Discrete Fractional Cosine Research Articles

Color Image Encryption Based on the Multiple-Order Discrete Fractional Cosine Transform and Chaos in YCbCr Space

Based on the theory that a color image can be decomposed into three primary components and each one can be seen as a gray image, we propose a color image encryption method with multiple-order discrete fractional cosine transform (MODFrCT), which is a kind of encryption with the secrecy of pixel value and pixel position simultaneously. The complex number mode that has a real part and an imaginary one is used in this encryption method to save the transmission channel. Human vision is more sensitive to the Y component than to other two components in YCbCr color space and this color format is used for encrypting the color image. Chaos is introduced to scramble the image phases both in spatial and transformation domains. The numerical simulations demonstrate the validity and efficiency of this scheme and the robustness of the method against occlusion attack is examined.

Multiple and color images compression-encryption schemes with balanced qualities based on the multiple-order discrete fractional cosine transform

By making use of the discrete fractional cosine transform, spectrum cutting and combining, rate-distortion control and color space transform, a joint compression and encryption scheme for multiple images and color images is proposed based on the multiple-order discrete fractional cosine transform (MODFrCT). The spectra coefficients of images gotten by the discrete cosine transform are scanned in a way of zigzag, cut at an appropriate position, and combined into a single spectrum image sequentially encrypted by the MODFrCT. Rate-distortion control is utilized during the spectrum cutting to balance the qualities of the multiple reconstructed images. A color image can be decomposed into Y, Cb, and Cr components prior to the encryption, and these three components are then encrypted in the same way as that for multiple images. The numerical simulations demonstrate the validity and efficiency of these schemes, and the robustness of the schemes against occlusion attack is examined.

Image encryption based on the multiple-order discrete fractional cosine transform

A novel scheme for image encryption based-on the multiple-order discrete fractional cosine transform (MODFrCT) is proposed. The DFrCT has a similar relationship with the discrete fractional Fourier transform (DFrFT). Not only has the DFrCT many useful properties similar to the conventional discrete cosine transform, but it also has another property, namely its fraction, or its transform order. The image to be encrypted is transformed with the multiple-order DFrCT using a random row cipher key vector and a random column key vector successively, and the corresponding cipher key vectors of decryption are also very sensitive. The transmission of the encrypted image with the algorithm of the multiple-order DFrCT is faster due to its reality. The digital simulation results proved the validity and safety of this algorithm.

Tridiagonal Commuting Matrices and Fractionalizations of DCT and DST Matrices of Types I, IV, V, and VIII

In this paper, we first establish new relationships in matrix forms among discrete Fourier transform (DFT), generalized DFT (GDFT), and various types of discrete cosine transform (DCT) and discrete sine transform (DST) matrices. Two new independent tridiagonal commuting matrices for each of DCT and DST matrices of types I, IV, V, and VIII are then derived from the existing commuting matrices of DFT and GDFT. With these new commuting matrices, the orthonormal sets of Hermite-like eigenvectors for DCT and DST matrices can be determined and the discrete fractional cosine transform (DFRCT) and the discrete fractional sine transform (DFRST) are defined. The relationships among the discrete fractional Fourier transform (DFRFT), fractional GDFT, and various types of DFRCT and DFRST are developed to reduce computations for DFRFT and fractional GDFT.

Sliding discrete fractional transforms

Fractional transforms are useful tools for processing of non-stationary signals. The methods of implementing sliding discrete fractional Fourier transform (SDFRFT), sliding discrete fractional cosine transform (SDFRCT) and sliding discrete fractional sine transform (SDFRST) for real time processing of signals are presented. The performances of these sliding transforms, with regard to computational complexity, variance of quantization error and signal-to-noise ratio (SNR), are presented and compared. The three sliding discrete fractional transforms are compared with sliding discrete Fourier transform (SDFT) in terms of SNR. Computational complexity in the case of sliding discrete fractional transform is less than that in the case of discrete fractional transform when a particular time–frequency bin is to be observed. In comparison with SDFT, the sliding discrete fractional transforms require less number of bits for representing coefficients. The SDFRST performs better in comparison with SDFRFT and SDFRCT.

The fractional discrete cosine transform

The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed out.

The discrete fractional cosine and sine transforms

This paper is concerned with the definitions of the discrete fractional cosine transform (DFRCT) and the discrete fractional sine transform (DFRST). The definitions of DFRCT and DFRST are based on the eigen decomposition of DCT and DST kernels. This is the same idea as that of the discrete fractional Fourier transform (DFRFT); the eigenvalue and eigenvector relationships between the DFRCT, DFRST, and DFRFT can be established. The computations of DFRFT for even or odd signals can be planted into the half-size DFRCT and DFRST calculations. This will reduce the computational load of the DFRFT by about one half.