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Discrete Fourier transform and permutations

Hui S., Żak S. H.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 6

995--1005

EN

It is well known that the magnitudes of the coefficients of the discrete Fourier transform (DFT) are invariant under certain operations on the input data. In this paper, the effects of rearranging the elements of an input data on its DFT are studied. In the one-dimensional case, the effects of permuting the elements of a finite sequence of length N on its Discrete Fourier transform (DFT) coefficients are investigated. The permutations that leave the unordered collection of Fourier coefficients and their magnitudes invariant are completely characterized. Conditions under which two different permutations give the same DFT coefficient magnitudes are given. The characterizations are based on the automorphism group of the additive group ZN of integers modulo N and the group of translations of ZN. As an application of the results presented, a generalization of the theorem characterizing all permutations that commute with the discrete Fourier transform is given. Numerical examples illustrate the obtained results. Possible generalizations and open problems are discussed. In higher dimensions, results on the effects of certain geometric transformations of an input data array on its DFT are given and illustrated with an example.

2

Discrete Fourier transform based pattern classifiers

Hui S., Żak S. H.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2014

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Vol. 62, nr 1

15--22

EN

A technique for pattern classification using the Fourier transform combined with the nearest neighbor classifier is proposed. The multidimensional fast Fourier transform (FFT) is applied to the patterns in the data base. Then the magnitudes of the Fourier coefficients are sorted in descending order and the first P coefficients with largest magnitudes are selected, where P is a design parameter. These coefficients are then used in further processing rather than the original patterns. When a noisy pattern is presented for classification, the pattern’s P Fourier coefficients with largest magnitude are extracted. The coefficients are arranged in a vector in the descending order of their magnitudes. The obtained vector is referred to as the signature vector of the corresponding pattern. Then the distance between the signature vector of the pattern to be classified and the signature vectors of the patterns in the data base are computed and the pattern to be classified is matched with a pattern in the data base whose signature vector is closest to the signature vector of the pattern being classified.

3

Image Recall Using a Large Scale Generalized Brain-state-in-a-box Neural Network

Oh Ch., Żak S. H.

International Journal of Applied Mathematics and Computer Science

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2005

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Vol. 15, no 1

99-114

EN

An image recall system using a large scale associative memory employing the generalized Brain-State-in-a-Box (gBSB) neural network model is proposed. The gBSB neural network can store binary vectors as stable equilibrium points. This property is used to store images in the gBSB memory. When a noisy image is presented as an input to the gBSB network, the gBSB net processes it to filter out the noise. The overlapping decomposition method is utilized to efficiently process images using their binary representation. Furthermore, the uniform quantization is employed to reduce the size of the data representation of the images. Simulation results for monochrome gray scale and color images are presented. Also, a hybrid gBSB-McCulloch-Pitts neural model is introduced and an image recall system is built around this neural net. Simulation results for this model are presented and compared with the results for the system employing the gBSB neural model.