# Fractional Fourier Transform and Fractional-Order Calculus-Based Image Edge Detection

## Abstract

Edge detection is an integral component of image processing to enhance the clarity of edges in an image. Detection of edges for an image may help for image segmentation, data compression, and image reconstruction. Edges of an image are considered a type of crucial information that can be extracted by applying detectors with different methodologies. Its main purpose is to simplify the image data in order to minimize the amount of data to be processed. There exist many rich classical edge detection techniques which make use of integer-order differentiation operators and can function in both spatial and frequency domains. In the case of integer-order differentiation operators, the gradient operator is identified by order ‘one’ and the Laplacian by order ‘two.’ This paper demonstrates a new kind of edge detector based on the ‘fractional’ (‘non-integer’)-order differentiation operation and through the usage of the ‘fractional Fourier transformation’ tool, so as to perform it in the fractional Fourier frequency domain, known as the *edge detection based on fractional signal processing approach*. It is shown through computer simulations that this approach can detect the edges precisely and efficiently. Finally, the performance of the proposed methodology is illustrated from the quantitative aspects of mean square error and peak signal-to-noise ratio through simulations. The experiments show that, for any grayscale image, this method can obtain better edge detection performance to satisfy human visual sense. Moreover, comparisons are also provided to prove that the proposed method outperforms the classical edge detection operators, interpreted in terms of robustness to noise.

## Keywords

Edge detection Fractional differential mask Fractional-order calculus Fractional Fourier transform Fractional Fourier frequency domain filtering Grünwald–Letnikov fractional derivative## Notes

### Acknowledgments

The authors thank the anonymous reviewers for their rigorous reviews, constructive comments, and valuable suggestions which greatly improved the quality and clarity of manuscript presentation.

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