They cover complex and hypercomplex numbers; octonion numbers; quaternions and color images; color images as two-dimensional grayscale images; one-dimensional and two-dimensional quaternion and octonion discrete

Fourier transforms; color image enhancement and quaternion discrete

Fourier transforms; gradients, face recognition, visualization, and quaternions; and color image restoration and quaternion discrete

Fourier transforms.

The two-sided, right-sided, and left-sided quaternion

Fourier transforms (QFTs) of f [member of] [L.sup.1]([R.sup.2]; H) are given by, respectively,

where [gamma] is a small real value related to SNR, representing the ratio of the power spectral density of the noise and that of the signal [9], [X.sup.*] (f) is the complex conjugate of X(f), F([R.sub.yx]([tau])), and F([R.sub.xx]([tau])) are

Fourier transforms of cross- correlation and autocorrelation functions, respectively [10].

Ding, "Eigenfunctions of fourier and fractional

fourier transforms with complex offsets and parameters," IEEE Transactions on Circuits and Systems.

The quaternion

Fourier transform (QFT) is a nontrivial generalization of the real and complex classical

Fourier transforms (FT) using quaternion algebra.

If two functions are shifted in arguments, that is, [f.sub.2](x; y) = [f.sub.1](x - [x.sub.0]; y - [y.sub.0]), their

Fourier transforms are shifted in phase; that is,

VETTERLI, Fast

Fourier transforms: a tutorial review and a state of the art, Signal Process., 19 (1990), pp.

The spectral analysis through

Fourier transform obtained an extensive use, especially when switching to discrete and fast

Fourier transforms.

Younis ([7], Theorem 5.2) characterized the set of functions in [L.sup.2](R) satisfying the Dini-Lipschitz condition by means of an asymptotic estimate growth of the norm of their

Fourier transforms, namely we have

The discrepancy for the

Fourier transforms corrected by theoretical and experimental phase and amplitude function could be caused by the side lobe generated from the finite range of the

Fourier transform or the difference of amplitude functions obtained from theoretical and experimental methods.

E) Transformation of relations derived in eqs.(3)-(4) to frequency domain If the

Fourier transforms of auto and cross correlation functions are represented by SXX(f) and SXY(f), then we may consider SXX(f), SYY(f), Shh(f) and H(f) as energy spectral densities present in the input process, output process and the filter and frequency response of the filter respectively.