3D Clifford Analytic Signal for 3D Envelope Detection on Ultrasound Volume

Envelope detection is one of the essential steps for ultrasound B-mode image reconstruction. Typically, this reconstruction is based on one dimensional (1D) radio frequency (RF) signal demodulation. In the literature, 1D envelopes are usually stacked to achieve two or three dimensional (2D or 3D) envelope detection, which leads to possible losses of multidimensional context information. In this paper, we propose a hypercomplex analytical signal in the form of Clifford biquater-nion, which is called 3D Clifford analytic signal (3D CAS). Firstly, the convolution is utilized to define the 3D CAS in terms of biquaternion. Next, the relation between the 3D CAS and the classical Hahn's analytical signals in the 3D case is illustrated with the help of Hilbert transforms, which enables a reliable numerical implementation of 3D CAS. Then, the modulus of 3D CAS is proposed and applied to a straightforward 3D ultrasonic RF envelope detection by taking into consideration the information of amplitude of all three directions of space. Experiments were carried out on the ultrasound volume from a phantom with a biopsy needle inserted. The experiments provide a better visual experience from envelope images of RF volume than the compared 1D and 2D methods. The 3D and 2D envelopes present 36.5% and 15.2% improvement to 1D, respectively, from the contrast to noise ratio between the needle region and the adjacent background. From an independent public 2D liver tumor RF sequences of 29 mice, the average structural index similarity (SSIM) and peak signal to noise ratio (PSNR) were improved by 53.0% and 39.0%, respectively, from 2D envelopes comparing to 1D envelopes.