Quaternion ridgelet transform and curvelet transform (Q1990956)

The relationships between the Fourier, Radon, wavelet, ridgelet, curvelet transforms for real-valued functions have been extensively studied and are well known. The paper under review extends some of these relationships to quaternion-valued functions. A quaternion \(a\) can be represented as \[ a=a_0+a_1 i+a_2 j+a_3 k, \] with \[ ij=k,\; jk=i,\; ki=j,\; i^2=j^2=k^2=-1, \] and the product is anti-commutative. Any quaternion-valued functions can be written as \[ f=f_0 +f_1i+f_2 j+f_3 k, \] where \(f_0,f_1,f_2, f_3\) are real-valued functions. The paper begins by defining the quaternion Fourier transform or more precisely the so-called right-side quaternion Fourier transform which differs from the conventional Fourier transform in that the imaginary number \(i=\sqrt{-1}\) in the exponent \(e^{ixt}\) is replaced by a unit quaternion \(\mu=(i+j+k)/\sqrt{3}.\) By setting \[ \mu'=(i-j)/\sqrt{2}, \mu''(i+j-2k)/\sqrt{6}, \] each quaternion-valued function can be expressed as \[ f=f_0+f_1 \mu+f_2 \mu' +f_3 \mu''. \] Therefor, the right-side quaternion Fourier transform of \(f\) is defined as \[ \hat{f}(x)=\int_{\mathbb{R}^2}f(t) e^{-2\pi \mu (x\cdot t)}dt. \] Using the definition of the quaternion Fourier transform, the author proceeds to define the quaternion Radon transform and hence the quaternion ridgelet transform as in the classical case. An uncertainty principle for the quaternion ridgelet transform is derived. Continuing along the same path, the author extends the definition of the curvelet transform to quaternion-valued functions. A reconstruction formula for quaternion-valued functions from their curvelet transform is also obtained. Overall, the results look similar to those for real-valued functions but with some obvious modifications.