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\begin{document}
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\begin{textblock}{12}(0,-0.2)
\baselineskip=3\baselineskip \Title{ERROR-FREE STEREO REACHES OPTIMAL THEORETICAL ACCURACY}
\end{textblock}
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\begin{textblock}{12}(0,0.4)
\Author{Neus Sabater
\hfil\break
\texttt{\normalsize Sabater@cmla.ens-cachan.fr}}
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\begin{textblock}{2}(12.5,-0.5)
% logos
$$
\begin{array}{cp{8mm}c}
\includegraphics[height=2cm]{Images/logoCMLA.eps} &
\includegraphics[height=2cm]{Images/logoENS.eps} \\
\end{array}
$$
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\begin{textblock}{7.5}(0,1.5)
\BigHead{Summary}

A new parameterless method for detecting matches in pairs of stereo images with virtually no false matches \cite{sabater08} is proposed. A very precise disparity map is provided avoiding the fattening effect. Matches are found \textit{a contrario} and Principal Component Analysis (PCA) is used to extract a feature vector in each point. The subpixel accuracy is obtained by interpolating and minimizing the quadratic distance under Shannon conditions.

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\begin{textblock}{7.5}(0,4.3)
\BigHead{Full Control of False Matches}
\begin{itemize}
\item {\color{grana} Feature Extraction: Local PCA.} The patch features are the PCA coefficients trained separately for patches with similar mean and variance. ADVANTAGE: More precise results in regions with lack of information (shadows and saturated regions).
\smallskip
\item  {\color{grana} The \textit{a contrario} model} is a statistical model for image patches deriving from the {\color{green} Helmholtz Principle}: A geometric structure is meaningful if the expected number of occurrences is very small in white noise. In our case: An empirical background model is fixed from the images. The relevant matches are detected as rare events for this model.
%A match is meaningful if according to probabilistic measures it would not happen by chance in these images.
\begin{description}
\item [-] Let $B_{q}$ be a block in the reference image. The empirical probability that an observed block $B_{q'}$ in the secondary image be similar to $B_{q}$  for the feature $i$ is
$$
    \widehat{ \, p^{i} (q,q)}=2 \cdot |H_{i}(q)-H_{i}(q')| \,,
$$
where $H_{i}$ is the cumulative histogram of the principal components coefficients for the secondary image.
\smallskip
\item [-]The {\color{green} Number of False Alarms (NFA)} is the expected number of appearances of the match $(q,q')$ :
$$
     NFA_{q,q'}= N_{test}  \cdot \prod_{i=1}^{N}p_{q,q'}^{i} \,,
$$
where $N_{test}$ is the number of matches to be tested, $N$ is the number of principal components considered and  $p_{q,q'}^{i}$ is a quantification of $\widehat{ \, p^{i}_{q,q'}}$ and is imposed to be non-decreasing in $i$.
%(Probability thresholds: $1/2^{j-1}$, $j=1,\ldots,Q$.)
\smallskip
\item[-] A match $(q,q')$  is {\color{green} $\epsilon$-meanigful} when $NFA_{q,q'}<\epsilon$.
\smallskip
\end{description}
%\item {\color{grana} Self-similarity threshold}: sanity check avoiding %{\color{green} stroboscopic effects} (self-similar repeated structures)
\end{itemize}
\begin{theorem} Under the {\it a contrario } assumption that all
principal components are independent, ${\mathbb E}\#\{\varepsilon-meaningful\;matches\}\leq\varepsilon.$\end{theorem}

\begin{center}
% use packages: array
\begin{tabular}{|c|c|}
\hline
CAUSE OF FALSE MATCHES & PROPOSED SOLUTIONS \\
\hline
\hline
Occlusions , moving objects and noise &  \textit{a contrario} model\\
\hline
Poor textured regions & Local PCA \\
\hline
Stroboscopic effects  & Self-similarity threshold\\
\hline
\end{tabular}
\end{center}
\end{textblock}
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\begin{textblock}{7.5}(0,17.1)
\BigHead{Subpixel Accuracy:}
\begin{itemize}
\item  Once a meaningful match have been found in $x_{0}$ its disparity is refined.
%An interpolation under Shannon conditions and minimization of the $L^{2}$ distance is done in this phase.
%\item Experimental results: $1/20$ pixel accuracy for more than half of the image points.
\end{itemize}
$$ \mu^d(x_0)   := \underset{\mu \in \R}{\arg\min} \; e^d_{x_0}(\mu), \quad e^d_{x_0}(\mu):=\normphi{ \tau_\mu u - \tilde{u} }^2  \; ,$$
where $\scalar{u}{v}{\w}$ is the weighted discrete scalar product and $\normphi{\cdot}$ the corresponding weighted norm. We write $\tau_\mu u(x)=u(x+\mu)$, and $\varphi_{x_{0}}=\varphi(x-x_{0})$ is a symmetric and normalized window in a compact support (e.g. prolate).

\begin{theorem}\label{theoremone}
Assuming that  $\utilde (x) = u(x +\varepsilon(x)) + n(x)$ where the noise $n\sim N(0,\sigma)$ . Then
$$\mu^{d} (x_{0}) - \varepsilon(x_{0}) =
\underbrace{\dfrac{\scalarphi{u'^{2}}{\varepsilon}}{ \normphi{u'}^{2} } - \varepsilon(x_{0})}_\text{fattening error} \; +
\underbrace{\dfrac{\scalarphi{u'}{n}}{\normphi{u'}^{2}}}_{ \text{noise error} \sim N\big(0,{\color{green} \frac{\sigma \| u' \|_{\varphi^{2}_{x_{0} }}}{\normphi{u'}^{2}}}\big)}  + o(|\mu^d-\varepsilon|^2)$$
\end{theorem}\vspace{-0.2cm}
{\color{grana} Remark} By Thm \ref{theoremone} the noise error is estimated at each point $x_{0}$.
%\begin{theorem}[Sub-pixel accuracy requires x2 zoom]
%If  $u$ is $\pi$-bandlimited, then the $\ell^2$ distance $e^d_{x_0}(\mu)$ is %$2\pi$-bandlimited. Hence
%the $\half$-sampling of $e^d_{x_0}$ allows the exact reconstruction (via
%$\sinc_{\half}$ interpolation) of $e^d_{x_0}(\mu), \; \forall\mu \in \R$.
%\end{theorem}
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% 			Segona columna
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\begin{textblock}{6.5}(8.5,1)
\BigHead{Correcting the Fattening Effect}
\begin{itemize}
\item Dilation of the size of the patch of structures with boundaries coinciding with depth discontinuities.
\item PROPOSED APPROACH: Compute a new disparity map which considers the disparity computed at pixel x as a feasible disparity for a whole set of pixels: namely, all pixels in the patch with best matching gradient angle. Reject the match when the two solutions differ by  more than the allowed precision.
\end{itemize}
\end{textblock}

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\begin{textblock}{6.5}(8.5,4.3)
\BigHead{Experiments}

\SubHead{Simulated Pair:}

\begin{minipage}{0.48\linewidth}
\begin{center}
\includegraphics[width=6.7cm, height=6.7cm]{Images/u_ref_crop.eps}\\
\small{First image}\\
\includegraphics[width=6.7cm, height=6.7cm]{Images/ref_preso_0_crop_color.eps}\\
\small{Disparity. Red pixels aren't matched (rejected by the \textit{a contrario} method or the fattening correction.)}\\
\end{center}
\end{minipage}
\begin{minipage}{0.48\linewidth}
\begin{center}
\includegraphics[width=6.7cm, height=6.7cm]{Images/GTP_crop.eps}\\
\small{Groundtruth}\\
\includegraphics[width=6.7cm, height=6.7cm]{Images/noise_error.eps}\\
\small{Noise error estimation at each point. The darker the pixel, the higher the error.}\\
\end{center}
\end{minipage}

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
SNR & RMSE & Matches & Bad matches & \small{Predicted noise error}\\
\hline
\hline
$\infty$ & 0.023 & 70.6\% & 0\% & 0\\
\hline
357.32  &  0.033 & 63.3\% & 0\% & 0.023\\
\hline
125.06 & 0.058 & 41.5\% & 0.02\% & 0.065\\
\hline
\end{tabular}
\end{center}
\begin{center}
\small{Root Mean Squared Error for different Signal to Noise Ratios.}
\end{center}


\SubHead{Lion Experiment:}

\begin{minipage}{0.42\linewidth}
\begin{center}
\includegraphics[width=8cm, height=6cm]{Images/LION.res.eps}\\
\small{First image}\\
\includegraphics[width=8cm, height=6cm]{Images/LION_Med-2Drendering_crop.eps}\\
\small{Upper view of the 3D rendering of the computer surface}
\end{center}
\end{minipage}
\hfil
\begin{minipage}{0.42\linewidth}
\begin{center}
\includegraphics[width=8cm, height=6cm]{Images/LION_Med.eps}\\
\small{Disparity map}\\
\includegraphics[width=8cm, height=6cm]{Images/LION_Med-3Drendering-plus-texture_crop.eps}\\
\small{Slanted view of the computed surface with the reference image}
\end{center}
\end{minipage}


\SubHead{Stereo benchmark:} The Middlebury dataset \cite{middle} provides  9 orthorectified views at uniform intervals of piecewise planar scenes. A cross-validation  is done taking the central view as reference:
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
 		& \small{Matches} &  \small{Bad Matches}   & \small {RMSE} & \small{Predicted noise error}\\
\hline
\small{Sawtooth} & 45.2 \% & 0.1 \%  & 0.090 & 0.081\\
\hline
\small{Venus}    & 47.2 \% & 0.1 \%  & 0.056 & 0.061\\
\hline
\end{tabular}
\end{center}
%\begin{center}
%\small{RMSE computed by cross-validation and estimated error noise}
%\end{center}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=6cm, height=5cm]{Images/im2.eps}\\
First image
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=6cm, height=5cm]{Images/disp2.eps}\\
Groundtruth
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=6cm, height=5cm]{Images/result_saw_color.eps}\\
Disparity map
\end{minipage}


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\begin{textblock}{6.5}(8.5,23.1)
\rm
\begin{thebibliography}{7}
\bibitem{sabater08} \small{ N. Sabater, A. Almansa et J.-M. Morel, \emph{Rejecting Wrong Matches in Stereovision.}  CMLA preprint  2008-28.}
\bibitem{middle} \small{Middleburry dataset. \emph{http://vision.middlebury.edu/stereo/}}.
\end{thebibliography}


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