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\title{A fully affine invariant image comparison method}
\twoauthors {Jean-Michel Morel\sthanks{We thank ONR and CNES for
their support.}}
    {CMLA, ENS Cachan, 61 av. du\\
     President Wilson, Cachan 94235, France}
  {Guoshen Yu}
   {CMAP, Ecole Polytechnique, \\
   91128 Palaiseau Cedex, France}
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\begin{abstract}
A fully affine invariant image comparison method, Affine SIFT
(A-SIFT) is described and tested. While SIFT is fully invariant with
respect to only four parameters, the new method adds up the two left
over parameters : the angles (a longitude and a latitude) 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 transition tilts of large magnitude, up to 40
and more, while state-of-the-art affine normalization methods hardly
exceed transition tilts of 2.5 (SIFT), 5 (Harris Affine) and 10
(MSER-SIFT).
\end{abstract}

\begin{keywords} Scale invariance, Affine invariance, SIFT method, transition tilt, absolute
tilt.
 \end{keywords}
 \vspace{-2ex}
 \section{Introduction}
\vspace{-2ex} Local image detectors used for image comparison can be
classified by their incremental invariance properties. All of them
are translation invariant. The Harris point
detector~\cite{harris88combined} is also rotation invariant. The
Harris-Laplace, Hessian-Laplace and the DoG (Difference-of-Gaussian)
region detectors~\cite{MikolajczykS01,
MikolajczykS04,lowe04:_distin, Fevrier07} are invariant to rotations
and changes of scale. Some moment-based region
detectors~\cite{DBLP:conf/eccv/LindebergG94, baumberg00:_reliab}
including the Harris-Affine and Hessian-Affine region
detectors~\cite{MikolajczykS02, MikolajczykS04}, an edge-based
region detector~\cite{TuytelaarsG04}, an entropy-based region
detector~\cite{KadirZB04}, and two level line-based region detectors
MSER (``maximally stable extremal region'')~\cite{matas02mser} and
LLD (``level line descriptor'')~\cite{muse04:_acontrario} are
designed to be invariant to affine transformations. MSER, in
particular, has been demonstrated to have often better performance
than other affine invariant detectors, followed by Hessian-Affine
and Harris-Affine~\cite{mikolajczyk05, MikolajczykS01,
MikolajczykS04}. These  methods proceed by {\it normalizing} local
patches,  regions, or  level lines that have undergone an unknown
affine transform. Normalization transforms them into a standard
object, where the effect of the affine transform has been
eliminated.
   However, when a strong change of scale is present (in practice larger than 3), SIFT still beats all other
methods~\cite{lowe04:_distin}. Indeed, as proved
mathematically~\cite{GuoshenMorelSIFT}, SIFT is fully scale
invariant and, as pointed out in~\cite{lowe04:_distin} none of the
normalization methods is fully scale or affine invariant:
   {\it However, 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 due to the
prohibitive cost of exploring the full affine space. }

\vspace{-2ex}

\begin{figure}[htbp]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=7cm]{./figures/MagaMIT_tilt6p37vs0p17.jpg}
\vspace{-3ex}
\\
\includegraphics[width=7cm]{./figures/MagaMIT_tilt6p37vs0p17_CAffine2sides_tmax8_tk_sqrt2_MR_fs1_ratio0p4_0p55_94matches.jpg}
\vspace{-5ex}
\\
\end{tabular}
\end{center}
\caption{Image pair with high transition tilt $t \approx 37$.
Bottom: A-SIFT finds 116 correct matches out of 120. SIFT,
Harris-Affine, Hessian-Affine, and MSER fail completely.}
\label{fig:numeric:CSIFT:extreme:tilt} \vspace{-2ex}
\end{figure}


\section{The affine camera model\label{AFastpresentation}}
\vspace{-1ex}
Image distortions arising from viewpoint changes can
be locally modeled by  affine  planar transforms, provided the
object's boundaries are piecewise smooth~\cite{mikolajczyk05}. Thus,
the (local) image deformation model under a camera motion is
$$u(x,y)\to u(ax+by+e, cx+dy+f),$$
where the mapping $A\left(\begin{array}{c}x\\y
\end{array} \right) =
\left[\begin{array}{cc}
a & b  \\
c & d
\end{array} \right]
\left(\begin{array}{c} x\\y
\end{array} \right)$ is any linear planar map with positive
 singular values.
\begin{figure}[htbp]
\begin{center}
\vspace{-3ex}
\includegraphics[width=2.3cm]{./figures/camera.jpg}
\vspace{-3ex}
\end{center}
\caption{ Geometric interpretation of formula
\eqref{decompositionofaffine}.} \label{fig:decomp:RTR} \vspace{-2ex}
\end{figure}
 Any such  map has a {\it unique}  decomposition
{\small \begin{equation} \label{decompositionofaffine}A=  \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]
\end{equation}}
 which we note $A=\lambda R(\psi)T_tR(\phi),$ where $\lambda>0$,  $\lambda t$ is the determinant of $A$,
 $\phi\in [0, 180^\circ[$,  $R(\psi)$ denotes the planar rotation with angle $\psi$, and $T_t$ ($t\geq 1$)
 is called the {\it  tilt}. Fig.~\ref{fig:decomp:RTR}
shows a camera motion interpretation of
\eqref{decompositionofaffine}: $\phi$ and $\theta = \arccos 1/t$ are
the camera viewpoint angles and $\psi$ parameterizes the camera
spin. In this affine model the camera stands far away from a planar
object. Starting from a frontal position, a camera motion parallel
to the object's plane induces an image translation. The camera can
rotate around its optical axis inducing an image rotation $R(\psi$).
In oblique views, the camera optical axis takes a $\theta$ angle
with respect to the normal to the object's plane, which
  we call {\it latitude}. The tilt $t\geq 1$ is defined by $t\cos\theta=1$. The plane containing the normal
to the object's plane and the new position of the optical axis makes an angle $\phi$ with a fixed vertical plane. This
angle is called {\it longitude}. Last but not least, the camera can move forward or backward. This induces an image
zoom (actually a blur) with scale parameter $\lambda$. In short,  \eqref{decompositionofaffine} models the image
deformation
 $\bu(x,y)\to
\bu(A(x,y))$ induced by a camera motion from a frontal view $\lambda_0=1$, $t_0=1$, $\phi_0=\psi_0=0$ to an oblique
view characterized by $\lambda, t, \phi, $ and $\psi$.

\vspace{-3ex}
\section{High transition tilts\label{HighTilts}}
\vspace{-1ex} Equation \eqref{decompositionofaffine} defines the
{\it absolute tilt}, namely the image deformation ratio when the
camera passes from a frontal view to an oblique view.
 But the compared images $\bu_1(x,y)= \bu(A(x,y))$ and $\bu_2(x,y)=\bu(B(x,y))$
  are in general obtained from {\it two} oblique camera positions.
\begin{figure}[htbp]
\begin{center}
\vspace{-2ex}
\includegraphics[width=7.2cm]{./figures/brevet_fig1_english.jpg}
\vspace{-3ex}
\end{center}
\caption{Illustration of the difference between \textit{absolute}
tilt and \textit{transition} tilt.} \label{fig:tilt:transition}
\vspace{-3ex}
\end{figure}
\begin{definition}  \label{decompositionofaffinerelative}Given two  views of a planar image, $\bu_1(x,y)=\bu(A(x,y))$
 and $\bu_2(x,y)=\bu(B(x,y))$,
we call transition tilt $\tau (\bu_1,\bu_2)$  and transition
rotation $\phi(\bu_1,\bu_2)$   the unique  parameters such that
$BA^{-1}=H_\lambda R_1(\psi)T_\tau R_2(\phi),$
 with the notation of Formula \eqref{decompositionofaffine}.
\end{definition}
  Fig.~\ref{fig:tilt:transition} illustrates the
affine transition between two images taken from different
viewpoints, and in particular the difference between absolute tilt
and transition tilt. With the two tilts $t$ and $t'$ made in two
orthogonal directions $\phi = \phi'+{\pi/2}$, one can verify that
the transition tilt between $\bu_1$ and $\bu_2$ is the product $\tau
= tt'$. Thus, {\it two moderate absolute tilts can lead to a large
transition tilt!} Since in realistic cases the tilt can go up to 6
or even 8, it  is easily understood that the {\it transition tilt}
can go up to 36, 64, and more.
Fig.~\ref{fig:numeric:CSIFT:extreme:tilt} shows the A-SIFT results
for an image pair under orthogonal viewpoints (transition rotation
$\phi=90^\circ$, absolute tilt $t\simeq 6$) that leads to a
transition tilt $\tau \approx 37$. This is not at all an exceptional
situation.
 The relevance of the notion of transition
 tilt is  corroborated by the fact that the highest transition tilt $\tau_{\max}$ permitting to match
 two images with absolute tilts $t$ and $t'$ is fairly independent from $t$ and $t'$.
 It has been experimentally checked that  SIFT works up to $\tau_{\max}\simeq 2.5$. The attainable absolute
  tilts for Harris Affine and Hessian Affine are also close to 2.5, which allows them to attain transition tilts
  close to $6$.   For MSER the absolute tilt nears  $t_{\max}\simeq 4$, but {\it only in
  well-contrasted images with comparable scales}. Thus, MSER attains transition tilts
  close to 16, but only in the best cases. Usually it
  finds matches for transition tilts lower than 10.
  A-SIFT attains regularly transition tilts larger than 40, and matches images in cases far beyond human performance
  (see Fig. \ref{fig:numeric:CSIFT:extreme:tilt}).

\vspace{-2ex}
\section{The A-SIFT algorithm\label{thealgorithm}}
\vspace{-1ex}
  The idea of combining simulation and normalization is  the main
successful ingredient of the SIFT method. Indeed, scale changes amount to blur and cannot be normalized.  Thus SIFT
normalizes rotations and translations, but simulates all zooms out of the query and of the search images.
  David Pritchard's extension of SIFT \cite{pritchard2003cmc} is actually a first step toward  the method
developed here. This author simulated four additional tilts to get
more affine invariance. As illustrated in
Fig.~\ref{fig:CAffine:algo}, The A-SIFT algorithm that simulates two
additional parameters (longitude and tilt) is now
 described.
\begin{figure}[htbp]
\begin{center}
\vspace{-2ex}
\begin{tabular}{c}
\includegraphics[width=4cm]{./figures/brevet_fig5.jpg} \\
\vspace{-6ex}
\end{tabular}
\end{center}
\caption{Overview of the A-SIFT algorithm. All pairs of rotated and
tilted images obtained from $A$ and $B$ are compared with SIFT.}
\label{fig:CAffine:algo} \vspace{-3ex}
\end{figure}
  \begin{enumerate}

 \item Each image is transformed by  simulating all possible linear distortions caused by the change of orientation of
 the camera axis from a frontal position. These distortions depend upon two parameters: the longitude
 $\phi$ and the latitude $\theta$. The images undergo $\phi$-rotations followed by tilts  with parameter
 $t=|\frac 1{\cos \theta}|$.     For
 digital images, the tilt is performed as a $t$-subsampling, and therefore requires the previous application of
  an antialiasing filter in the direction of $x$, namely the convolution by a gaussian with standard
 deviation $c\sqrt{t^2-1}$~\cite{GuoshenMorelSIFT}.
\vspace{-1ex}
 \item These rotations and tilts are performed for a finite and small number of latitudes and longitudes, the sampling
 steps of these parameters ensuring that the simulated images keep close to any other possible view generated by
 other values of $\phi$ and $\theta$.

\vspace{-1ex}
\item All simulated images are compared to each other by some scale invariant, rotation invariant, and translation
invariant algorithm (typically SIFT). Since SIFT normalizes the
translation of the camera parallel to its focal plane and the
rotation of the camera around its optical axis, but simulates the
scale change, all six camera parameters are either normalized or
simulated by A-SIFT.
\vspace{-1ex}
\item The simulated latitudes $\theta$ correspond to tilts $t=$ $1$, $a$, $a^2$, $\dots$, $a^n$, with $a>1$.
Taking $a=\sqrt{2}$ is a good compromise between accuracy and
sparsity. The value $n$ can go up to 6 or more. That way, all
transition  tilts from 1 to 36 and more are  explored.
\vspace{-1ex}
\item The longitudes $\phi$ follow for each tilt $t$ an arithmetic series $0$,  $b/t$, $\dots$, $kb/t$, where $b\simeq
72^\circ$ seems again a good compromise, and $k$ is the last integer such that $kb/t<180^\circ$.

\vspace{-1ex}
\item Complexity: Since each tilt is a $t$ sub-sampling, the image area is divided by $t$. Counting all rotations associated with
 a tilt,
the overall simulated image area  for each tilt is $(180/72) t=2.5 t$. This implies that {\it the method complexity is
proportional to the number of tilts}. Controlling the overall area of the simulated images  is equivalent to
controlling the algorithm complexity. Indeed, the SIFT search time and  memory size are proportional to the image area.
 This complexity can be further downgraded by a) sub-sampling the query and search images; b) identifying
the pairs $(t, \phi)$ that give positive results; c) going back to
the original resolution only for these pairs.
\vspace{-1ex}
\item This description ends with a concrete example of how the multi-resolution search strategy can actually make the algorithm
only twice slower than SIFT. Take $a=\sqrt{2}$, $n=5$. The maximal
absolute tilt for each image is $5.7$ and the maximal transition
tilt goes up to $\simeq 32$. The simulated image area is $5\times
2.5=12.5$ times the original area. By
 a 3-subsampling of the original, this area is reduced to $1.4$ times the one of the original image. If this
reduction is applied to both the query and the search image, {\it
the overall comparison complexity is equivalent to twice the SIFT
complexity. } Fig.~\ref{fig:CAffine:simulation} shows the relatively
sparse sampling of the longitude-latitude sphere needed to perform a
fully affine recognition.
\end{enumerate}

\begin{figure}[htbp]
\begin{center}
\vspace{-5ex}
\begin{tabular}{c}
\includegraphics[width=3cm]{./figures/brevet_fig4b_english1.jpg}
\vspace{-4ex}
\end{tabular}
\end{center}
\caption{Sampling of the parameters $\theta = \arccos 1/t$ and $\phi$ in a
 zenith view
of the observation half sphere. } \label{fig:CAffine:simulation}
\vspace{-3ex}
\end{figure}
\vspace{-1ex}
\section{Experiments and Results}
\vspace{-1ex} In the experiments ASIFT is compared with SIFT,
Hessian-Affine, Harris-Affine and MSER detectors, all coded with the
SIFT descriptor.~\footnote{An online demo and the a software are
available at http://www.cmap.polytechnique.fr/\~{\hspace{0.1ex}}yu.}
\vspace{-2ex}
\subsubsection*{Testing absolute tilts\label{Experiments}}

Fig.~\ref{camerapositions} shows the setting adopted for evaluating the maximum absolute tilt and transition tilt
attained by each algorithm. A magazine and a poster were photographed for the experiments. Unlike SIFT and A-SIFT, the
Hessian-Affine, Harris-Affine and MSER detectors are not robust to scale changes. Thus, to focus on tilts, the pairs of
images under comparison were chosen free of scale changes.
\begin{figure}[htbp]
\begin{center}
\vspace{-2ex}
\begin{tabular}{cc}
\includegraphics[width=4.5cm]{./figures/fig1.jpg}
\vspace{-4ex}
\end{tabular}
\end{center}
\caption{Camera positions for systematic comparison.  }
\label{camerapositions} \vspace{-3ex}
\end{figure}
The poster shown in Fig.~\ref{fig:numeric:CSIFT:AdamZ10:65deg} was
photographed with a reflex camera with viewpoint angles  between the
camera axis and the normal to the poster varying from $\theta=
0^\circ$ (frontal view) to $\theta=80^\circ$. It seems physically
unrealistic to insist on larger latitudes. Table~\ref{tab:numerical}
compares A-SIFT with the performance of three state-of-the art
algorithms in terms of number of correct matches. One of these
matching results is illustrated in Fig.
\ref{fig:numeric:CSIFT:AdamZ10:65deg}. For these images SIFT works
with angles smaller than $45^\circ$. The performance of
Harris-Affine and Hessian-Affine plummets when the angle goes from
45 to $65^\circ$.
 Beyond this value, they fail completely. MSER struggles at the angle of $45^\circ$ and fails at $65^\circ$ degrees. A-SIFT
functions until $80^\circ$.

\begin{table}[htb]
\begin{center}
\vspace{-1ex}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$\theta$/$t$ & SIFT & HarAff & HesAff& MSER & A-SIFT \\
\hline $80^\circ$/5.8 & 3 & 0 & 0 & 2 & 110 \\
\hline $75^\circ$/3.9 & 1 & 1& 0 &  4 & 152 \\
\hline $65^\circ$/2.3 & 5 & 12 & 5 &   6 & 468 \\
\hline $45^\circ$/1.4 & 171 & 54 & 26 &  15 & 707 \\
\hline
\end{tabular}
\vspace{-3ex}
\end{center}
\caption{Absolute tilt invariance comparison  for viewpoint angles
between 45 and $80^\circ$.   The latitude angles and the absolute
tilts are listed in the left column.   } \label{tab:numerical}
\vspace{-3ex}
\end{table}

%AdamZ1:45deg
\begin{figure}[htbp]
\begin{center}
\vspace{-1ex}
\begin{tabular}{ccc}
\includegraphics[width=2.7cm]{./figures/adam_zoom10_front-adam_zoom10_84deg_ASIFT_77matches_b.jpg} &
\includegraphics[width=2.7cm]{./figures/adam_zoom10_front-adam_zoom10_84deg_MserSift_2matches_b.jpg} \vspace{-4ex}  \\
\vspace{-2ex}
\end{tabular}
\end{center}
\caption{Correspondences between the poster at frontal view and at
$80^\circ$ angle, absolute tilt $t=5.8$. A-SIFT (shown), SIFT,
Harris-Affine, Hessian-Affine and MSER (shown) find respectively
110, 3, 0, 0 and 2 correct matches.}
\label{fig:numeric:CSIFT:AdamZ10:65deg} \vspace{-1ex}
\end{figure}

The above experiments and other many lead to the following
conclusion for maximal absolute tilts. SIFT  hardly exceeds a
$t_{\max}=2$ absolute tilt. The limit is $t_{\max}\simeq 2.5$ for
Harris-Affine and Hessian-Affine. The performance of MSER depends
heavily on the type of image. For images with highly contrasted
regions, MSER reaches an absolute tilt $t\simeq 4$. However, if the
images do not contain highly contrasted regions or if the scale
change is larger than 3, the performance of MSER decays  strongly,
even under small tilts. For A-SIFT, an absolute tilt of
$t_{\max}\simeq 5.8$
 corresponding to the extreme viewpoint angle of $80^\circ$ is always attained.

\vspace{-2ex}
\subsubsection*{Transition Tilt Tests}
%The attainable transition tilt for all normalization methods and for A-SIFT
%should be the square of the absolute tilt. This is not  so, due to the limiting factors due to scale and contrast.
Fig.~\ref{fig:numeric:CSIFT:H8} looks striking: SIFT, Harris-Affine
and Harris-Affine fail completely on a seemingly easy example.
Indeed the small absolute tilts $t_1=t_2=2$ with the longitude
angles $\phi_1=0^\circ$ yield a moderate transition tilt $\tau=3$
that is out of scope of the above methods. A-SIFT works perfectly.
MSER works well under its optimal condition: highly contrasted
images and no scale change.

\begin{figure}[htbp]
\begin{center}
\vspace{-3ex}
\begin{tabular}{ccc}
\includegraphics[width=2.7cm]{./figures/H8A0vsH8A5_ASIFT_881matches_b.jpg} &
\includegraphics[width=2.7cm]{./figures/H8A0vsH8A5_SIFT_3matches_b.jpg} \vspace{-4ex}  \\
\vspace{-2ex}
\end{tabular}
\end{center}
\caption{Correspondences between the magazine images taken with
absolute tilts $t_1=t_2=2$ with longitude angles $\phi_1=0^\circ$
and $\phi_2=50^\circ$, transition tilt $\tau=3$. A-SIFT (shown),
SIFT (shown), Harris-Affine, Hessian-Affine and MSER find
respectively 881, 3, 0, 2, 87 correct matches.}
\label{fig:numeric:CSIFT:H8} \vspace{-3ex}
\end{figure}

Table~\ref{tab:comparison:transtilt} compares the performance of the
algorithms for the set of magazine images, matching those that have
an absolute tilt $t=4$. For these images, SIFT, Harris-Affine and
Hessian-Affine struggle with a $1.9$ transition tilt. They fail
completely over this value. MSER works stably up to a $\tau=7.7$
transition tilt. Over this value, the number of correspondences is
too small for reliable recognition. A-SIFT works perfectly up to the
transition tilt $16$. As shown in
Fig.~\ref{fig:numeric:CSIFT:extreme:tilt}, ASIFT actually attains
the transition tilt large than 37.

Fig.~\ref{fig:numeric:CSIFT:TR:casert} illustrates a round building.
After a viewpoint change, the left and right sides sustain big
transition tilts. ASIFT finds 123 correspondences covering the
graffiti on all the left, central and right parts of the building.
The other methods either fail or find a small number of matches in
the central part.\vspace{-1ex}

\begin{figure}[htbp]
\begin{center}
\vspace{-1ex}
\begin{tabular}{ccc}
\includegraphics[width=2.7cm]{./figures/casert1vs2_ASIFT_123matches_b.jpg} &
\includegraphics[width=2.7cm]{./figures/casert1vs2_MserSift_13matches_b.jpg}  \\
\end{tabular}
\vspace{-4ex}
\end{center}
\caption{Image matching: round building. Transition tilt
$\tau\in[1.8, \infty]$. A-SIFT (shown), SIFT, Harris-Affine,
Hessian-Affine and MSER (shown) find respectively 123, 19, 5, 7 and
13 correct matches.} \label{fig:numeric:CSIFT:TR:casert}
\vspace{-3ex}
\end{figure}
\vspace{-4ex}
\section{Conclusion}
\vspace{-2ex} Our conclusion is  illustrated by Fig.
\ref{fig:numeric:CSIFT:TR:direction} where all methods fail except
A-SIFT,
 on an image pair with moderate transition tilts. This happens because {\it normalization} methods, ideal
 in principle,
do not deal in practice correctly with small shapes or with low contrast.  {\it Simulation} methods are by far more
precise.  They might have looked prohibitive at first sight but turn out to be feasible, thanks to the very sparse
sampling of the affine space that they require (see Fig. \ref{fig:CAffine:simulation}). The surprising robustness of
the SIFT method to moderate transition tilts is key to this sparse sampling.

\begin{table}
\vspace{-1ex}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
 $\phi_2$/$\tau$ & SIFT & HarAff & HesAff & MSER & A-SIFT \\
\hline $10^\circ$/1.9 & 22 & 32 &14 & 49 & 1054 \\
\hline $20^\circ$/3.3 & 4 & 5 & 1 & 39 & 842 \\
\hline $30^\circ$/5.3 & 3 & 2 & 1 & 32 & 564 \\
\hline $40^\circ$/7.7 & 0 & 0 & 0&  28 & 351 \\
\hline $50^\circ$/10.2 & 0 & 0 &0 & 19 & 293 \\
\hline $60^\circ$/12.4 & 1 & 0 &0 & 17 & 145 \\
\hline $70^\circ$/14.3 & 0 & 0 &0 & 13 & 90 \\
\hline $80^\circ$/15.6 & 0 & 0 &0 & 12 & 106 \\
\hline $90^\circ$/16.0 & 0 & 0  &0 & 9 & 88 \\
\hline
\end{tabular}
\vspace{-3ex}
\end{center} \caption{Transition tilt performance. For the first
image $\phi_1=0^\circ$ and for both images the absolute tilt is
$t_1=t_2=4$. The longitude of the second image $\phi_2$ and the
resulting transition tilt $\tau$ are given in the first column.}
\label{tab:comparison:transtilt} \vspace{-2ex}
\end{table}

% Direction
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=7cm]{./figures/direction1vs3-AISFT-50matches_bis.jpg}
\vspace{-5ex}
\end{tabular}
\end{center}
\caption{Image matching: road signs. Transition tilt $\tau \approx
2.6$.   A-SIFT (shown), SIFT, Harris-Affine, Hessian-Affine and MSER
find respectively 50, 0, 0, 0 and 1 correct matches.   }
\vspace{-3ex}\label{fig:numeric:CSIFT:TR:direction}
\end{figure}

\vspace{-2ex}
\bibliographystyle{IEEEbib}
\begin{thebibliography}{99}
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\end{thebibliography}

\end{document}



% OutdoorX
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\hspace{-5.5ex}
\includegraphics[width=7.9cm]{./figures/Xlab1vs14_CAffine2_71match.jpg}
%\hspace{-0.5ex}
%\includegraphics[width=9cm]{./figures/outdoor1vs14_MserSift_2matches.jpg}
\\
\end{tabular}
\end{center}
\caption{Image matching: Facade. Absolute (and transition) tilt $t = 3.8$ ($\theta=75 ^\circ$). A-SIFT (shown), SIFT,
Harris-Affine, Hessian-Affine, MSER+Corr and MSER+SIFT find respectively 68, 0, 1, 1, 0 and 2 correct matches.  }
\label{fig:numeric:CSIFT:TR:facade1vs8}
\end{figure}
