Many applications of digital image processing require the evaluation of fast Fourier transforms. Therefore, for the more conventional rectangular grid image systems, FFT algorithms have been largely developed so far. For users of hexagonal grid image systems, unfortunately, life is less easier since they generally have to write the hexagonal FFT codes by themselves. This complexity tends to hinder the development and use of the hexagonal imaging system. In this short paper, we propose, without a mathematical proof, a method to simulate hexagonal FFTs based on the relations between the two grid systems. And this is done with only the use of regular rectangular FFT schemes. By this method, a hexagonally sampled image can be easily transformed via the many FFT programs available in the market.

Both the Hopfield network and the genetic algorithm are powerful tools for object recognition tasks, e.g., subgraph matching problems. Unfortunately, they both have serious drawbacks. The Hopfield network is very sensitive to its initial state, and it stops at a local minimum if the initial state is not properly given. The genetic algorithm, on the other hand, usually only finds a near-global solution, and it is time-consuming for large-scale problems. In this paper, we propose an integrated scheme of these two methods, while eliminating their drawbacks and keeping their advantages, to solve object recognition problems under affine transformations. Some arrangements and programming strategies are required. First, we use some specialized 2-D genetic algorithm operators to accelerate the convergence. Second, we extract the "seeds" of the solution of the genetic algorithm to serve as the initial state of the Hopfield network. This procedure further improves the efficiency of the system. In addition, we also include several pertinent post matching algorithms for refining the accuracy and robustness. In the examples, the proposed scheme is used to solve some subgraph matching problems with occlusions under affine transformations. As shown by the results, this integrated scheme does outperform many counterpart algorithms in accuracy, efficiency, and stability.