\documentclass{revtex4-2}
\usepackage{graphicx}
\usepackage{float}
\usepackage[export]{adjustbox}

\begin{document}
\title{Module 7: Imaging Artifacts}
\author{Sebastian Koranda}
\date{28th November 2025}
\noaffiliation
\maketitle

\section{Sound Speed}

\begin{figure}[H]
\centering
\includegraphics[width=0.26\textwidth,trim={50mm 0 60mm 0},clip,valign=c]{figs/Figure_3_1460.pdf}
\includegraphics[width=0.33\textwidth,valign=c]{figs/Figure_2_1460.pdf}
\caption{1460 m/s (fat)}
\label{fig:1460_1_2}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.26\textwidth,trim={50mm 0 60mm 0},clip,valign=c]{figs/Figure_3_1540.pdf}
\includegraphics[width=0.33\textwidth,valign=c]{figs/Figure_2_1540.pdf}
\caption{1540 m/s (typical mean)}
\label{fig:1540_1_2}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.26\textwidth,trim={50mm 0 60mm 0},clip,valign=c]{figs/Figure_3_1600.pdf}
\includegraphics[width=0.33\textwidth,valign=c]{figs/Figure_2_1600.pdf}
\caption{1600 m/s (muscle)}
\label{fig:1600_1_2}
\end{figure}

Three different sound speeds were used to beamform the plane wave image: 1460 m/s (fat), 1540 m/s (typical mean), and 1600 m/s (muscle). The choice of sound speed significantly affects the reconstructed image in several ways, for example:
\begin{itemize}
    \item \textbf{Object positioning:} When the sound speed changes, objects appear at different depths in the reconstruction. Lower sound speed (1460 m/s) makes objects appear shallower, while higher sound speed (1600 m/s) makes them appear deeper. This occurs because beamforming calculates depth as $z = ct/2$, where $c$ is the assumed sound speed and $t$ is the time-of-flight.
    \item \textbf{Point scatter resolution:} The lateral resolution of the point scatter is most affected by sound speed selection. With incorrect sound speed (either 1460 m/s or 1600 m/s), the point scatter appears broader with reduced lateral resolution. The correct sound speed (1540 m/s) produces the sharpest, most focused point scatter with the best lateral resolution.
\end{itemize}
The point scatter resolution serves as a key indicator for sound speed accuracy: the sound speed that produces the sharpest point scatter is closest to the actual sound speed of the medium.

\section{Adjusting Z-axis}

To enable pixel-by-pixel comparison of images reconstructed with different sound speeds, the z-axis must be scaled appropriately with sound speed. The solution is to express depth in units of wavelength ($\lambda$) rather than meters.

\textbf{The problem:} When sound speed changes, the time-of-flight calculations in beamforming change, causing objects to appear at different depths. This occurs because depth is calculated as $z = ct/2$, where $c$ is the sound speed. When $c$ increases, the same time delay is interpreted as a greater depth, shifting objects in the reconstructed image.

\textbf{The wavelength solution:} The wavelength is defined as $\lambda = c/f$, where $f$ is the frequency. By expressing depth in wavelengths instead of meters, we create a coordinate system that automatically adjusts with sound speed changes. This can be implemented by scaling the z-axis proportionally:
\begin{equation}
z_{\text{scaled}} = z_{\text{meters}} \times \frac{c}{c_{\text{ref}}}
\end{equation}
where $c_{\text{ref}}$ is a reference sound speed (e.g., 1540 m/s).

\textbf{Why it works:} When sound speed increases, two effects occur simultaneously:
\begin{enumerate}
\item Objects appear deeper in the reconstruction (larger $z$)
\item The wavelength increases proportionally ($\lambda \propto c$)
\end{enumerate}

These effects cancel out when depth is expressed in wavelengths. Mathematically:
\begin{equation}
z_{\text{wavelengths}} = \frac{z_{\text{meters}}}{\lambda} = \frac{z_{\text{meters}}}{c/f} = \frac{z_{\text{meters}} \times f}{c}
\end{equation}

Since the time-of-flight $t$ remains constant for a given scatterer, and $z_{\text{meters}} = ct/2$, we have:
\begin{equation}
z_{\text{wavelengths}} = \frac{ct/2 \times f}{c} = \frac{ft}{2}
\end{equation}

This expression is independent of sound speed $c$, ensuring that objects remain at the same position in wavelength coordinates regardless of the sound speed used for reconstruction. This enables direct pixel-by-pixel comparison across images reconstructed with different sound speeds.

\begin{figure}[H]
\centering
\includegraphics[width=0.26\textwidth,trim={50mm 0 60mm 0},clip,valign=c]{figs/Figure_6_1460.pdf}
\includegraphics[width=0.33\textwidth,valign=c]{figs/Figure_5_1460.pdf}
\caption{1460 m/s (fat) with adjusted z-axis}
\label{fig:1460_1_2}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.26\textwidth,trim={50mm 0 60mm 0},clip,valign=c]{figs/Figure_6_1540.pdf}
\includegraphics[width=0.33\textwidth,valign=c]{figs/Figure_5_1540.pdf}
\caption{1540 m/s (typical mean) with adjusted z-axis}
\label{fig:1540_1_2}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.26\textwidth,trim={50mm 0 60mm 0},clip,valign=c]{figs/Figure_6_1600.pdf}
\includegraphics[width=0.33\textwidth,valign=c]{figs/Figure_5_1600.pdf}
\caption{1600 m/s (muscle) with adjusted z-axis}
\label{fig:1600_1_2}
\end{figure}

With the z-axis adjusted according to sound speed, all three images now align perfectly in terms of object positioning and size.

\section{Sound Speed Evaluation Criteria}

Based on the observations from the previous experiments, the correct sound speed for reconstruction can be determined using the following criterion: \textbf{the sound speed that produces the sharpest point scatter with maximum lateral resolution is the correct one}.

\end{document}
