domain) or as a sum of wavelet functions, etc.

As can be seen, different representations of the same signal differ by the “dictionary” used. For certain purposes the choice of dictionary can be crucial (analysis, compression, de-noising…). Nowadays, the theoretical and application area of the orthogonal dictionaries is becoming exhausted and, therefore, attention begins to be directed to non-orthogonal representations of signal. Orthogonality of dictionary elements (e.g. the Fourier transform) means that the elements are mutually “orthogonal”, uncorrelated, which is advantageous from the computational and numerical viewpoints (the representation can be computed quickly and without error propagation), but adapting to the nature of the signal (and thus understanding it better is limited. Non-orthogonal representations can be tailored to a particular type of signal, but at the cost of increased computational complexity and the risk of numerical instability.

As a typical example of such dictionaries the Gabor systems can be mentioned, which are used for short-term spectral analysis of signals or bi-orthogonal wavelets used in the image compression format for JPEG2000. One of the directions, which currently attract considerable attention, deals with situations where the representation of the signal using a dictionary is “sparse.” This means that the expression of the signal requires only a few elements from the dictionary; the other elements are not involved. It can be shown that such signals (especially images come close to this property ) can be processed in other ways than it has been customary (e.g., the sampling may be much slower than required by the Nyquist criterion, the compression is more efficient, etc.). This direction is collectively called Compressed Sensing. Other important applications include noise reduction, filling in the missing parts of the signal (inpainting), superresolution, pattern recognition, and morphological component analysis.

The group is dedicated to tracking these modern trends, both generally and specifically by focusing on finding new applications such as methods for finding optimal dictionary methods for adding errors in the image, algorithms for signal extension at the edge (especially useful for images). The research interests of the group also include methods for the parallelization of the discrete wavelet transform (orthogonal and bi-orthogonal, lifting and pyramid algorithm) by segmenting the signals (1D and 2D) without artifacts at the edges of the segments (SegDTWT algorithm).

# Companies and universities we are cooperating with

# Relation to the study subjects

- Signals and systems analysis (BASS)
- Digital signal processing (BCZS)
- Introduction to computer typography and graphics (BZSG)
- Digital signal processing (MCSI)
- Graphic and Multimedia Processors (MGMP)
- Speech processing (MZPR)

# Information for potential applicants

The group is open to cooperation, both on topics already in progress (see above) or new ones. In this research it is advisable to be knowledgeable about linear algebra, to have algorithmic thinking. Experience of any programming language is also welcome. Applicants, either students or researchers should not hesitate to contact the group leader.

# Important publications

PRŮŠA, Z., RAJMIC, P., MALÝ, J.: Segmentwise Computation of 2D Forward Discrete Wavelet Transform. In Proceedings of the 33rd International Conference on Telecommunications and Signal Processing. Baden near Vienna, Austria: ASSZISZTENCIA Congress Bureau, 2010. pp. 1-4. ISBN 978-963-88981-0-4.

RAJMIC, P.: Algorithms for Segmentwise Computation of Forward and Inverse Discrete-time Wavelet Transform. Journal of Concrete and Applicable Mathematics, Special Issues on Applied Mathematics and Approximation Theory, Vol. 8, No. 3, pp. 393-406. Eudoxuspress, Memphis, TN USA, 2010. ISSN 1548-5390.

ŠPIŘÍK, J., RAJMIC, P., VESELÝ, V.: Reprezentace signálů: od bází k framům, Elektrorevue, 2010. ISSN 1213-1539.

RAJMIC, P., PRŮŠA, Z.: Podrobná studie algoritmu pro výpočet waveletové transformace v diskrétním čase. Elektrorevue, 2010. ISSN 1213-1539.