## Materials and Methods  
### Theory of H-Scan Ultrasound Imaging  
The backscattered ultrasound signal \(e(t)\) can be modeled as [26]:  
\[e(t)=A\Big{\{}p(t)s(x,y)***R(x,y,\frac{ct}{2})\Big{\}} \tag{1}\]  
where \(A\) is a signal amplitude consistent, \(c\) is the speed of ultrasound, \(p(t)\) is the propagating ultrasound pulse and \(s(x,y)\) is the beam pattern (beam width in the transverse and elevational axes). \(R(x,y,ct/2)\) represents the 3D pattern of acoustic reflectors or spreading objects in the medium. The symbol *** stands for 3D convolution. With an assumption of tiny spatial variants in medium density and compressibility, the function \(R\) can be related to the spatial derivative of acoustic impedance \(Z\) in the direction \(z\) of the ultrasound imaging pulse [26]. \[R(z)\approx\Big{(}\frac{1}{2Z}\Big{)}\frac{\mathrm{d}Z(z)}{\mathrm{d}z} \tag{2}\]  
For H-scan US imaging, we considered three types of reflections in the one-dimensional convolution model:  
1. Large layer:  
Large layer scattering occurs when the scattering object is larger than the wavelength of the transmitted ultrasound signal. With a small step increase in acoustic impedance at position \(z_{0}\), the acoustic impedance of the large layer at \(z_{0}\) is similar to a step function \(u(z\)-\(z_{0})\). Since the spatial derivative of a step function is the Dirac delta function \(\delta(z)\), the spatial derivative of acoustic impedance at \(z_{0}\) can be expressed as:  
\[\frac{\mathrm{d}Z(z)}{\mathrm{d}z}=\Delta Z\cdot\delta(z-z_{0}) \tag{3}\]  
The reflection coefficient can be rewritten as:  
\[R_{\mathrm{L}}(z)=\frac{\Delta Z}{2Z}\delta(z-z_{0}) \tag{4}\]  
The received signal is then the 1D convolution * of the reflection coefficient at the large interface and the transmitted ultrasound signal \(p(t)\):  
\[e_{\mathrm{L}}(t)=p(t)*R_{\mathrm{L}}(z)=\frac{\Delta Z}{2Z}p(t-t_{0}) \tag{5}\]  
2. Thin layer:  
Thin layer scattering corresponds to an acoustic scattering source with a thin interface and higher acoustic impedance, such as an arterial wall. We can approximate the impedance profile as:  
\[Z_{T}(z)=Z+\Delta Z\delta(z-z_{0}) \tag{6}\]Therefore, the reflection function is:  
\[R_{T}(z)=\left(\frac{\Delta Z}{2Z}\right)\delta^{\prime}(z-z_{0}) \tag{7}\]  
where \(\delta^{\prime}(\cdot)\) is the derivative of the Dirac delta function. The received signal is then the convolution of the transmitted ultrasound signal \(p(t)\) with the reflection coefficient of the thin layer:  
\[\mathbf{e}_{T}(t)=p(t)*R_{T}=\left(\frac{\Delta Z}{2Z}\right)p^{\prime}(t-t_{0}) \tag{8}\]  
where \(p^{\prime}(t-t_{0})\) is the first derivative with respect to time of \(p(t-t_{0})\). 3 Rayleigh scatterers:  
Rayleigh scattering takes place when the acoustic scattering source is smaller than the ultrasound wavelength [27]. For small spherical scatterers, the Born approximation gives a leading term for backscattered reflection that has pressure dependence with a frequency-squared weighting [28]. Additionally, the position correlation and multiple reflections can be ignored [26; 27]. For Rayleigh scattering (single scatterer or cloud of scatterers), the received signal can be given as [20; 27]:  
\[\mathbf{e}_{s}(t)=p(t)*=S_{c}\left(\frac{\Delta Z}{2Z}\right)p^{\prime\prime}( t-t_{0}) \tag{9}\]  
where \(p^{\prime\prime}(t)\) is the second derivative of \(p(t)\) to time, obtained from the convolution property of \(\delta^{\prime\prime}(\cdot)\), and where \(S_{c}\) incorporates all geometric factors and constants of proportionality in the Rayleigh backscatter equation [27]. To identify these 3 classes of ultrasound scattering through the relationship with transmitted signals and the derivative in the direction of propagation, a set of functions related to Gaussian-weighted Hermite polynomials was employed in [19].