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 \begin{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% perque quedi un poc millor

\begin{slide}
\begin{center}

\vspace*{3 cm}

\textbf{\Large \blue Fully affine invariant image comparison} \\
\end{center}
 \vspace*{3 cm}

\begin{center}
Guoshen Yu, Jean-Michel Morel
\end{center}

\addtocounter{slide}{-1} \slidepagestyle{empty}
\end{slide}


\begin{slide} {\blue \large Introduction:}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
%\includegraphics[width=4cm]{./figures/MIT_Maga1_RGB.jpg} &
%\hspace{-2ex} \includegraphics[width=4cm]{./figures/MIT_Maga2_RGB.jpg} \\
\includegraphics[width=9cm]{./figures/MagaMIT_tilt6p37vs0p17.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\vspace{-7ex}
\begin{center} \small{{The same magazine from two viewpoints.}}
\end{center}
%\vspace{-1ex} {Some requirements:}
\vspace{4ex}
\begin{enumerate}
 \item Independence of \textbf{viewpoint}.
 \item Robustness to \textbf{subsampling (zooms)}.
 %\item Invariance to \textbf{illuminance changes}.
% \item Robustness to \textbf{partial
%occlusions}: \textbf{locality}.
\end{enumerate}
\end{slide}

\begin{slide} {\blue \large Introduction:}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=9cm]{./figures/MagaMIT_tilt6p37vs0p17_CAffine2sides_tmax8_tk_sqrt2_MR_fs1_ratio0p4_0p55_94matches.jpg} \\
\end{tabular}
\end{center}
\end{figure}
%\vspace{-1ex} {Some requirements:}
\vspace{-7ex}
\begin{center} \small{{The same magazine from two viewpoints.}}
\end{center}
%\vspace{-1ex} {Some requirements:}
\vspace{4ex}
\begin{enumerate}
 \item Independence of \textbf{viewpoint}.
 \item Robustness to \textbf{subsampling (zooms)}.
 %\item Invariance to \textbf{illuminance changes}.
 %\item Robustness to \textbf{partial
%occlusions}: \textbf{locality}.
\end{enumerate}
\end{slide}



\begin{slide} {\blue \large Camera Model} \vspace{-3ex} \\
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=5cm]{./figures/image_formation.jpg}
\end{tabular}
\end{center}
\end{figure}
\vspace{-3ex}
%\begin{center}
{\small The projective camera model $u=\bS_1 \bG_1 \mathbf{\mathcal A} \bu_0$. \\
- $\mathcal A$ is a planar projective transform (homography) . \\
- $\bG_1$ is an anti-aliasing gaussian filtering. \\
- $\bS_1$ is the CCD sampling. Eliminated from the following
discussion because we can always reinterpolate: $u = \bS_1 \bG_1
\mathbf{\mathcal A} \bu_0 \longrightarrow \bu = \bG_1 \mathcal A
\bu_0$.}
%\end{center}
\end{slide}


\begin{slide} {\blue \large Affine Simplification} \vspace{2ex} \\
If the object's shape is locally smooth, local deformations in
different views can be be approximated by local affine transforms.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=5cm]{./figures/affine_simplification2.jpg}
\end{tabular}
\end{center}
\end{figure}
\vspace{-3ex}
\begin{center}
{\small  Local planar homographies equivalent to \\ multiple local
cameras at infinity.}
\end{center}
\end{slide}

\begin{slide} {\blue \large Camera at Infinity} \\
%$${\scriptsize u(x,y)\to u(ax+by+e, cx+dy+f)} $$
$${\scriptsize\left(\begin{array}{c}x\\y
\end{array} \right) \to
\left[\begin{array}{cc}
a & b  \\
c & d
\end{array} \right]
\left(\begin{array}{c} x\\y
\end{array} \right)
+\left(\begin{array}{c} e\\f\end{array}\right)}$$
 {\scriptsize $$
\hspace{-10ex}\label{decompositionofaffine}
A=\left[\begin{array}{cc}
a & b  \\
c & d
\end{array} \right]=H_\lambda
R_1(\phi)T_tR_2(\psi) = \lambda \left[
\begin{array}{cc}
\cos \phi & -\sin \phi \\
\sin \phi & \cos \phi
\end{array} \right]
\left[
\begin{array}{cc}
t & 0 \\
0 & 1
\end{array} \right]
\left[
\begin{array}{cc}
\cos \psi & -\sin \psi \\
\sin \psi & \cos \psi
\end{array} \right]
$$}


\begin{figure}
\begin{minipage}[b]{0.2\linewidth}
\centering
\includegraphics[width=2cm]{./figures/camera.jpg}
\end{minipage}
\begin{minipage}[b]{0.8\linewidth}
{\begin{itemize}
\item $\psi$: rotation angle of camera around optical axis.
\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 $\phi$: {\it longitude} angle between optical axis and a fixed vertical plane.
\item $\lambda$: \textit{zoom} parameter.
\item $\mathcal T=(e,f)^T$: translation, not presented here.
\end{itemize}}
\end{minipage}
\end{figure}
\end{slide}




\begin{slide} {\blue \large High Transition Tilts}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3.5cm]{./figures/absolute_relative_tilt.jpg}
& \includegraphics[width=4cm]{./figures/relative_tilt_symbol.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\vspace{-2ex}
\begin{itemize}
\item Absolute tilts $t = 1/\cos\theta$: frontal view v.s. slanted view.
\item Transition tilts $\tau(t, t', \phi-\phi')$: slanted view v.s. slanted view.
\end{itemize}
\begin{figure}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=4cm]{./figures/curve_tau_vs_phi_t4.jpg}
\end{minipage}
\begin{minipage}[b]{0.6\linewidth}
{
$\phi=\phi'$, $\tau(t, t', 0) = t/t'.$ \\
$\phi=\phi'+\pi/2$, $\tau(t, t', \pi/2) = tt'.$ \\
      \textbf{$\tau$ can easily be high!} \\
      \newline
}
\end{minipage}
\end{figure}
\end{slide}

\begin{slide} {\blue \large High Transition Tilts}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=10cm]{./figures/relative_tilt_MITMaga.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide} {\blue \large High Transition Tilts}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{ccc}
\hspace{-10ex}
\includegraphics[width=5cm]{./figures/RecogSphere_theta75deg_RelativeTilts.jpg}
&
\hspace{-4ex}\includegraphics[width=3.5cm]{./figures/RecogSphere_theta75deg_tiltmax3.jpg}
&
\hspace{-4ex}\includegraphics[width=3.5cm]{./figures/RecogSphere_theta75deg_tiltmax40.jpg}
\\
\hspace{-10ex}
\includegraphics[width=4cm]{./figures/RecogSphere_theta75deg_RelativeTilts_zenith.jpg}
&
\hspace{-4ex}\includegraphics[width=3cm]{./figures/RecogSphere_theta75deg_tiltmax3_zenith.jpg}
&
\hspace{-4ex}\includegraphics[width=3cm]{./figures/RecogSphere_theta75deg_tiltmax40_zenith.jpg}
\\
{\small $\tau(4, t', \phi-\phi')$} & {\small $\tau(4, t',
\phi-\phi') < 3$} & {\small
$\tau(4, t', \phi-\phi') < 40$} \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide} {\blue \large Affine Invariance: Simulation v.s. Normalization}
\begin{itemize}
\item Simulation.
\begin{itemize}
\item all 6 parameters impossible, e.g. $10^6$.
\end{itemize}
\item Normalization.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=3cm]{./figures/normalization_translation.jpg} &
\includegraphics[width=3cm]{./figures/normalization_rotation.jpg} &
\includegraphics[width=3cm]{./figures/normalization_tilt.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\begin{itemize}
\item Translation and rotation can be normalized.
\item Zoom, latitude and longitude cannot be normalized \textit{stricto sensu}.
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide} {\blue \large Normalization and Commutation} \vspace{-3ex}\\
\begin{itemize}


\item Rotation, translation commute with $\bG_1$ ($\bG_1 \bR = \bR \bG_1$) $\Longrightarrow$
normalizable.

\item Zoom weakly commute  with $\bG_1$ => normalizable after
zoom simulations (SIFT). Indeed, $\bG_1 H_\lambda= H_\lambda
\bG_\lambda$. If $\lambda>1$, one must SIMULATE $\bG_\lambda$
starting from $\bG_1$

\item Directions changes (latitude, longitude) DO NOT COMMUTE with
$\bG_1$ => not normalizable => MSER that does the normalization
anyway is limited in performance.

\item Affine SIFT: SIMULATES direction changes to achieve full
performance.




\item An image operator $\bB$ \textit{weakly commutes} with blur $\bG_1$ if $\bG_1 \bB=\bB'\bG_1$,
where $\bB'$ is of the same class of image operator as $\bB$. These
image operators can be {\it normalized}.
\item Rotations and translations weakly commute with $\bG_1$: they
can be normalized.
\item Zoom, latitude and longitude change \textit{do not} weakly
commute with $\bG_1$: they cannot be normalized \textit{stricto
sensu}. \vspace{-1ex}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=1.2cm]{./figures/graf1_blur_comp.jpg} &
\includegraphics[width=1.2cm]{./figures/graf1_comp_blur.jpg} &
\hspace{1.3ex} \includegraphics[width=1.2cm]{./figures/graf1_comp_blur_absdif_blur_comp.jpg} \vspace{-1ex}\\
$\bu = \bG_1 \bA \bu_0$ & $\bv = \bA \bG_1 \bu_0$ & \hspace{1ex} $|\bu - \bv|$ \\
\end{tabular}
\end{center}
\end{figure}
\vspace{-3ex}
\begin{itemize}
{\small \item $\bu_0$ is a frontal-view image.
\item $\bA$ is a tilt (latitude change).
\item $\bG_1$ is a Gaussian blur.}
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide} {\blue \large State-of-the-art}
\begin{itemize}
\item SIFT (Scale-Invariant Feature Transform) [Lowe 99, 04]:
\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: modest robustness.
\end{itemize}


\item MSER (Maximally Stable Extremal Region) [Matas et al. 02] and LLD
(Level Line Descriptor) [Mus\'e et al. 06]
\begin{itemize}
\item Zoom, latitude, longitude are \textit{normalized} (but only an
approximation: these operators DO NOT weakly commute with blur).
\item Rotation invariant is used (rotation is normalizable).
\item Weakness: small number of features, limited zoom invariance.
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide} {\blue \large State-of-the-art}
\begin{itemize}
\item Other methods: {\small[Baumberg, 00; Tuytelaars and Van Gool,
00, 04; Mikolajczyk and Schmid, 02, 04, 05; Schaffalitzky and
Zisserman, 02; Brown and Lowe, 02, S. Belongie, J. Malik, and J.
Puzicha, 02, Kadir, Zisserman, Brady, 04, Ke and Sukthankar, 04]}

\item Evaluations.
\begin{itemize}
\item {\small [Mikolajczyk and Schmid 03, 05, K. Mikolajczyk, T. Tuytelaars, C. Schmid, A. Zisserman, J. Matas, F. Schaffalitzky, T. Kadir,
and L. Van Gool, 05]}
\begin{itemize}
\item SIFT-based descriptors perform best.
\item MSER outperforms other affine invariant detectors.
\end{itemize}
\item {\small {\it ``none of these approaches are yet
fully affine invariant, as they start with initial feature scales
and locations selected in a non-affine-invariant manner''}.} --- D.
Lowe
\item {\small {\it ``exploring the full affine space''} would have a {\it ``prohibitive
cost''}.} --- D. Lowe
\end{itemize}
\end{itemize}
\end{slide}


\begin{slide} {\blue \large Consistency of SIFT}

\textbf{Lemma}
\begin{itemize}
\item \textbf{Rotation and Translation Invariance} \\
For any rotation $\bR$  and any translation
   $\mathcal T$, the SIFT descriptors of
  $\bG_1\bR\mathcal T\bu_0$
   are identical to those of $\bG_1\bu_0.$

\item \textbf{Scale Invariance} \\
For any rotation $\bR$, translation $\mathcal T$ and homothety
$\bH_\lambda $, the SIFT descriptors of  $v=\bG_1\bH_\lambda
\bR\mathcal T\bu_0$ are similar to those of $u=\bG_\delta\bu_0$, as
soon as the scales of the SIFs in $u$ are significantly larger than
$\lambda$.
   \end{itemize}
\end{slide}

\begin{slide} {\blue \large Affine-SIFT (A-SIFT) Overview}
\begin{itemize}
\item Simulate latitude (tilt), longitude.
\item Simulated images are compared by a rotation-, translation- and zoom-invariant algorithm, e.g., SIFT. (SIFT normalizes translation and rotation and simulates zoom.)
\end{itemize}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=7cm]{./figures/brevet_fig5.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide} {\blue \large A-SIFT Algorithm} \\
$$ \bu = \bG_{1} \bR_{1} \bT_{t}
\bH_{\lambda} \bR_{2} \mathcal T\bu_0~\textrm{and}~\bv = \bG_{1}
\bR_1' \bT_{t'} \bH_{\lambda'} \bR_2' \mathcal T'\bu_0 $$
\begin{enumerate}
\item Change of camera axis  on $\bu$.
\begin{enumerate}
\item Simulate rotations $\tilde{\bR}_1$ on $\bu$ and obtain
$\bu_{\tilde{\phi}}$.
\item Simulate tilts $\tilde{\mathbb T}_{\tilde{t}}=:\tilde{\bT}_{\tilde{t}} \bG_{\sqrt{{\tilde{t}}^2-1}}\ast_1$
on $\bu_{{\tilde{\phi}}}$ and obtain
$\bu_{{\tilde{\phi}},{\tilde{t}}}$.
$$\bu_{{\tilde{\phi}},{\tilde{t}}}  = \bG_1
\tilde{\bT}_{\tilde{t}} \tilde{\bR}_1 \bR_{1} \bT_{t} \bH_{\lambda}
\bR_{2}\mathcal T \bu_0.$$
\end{enumerate}
\item Similar camera motion simulation on $\bv$:
$$\bv_{{\tilde{\phi}}',{\tilde{t}}'} = \bG_1 \tilde{\bT}_{{\tilde{t}}'}
\tilde{\bR}_1' \bR_{1}' \bT_{t'} \bH_{\lambda'} \bR_{2}'\mathcal T'
\bu_0.$$
\item Compare $\bu_{{\tilde{\phi}},{\tilde{t}}}$ with $\bv_{{\tilde{\phi}}',{\tilde{t}}'}$ with
SIFT.
\end{enumerate}
\end{slide}

\begin{slide} {\blue \large Tilt Correction} \\
\begin{itemize}
\item Tilt = 1D blur + 1D subsampling.
$$\mathbb T_t^x =\bS_1^x \bG_{\sqrt{t^2-1}}\ast_1^x.$$
\textbf{Guoshen: It seems that this is the definition of a
``digital'' tilt. The tilted version of $\bu_0$, an image of
infinite resolution without blur, seems to be simply $\bS_t^x \bu_0$
(not $\bS_t^x \bG_t^x \bu_0$, otherwise I have difficulty to deduce
the following ``pseudo inverse'') Am I correct?}
\item Tilt cannot be inverted because of the 1D-blur.
\item But there is a pseudo inverse: ($\bu = \bG_1 \bS_t^y
\bu_0$, $\bv = \bG_1 \bu_0$)
$$\mathbb T_t^x \bu = \mathbb T_t^x \bG_1 \bS_t^y
\bu_0 = \bS_1^x \bG_{\sqrt{t^2-1}}^x \bG_1^x \bG_1^y \bS_t^y \bu_0 =
\bS_1^x \bG_t^x \bS_t^y \bG_t^y \bu_0 = \bS_t \bG_t \bu_0.
$$
The result of the pseudo inversion is a $t$ zoom out of $\bv$. \\
\textbf{Guoshen: in the A-SIFT paper, page 17, lemma 3, we show the
tilt correction in another way: there we CORRECT the tilt by making
an inverse tilt $\bT_{1/t}$ and then convolving the image in the
orthogonal direction to make the blur isotropic. The resulting image
is $\bG_t \bu_0$, a more blurred version of $\bG_1 \bu_0$. What is
the relation between these two versions (``pseudo inverse'' and
``tilt correction'')? It seems that the former is more exact than
the latter, as the latter generates a more blurred version. (SIFT is
not invariant to blur.) Am I right?}
\end{itemize}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=7cm]{./figures/tilt_pseudo_inverse.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}


\begin{slide}
\textbf{\large \blue Consistency of A-SIFT} \\
\begin{theorem}
A-SIFT, applied with a dense set of latitudes and longitudes,
simulates two views of $\bu$ and $\bv$ 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}
\vspace{-3ex}
\paragraph{Proof}
$$\bu_{{\tilde{\phi}},{\tilde{t}}}  = \bG_1 \tilde{\bT}_{\tilde{t}}
\tilde{\bR}_1 \bR_{1} \bT_{t} \bH_{\lambda} \bR_{2}\mathcal T
\bu_0,$$
$$\bv_{{\tilde{\phi}}',{\tilde{t}}'} = \bG_1
\tilde{\bT}_{{\tilde{t}}'} \tilde{\bR}_1' \bR_{1}' \bT_{t'}
\bH_{\lambda'} \bR_{2}'\mathcal T' \bu_0$$ are SIFT-matched if
$$\tilde{\bT}_{{\tilde{t}}} \tilde{\bR}_1 \bR_{1} \bT_{t}  =
\bH_{\mu}~~\textrm{and}~~ \tilde{\bT}_{{\tilde{t}}'} \tilde{\bR}_1'
\bR_{1}' \bT_{t'} = \bH_{\mu'},~\textrm{i.e., if}$$
$$
\bu_{{\tilde{\phi}},{\tilde{t}}} = \bG_1 \bH_{\mu \lambda} \bR_{2}
\mathcal T\bu_0~~\textrm{and}~~\bv_{{\tilde{\phi}}',{\tilde{t}}'} =
\bG_1 \bH_{\mu' \lambda'} \bR_{2}'\mathcal T' \bu_0,
$$
which means that $\bu_{{\tilde{\phi}},{\tilde{t}}}$ and
$\bv_{{\tilde{\phi}}',{\tilde{t}}'}$ are identical up to similarity
+ gaussian blur (just what SIFT likes).
\end{slide}

\begin{slide} {\large \blue Parameter Sampling Range}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=2.5cm]{./figures/camera.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\begin{itemize}
\item Longitude angle $\phi \in [0, \pi)$.
\begin{itemize}
\item $R_1(\phi+\pi) T_t R_2(\psi) = R_1(\phi) T_t R_2(\psi+\pi)$.
\end{itemize}
\item Tilt $t= 1/ \cos \theta \in [1, t_{\max}]$.
\begin{itemize}
\item Physical limitation: planar and Lambertian, structures are not squashed.
\item $t_{\max} = 4 \sqrt{2}$ obtained experimentally.
\item The resulting $\tau_{\max}=32$.
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide}  {\large \blue Parameter Sampling Range}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-8ex} \includegraphics[width=12cm]{./figures/lepoint1vs3_tilt3_CAffine2sides_Tmax2sqrt2_tk_sqrt2_MR_107match.jpg} \vspace{-3ex}\\
t = 3 ($\theta=70.5^\circ$), 107 A-SIFT matches (3 false). \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide}  {\large \blue Parameter Sampling Range}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-8ex}  \includegraphics[width=12cm]{./figures/lepoint1vs5_tilt5_CAffine2sides_Tmax2sqrt2_tk_sqrt2_MR_25match.jpg} \vspace{-3ex} \\
t = 5.2 ($\theta=78.9^\circ$), 25 A-SIFT matches (7 false). \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide}  {\large \blue Parameter Sampling Range}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-8ex}  \includegraphics[width=12cm]{./figures/lepoint1vs6_tilt8_CAffine2sides_Tmax2sqrt2_tk_sqrt2_MR_7match_fail.jpg} \vspace{-3ex} \\
t = 8 ($\theta=82.8^\circ$), 7 A-SIFT matches (all false). \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}


\begin{slide}  {\large \blue Parameter Sampling Range}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-8ex} \includegraphics[width=12cm]{./figures/Xlab1vs14_CAffine2_71match.jpg} \vspace{-3ex}\\
t = 3.8 ($\theta=74.7 ^\circ$), 71 A-SIFT matches (4 false). \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide}  {\large \blue Parameter Sampling Range}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-8ex}  \includegraphics[width=12cm]{./figures/outdoorX1vs15_CAffine2sides_tmax2sqrt2_tk_sqrt2_MR_33match.jpg} \vspace{-3ex} \\
t = 5.6 ($\theta=79.7^\circ$), 33 A-SIFT matches (4 false). \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide}  {\large \blue Parameter Sampling Range}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-8ex}  \includegraphics[width=12cm]{./figures/outdoorX1vs16_CAffine2sides_tmax2sqrt2_tk_sqrt2_MR_8match_fail.jpg} \vspace{-3ex} \\
t = 8 ($\theta=82.8^\circ$), 8 A-SIFT matches (all false). \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide} {\large \blue Parameter Sampling Step: $\triangle t$}

\begin{figure}
\begin{minipage}[b]{0.2\linewidth}
\centering
\includegraphics[width=2cm]{./figures/camera.jpg}
\end{minipage}
\begin{minipage}[b]{0.8\linewidth}
{\begin{itemize}
\item $t = 1/\cos\theta$, $\theta$ is the latitude angle.
\item $\theta$ sampled with higher precision when $\theta \rightarrow
90^{\circ}$.
\item Geometric sampling of $t$: $\triangle t = t_{k+1} / t_k$.
\item $\triangle t = \sqrt{2}$ is obtained experimentally: \\ compare
$\bu = \mathbf{T}_{t_1} \bu_0$ and $ \bv = \mathbf{T}_{t_2} \bu_0$
with SIFT.
\end{itemize}}
\end{minipage}
\end{figure}
\vspace{-3ex}
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=4cm]{./figures/CSIFT_longitude_sampling_rate_tmax8.jpg}
&
\includegraphics[width=4cm]{./figures/CSIFT_longitude_sampling_rate_1over4_grid_tmax8.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\vspace{-5ex}
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
{ \scriptsize t} & \scriptsize 1 & \scriptsize \sqrt{2} & \scriptsize 2 & \scriptsize 2\sqrt{2} & \scriptsize 4 & \scriptsize 4\sqrt{2} \\
\hline { \scriptsize \theta} & { \scriptsize 0^{\circ}} & {
\scriptsize 45^{\circ}} & { \scriptsize 60^{\circ}} & { \scriptsize 69.3^{\circ}} & { \scriptsize 75.5^{\circ}} & { \scriptsize 79.8^{\circ}}  \\
\hline
\end{array}
$$
\end{slide}


\begin{slide} {\large \blue Parameter Sampling Step: $\triangle \phi$}

\begin{figure}
\begin{minipage}[b]{0.2\linewidth}
\centering
\includegraphics[width=2cm]{./figures/camera.jpg}
\end{minipage}
\begin{minipage}[b]{0.8\linewidth}
{\begin{itemize}
\item $\phi$: longitude angle.
\item $\phi$ sampled with higher precision when $\theta \rightarrow
90^{\circ}$: \\
$t \uparrow \Rightarrow \triangle \phi \downarrow$.
\item Arithmetical sampling of $\phi$: $\triangle \phi = \phi_{k+1} - \phi_k$.
\item $\triangle \phi =
2\times \frac{36^{\circ}}{t} = \frac{72^{\circ}}{t}$ is obtained
experimentally: \\ compare $\bu= \mathbf{T}_t \mathbf{R}_1(\phi)
\bu_0$ and $\bv=\mathbf{T}_t \bu_0$ with SIFT.
\end{itemize}}
\end{minipage}
\end{figure}
\vspace{-3ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=6cm]{./figures/longitude_sampling_graf_tilt_compression_0p2.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide} {\large \blue Parameter Sampling}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\hspace{-3ex}
\includegraphics[width=5cm]{./figures/brevet_fig2b.jpg} &
\includegraphics[width=5cm]{./figures/brevet_fig4b_english1.jpg} \\
Perspective view & View from the zenith\\
\end{tabular}
\end{center}
\end{figure}

\end{slide}

\begin{slide} {\large \blue Acceleration: Multi-resolution A-SIFT}
\begin{enumerate}
\item A-SIFT on low-resolution images ($r\times r$ sub-sampled) .
\item A-SIFT on high-resolution images obtained with the M best affine
transforms.
\end{enumerate}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=1cm]{./figures/77Mass_LRCAffine_26matches.jpg}
&
\includegraphics[width=3cm]{./figures/77Mass_HRCAffine_329matches.jpg}
\\
\end{tabular}
\end{center}
\end{figure}

\end{slide}

\begin{slide} {\large \blue A-SIFT Complexity}
\begin{itemize}
\item Complexity $\sim$ number of simulated tilts.
\begin{itemize}
\item SIFT Complexity $\sim$ simulated image surface.
\item $t = {2}^{k/2}$,~~$k=0, \ldots, K$.
\item $\phi \in [0^{\circ}, 180^{\circ})$,~~$\triangle \phi =
\frac{72^{\circ}}{t}$: $|\{\phi(t)\}| \sim t$.
\item At tilt $t$, image surface $\sim 1/t$.
\end{itemize}
\item Example: $t_{\max}=4\sqrt{2}$ (i.e. $K=5$), $r \times r = 3 \times
3$ subsampling, $M=5$ good affine transforms.
\begin{itemize}
\item Affine identification (A-SIFT on low-resolution images)
$$ \frac{1+K\frac{180^{\circ}}{72^{\circ}}}{r^2} = 1.5 \times \textrm{SIFT}$$
\item Full-resolution A-SIFT: $\left\{ \begin{array}{cc} \sim 2 \times \textrm{SIFT} & \textrm{if images match} \\ 0 & \textrm{if images do not match} \end{array} \right.$
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide} {\large \blue Experiments: Image Matching}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-4ex}  \includegraphics[width=10cm]{./figures/fig1.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}


\begin{slide} {\large \blue Experiments: Image Matching}
\vspace{8ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-4ex}  \includegraphics[width=8cm]{./figures/Adam1.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}

\begin{slide} {\large \blue Experiments: Image Matching}
\vspace{-2ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-5ex}  \includegraphics[width=10cm]{./figures/fig1c_v3.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\vspace{-4ex}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\hspace{-5ex}  \includegraphics[width=9.5cm]{./figures/Adam_results.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\end{slide}


\begin{slide}  {\large \blue Experiments}
\vspace{-1ex}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_45deg.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_45deg_SIFT.jpg}
\\
\end{tabular}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_45deg_MSER1.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_45deg_MSER2.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\vspace{-3ex} {\scriptsize Frontal v.s. $-45^\circ$ angle, zoom
$\times 1$: absolute tilt $t=2$ (middle), $t\prec2$ (left part),
$t\succ2$ (right part). A-SIFT (top left): 562 matches (3 false).
SIFT (top right): 246 matches (1 false). MSER (bottom): 29 matches
(16 false).}
\end{slide}

\begin{slide}  {\large \blue Experiments: Image Matching}
\vspace{-1ex}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_77degR.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_77degR_SIFT.jpg}
\\
\end{tabular}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_77degR_MSER1.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom1_77degR_MSER2.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\vspace{-3ex} {\scriptsize Frontal v.s. $75^\circ$ angle, zoom
$\times 1$:absolute tilt $t=4$ (middle), $t\prec4$ (right part),
$t\succ4$ (left part). A-SIFT (top left): 203 matches (0 false).
SIFT (top right): 15 matches (0 false). MSER (bottom): 10 matches (3
false).}
\end{slide}


\begin{slide}  {\large \blue Experiments: Image Matching}
\vspace{-1ex}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom10_65degR.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom10_65degR_SIFT.jpg}
\\
\end{tabular}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom10_65degR_MSER1.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom10_65degR_MSER2.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\vspace{-3ex} {\scriptsize Frontal v.s. $65^\circ$ angle, zoom
$\times 10$: absolute tilt $t=2.4$. A-SIFT (top left): 469 matches
(1 false). SIFT (top right): 5 matches (0 false). MSER (bottom): 9
matches (0 false).}
\end{slide}

\begin{slide}  {\large \blue Experiments: Image Matching}
\vspace{-1ex}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom10_84deg.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_front-adam_zoom10_84deg_SIFT.jpg}
\\
\end{tabular}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/adam_zoom10_front.jpg}
&
\includegraphics[width=3cm]{./figures/adam_zoom10_84deg.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\vspace{-3ex} {\scriptsize Frontal v.s. $80^\circ$ angle, zoom
$\times 10$: absolute tilt $t=5.6$. A-SIFT (top left): 118 matches
(116 false). SIFT (top right): 2 matches (1 false). MSER (bottom): 0
match.}
\end{slide}



\begin{slide} {\large \blue Experiments: Image Matching}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=8cm]{./figures/MagaMIT_tilt6p37vs0p17.jpg}
\vspace{-2ex}
\\
\includegraphics[width=8cm]{./figures/MagaMIT_tilt6p37vs0p17_CAffine2sides_tmax8_tk_sqrt2_MR_fs1_ratio0p4_0p55_94matches.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\vspace{-4ex} {\scriptsize Top: Image pair with absolute tilts
$t=6.4$ (left) and $t'=5.9$ (right), transition tilt $\tau=37$.
Bottom: A-SIFT finds 120 matches out which 4 are false. (SIFT and
MSER fail completely.)}
\end{slide}


\begin{slide}  {\large \blue Experiments: Image Matching}
\begin{figure}
\begin{minipage}[b]{0.7\linewidth}
\begin{center}
\begin{tabular}{c}
%\includegraphics[width=14cm]{./figures/H1A1vsH1A2_CAffine_271match.jpg}
\includegraphics[width=7cm]{./figures/H1A1vsH1A2_CAffine2_ratio0p4_1667match.jpg}
\vspace{-2ex}
\\
\includegraphics[width=7cm]{./figures/xpassionH1A1vsH1A2_SIFT_3match.jpg} \vspace{-1.5ex}
\\
\end{tabular}
\begin{tabular}{cc}
\hspace{.4ex}
\includegraphics[width=3cm]{./figures/H1A1vsH1A2_H1A1.jpg} &
\hspace{-2.7ex}
\includegraphics[width=3cm]{./figures/H1A1vsH1A2_H1A2.jpg}
\\
\end{tabular}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
{{\scriptsize Absolute tilts $t=t'=2.1$. transition tilt:
$\tau=3.0$. \vspace{5ex}
\\
Top: A-SIFT finds 1667 correspondences, all correct.\vspace{3ex}
\\
Middle: SIFT finds 3 correspondences. \vspace{3ex} \\
Bottom: MSER finds 46 correspondences, out of which 35 are
correct.}}
\end{minipage}
\end{figure}
\end{slide}

\begin{slide}  {\large \blue Experiments: Image Matching}
\begin{figure}
\begin{minipage}[b]{0.7\linewidth}
\begin{center}
\begin{tabular}{c}
%\includegraphics[width=14cm]{./figures/H1A1vsH1A2_CAffine_271match.jpg}
\includegraphics[width=7cm]{./figures/H1A1vsH3A1_CAffine2_ratio5p5_388match.jpg}
\vspace{-2ex}
\\
\includegraphics[width=7cm]{./figures/xpassionH1A1vsH3A1_SIFT_5match.jpg} \vspace{-1.5ex}
\\
\end{tabular}
\begin{tabular}{cc}
\hspace{.4ex}
\includegraphics[width=3cm]{./figures/H1_A1.jpg} &
\hspace{-2.7ex}
\includegraphics[width=3cm]{./figures/H3_A1.jpg}
\\
\end{tabular}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
{{\scriptsize Absolute tilts $t=2.1$ (left), $t'=6.0$ (right). transition tilt: $\tau=2.9$. \vspace{1ex} \\
Top: A-SIFT finds 338 correspondences, out of which 2
are false. \vspace{1ex} \\
Middle: SIFT finds 5 correspondences.  \vspace{1ex} \\
Bottom: MSER finds 3 false correspondences in total that have been
rejected.}}
\end{minipage}
\end{figure}
\end{slide}


\begin{slide}  {\large \blue Experiments: Image Matching}
\begin{figure}
\begin{minipage}[b]{0.7\linewidth}
\begin{center}
\begin{tabular}{c}
%\includegraphics[width=14cm]{./figures/H1A1vsH1A2_CAffine_271match.jpg}
\includegraphics[width=7cm]{./figures/Office1vs8_CAffine2_62match.jpg}
\vspace{-2ex}
\\
\includegraphics[width=7cm]{./figures/office1vs8_SIFT_0match.jpg} \vspace{-1.5ex}
\\
\end{tabular}
\begin{tabular}{cc}
\hspace{.4ex}
\includegraphics[width=3cm]{./figures/office1vsoffice8_office1.jpg} &
\hspace{-2.7ex}
\includegraphics[width=3cm]{./figures/office1vsoffice8_office8.jpg}
\\
\end{tabular}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
{{\scriptsize Transition tilt: $\tau \approx 3$. \vspace{5ex} \\
 Top: A-SIFT finds
62 correspondences, out of which 4
are false.  \vspace{6ex} \\
Middle: SIFT finds 0 correspondences.  \vspace{6ex} \\
Bottom: MSER finds 13 correspondences, out of which 2 are correct.}}
\end{minipage}
\end{figure}
\end{slide}

\begin{slide}  {\large \blue Experiments: Image Matching}
\begin{figure}
\begin{minipage}[b]{0.7\linewidth}
\begin{center}
\begin{tabular}{c}
%\includegraphics[width=14cm]{./figures/H1A1vsH1A2_CAffine_271match.jpg}
\includegraphics[width=7cm]{./figures/Graf1vs6_CAffine2_724match.jpg}
\vspace{-5ex}
\\
\includegraphics[width=7cm]{./figures/graf1vs6_SIFT_6match.jpg} \vspace{-2.5ex}
\\
\end{tabular}
\begin{tabular}{cc}
\hspace{.4ex}
\includegraphics[width=3cm]{./figures/graf1vsgraf6_graf1.jpg} &
\hspace{-2.7ex}
\includegraphics[width=3cm]{./figures/graf1vsgraf6_graf6.jpg}
\\
\end{tabular}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
{{\scriptsize Transition tilt: $\tau \approx 3.2$. \vspace{5ex} \\
Top: A-SIFT finds 724 correspondences, out of which 3
are false.  \vspace{5ex} \\
Middle: SIFT finds 6 correspondences.   \vspace{5ex} \\
Bottom: MSER finds 127 correspondences, out of which 50 are
correct.}}
\end{minipage}
\end{figure}
\end{slide}


\begin{slide}  {\large \blue Experiments: Image Matching}
\begin{figure}
\begin{minipage}[b]{0.7\linewidth}
\begin{center}
\begin{tabular}{c}
%\includegraphics[width=14cm]{./figures/H1A1vsH1A2_CAffine_271match.jpg}
\includegraphics[width=7cm]{./figures/MSERimage1_CAffine2sides_MR1_M10_ratio0p5_255match.jpg}
\vspace{-5ex}
\\
\includegraphics[width=7cm]{./figures/MSERimage1_SIFT_16match.jpg} \vspace{-2.5ex}
\\
\end{tabular}
\begin{tabular}{cc}
\hspace{.4ex}
\includegraphics[width=3cm]{./figures/MSER_img2.jpg} &
\hspace{-2.7ex}
\includegraphics[width=3cm]{./figures/MSER_img1.jpg}
\\
\end{tabular}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
{{\scriptsize Top: A-SIFT finds 255 matches out of which 1 is false. \vspace{5ex} \\
Middle: SIFT finds 16 matches out of which 6 are false.    \vspace{5ex} \\
Bottom: MSER finds 70 tentative correspondences out of which there
are 51 inliers.}}
\end{minipage}
\end{figure}
\end{slide}

\begin{slide} {\large \blue Experiments: Video Object Tracking}


\begin{figure}[htbp]
\begin{center}
\begin{tabular}{ccc}
\hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_lepoint-000000001.jpg}
& \hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_lepoint-000000081.jpg}
& \hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_lepoint-000000161.jpg}
\vspace{-2ex}\\
\hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_SIFT_lepoint-000000001.jpg}
& \hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_SIFT_lepoint-000000081.jpg}
& \hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_SIFT_lepoint-000000161.jpg}
\vspace{-2ex}\\
\hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_lepoint-000000301.jpg}
& \hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_lepoint-000000421.jpg}
& \hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_lepoint-000000481.jpg} \vspace{-2ex}\\
\hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_SIFT_lepoint-000000301.jpg}
& \hspace{-7ex}
\includegraphics[width=4cm]{./figures/tracking_SIFT_lepoint-000000421.jpg}
& \hspace{-7.7ex}
\includegraphics[width=4cm]{./figures/tracking_SIFT_lepoint-000000481.jpg} \\
\end{tabular}
\end{center}
\end{figure}
\vspace{-4ex} {\scriptsize Object tracking: one looks for
correspondences between video frames. First and third row: A-SIFT
succeeds. Second and fourth row: SIFT fails.}
\end{slide}

\begin{slide} {\large \blue Symmetry Detection in Perspective} \\
\vspace{-3ex}
\begin{itemize}
\begin{figure}
\begin{minipage}[b]{0.6\linewidth}
{\item Symmetry detection  = image comparison with the flipped
version.
\item Symmetric object not in frontal view $\Rightarrow$ a viewpoint
change between the flipped images.}
\end{minipage}
\begin{minipage}[b]{0.4\linewidth}
\begin{center}
\begin{tabular}{cc}
\vspace{15ex}
\includegraphics[width=2cm]{./figures/RedAdmiralS.jpg} &
\hspace{-3ex}
\includegraphics[width=2cm]{./figures/RedAdmiralS1.jpg}
\\
\end{tabular}
\end{center}
\end{minipage}
\end{figure}
\vspace{-15ex}
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=3cm]{./figures/Admiral_sym_CAffine_v2_crop.jpg} &
\hspace{-3ex}
\includegraphics[width=2cm]{./figures/ButterflyBK_sym_CAffine_v2_crop.jpg}
\\
\end{tabular}
\end{center}
\end{figure}
\vspace{-4ex}
\begin{itemize}
\item {\scriptsize SIFT fails: big viewpoint change.}
\item {\scriptsize MSER fails: two wings confused.}
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\vspace*{2 cm} \hspace{-3ex} \textbf{\Large \blue \textit{Conclusion: Affine SIFT is A Fine SIFT}} \\
\vspace*{1cm} \vspace*{1cm}
\textbf{\Large \blue  Questions? Comments?} \\
\end{center}
\end{slide}

\end{document}
