In this note, we propose necessary and sufficient conditions for the asymptotic stability analysis of two-dimensional (2-D) systems in terms linear matrix inequalities (LMIs). By introducing a guardian map for the set of Schur stable complex matrices, we first reduce the stability analysis problems into nonsingularity analysis problems of parameter-dependent complex matrices. Then, by means of the discrete-time positive real lemma and the generalized-procedure, we derive LMI-based conditions that enable us to analyze the asymptotic stability in an exact (i.e., nonconservative) fashion. It turns out that, by employing the generalized-procedure, we can derive smaller size of LMIs so that the computational burden can be reduced.
Given a 2-D binary image of size n × n, Euclidean Distance Map (EDM) is a 2-D array of the same size such that each element is storing the Euclidean distance to the nearest black pixel. It is known that a sequential algorithm can compute the EDM in O(n 2 ) and thus this algorithm is optimal. Also, work-time optimal parallel algorithms for shared memory model have been presented. However, the presented parallel algorithms are too complicated to implement in existing shared memory parallel machines. The main contribution of this paper is to develop a simple parallel algorithm for the EDM and implement it in two different parallel platforms: multicore processors and Graphics Processing Units (GPUs). We have implemented our parallel algorithm in a Linux server with four Intel hexad-core processors (Intel Xeon X7460 2.66GHz). We have also implemented it in the following two modern GPU systems, Tesla C1060 and GTX 480, respectively. The experimental results have shown that, for an input binary image with size of 9216 × 9216, our implementation in the multicore system achieves a speedup factor of 18 over the performance of a sequential algorithm using a single processor in the same system. Meanwhile, for the same input binary image, our implementation on the GPU achieves a speedup factor of 26 over the sequential algorithm implementation.
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