Fractional convolution, fractional correlation and their translation invariance properties

A Convolution and Correlation Theorem for the Linear Canonical Transform and Its Application

“…Moreover, convolution and correlation are widely used in signal processing, as well as in optics, in pattern recognition or in the description of image formation with incoherent illumination [2,4,11,12,14,15,22,27]. The convolution and correlation theorems in FT domain are defined as…”

“…The output of any continuous time LTI system is found via the convolution of the input signal with the system impulse response. Correlation, which is similar to convolution, is another important operation in signal processing, as well as in optics, in pattern recognition, especially in detection applications [2,11,12,22]. As the LCT has found wide applications in signal processing fields, it is theoretically interesting and practically useful to consider the convolution and correlation theory in the LCT domain.…”

Generalized formulation of an encryption system based on a joint transform correlator and fractional Fourier transform

“…In this paper, we propose a generalization of the JTC-based encryption systems described in [14] using the fractional Fourier operators, such as the FrFT, fractional traslation, and the new definitions for: fractional convolution and fractional correlation [34], with the purpose of improving the quality of the decrypted images and increasing the security of the encryption system in comparison with the previous encryption systems based on a JTC architecture [11,12,13,14,15,18,19,20]. We explain the main causes of the low quality of the decrypted images obtained in [18,19,20] and propose two approaches to improve the quality of the decrypted images.…”

“…In addition, the two approaches used to improve the quality of the decrypted image do not increase the amount of information to be transmitted because the resulting encrypted function has the same size as the original version. The proposed JTC-based encryption-decryption system in the FrFD preserves the shift-invariance property with respect to lateral displacements of both the key random mask in the decryption process and the retrieval of the primary image [1,34].…”

A generalized convolution theorem for the special affine Fourier transform and its application to filtering

“…Simultaneously, as the generalization of FT, the relevant theory of SAFT has been developed including the convolution theorem, uncertainty principle, sampling theory and so on [16][17][18], which are generalizations of the corresponding properties of the FT, FRFT and LCT [3,9,[19][20][21][22][23][24][25][26][27][28]. Conventional convolution operations for FT are fundamental in the theory of linear timeinvariant (LTI) system [9].…”