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\pagestyle{empty}
\begin{document}
\begin{landscape}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\commentout{
\begin{slide}
\header{ Seeing Extended Target Behind Screen and Wall with Multi-Frequency Signals}
\begin{center}
\vfill

\textbf{ A. Fannjiang, M. Yan}

Department of Mathematics\\
University of California, Davis\\

\vspace{2cm}
{in collaboration with}

\textbf{UC Irvine Team}

\rule{0.4\hsize}{0.5pt}

\end{center}

\vfill\noindent\rule{\hsize}{1pt}\vspace{-.5em}


\end{slide}
}

\begin{slide}
\header{Foldy-Lax simulation with point scatterers}
\begin{itemize}
\item Domain: $[-5000, 5000]\times [0, 5000]$.
\item 3000 point scatterers are randomly distributed in  the right half domain. 
\item Passive or active array of 5 antennas. 
\item Wavelengths: $52\sim 90$.

\item  Lippmann-Schwinger equation
\beqn
G(\bx,\by_n;\omega_l)=G_0(\bx,\by_n;\omega_l)
+\sum_{j=1}^J \tau_j(\omega_l) G_0(\bx,\bx_j;\omega_l) G(\bx_j,\by_n;\omega_l),\quad n=1,...,N
\eeqn
where the scattering strength $\tau_j$ of the $j$-th scatterer
is given by 
\beqn
\label{strength}
\tau_j(\omega_l)=\omega_l^2(n_j^2-1).
\eeqn
\end{itemize}
\end{slide}

\begin{slide}
\header{Theory of  stability}
\begin{itemize}
\item $N$-element array located at $\by_j$; $M$ hidden targets located at $\bx_i$.
 \item $H(\bx,\by;\omega)$: the transfer function of
  the propagation channel at the frequency $\omega$
   from point $\by$ to $\bx$. 
  \item $\bH(\bx_i,\by_j;\ks)=[H_{ij}(\ks)]$: the transfer matrix between
    the antenna located at $\by_j$ and the target located at
    $\bx_i$ where
     $
      H_{ij}(\ks)=H(\bx_i,\by_j;\ks).$  
      \item         
We decompose the random transfer function into the mean 
(coheren) field $\bar H$ and the fluctuation (diffuse) field
$h, \IE h=0,$ as $H(\bx,\by)=\bar H(\bx,\by)+h(\bx,\by)$.

\item 
We write $\bar\bH(\omega)=[\bar H_{ij}(\omega)]$
and $\bh(\omega)=[ h_{ij}(\omega)]$ with $
h_{ij}(\omega)=h(\bx_i,\by_j;\omega)$.
\end{itemize}
\end{slide}

\begin{slide}
\header{Imaging field}
\begin{itemize}
\item Active array: differential response
\[
 u(\bx)=\sum_{l=1}^B\sum_{i=1}^M\sum_{j=1}^N P(\bx,\by_j;\omega_l)H^2_{ij}(\omega_l)P(\by_j,\bx;\omega_l).
 \]
 \item $P$: the imaging propagator, e.g.  the phase factor
 of the mean
 Green function $\bar H$.
 
 
\item Our main assumptions  are
\begin{itemize}
\item  the separations of the frequencies $\omega_j, j=1,2,...,B$
 are
larger than the coherence bandwidth $\beta_c$
\item 
the spacings of the antennas are wider than the coherence length $\ell_c$. 
\end{itemize}
\item Criterion for stability
 the normalized variance 
\[\red
\Var_m(\bx)= \frac{\IE|u|^2(\bx)-|\IE u(\bx)|^2}{|\IE u(\bx_m)|^2},\quad j=1,...,M
\]
tends to zero
where $u(\bx)$ is the imaging field. 
\item $\Var^{-1}_m(\bx_m)$ is the signal-to-inteference-ratio.

\item Effective medium theory:
\beqn
\label{mean2}
\bar H(\bx,\by;k)\sim e^{-\frac{r}{2\ell}} \frac{e^{ik r}}{r}, \quad r=|\bx-\by|,
\eeqn
where  the  mean-free path $\ell$ is related to
the extinction cross section $\sigma_t$  by
\[
\ell=\frac{1}{\rho\sigma_t}.
\]
In the case of  Rayleigh scattering
without absorption, the scattering cross section 
scales like $k^4$  
\beq
\label{ray}
\ell\sim  \rho^{-1} k^{-4} a^{-6}.
\eeq


\end{itemize}
\end{slide}

\begin{slide}
\header{\red Stability condition}
\begin{itemize}
\item Rician $K$-factor is defined as
 the ratio of the mean power to the fluctuating power. 
\item $\red K BN\gg M$: Rician factor $\times$ Number of independent array elements $\times$ number of independent
frequencies $\gg$ number of
targets.




\item Independent frequencies with larger than {\em coherence
bandwidth} separation; independent antennas with larger than
{\em coherence length} spacing.

\item {\green Convergence of the imaging field $u(\bx)$ to
$\IE u(\bx)$.}

\item {\blue Limitation: point targets and weakly scattering media.}

\end{itemize}
\end{slide}
\includegraphics[width=.7\textwidth]{le00005.pdf}

{Longitudinal (left) and transverse (right)  resolution with the free-space propagator
 as filter
 (blue curve) and the mean phase-matched filter (red curve).  
The effective medium is defined by (\ref{mean2})
 with $\ell=5\times 10^{-5} \lambda^4$. The horizontal axis is the antenna spacing. }



\includegraphics[width=1\textwidth]{Fourplots_active11_ray_T1_S50_diag.pdf}

 {Synthetic aperture imaging with 
1, 5, 10, and 20 frequencies.  $n^2-1$ of the scatterers and the target equal to $50$ and $1$, respectively. }

 
 
\includegraphics[width=1\textwidth]{Fourplots_active11_ray_T1_S70_diag.pdf}

{Synthetic aperture imaging with 
1, 5, 10, and 20 frequencies.  $n^2-1$ of the scatterers and the target equal to $70$ and $1$, respectively. }

\includegraphics[width=1\textwidth]{7targets_30.pdf}

{SA imaging of seven targets with 
 20 frequencies .  $n^2-1$ of the scatterers and the target equal to $30$ and $1$, respectively. }
 
 \includegraphics[width=1\textwidth]{active11_7targets_noisearound_3000scatterers.pdf}

{Imaging with the full response matrix.
$n^2-1$ of the scatterers and the targets equals $50$ and 1, respectively.}


 
\includegraphics[width=1\textwidth]{SAR_vs_full.pdf}

{Comparison of longitudinal (bottom) and transverse (top)
 resolutions with synthetic aperture (red) and full response matrix (blue).  $n^2-1$ of the scatterers and the target equal to $50$ and $1$, respectively. The horizontal axis is the antenna spacing. }
 


 
 \includegraphics[width=1\textwidth]{le0025.pdf}

{Longitudinal (left) and transverse (right) resolutions in the clutter (blue curve) and
in the effective medium with $\ell=0.0025 \lambda^4$  (red curve) as a function of the aperture. $n^2-1$ of the scatterers and the targets equals $70$ and 1, respectively. The horizontal axis is the antenna spacing. }
 
\end{landscape}
\end{document}
 
