Алгоритм быстрого преобразования Фурье для восстановления изображений из голограмм, зарегистрированных с помощью фотоматриц произвольного размера

Abstract

The paper considers the possibility of practical realization of the FFT used for the restoration of rectangular images obtained during the restoration of digital holograms registered in the Fraunhofer zone by photodetector matrices whose sizes have bases not multiple of two. The algorithm is based on splitting the rows and columns of the matrix into submatrices with different bases. It is shown that the main problem in the development of algorithms of this class is the complexity of the structure of algorithms. A block diagram of the use of composite multipliers is demonstrated on the example of a row of a photodetector matrix of size 8000 points, from which the partitioning into composite multipliers becomes clear. Matrixes of photodetectors with such number of pixels have the size of one pixel 1,3 × 1,3 microns which allows registering digital holograms directly on the matrix of photodetector as to register holograms increased spatial resolution is necessary, therefore photodetectors with the size of 8000 × 6000 and 16 384 × 12 288 pixels are used for them. Note that often recommended addition of the transformed sequence to a multiple of a power 2 (zero padding) leads to oversampling and distortion of the phase spectrum, which is not applicable for phase measurements. For practical realization we interpret the composite length FFT algorithm as a two-dimensional transform. The step-by-step algorithm of FFT realization for image reconstruction from digital holograms of 8000 × 6000 size is considered. The efficiency of the proposed algorithm is evaluated. It is shown that the efficiency of this algorithm is comparable with the efficiency of the FFT algorithm, but unlike the latter, the restriction on the multiplicity of the basis is removed. The computational complexity of the transformation for 6000 elements in the column was 123 200, and for the FFT on base two (8192) was 159 744, taking into account padding a row of size 6000 to the nearest power of size 2 (8192). Recommendations for splitting bases with arbitrary sizes are given.