Quaternion wavelets for image analysis and processing

Using the concepts of two-dimensional Hilbert transform and analytic signal, we construct a new quaternion wavelet transform (QWT). The QWT forms a tight frame and can be efficiently computed using a 2-D dual-tree filter bank. The QWT and the 2-D complex wavelet transform (CWT) are related by a unitary transformation, but the former inherits the quaternion Fourier transform (QFT) phase properties, which are desirable for image analysis. The quaternion magnitude-phase representation of the QWT directly leads to near shift-invariance and the ability to encode phase shifts in an absolute xy-coordinate system, which we can use for applications such as edge estimation and statistical image modeling.