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A linear programming based analysis of the CP-rank of completely positive matrices

100%
Li Y., Kummert A., Frommer A.
International Journal of Applied Mathematics and Computer Science
|
2004
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tom 14
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nr 1
25-31
EN
A real matrix A is said to be completely positive (CP) if it can be decomposed as A = BB^T, where the real matrix B has exclusively non-negative entries. Let k be the rank of A and Φ_k the least possible number of columns of the matrix B, the so-called completely positive rank (cp-rank) of A. The present work is devoted to a study of a general upper bound for the cp-rank of an arbitrary completely positive matrix A and its dependence on the ordinary rank k. This general upper bound of the cp-rank has been proved to be at most k(k + 1)/2. In a recent pioneering work of Barioli and Berman it was slightly reduced by one, which means that Φ_k ≤ k(k + 1)/2 - 1 holds for k ≥ 2. An alternative constructive proof of the same result is given in the present paper based on the properties of the simplex algorithm known from linear programming. Our proof illuminates complete positivity from a different point of view. Discussions concerning dual cones are not needed here. In addition to that, the proof is of constructive nature, i.e. starting from an arbitrary decomposition A = B_1B^T_1 (B_1 ≥ 0) a new decomposition A = B_2B^T_2 (B_2 ≥ 0) can be generated in a constructive manner, where the number of column vectors of B_2 does not exceed k(k + 1)/2 − 1. This algorithm is based mainly on the well-known techniques stemming from linear programming, where the pivot step of the simplex algorithm plays a key role.
2
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Robust stabilization of discrete linear repetitive processes with switched dynamics

100%
Bochniak J., Galkowski K., Rogers E., Kummert A.
International Journal of Applied Mathematics and Computer Science
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2006
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tom 16
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nr 4
441-462
EN
Repetitive processes constitute a distinct class of 2D systems, i.e., systems characterized by information propagation in two independent directions, which are interesting in both theory and applications. They cannot be controlled by a direct extension of the existing techniques from either standard (termed 1D here) or 2D systems theories. Here we give new results on the design of physically based control laws. These results are for a sub-class of discrete linear repetitive processes with switched dynamics in both independent directions of information propagation.
3
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Detection of moving objects in image sequences using 3D velocity filters

100%
Schauland S., Velten J., Kummert A.
International Journal of Applied Mathematics and Computer Science
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2008
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tom 18
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nr 1
21-31
EN
A movement analysis of objects contained in visual scenes can be performed by means of linear multidimensional filters, which have already been analyzed in the past. While the soundness of the results was convincing, interest in those systems declined due to the limited computational power of contemporary computers. Recent advances in design and implementation of integrated circuits and hardware architectures allow realizing velocity filters if the n-D system is carefully adapted to the analyzed problem. In this paper, the fundamental principles of visual scene analysis by linear multidimensional filters are examined with respect to possible sources of degradation. The extraction of movement information and its practical use are demonstrated using a wave digital filter (WDF) implementation.
4
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Interpolation-based reconstruction methods for tomographic imaging in 3D Positron Emission Tomography

100%
Li Y., Kummert A., Boschen F., Herzog H.
International Journal of Applied Mathematics and Computer Science
|
2008
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tom 18
|
nr 1
63-73
EN
Positron Emission Tomography (PET) is considered a key diagnostic tool in neuroscience, by means of which valuable insight into the metabolism function in vivo may be gained. Due to the underlying physical nature of PET, 3D imaging techniques in terms of a 3D measuring mode are intrinsically demanded to assure satisfying resolutions of the reconstructed images. However, incorporating additional cross-plane measurements, which are specific for the 3D measuring mode, usually imposes an excessive amount of projection data and significantly complicates the reconstruction procedure. For this reason, interpolation-based reconstruction methods deserve a thorough investigation, whose crucial parts are the interpolating processes in the 3D frequency domain. The benefit of such approaches is apparently short reconstruction duration, which can, however, only be achieved at the expense of accepting the inaccuracies associated with the interpolating process. In the present paper, two distinct approaches to the realization of the interpolating procedure are proposed and analyzed. The first one refers to a direct approach based on linear averaging (inverse distance weighting), and the second one refers to an indirect approach based on two-dimensional convolution (gridding method). In particular, attention is paid to two aspects of the gridding method. The first aspect is the choice of the two-dimensional convolution function applied, and the second one is the correct discretization of the underlying continuous convolution. In this respect, the geometrical structure named the Voronoi diagram and its computational construction are considered. At the end, results of performed simulation studies are presented and discussed.
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