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%%%%%%%%%%%% Her begynner selve dokumentet %%%%%%%%%%%%%%%
\begin{document}
\vspace{-3ex} \hspace{5ex}
\includegraphics[width=5cm]{./figures/logox.jpg} \hspace{3ex}
\includegraphics[width=5cm]{./figures/cmap.jpg} \hspace{4ex}
\hspace{50cm}
\includegraphics[width=5cm]{./figures/logoCachan.jpg} \hspace{4ex}
\includegraphics[width=5cm]{./figures/logoCMLA.jpg} \vspace{4ex}


\vspace{-2ex} \hspace{3ex}
\begin{center}
{ \Huge \color{blue}{ASIFT: A  New Framework for Fully Affine Invariant Image Comparison}} \\
\vskip 2cm
\begin{tabular}{cc}
{\LARGE  Guoshen Yu} &~~~~~~~~~~~~~~~ {\LARGE  Jean-Michel Morel} \\
{\Large CMAP, Ecole Polytechnique} &~~~~~~~~~~~~~~~ {\Large CMLA,
ENS Cachan, 61
avenue du President Wilson,} \\
{\Large 91128 Palaiseau Cedex, France} &~~~~~~~~~~~~~~~ {\Large
94235 Cachan Cedex, France}
\end{tabular}

\end{center}
~\\
\vskip 1cm

\begin{multicols}{3}
 \noindent {\Large \color{blue} {Abstract}}
 \vskip 0.1cm \noindent
 A fully affine invariant image comparison method, Affine-SIFT
(ASIFT) is introduced. While SIFT is fully invariant with respect to
four parameters namely zoom, rotation and translation, the new
method treats the two left over parameters :  the angles defining
the camera axis orientation. Against any prognosis, simulating all
views depending on these two parameters is feasible with no dramatic
computational load. The method permits to reliably identify features
that have undergone large transition tilts, up to 36 and more, while
state-of-the-art methods hardly exceed transition tilts of 2 (SIFT),
2.5 (Harris-Affine and Hessian-Affine) and 10 (MSER).


\section{The Affine Camera Model}
\subsection{Image Formation Model}
\begin{center}
\includegraphics[width=12cm]{./figures/image_formation.jpg}
\end{center}
\subsection{Affine Simplification}
Local perspective effects can be modeled by local affine transforms
$u(x,y)\to u(ax+by+e, cx+dy+f)$ in each image region.
\begin{center}
\includegraphics[width=10cm]{./figures/UccelloHostie.jpg}\\
\textit{The global deformation of the ground is strongly projective
(a rectangle becomes a trapezoid), but the local deformation is
affine: each tile on the pavement is almost a parallelogram.}
\end{center}
\subsection{Affine Camera Model}
\begin{center}
\includegraphics[width=9cm]{./figures/camera_phi2_nikon.jpg}
\end{center}
$$ \label{decompositionofaffine}A=\left[\begin{array}{cc}
a & ~b  \\
c & ~d
\end{array} \right]\!=\!H_\lambda
R_1(\psi)T_tR_2(\phi)\! = \!\lambda \left[
\begin{array}{cc}
\cos \psi & ~-\sin \psi \\
\sin \psi & ~\cos \psi
\end{array} \right]
\left[
\begin{array}{cc}
t & ~0 \\
0 & ~1
\end{array} \right]
\left[
\begin{array}{cc}
\cos \phi & ~-\sin \phi\\
\sin \phi & ~\cos \phi
\end{array} \right]
$$
\begin{itemize}
\item A: affine map with strictly positive determinant.
\item $\mathbf{\phi}$: {{\it {longitude}} angle between optical axis and a fixed vertical plane}.
\item $\theta=\arccos(1/t)$:  {\it latitude} angle between optical axis and the normal to the image
plane. \textbf{Tilt} $t>1 \leftrightarrow \theta \in
[0^\circ,90^\circ]$.
\item $\psi$: rotation angle of camera around optical axis.
\item $\lambda$: \textit{zoom} parameter.
\end{itemize}
\subsection{High Transition Tilts}
Both compared images $u_1(x,y)=u(A(x,y))$  and $u_2(x,y)=u(B(x,y))$
are usually slanted views. The~\textit{transition tilt} quantifies
the tilt between two such images. $$
\label{decompositionofaffinerelative}BA^{-1}=H_\lambda
R_1(\psi)T_\tau R_2(\phi).
$$
The \textbf{transition tilt} $t_1/t_2 \leq\tau \leq t_1 t_2$.
\begin{center}
\includegraphics[width=12cm]{./figures/relative_tilt_MITMaga.jpg}
\end{center}

\section{State-of-the-art}
\subsection{Simulation v.s. Normalization}
\begin{itemize}
\item Simulation: all 6 parameters impossible, e.g. $10^6$.
\item Normalization:\\
\includegraphics[width=6cm]{./figures/normalization_translation.jpg}
\includegraphics[width=6cm]{./figures/normalization_rotation.jpg}
\includegraphics[width=6cm]{./figures/normalization_tilt.jpg}
\begin{itemize}
\item Translation $\mathcal{T}$ and rotation $\bR$: OK.
$\bG_1 \bR u_0 = \bR \bG_1 u_0$.
\item Zoom $\bH_\lambda$ and tilt $\bT$: not exact.
$\bH_\lambda \bG_1 u_0 \neq \bG_1 \bH_\lambda u_0$.
\end{itemize}
\end{itemize}

\subsection{SIFT}
\begin{itemize}
\item Rotation and translation  are \textit{normalized}.
\item Zoom is \textit{simulated} in the scale space.
\item No treatment on latitude and longitude: $\tau_{\max} < 2 $.
\end{itemize}
\subsection{MSER, Harris-Affine, Hessian-Affine}
\begin{itemize}
\item Normalize all affine parameters.
\item Limited performance on scale- and tilt-invariance.
\item MSER: $\tau_{\max} < 10 $ in optimal conditions.
\end{itemize}

\section{ASIFT}
\subsection{Algorithm}
\begin{center}
\includegraphics[width=15cm]{./figures/brevet_fig5.jpg}
\end{center}
\begin{enumerate}
\item Simulate latitude, longitude to achieve full affine invariance.
\item Simulated images are compared by a rotation-, translation- and zoom-invariant algorithm, e.g., SIFT.
\end{enumerate}

\subsection{Full Affine Invariance}
\begin{theorem}\label{proofthatCSIFTworks} Let
$u= \bbG_{1} A\mathcal T_1u_0 $ and $v= \bbG_{1} B\mathcal T_2u_0$
be two images obtained from an infinite resolution image $u_0$ by
cameras at infinity with arbitrary position and focal lengths. Then
ASIFT, applied with a dense set of tilts and longitudes, simulates
two views of $u$ and $v$ that are obtained from each other by a
translation, a rotation, and a camera zoom. As a consequence, these
images match by the SIFT algorithm.
\end{theorem}

\subsection{Why it works?~---~Inverting Tilts}
A tilt in one direction is reversed by simulating a tilt of same
amount in the orthogonal direction, up to a zoom-out scale change.

\subsection{Parameter sampling}
\begin{center}
\includegraphics[width=7cm]{./figures/brevet_fig2b.jpg}
\includegraphics[width=7cm]{./figures/brevet_fig4b_english1.jpg}
\end{center}
Denser sampling when the latitude angle $\theta$ (or tilt $t$)
increases.

\subsection{Two-resolution Acceleration}
\begin{enumerate}
\item ASIFT on low-resolution images ($r\times r$ sub-sampled) .
\item ASIFT on high-resolution images obtained with the identified good affine
transforms (only in case of success in 1.).
\end{enumerate}

\subsection{ASIFT Complexity}
\begin{itemize}
\item About twice SIFT. ($t_{\max} = 4\sqrt{2} \Rightarrow
\tau_{\max} = 32$, $r=3$)
\item SIFT subroutines fully parallizable.
\end{itemize}

\section{Experiments}

\begin{center}
\includegraphics[width=12cm]{./figures/MagaMIT_tilt6p37vs0p17_crop_contrast.jpg}
\includegraphics[width=12cm]{./figures/MagaMIT_tilt6p37vs0p17_CAffine2sides_tmax8_tk_sqrt2_MR_fs1_ratio0p4_0p55_94matches_crop_contrast.jpg}
\end{center}
{\small  \textit{Transition tilt $t \approx 36$. Bottom: ASIFT
(shown) -- 116 correct matches. SIFT, Harris-Affine, Hessian-Affine
and MSER fail completely.}} \vspace{1ex}

\begin{center}
\includegraphics[width=8cm]{./figures/adam_zoom10_front-adam_zoom10_84deg_ASIFT_77matches_crop.jpg}
\includegraphics[width=8cm]{./figures/adam_zoom10_front-adam_zoom10_84deg_HarAff_1matches_crop.jpg}
\includegraphics[width=8cm]{./figures/adam_zoom10_front-adam_zoom10_84deg_MserSift_2matches_crop.jpg}
\end{center}
{\small \textit{Transition $\tau \approx 5.8$. ASIFT (shown) -- 116,
SIFT -- 1, Harris-Affine (shown) -- 1, Hessian-Affine -- 0, and MSER
(shown) -- 2 correct matches.}} \vspace{1ex}

\begin{center}
\includegraphics[width=8cm]{./figures/H8A0vsH8A5_ASIFT_881matches_crop.jpg}
\includegraphics[width=8cm]{./figures/H8A0vsH8A5_SIFT_3matches_crop.jpg}
\includegraphics[width=8cm]{./figures/H8A0vsH8A5_MserSift_87matches_crop.jpg}
\end{center}
{\small \textit{Transition $\tau \approx 3$. ASIFT (shown) -- 881,
SIFT (shown) -- 3, Harris-Affine  -- 1, Hessian-Affine -- 3, and
MSER (shown) -- 87 correct matches.}} \vspace{1ex}

\begin{center}
\includegraphics[width=12cm]{./figures/MSERimage1_CAffine2sides_MR1_M10_ratio0p5_255match_crop.jpg}
\includegraphics[width=12cm]{./figures/MSERimage1_MserSift_22matches_crop.jpg}
\end{center}
{\small \textit{Transition tilt: $\tau \in [1.6, 3.0]$ (images
proposed by the authors of MSER). ASIFT (shown) -- 254, SIFT--10,
Harris-Affine--23, Hessian-Affine--11 and MSER (shown) -- 22
correct matches.}} \vspace{1ex}

\begin{center}
\includegraphics[width=12cm]{./figures/direction1vs3-AISFT-50matches_bis_crop.jpg}
\includegraphics[width=12cm]{./figures/direction1vs2_MserSift_1matches_crop.jpg}
\end{center}
{\small \textit{Transition tilt: $\tau \approx 2.6$. ASIFT (shown)
-- 50, SIFT -- 0, Harris-Affine -- 0, Hessian-Affine -- 0 and MSER
(shown) -- 1 correct matches.}} \vspace{1ex}


\begin{center}
\includegraphics[width=8cm]{./figures/pentagon10vs11_ASIFT_378matches.jpg}
\includegraphics[width=8cm]{./figures/pentagon10vs11_SIFT_7matches.jpg}
\includegraphics[width=8cm]{./figures/MserSift_pentagon10vs11_17matches.jpg}
\end{center}
{\small \textit{Transition $\tau \in [1.3, \infty)$. ASIFT (shown)
-- 378, SIFT (shown)-- 6, Harris-Affine  -- 2, Hessian-Affine -- 8,
and MSER (shown) -- 17 correct matches.}} \vspace{1ex}


\begin{center}
\includegraphics[width=10cm]{./figures/liberty1vs3_ASIFT_22matches.jpg}
\includegraphics[width=10cm]{./figures/liberty1vs3_SIFT_1matches.jpg}
\end{center}
{\small \textit{Transition $\tau \approx 5.8$. ASIFT (shown) -- 22,
SIFT (shown)-- 16, Harris-Affine  -- 0, Hessian-Affine -- 0, and
MSER -- 0 correct matches.}} \vspace{1ex}

\begin{center}
\includegraphics[width=8cm]{./figures/flag_ASIFT_141matches.jpg}
\includegraphics[width=9.6cm]{./figures/r06_HRASIFT_191matches.jpg}
\end{center}
{\small \textit{Object deformation (images proposed by Ling and
Jacobs). Left: flag. ASIFT (shown) -- 141, SIFT -- 31, Harris-Affine
-- 15, Hessian-Affine -- 10 and MSER -- 2 correct matches. Right:
SpongeBob. ASIFT (shown) -- 370, SIFT -- 75, Harris-Affine -- 8,
Hessian-Affine -- 6 and MSER -- 4correct matches.}} \vspace{1ex}

 \noindent {\Large \color{blue} {References:}}
\begin{itemize}
\item G. Yu and J.M. Morel,  A Fully Affine Invariant Image Comparison Method, \textit{IEEE ICASSP, Taipei},
2009.
\item J.M. Morel and G.Yu,  ASIFT: A New Framework for Fully Affine
Invariant Image Comparison, to appear in \textit{SIAM Journal on
Imaging Sciences}, 2009.
\end{itemize}


\end{multicols}
\vspace{4ex}
\begin{center}
{\Large \color{blue}{\textbf{For more information,}}} \textbf{\Huge
 \color{blue}{G}\color{red}{o}\color{yellow}{o}\color{blue}{g}\color{green}{l}\color{red}{e} \color{red}{ASIFT}}
\end{center}

\end{document}
