


\def\PPRECOMPRESSION{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/jpeg/crop_victor_b.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/jpeg/jpeg7.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/jpeg/jpeg10.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/jpeg/jpeg30.ps}
}}
\caption[Compression]{\Pstyle \label{IMGPREFACE1} Compression. From left to right :
 An original image and its more and more compressed versions : compression factor 7, 10 and 25 respectively.
One of the first goals of image processing is the definition of algorithms permitting high compression factors without visible alteration.  Compression may, however alter the image.}
\end{figure}
}

\def\PPREDEBRUITE{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/debruite/bruite.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/debruite/Mbruite.ps}
}}
\caption[Denoising]{\Pstyle \label{IMGPREFACE2}  Denoising.  A second goal of image processing : the restoration. Left: original noisy image (simulated salt and pepper noise up to 75\%), right: denoised version}
\end{figure}
}


\def\PPREANALYSE{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/analyse/set_orig.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/analyse/S103.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/analyse/smoothed.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/analyse/S127.ps}
}}
\caption[Analysis of a shape]{\Pstyle \label{IMGPREFACE3} Analysis of a shape. Left : Original scanned shape, then some simplified versions : the aim here is not restoration, but analysis, that is, to define more and more sketchy versions of the shape. Those sketchy versions may permit a very short and invariant encoding of the shape. Notice how the number of inflexion points of the shape has decreased in the simplification process  (Chapter \ref{chapter20}).}
\end{figure}
}

\def\PSHANNON{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Shannon/victor.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Shannon/smoothed_victor.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Shannon/Z01.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Shannon/PZ01.ps}
}}
\caption[Shannon Theory and Subsampling]{\Pstyle \label{IMGPREFACE4} Shannon theory and subsampling. From left to right: original image, smoothed image, subsampling of the original image and subsampling of the smoothed image. In the subsampling, one point of each 4 is taken in the horizontal and vertical directions. In order to make the reduced image still visible, we have zoomed back the subsampled versions by a zoom factor 4. We clearly see that subsampling an image without previous smoothing  creates aliasing : high frequencies are projected onto lower frequencies and therefore generate new patterns. Shannon theory tells us how  to remove those potentially parasite high frequencies before subsampling.  This results in the necessity of smoothing the image before subsampling.}
\end{figure}
}

\def\PVICBLUR{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/cvictor.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/R22.ps}
}}
\caption[Heat equation and blur]{\Pstyle \label{IMGPREFACE5} Heat equation and blur. Left : original image, right : the  heat equation has been applied to some scale and  the resulting image is blurred (Chapter \ref{chapter1}).}
\end{figure}
}

\def\PVICDEBLUR{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/cvictor.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/D13.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/D22.ps}
}}
\caption[Gabor restoration]{\Pstyle  \label{IMGPREFACE6} Gabor's deblurring.  Gabor proposed in 1960 to deblur an image by  substracting  its laplacian : this means inverting the heat equation ! Left : original image, middle :  three  iterations  of Gabor's  algorithm, right : ten iterations. As is  well known, and can be  observed in the  right image, the inverse heat equation blows up. A few iterations can, as we see in the middle, nicely enhance the image. \label{vicdeblur}}
\end{figure}
}

\def\PVICBLURDEBLUR{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/R22.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/DR17.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Deblur/DR22.ps}
}}
\caption[Gabor restoration II]{\Pstyle \label{IMGPREFACE7} Gabor's deblurring again. Same deblurring experiment as  in Figure \ref{vicdeblur}, but applied on a much more blurred image}
\end{figure}
}


\def\PWAVELETS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/wavelets/crop_victor_b.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/wavelets/blur.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/wavelets/DoG.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/wavelets/SDoG.ps}
}}
\caption[The laplacian pyramid]{\Pstyle \label{IMGPREFACE8} The ``laplacian pyramid'' of Burt and Adelson. From left to right : original image,  image blurred  by gaussian convolution, then difference between the original image and the blurred version, which simulates the laplacian of the original image.  In  black in the last image, points where this laplacian image is large. This experiment simulates the first step of the laplacian pyramid. The laplacian image is, for most digital images, a sparse representation, therefore well adapted to  compression.}
\end{figure}
}

\def\PBOUNDHEAT{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=7.8cm
\epsfbox{\DDIR/preface/PeronaHeat/H1.ps}
\epsfxsize=7.8cm
\epsfbox{\DDIR/preface/PeronaHeat/H2.ps}
}}
\caption[\label{heatequationandCanny}Heat equation and Canny's edge detector]{\Pstyle \label{IMGPREFACE9}
Heat equation and Canny's edge detector.  Boundaries, or ``edges'' of the image can be defined as points where the gradient attains a maximal and large value along the gradient lines.  This amounts to say that edge points are points where $D^2u(Du, Du)$ crosses zero and $|Du|$ is large. Canny's edge detector computes those points. On the left, the original image, followed by the edge points found.  They make a very dense set, because of the oscillatory character of the image.  Next, the image blurred by the Gauss kernel (heat equation) and the Canny edges found.  The heat equation has removed the ``irrelevant'' edges (Chapter \ref{chapter2})}
\end{figure}
}


\def\PVICKRAMER{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Kramer/blur.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Kramer/Kramer.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Kramer/RudinOsher.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Kramer/deblur.ps}
}}
\caption[Image deblurring by shock filters and by a variational method]{\Pstyle 
\label{IMGPREFACE10}
 Image deblurring by shock filters and by a variational method.  From left to right : blurred image, Rudin-Osher shock filter which is a pseudoinverse of the heat equation attaining a steady state, Kramer's improved shock filter, also attaining a steady state and the Rudin, Osher, Fatemi restoration method, obtained by deblurring with a controlled image total variation.  This last method is very efficient when the noise and blur models are known. It is currently being used by the French Space Agency (CNES) for satellite image restoration (Chapter \ref{chapter14}). }
\end{figure}
}


\def\PBOUNDPERONA{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=7.8cm
\epsfbox{\DDIR/preface/PeronaHeat/P1.ps}
\epsfxsize=7.8cm
\epsfbox{\DDIR/preface/PeronaHeat/P2.ps}
}}
\caption[Perona and Malik equation and edge detection]{\Pstyle  \label{IMGPREFACE11}
 Perona-Malik equation and edge detection : same experiment as in Figure 
\ref{heatequationandCanny}, but the heat equation has been replaced by the Perona-Malik equation. Notice that the edge map looks slightly better localized as with the heat equation.}
\end{figure}
}


\def\PFIRSTJUNGLEDIFFUSION{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/lenna.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/perona.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/carmona1.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/maladi.ps}
}}
\caption[A proliferation of diffusion models I]{\Pstyle \label{IMGPREFACE12} A proliferation of diffusion models. From left to right: Original image, Perona and Malik equation 1987, Zhong, Carmona 1998 (diffusion along the least eigenvector of $D^2u$) and Sochen, Kimmel and Malladi, 1998 (minimization  of the image graph area.} 
\end{figure}
}


\def\PSECONDJUNGLEDIFFUSION{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/mcm1.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/TVdenoise75.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/alm.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Diffusions/coherence100.ps}
}}
\caption[Proliferation of diffusion models II]{\Pstyle \label{IMGPREFACE13} A proliferation of diffusion models (II). From left to right: Osher, Sethian 1988: curvature equation, Rudin, Osher and Fatemi 1992: minimization of the image total variation, Alvarez, Lions et al. : nonlocal variant of the preceding, 1992, Weickert 1994 : nonlocal variant of the curvature equation.  All of these models only  diffuse in the direction orthogonal to the gradient, with a more or less local estimate of this direction.}
\end{figure}
}


\def\PTHIRDJUNGLEDIFFUSION{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Caselles/lenna.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Caselles/lennaQ.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/Caselles/lennaQ_inter.ps}
}}
\caption[Proliferation of diffusion models III]{\Pstyle  \label{IMGPREFACE14} A proliferation of diffusion models (III)
Here, the diffusion is made in the direction of the gradient and the model is applied for image interpolation when level lines are sparse. From left to right : original image, quantized image (only 10 levels are kept - 3.32 bits/pixel) and reinterpolated image by the Caselles and Sbert (1998)algorithm. They apply a diffusion  on the quantized image, with values on the remaining level lines as boundary conditions. }
\end{figure}
}


\def\PPREFEXTKILL{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/ExtKill/chateau.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/ExtKill/DC.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/ExtKill/Bchateau.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/ExtKill/DBC.ps}
}}
\caption["Extrema killer"]{\Pstyle \label{IMGPREFACE15} The "extrema killer" filter: all connected components of the upper or lower level sets with small area are removed from the image. From left to right : original image, extrema killer applied with area 80 pixels, then  75\% salt and pepper noise added to the original image and the same filter applied (Chapter \ref{chapter5}). }
\end{figure}
}

\def\PPRELEVELSET{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/LevelSet/lenna.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/LevelSet/S140.ps}
}}
\caption[Level set of an image]{\Pstyle \label{IMGPREFACE16} An image and one of its level sets.  Right : level set 140 of the left image.  This experiment illustrates Matheron's thesis that the main shape information is contained in the level sets of the image. Level sets are contrast invariant (Chapter \ref{chapter3}).}
\end{figure}
}

\def\PPRELEVELLINES{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/LevelLines/chateau.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/LevelLines/L283.ps}
}}
\caption[Level lines of an image]{\Pstyle \label{IMGPREFACE17} Level lines of an image. Level lines, defined as the boundaries of level sets, can be defined to be a nested set of Jordan curves.  They give a contrast invariant representation of the image.  Right : level lines with level 183 of the left image (Chapter \ref{chapter4}).}
\end{figure}
}


\def\PLEVELSETLEVELLINES{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/LSLL/LS.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/LSLL/LSLS.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/LSLL/LSLL.ps}
}}
\caption[Contrast invariant localization]{\Pstyle \label{IMGPREFACE18} Contrast invariance localization. The image is decomposed into more and more local contrast invariant objects : first into level sets, then connected components of level sets, finally into the level curves bounding these connected components. These level curves are nested (they do not cross). Left : original image, middle : its unique level set, right : its three level curves (Chapter \ref{chapter4}).}
\end{figure}
}

\def\PPREAMSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/amss/amss101.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/amss/lignes101.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/amss/amss115.ps}
\epsfxsize=3.9cm
\epsfbox{\DDIR/preface/amss/lignes115.ps}
}}
\caption[The Affine and Morphological Scale-Space]{\Pstyle \label{IMGPREFACE19} The Affine and Morphological Scale Space (AMSS model). From left to right : original image, level lines of the images (16 levels only), smoothed image by using the affine and morphological scale space, and its level lines  (Chapter \ref{chaptermainequations}).}
\end{figure}
}

\def\PPRELISANI{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=8cm
\epsfbox{\DDIR/preface/lisani/r1.ps}
\epsfxsize=8cm
\epsfbox{\DDIR/preface/lisani/r_dominant.ps}
}}
\caption[Level lines based shape parser]{\Pstyle \label{IMGPREFACE20} Level lines based shape parser. Shape extraction has ever been the bottleneck of shape recognition algorithms. With the presented algorithm, this problem is solved by a brute force method : it compares all level lines of the images to be compared.  Left pair of images : two images of a desk taken from different angles.  In the left hand desk image, one level line has been put in white.  We display, also in white, in the right image  of the pair, the matching level lines.  The match is ambiguous, as must be expected when the same object is repeated twice in the scene !  In the right pair of images, we display in white all matching pairs of level lines.
(Experiment~: J.-L. Lisani). }
\end{figure}
}


\def\PPRELIONEL{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=5.2cm
\epsfbox{\DDIR/preface/lionel/orig.ps}
\epsfxsize=5.2cm
\epsfbox{\DDIR/preface/lionel/mcm.ps}
\epsfxsize=5.2cm
\epsfbox{\DDIR/preface/lionel/amss.ps}
}}
\caption[Experimental check of the affine invariance of the affine shortening (AMSS)]{\Pstyle
\label{IMGPREFACE21} Experimental check of the affine invariance of the affine shortening (AMSS).  We display on the left image three shapes. The second one is an affine transform  $AS$ of the first shape $S$. The third one is obtained from the second by the inverse affine transform. It therefore initially is $A^{-1}AS=S$.  On the right image : result after application of AMSS to the two first shapes : are viewed $S(t),$, $(AS)(t)$ and $A^{-1}((AS)(t))$. If the numerical scheme  is affine invariant, this third shape must coincide with $S(t)$, which is indeed the case.   Middle : the same procedure applied with the curvature equation, which proves not to be affine invariant, as expected (Chapter \ref{chapter22}). }
\end{figure}
}

\def\PPRESNAKE{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=3.8cm
\epsfbox{\DDIR/preface/snake/brain.ps}
\epsfxsize=3.8cm
\epsfbox{\DDIR/preface/snake/masque.ps}
\epsfxsize=3.8cm
\epsfbox{\DDIR/preface/snake/out50.ps}
\epsfxsize=3.8cm
\epsfbox{\DDIR/preface/snake/out50s.ps}
}}
\caption[Active Contour]{\Pstyle \label{IMGPREFACE22}Active contour, or "snake" From left to right : original image, initial contour, evolved distance function, final contour (Chapter \ref{chapter17}).}
%tres bien comme �a
\end{figure}
}


%%8888888888888888888888888888

%%FIN DES IMAGES DE LA PREFACE 

%%7777777777777777777777777777


%%LES SNAKES DU CHAPITRE SNAKES :


\def\PIMGSNAKEUN
{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=6cm
\epsfbox{\FDIR/snake.eps}
}}
\caption[Convection term of the active contour equation]{\label{snakefigure}\Pstyle  Convection term of the active contour equation. The convection term of the active contour equation tends to create shocks aroung values where $g$ is small. Indeed, the level lines of $u$ are moved in the direction opposite to the gradient of $g$.}
\end{figure}
}

\def\PIMGSNAKEDEUX{
\begin{figure}
\centerline{\hbox{
\epsfxsize=6cm
\epsfbox{\FDIR/star.eps}
}}
\caption[A difficulty : the local minima of the active contour energy]{\label{snakestarfigure}\Pstyle A difficulty : the local minima of the active contour energy. Left image: assume that $g$ is  null on the shape (drawn in bold) and that the initial contour is the line \#0. From the initial line to the contour of the shape, the intermediate state \#1 consisting of the convex hull of the polygon shows a smaller energy than the intermediate state \#2 (drawn on the right). This illustrates the difficulties arising with such an equation when we wish to land the active contour onto concave parts of the desired contour.
 }
\end{figure}
}



\def\PIMGSNAKETROIS
{
\begin{figure}
\centerline{\hbox{
\begin{tabular}{ccc}
\epsfxsize=4cm
\epsfbox{\DDIR/snake/bird.ps}
&\epsfxsize=4cm
\epsfbox{\DDIR/snake/label.ps}
&\epsfxsize=4cm
\epsfbox{\DDIR/snake/snake_bird.ps}
\end{tabular}
}}
\caption[Silhouette  of a bird by active contour.]{\Pstyle Silhouette of a bird by active contour. Left: original image, middle: initial contour, right: final contour (steady state of the snake equation).}
\end{figure}
}

\def\PIMGSNAKEQUATRE{
\begin{figure}
\centerline{\hbox{
\begin{tabular}{ccc}
\epsfxsize=4cm
\epsfbox{\DDIR/SNAKE2/boules.ps}
&\epsfxsize=4cm
\epsfbox{\DDIR/SNAKE2/label.ps}
&\epsfxsize=4cm
\epsfbox{\DDIR/SNAKE2/inter1.ps} \\
\epsfxsize=4cm
\epsfbox{\DDIR/SNAKE2/inter2.ps}
&\epsfxsize=4cm
\epsfbox{\DDIR/SNAKE2/inter3.ps}
&\epsfxsize=4cm
\epsfbox{\DDIR/SNAKE2/inter4.ps}
\end{tabular}
}}
\caption[Active contour with topological change.]{\Pstyle Active contour with topological change. Top, left: original image, middle: initial contour, right: intermediate state. Down, left and middle: successive intermediate states, down-right: final contour (steady state). Original image is "Vue d'esprit 3", by courtesy of {\it e-on software}.}
\end{figure}
}





%%��������������������������

%%FIN DES SNAKES

%%88888888888888888888888888

\def\PCANNY{
\begin{figure}
%\fbox{
\centerline{\hbox{
\epsfxsize=\PPCANNY
\epsfbox{\DDIR/canny.ps}
}}
\caption[Canny's edge detector]{\Pstyle Canny's edge detector.  These images illustrate the Canny edge detector, which attempts to find boundaries in an image.
The Canny filter involves the following operations : \\ \\
1. Convolution of the image by a gaussian kernel (from top to bottom, scale = 0.1, 0.5, 1.0). \\
2. Estimation of $D^2u(Du,Du)$ (we use here a finite difference scheme). \\
3. Convolution of $D^2u(Du,Du)$ with a small Gauss kernel (sigma = 0.001). \\
4. Thresholding  of the gradient of the result of Step 1. \\
5. Zero-crossings of the result of Step 3,  only displayed when the threshold of Step 4 is achieved. \\ \\
Left Column : result of the Canny filter without the threshold on the gradient, (Step 4 removed) \\
Middle column : result with a visually `` optimal'' scale and an image depending threshold.
   (from top to bottom : 15, 0.5, 0.6) \\
Right column : result with a fixed gradient  threshold  equal to  0.5. \\
Notice that this edge detection "theory" depends upon not less than three parameters which 
have to be fixed by the user.
}
%}
\end{figure}
}

\def\PMEAN{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPMEAN
\epsfbox{\DDIR/meanvh.ps}
}}
\caption[Local averaging algorithm]{\Pstyle Local averaging  algorithm.  Left : original image,
 right : result of replacing the grey level  at each pixel by  the average grey level over the neighboring pixels. The shape of the neighborhood is defined by the black spot 
displayed up-right.
}
\end{figure}
}

\def\PBLUR{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPBLUR
\epsfbox{\DDIR/blur.ps}
}}
\caption[Convolution by  gaussian kernels (heat equation)]{\Pstyle Convolution by gaussian kernels (heat equation). From top-left to bottom-right,
we display the original image, and the results of the convolutions with gaussians of increasing variance.
A grey level representation of the convolution kernel is put on the right of each convolved image ; it gives an idea of the size of the involved neighborhood.  The resulting image is more and more blurry.}
\end{figure}
}

\def\PCONCENT{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCONCENT
\epsfbox{\DDIR/concent.ps}
}}
\caption[Kernel rescaling]{}
\hspace{4cm} {
\Pstyle
A kernel $g$ and its rescalings $g_t= \frac{1}{t^2} g(x/t)$, for t=2, 3, 4.
}
\end{figure}
}

\def\PGAUSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPGAUSS
\epsfbox{\DDIR/gaussm.ps}
}}
\caption[The Gauss  kernel]{\Pstyle The Gauss kernel in two dimensions.}
\end{figure}
}

\def\PLOCALINCL{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPLOCALINCL
\epsfbox{\DDIR/locincl.ps}
}}
\centerline{\hbox{
\epsfxsize=\PPLOCALINCL
\epsfbox{\DDIR/Gincl2.ps}
}}
\caption[Violation of the shape inclusion principle by the linear scale-space]{\Pstyle Violation of the inclusion by the linear scale-space.
The linear scale-space does not maintain the inclusion between objects. Top-left: an image that contains a black disk enclosed by a white disk. At a certain scale, the black and white circles mix together (top-right). The level lines
(middle) show that the inclusion is no longer preserved.
Bottom : three-dimensional representation of both  images, where the vertical coordinate corresponds
to the gray-level. For the shape inclusion principle, see Chapter \ref{chapter20}.
}
\end{figure}
}

\def\PCOMPGAUSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCONPGAUSS
\epsfbox{\DDIR/cpgauss.ps}
}}
\caption[Level lines and the heat equation]{\Pstyle Level lines and the heat equation.
On the first
row, we display on the  left  a  410$\times$270  grey level image, photograph of a sea bird  and on the right~: its level lines for levels multiple of
12.  On the second row, the heat equation, i.e. a convolution with a
gaussian, has been applied to the original image.  The standard deviation of the gaussian is 4,
which means that its spatial range is comparable to a disk with radius 4.  The image
gets blurred by the convolution, which mixes grey level values and
removes all sharp edges.  This can be appreciated on the right, where
we have displayed all level lines for levels multiple of 12.  We can
see how the level lines on the boundaries of the image have split
into parallel level lines which have gone  away from each other.  The image
has become smooth, but is losing its structure.}
\end{figure}
}

\def\PCONTRAST_GAUSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfysize=12cm
\epsfbox{\DDIR/ctgauss.ps}
}}
\caption[The heat equation is not contrast invariant]{\label{PCONTRAST_GAUSS} \Pstyle \label{linearscalespacenotcontrastinvariant}
The heat equation is not contrast invariant. This experiment shows that the linear scale-space is not contrast 
invariant. First row~: original image.  Second  row~:
two different contrast changes have been applied to this image. Third row~: a convolution
by a gaussian is applied to both images of the second row. Fourth  row~: the inverse 
contrast change is applied to the images of the third row. If the 
linear scale-space were contrast invariant, these images should be
equal~: this is not the case, since the difference (displayed in the fifth row)
is not null.}
\end{figure}
}




\def\PLINEAR{
\begin{figure}
\centerline{\hbox{
\epsfxsize=\PPLINEAR
\epsfbox{\DDIR/linear1.ps}
}}
\caption[Iterated linear smoothing converges towards the heat equation]{\Pstyle
 Iterated linear smoothing converges towards the heat equation.   In this
experiment on one-dimensional functions, it can be appreciated how
fast an iterated convolution of a positive kernel converges to a
gaussian.  On the left, we display nine iterations of the convolution of the
characteristic function of an interval with itself, with the adequate rescaling. 
On the right, the same experiment is repeated with a by far more irregular
kernel. Convergence to the gaussian is almost equally fast. (See Theorem \ref{convergenceverschaleur}).
}
\end{figure}
}

\def\PTDLAGAUSS{
\begin{figure}
\centerline{\hbox{
\epsfxsize=\PPTDLAGAUSS
\epsfbox{\DDIR/3dlpga.ps}
}}
\caption[The laplacian of the Gauss kernel]{\Pstyle Three-dimensional representation  of the laplacian of the Gauss kernel. This convolution kernel, actually a wavelet, is used to estimate the laplacian of the image at different scales of linear smoothing. (See Proposition \ref{convolL1CS}, and Section \ref{Edgedetection}).}
\end{figure}
} 


\def\PZEROCROSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPZEROCROSS
\epsfbox{\DDIR/zerocross.ps}
}}
\caption[Zero crossing of the laplacian at different scales]{\Pstyle
``Zero-crossings of the laplacian'' at different scales.
In this display, we
illustrate the original Scale Space theory, as it is for instance
developed in the founding book by David Marr, Vision.  In order to
extract more global structure, the image is convolved
with  gaussians whose variances  are powers of 2. 
One computes the laplacian of the resulting smooth images and displays
the lines along which this laplacian changes sign : the so called ``zero-crossings of the laplacian".  According to David Marr, those zero-crossings represent the ``raw primal sketch" of the image, that is, the
information on which further vision algorithms should  be based. \newline
Above~: from left to right, we display the results of the smoothing, and the associated gaussian kernels, of
scales 1,2 and  4 respectively.
Below~: we display the zero-crossings of the laplacian, and the corresponding  kernels, i.e. 
the laplacians of the gaussians used above.
}
\end{figure}
}


\def\PCARRECROSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCARRECROSS
\epsfbox{\DDIR/cacross.ps}
}}
\caption[Zero-crossings of the laplacian of a synthetic image]{\Pstyle Zero crossings of the laplacian of a synthetic image. 
From left to right~: the original image, the image linearly smoothed by a convolution with 
a Gauss function,
 the sign of the laplacian of the filtered image (the gray color corresponds to values close to 0, black to clearcut negative values, white to clearcut positive values)  and the zero-crossings of the Laplacian. This experiment clearly shows the drawbacks of the Laplacian as edge detector.
}
\end{figure}
}

\def\PDETAILCHAR{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPDETAILCHAR
\epsfbox{\DDIR/detchar.ps}
}}
\caption[Zoom on Noise]{\Pstyle Zoom on Noise.}
\end{figure}
}

\def\PTREE{
\begin{figure}
\centerline{\hbox{
\epsfxsize=\PPTREE
\epsfbox{\DDIR/tree.ps}
}}
\caption[A multiscaled word]{\Pstyle 
A
multiscaled world ... This series of images is an experiment in the
relative stability of perception of objects seen at different
distances.  Each photograph has been taken in a park by stepping
forward and taking each snapshot at a much closer distance than the
former one.  We display by a rectangle in each image the part of the
object which has been photographed in the next image.  As one can
appreciate,  when getting closer, the visual aspect of objects changes
and new structures arise.  Thus, the computing of primitives in an
image is always a scale-dependent task, quite dependent on the
distance to objects.  When we look at a certain distance at an object,
we do not perceive  the  too much finer structure : for instance, leaves cannot be seen in the two first photographs.  We do not see them either  in the two last
ones, since we got too close !  Multiscale smoothing tries to emulate
this physical and perceptual fact by defining a smoothing of a digital
image at different scales which gets rid of the finer structures, but
minimally modifies the image at scales above the desired scale.
}
\end{figure}
}

\def\PLEVELPRUEBA{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPLEVELPRUEBA
\epsfbox{\DDIR/lprueba.ps}
}}
\caption{\Pstyle Level lines as representatives of the shapes present in an image.  Left: noisy binary image with two apparent shapes, right: its two longest level lines. }
\end{figure}
}


\def\PLLIGNESBIRD{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=6.5cm
\epsfbox{\DDIR/LEVELLINESBIRD/bird1z4.ps} &
\epsfxsize=6.5cm
\epsfbox{\DDIR/LEVELLINESBIRD/l12bird1z4.ps} \\
\end{tabular}
\end{center}
\caption[Level lines]{\Pstyle Level lines as a  complete representation of the shapes present in 
an image.
All level lines of
the image of a sea bird for levels which are multiples of 12. Notice that  we do not need a previous smoothing in order to visualize the shape structures in an image : a quantization of the displayed levels is enough.  The extrema killer is also very efficient to that visualization aim. }
\end{figure}
}

\def\PQUANTIF{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPQUANTIF
\epsfbox{\DDIR/quantif.ps}
}}
\caption[Image reconstruction from its level sets]{\Pstyle Image reconstruction from its level sets. Illustration of  Proposition \ref{II.1}. 
 We reconstruct an image from its level sets. We used  top-left : all
level sets,  top-right : all level sets whose grey level is a multiple of 8, bottom-left : multiple of 16 and  bottom-right : multiple of 32. Notice the relative stability of the image shape 
content under these drastic quantizations of the grey levels.
}
\end{figure}
}

\def\PLSBIRD{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{ccc}
\epsfxsize=4.5cm
\epsfbox{\DDIR/LEVELSETSBIRD2/bird1z4.ps}&
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S225.ps}&
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S200.ps}\\
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S175.ps}&
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S150.ps}&
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S125.ps}\\
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S100.ps}&
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S75.ps}&
\epsfxsize=4.5cm \epsfbox{\DDIR/LEVELSETSBIRD2/S50.ps}\\
\end{tabular}
\end{center}
\caption[Level sets of a digital image]{\Pstyle  Level sets of a digital image. 
In this figure, we
first show a grey level  image, which has a range of grey levels from 0 to 255, and then eight level sets, in decreasing order from 225 to 50, the grey scale step being  25.
Notice how essential features of the shapes are contained in the boundaries of level
sets, the level lines.  Each level set (in gray) is contained in the next,
(Formula \ref{II.1}.i). 
}
\end{figure}
}



%% \def\PLSBIRD_OLD{\begin{figure}[!ht] \centerline{\hbox{ \epsfxsize=\PPLSBIRD
%%\epsfbox{\DDIR/lsbird.ps} }}
%%\caption[Level sets]{\Pstyle In this figure, we display the image %%``seabird2'', which has a range of grey levels from 0 to 255, and then all %%level sets whose level is a multiple of 32, in decreasing order from 224 to %%32. Notice how essential features of the shapes are contained in the %%boundaries of level sets, the level lines.  Each level set (in white) is %%contained in the next, (Formula \ref{II.1}.i). } \end{figure} }

\def\PCCONTRAST{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfysize=\PPCCONTRAST
\epsfbox{\DDIR/ccont.ps}
}}
\caption[Contrast changes and an equivalence class of images]{\Pstyle  Contrast changes, and an equivalence class of images.
 A grey level image 
with different contrasts.  The three displayed images have exactly the same
level sets and level lines, but are associated with  three different
scales of grey levels. The graphs on
the right are the graphs of the functions
$u\to g(u)$ which have been applied to the initial grey levels.  The
first one is convex and enhances the brighter parts of the image.  The
second one is the identity, leaving thus the image unaltered.  The
third one is concave and enhances the darker parts of the image.  When we dispose of a digital image,
we spontaneously arrange the constrast in order to see better such or such shape information.
Thus, from the image analysis viewpoint, the image data should be considered as an equivalence class under all possible contrast changes.
}
\end{figure}
}


\def\PCAMSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfysize=\PPCAMSS
\epsfbox{\DDIR/ccamss.ps}
}}
\caption[Contrast invariance of AMSS]{\Pstyle Contrast invariance of the  affine morphological scale space (AMSS).  First row : original image. Second row : two contrast changes applied to the original.  Third row : AMSS applied to both images of the second row, by the finite difference scheme (FDS) explained in Chapter \ref{chapter22}. Fourth row : inverse contrast change applied to the filtered images.  A visual check shows that they are almost identical. Bottom image : numerical check by taking the difference of the images in the fourth row. Compare with the same experiment performed with the linear scale space, Figure \ref{linearscalespacenotcontrastinvariant}.}
\end{figure}
}

\def\PMEDIAN{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPMEDIAN
\epsfbox{\DDIR/median1.ps}
}}
\caption[Iterated Median Filter]{\Pstyle 
Scale-space based on iterations
of the median filter. From left to right and up to down : original
shape, the size of the disk used for the median, and the result of the
iterated median filter for an increasing number of iterations.}
\end{figure}
}

\def\PMEDAMSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPMEDAMSS
\epsfbox{\DDIR/medamss.ps}
}}
\caption[Comparison of the iterated median filter and of the  curvature motion]{\Pstyle Comparison of the iterated median filter and of the  curvature motion.  Numerically, the iterated
median filter and the curvature motion are very close, when the
curvatures of the level lines are not too small. Indeed, the iterated median filter converges towards the viscosity solution of the curvature motion. Left : simple shape
smoothed by a finite difference scheme of the curvature motion,
middle : smoothing by a median filter a the same scale, right :
difference between left and middle images. The difference is not
larger that one pixel width.}
\end{figure}
}

\def\PMEDCURV{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPMEDCURV
\epsfbox{\DDIR/medcurv.ps}
}}
\caption[Median filter and the curvature of level lines]{\Pstyle Median filter and the curvature of level lines.
The median filter does a smoothing
linked to the curvature of the level lines. Left~: image of a simple
shape. Right~: difference of this image with itself after it has been smoothed by one
iteration of the median filter. We see, in black, the points which
have changed. The width of the difference is proportional to the
curvature, according to Theorem \ref{B1}.
}
\end{figure}
}

\def\PDYNAMSH{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPDYNAMSH
\epsfbox{\DDIR/dynsh.ps}
}}
\caption[Gauss convolution+threshold=median]{\Pstyle \label{dynamicshapeexperiment} "Gauss convolution+threshold=median".  A convolution with a gaussian followed by a threshold yields a weighted median filter.
The alternate iterations on a binary shape of the linear smoothing and of a $1/2$ threshold
emulate the mean curvature motion. Left : original picture,
then, result of the convolution with a small gaussian kernel, in continuation
threshold at the mid-level of the blurred image, right : result after
few iterations of this alternate process.
}
\end{figure}
}


\def\PTJONC{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPTJONC
\epsfbox{\DDIR/tjonc1.ps}
}}
\caption[Same geometric figures, different evolutions under smoothing]
{\Pstyle
Same geometric figures, different evolutions under smoothing.
1st row : four T-junctions that only differ by the grey-level order
between the three regions. \\
2nd row : result of a smoothing by the AMSS model. We see that
two different evolutions are possible : if the regions of the image keep the same order, then the geometric evolution is identical. If, instead, a non monotone contrast change has been applied, the evolutions  differ.  
\\
3rd row : result of a smoothing by the linear scale-space : all of the T-junctions give different evolutions. The evolution depends on the gray-level values of the three level sets, and not only on their order. \\
4rd row : quantization of the 3rd row in order to display the
shapes of some level lines.
}
\end{figure}
}

\def\PCOURBUN{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCOURB
\epsfbox{\DDIR/courb.ps}
}}
\caption[Curvature scale space I]{ \Pstyle \label{curvaturescalespaceexperiment1}
Computation of the
curvature of the original sea bird image  after it has been smoothed by
curvature motion at calibrated scale 1.  The first image displays the smoothed version
of the sea bird at a small scale.  The second image displays the absolute
value of the curvature, with the convention that the darkest points have
the largest curvature.  We have displayed the curvature only at points
where the gradient of the image was larger than 6. (The image grey
levels range from 0 to 255).  In continuation, we display on
separate images the positive part of the curvature and the negative part. 
The curvature motion can be used as a nonlinear means to compute a "multiscale" curvature of the original image. Compare with Figure \ref{curvaturescalespaceexperiment2}, where the calibrated scale of smoothing is  4.  (A calibrated scale $t$ means that at this scale a disk with radius $t$ disappears)
}
\end{figure}
}

\def\PCOURBDEUX{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCOURB
\epsfbox{\DDIR/Courbure3.ps}
}}
\caption[Curvature scale space II]{ \Pstyle \label{curvaturescalespaceexperiment2}
Computation of the
curvature of the original sea bird image  after it has been smoothed by
curvature motion.  The first image displays the smoothed version
of the sea bird at calibrated scale 4.  The second image displays the absolute
value of the curvature, with the convention that the darkest points have
the largest curvature.  We have displayed the curvature only at points
where the gradient of the image was larger than 6. (The image grey
levels range from 0 to 255).  In continuation, we display on
separate images the positive part of the curvature and the negative part. 
The curvature motion can be used as a nonlinear means to compute a "multiscale" curvature of the original image. Compare with Figure \ref{curvaturescalespaceexperiment1}, where the calibrated scale of smoothing is 1. 
}
\end{figure}
}



\def\PCMPED{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCMPED
\epsfbox{\DDIR/cmped.ps}
}}
\caption[Erosion and dilation of a real image]{ \Pstyle  Erosion and dilation of a real image. On the first row, a sea bird image, and its level lines for all
levels multiple of 12. On the second row, an erosion with radius 4 has been  applied. 
On the right, the resulting level lines where the circular shape of the
structuring element (a disk with radius 4) appears around each local
minimum of the original image.  Erosion removes local maxima (in
particular, all small white spots), but expands minima. Thus, 
all dark spots, like here  the eye of the bird, are expanded.  The third row displays the effect of a dilation with radius 4, and the resulting level
lines.   We see how local minima are removed (see e.g. the eye of the
bird) and how white spots on the tail expand.  Here, in turn, circular
level lines appear around all local maxima of the original image.
}
\end{figure}
}

\def\PCMPOC{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCMPOC
\epsfbox{\DDIR/opeclo.ps}
}}
\caption[Opening and closing of a real image]{ \Pstyle Opening and closing of a real image.
On the first row, the original image  and its level lines for all
levels multiple of 12. On the second row (resp. third row), an opening (resp. closing) with radius 4 has been  applied. 
In the level-lines displayed on the right, we can recognize  the circular shape of the
structuring element.
}
\end{figure}
}


\def\PCMPMS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPCMPMS
\epsfbox{\DDIR/cmpms.ps}
}}
\caption[Consistency of the median filter and of the Catt\'{e}-Dibos scheme with curvature motion]{ \Pstyle Consistency of the median filter and of the Catt\'{e}-Dibos scheme with curvature motion. On the first
row, the sea bird image and its level lines for all levels multiple of 12.
On the second row, a median filter on a disk with radius 2 has been
iterated twice. On the third row, an inf-sup and then a sup-inf
filter based on segments have been applied. On the right : the corresponding
level lines of the results, which, according to the theoretical results, must have moved at
a normal speed proportional to their curvature, (Theorems \ref{B1} and \ref {cattedibosconsistence}). The results are very close.  This yields a cross-validation of two very different numerical schemes implementing the curvature motion. 
}
\end{figure}
}


\def\PGPLUS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPGPLUS
\epsfbox{\DDIR/gplus.ps}
}}
\caption[Histogram and contrast change II]{\Pstyle
The two images (left) have the same set of level sets. The contrast change
function that makes the upper image become the lower image is displayed on the right. It corresponds to one of the possible $g$ functions whose existence is stated  in  Corollary  \ref{changementdecontrastenonbijectif}. The function $g$ may be locally constant on intervals where the histogram of the upper image is 0 (see
top-middle graph). Indeed, on such grey level intervals, the level-sets are steady.
}
\end{figure}
}


\def\PLOSTDATA{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPLOSTDATA
\epsfbox{\DDIR/lostdata.ps}
}}
\caption[Histogram  contrast change I]{
\Pstyle
The original image (top-left) has a strictly positive histogram (all grey levels between 0 and 255 are represented). Therefore,
if any non strictly increasing contrast change $g$ is applied, 
some data will be lost.  Every level set of the transformed image $g(u)$ is a level set of the original image. Now, the original image has more level sets than the transformed one.
}
\end{figure}
}

\def\PSLS{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cccc}
\framebox{\epsfxsize=2.8cm \epsfbox{\DDIR/LEVELSETSSIMPLE2/simple.ps}} &
\framebox{\epsfxsize=2.8cm \epsfbox{\DDIR/LEVELSETSSIMPLE2/S255.ps}} &
\framebox{\epsfxsize=2.8cm \epsfbox{\DDIR/LEVELSETSSIMPLE2/S225.ps}} &
\framebox{\epsfxsize=2.8cm \epsfbox{\DDIR/LEVELSETSSIMPLE2/S215.ps}} \\
& & & \\
 &
\framebox{\epsfxsize=2.8cm \epsfbox{\DDIR/LEVELSETSSIMPLE2/S192.ps}} &
\framebox{\epsfxsize=2.8cm \epsfbox{\DDIR/LEVELSETSSIMPLE2/S80.ps}} &
\framebox{\epsfxsize=2.8cm \epsfbox{\DDIR/LEVELSETSSIMPLE2/S0.ps}} \\
\end{tabular}
\end{center}
\caption[Level sets of a simple synthetic image]{\Pstyle
A simple synthetic image and all of  its level-sets (in gray) with  decreasing levels
from top to bottom  and from left to right.}
\end{figure}
}

\def\PPLERO{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPPLERO
\epsfbox{\DDIR/PL_ERO.ps}
}}
\caption[Dilation of a set]{ \Pstyle Dilation of a  set.
A set $X$ (in black), its dilation by a ball of radius 20, and the
difference set.
}
\label{Pfig1}
\end{figure}
}

\def\PPLDIL{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPPLDIL
\epsfbox{\DDIR/PL_DIL.ps}
}}
\caption[Erosion of set]{ \Pstyle Erosion of a set.
A set $X$ (in black), its erosion by a ball of radius 20, and the
difference set.
}
\label{Pfig2}
\end{figure}
}

\def\PPLDIST{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPPLDIST
\epsfbox{\DDIR/PL_DIST.ps}
}}
\caption[Distance image]{Distance image. }
\parbox[t]{12cm}
{\Pstyle Dilations and erosions as level sets of the signed distance function.
 Left : a set $X$, middle : the signed distance to $X$,
$u(\fxx) = d(\fxx, X)$ if $\fxx \in X^c$, $u(\fxx) = -d(\fxx, X)$, if $\fxx \in X$, right : by quantizing the grey levels of $u$, as done on the right, one can easily check that the level sets of $u$ are dilated or eroded of $X$. Compare with  Figures  \ref{Pfig1} and \ref{Pfig2}.
}
\end{figure}
}

\def\PPLCLOSING{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPPCLOSING
\epsfbox{\DDIR/PL_CLOSING.ps}
}}
\caption[Opening of a set as a curvature threshold]{ \Pstyle Opening of a set as curvature threshold from above.
A set $X$, its opening by a ball of radius 20, and the
difference between original and processed. This opening transforms $X$ into  the union of all balls of radius 20 contained in it. The resulting operation can be understood as a threshold from above of the curvature of the set boundary.
}
\end{figure}
}

\def\PPLOPENING{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPPOPENING
\epsfbox{\DDIR/PL_OPENING.ps}
}}
\caption[Closing of a  set as a curvature threshold]{ \Pstyle Closing of a set as a curvature threshold, from below.
A set $X$, its closing by a ball of radius 20, and the
difference between original and processed. The closing of $X$ is nothing but the opening of $X^c$. It can be viewed as a threshold from below of the curvature of the set boundary.
}
\end{figure}
}


\def\POPERATORUN{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPOPERATOR
\epsfbox{\DDIR/operator12.ps}
}}
\caption[Denoising based on opening and closing I]{ \Pstyle Opening and closing based denoising. 
On the first row : scanned picture of the word ``operator'' with black dots and a black line added, a dilation with a $2 \times 5$ rectangle (middle), then an erosion with the same structuring element. The resulting operator is a closing.
Small black structures are removed by a such process.
In the second row : the word ``operator'' with  a white line and white dots inside the letters, erosion with a rectangle $2 \times 5$, followed by a dilation. The resulting operator is a opening. Small white structures are removed.
}
\label{operaun}
\end{figure}
}

\def\POPERATORTROIS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPOPERATOR
\epsfbox{\DDIR/operator5.ps}
}}
\caption[Denoising based on the median filter I \label{medianofoperatorimage}]{ \Pstyle Example of denoising with the median filter :
scanned picture of the word ``operator'' with perturbations and
noise made of
black or white lines and dots. Middle : first iteration of the median
filter with a circular neighborhood of radius 2, right second iteration.
Compare with the denoising  by opening-closing experiment (Figure \ref{POPERATORDEUX}).
}
\label{operatrois}
\end{figure}
}



\def\POPERATORDEUX{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPOPERATQR
\epsfbox{\DDIR/operator34.ps}
}}
\caption[Denoising based on opening and closing (II)]{ \Pstyle
Examples of denoising based on opening or closing, as in Figure \ref{operaun}.
Here are added on the ``operator'' image some perturbations made of both
black and white lines or dots.
In the first column,  up-down : original picture, erosion with a $1 \times 3$ rectangle, then dilation with the same structuring element. In other terms opening with this rectangle. In continuation, a dilation is applied with a rectangle $3 \times 1$, and finally an erosion with the same rectangle.
In the second column, the same process is applied,  but with 
erosions and dilations exchanging their roles.  It does not work so well because
closing expands white perturbations and opening expands black perturbations : those operators do not commute.
See Figure \ref{medianofoperatorimage}, where the median filter applies with more success.
}
\label{POPERATORDEUX}
\end{figure}
}






\def\PLLS{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cccc}
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/man.ps} &
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/S136.ps} &
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/S127.ps} &
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/S117.ps} \\
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/S111.ps} &
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/S107.ps} &
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/S80.ps} &
\epsfxsize=3.5cm \epsfbox{\DDIR/LEVELSETSMAN2/S40.ps} \\
\end{tabular}
\end{center}
\caption[Level sets]{\Pstyle
A simple image and some of its level-sets (in gray) with  decreasing levels
from top to bottom and  left to right.}
\end{figure}
}



\def\PTJOND{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPTJOND
\epsfbox{\DDIR/3dtj.ps}
}}
\caption{ \Pstyle
Evolution of T-jonction by amss.
}
\end{figure}
}

\def\PPROLONGE{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPPROLONGE
\epsfbox{\DDIR/prolonge.ps}
}}
\caption[Image extension]{ \Pstyle Image extension by symmetry and then periodization.}
\end{figure}
}

\def\PHISTOINT{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPHISTOINT
\epsfbox{\DDIR/HistoInt.ps}
}}
\caption[Histogram of an image]{\Pstyle Histogram of an image : for each $i \in \{0, 1,..., 255\}$, we display (above, right) the function $h(i) = Card\{\fxx, u(\fxx) = i \}$. The function below is the repartition function of $u$, $g(i) = Card \{ \fxx, u(\fxx) \leq i \}$, that is, a primitive of $h$. It gives an indication on  the overall contrast of the image and on the contrast change imposed by the sensor.  The inverse function $g^{-1}$ can be used as inverse contrast change, in order to restore an image $g^{-1}(u)$ with flat histogram. On the left column, original image and, below, the result of this "histogram equalization".}
\end{figure}
}


\def\PAFFIS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPAFFIS
\epsfbox{\DDIR/affineInfSup.ps}
}}
\caption[Affine ``inf-sup'']{ \Pstyle Checking the affine invariance of an affine "inf-sup". We display the final outcomes of the same comparison process  as in Figure \ref{affineamss},   with an affine ``infsup".
The chosen structuring  set is an approximately affine invariant set  $\setB$ of   49 ellipses, all with same area. This implementation is heavy and proves to be less affine invariant as the one obtained by a finite difference scheme. This is due to grid effects.}
\end{figure}
}

\def\PAFFAMSS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPAFFAMSS
\epsfbox{\DDIR/affineamss.ps}
}}
\caption[Affine invariance of (AMSS)]{ \Pstyle Affine invariance of (AMSS). We check the affine invariance of
the affine and morphological scale space (AMSS). A simple shape (top-left) is smoothed
using a finite differences discretization of (AMSS) followed by thresholding  (bottom-left). We
apply on the same shape an affine transform, with determinant equal to 1, (top-right),
then the same smoothing process (middle-right), and finally the inverse of the affine transform
(down-right). The final results of both processes are experimentally equal.  \label{affineamss}}
\end{figure}
}


\def\P3DVERT{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PP3DVERT
\epsfbox{\DDIR/3DVERT.ps}
}}
\caption[Median filtering of a three-dimensional image]{ \Pstyle Median filtering of a three-dimensional image.  First image : representation of the horizontal slices of a  3-D level-surface of the 3-D image of a vertebra. Right-left and up-down : 1, 2, 5, 10, 20, 30, 60, 100 iterations of a three-dimensional median filter, supported by a ball with radius 3. We have just proved that this scheme is a possible implementation of the mean curvature motion, originally proposed as such by Bence, Merriman and Osher.}
\end{figure}
}


\def\PMEDIANFIX{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPMEDIANFIX
\epsfbox{\DDIR/MedianFix.ps}
}}
\caption[Fixed point property of the digital median filter]{ \Pstyle
Fixed point property of the median filter, proving its grid-dependence. 
On the left, original image.  On the right, result of 46 iterations of the median filter with a radius of 2. The resulting image turns out to be a fixed point of the median filter with radius 2. This is not in agreement with Theorem \ref{B1} showing the median filter moving images by their curvature : the image on the right clearly has non zero curvatures !  Now, the discrete, pixelized median filter which  we have applied here is grid-dependent and blind to small curvatures.
}
\end{figure}
}


%%\def\PMEDIANITTOONE{
%%\begin{figure}[!ht]
%%\centerline{\hbox{
%%\epsfxsize=\PPMEDIANITTOONE
%%\epsfbox{\DDIR/MedianITTO2.ps}
%%}}
%%\caption[Median filter and iterated median filter I]{ \Pstyle
%%A SUPPRIMER
%%Comparaison between iterated median filter and median filter :
%%top-middle : 4 iterations of the median filter with a radius 3, top-right :
%%one iteration of the median filter with a radius 6. According to the %%proposition ?, they opper a smoothing the same scale. In practice, they %%differ.
%%(bottom : all the level-lines multiple of 16.). \label{medianitto1}
%%}
%%\end{figure}
%%}



\def\PMEDIANITTOTWO{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPMEDIANITTOTWO
\epsfbox{\DDIR/MedianITTO1.ps}
}}
\caption[Median filter and iterated median filter II]{ \Pstyle
Comparison between iterated median filter and median filter :
Top-middle : 16 iterations of the median filter with a radius 2, top-right :
one iteration of the same median filter with a radius 8. 
Below : for each image,  the level-lines for grey levels multiple of 16. To iterate a small size median filter results in more accuracy and less shape-mixing than to apply a large size median one.  See the comparison between the Koenderink-Van Dorn  shape smoothing and  the Bence-Merriman-Osher iterated filter in Chapter \ref{chapter2}, in particular Figures \ref{DS1}, \ref{DS2}, \ref{DS3} .
}
\end{figure}
}


\def\PSINGMORPUN{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPSINGMORPUN
\epsfbox{\DDIR/SinguMorp1.ps}
}}
\caption[Erosions and dilations can create singularities]{ \Pstyle
Erosions and dilations can create singularities. 
Top-left : original $C^\infty$ image, and below : representation of 
the intensity along the horizontal axis. Middle : dilation with a circle
of 30 pixels : at the central point, the image is no more $C^\infty$ or even $C^1$. Right : erosion with the same circle ; we also see some loss of regularity : the solution is no more $C^2$.
}
\end{figure}
}

\def\PSINGMORPTWO{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPSINGMORPTWO
\epsfbox{\DDIR/SinguMorp2.ps}
}}
\caption[Erosion and dilation can generate singularities within the level lines]{ \Pstyle Erosion and dilation can generate singularities within
the level-lines. Left : original image where the only level-line, corresponding to the boundary of the black shape,  is $C^2$. Right : erosion with a circle of radius 10 : the 
resulting level-lines have angles.
}
\end{figure}
}

\def\PILLUSROT{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPILLUSROT
\epsfbox{\DDIR/IllusRot.ps}
}}
\caption[A non euclidean invariant filter]{ \Pstyle 
An euclidean invariant filter and a non-euclidean one. The left frame contains simple shapes that can be deduced from each other by rotations. The middle image is the
closing of the left image by a horizontal rectangle of size 6*2 pixels. 
This non euclidean filter
produces different results, depending on the shapes orientations.
The right image is the closing of the left image by a circle of radius 4 pixels (same area up to the pixel precision as the rectangle used in the middle image). 
This filter is euclidean, therefore the resulting shapes can also be deduced from each other by rotations.
}
\end{figure}
}

\def\PILLUSMORPH{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPILLUSMORPH
\epsfbox{\DDIR/IllusMorph.ps}
}}
\caption[The heat equation creates structure]{ \Pstyle The heat equation creates structure. 
 This experiment shows that the linear scale-space can create structure, i.e. increase the complexity of the image. On the left : original synthetic image, three grey levels, two black regional minima, one grey regional maximum, one white regional maximum.  Middle : result when applying the heat equation : the grey regional minimum is split into three regional minima. Compare with the application of a contrast invariant local filter, the iterated median filter introduced in Chapter \ref{chapter8}.
}
\end{figure}
}


\def\PTEXTURESPX13{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPTEXTURESPX
\epsfbox{\DDIR/SPX13.ps}
}}
\caption[Hyperdiscrimination of textures by nonlinear scale space]{\Pstyle
Hyperdiscrimination of textures by nonlinear scale space.  According to Julesz' theory of textons, textures are discriminable  to the human perception if their average behaviour in terms of "texton" density is different.  As shown in its mathematical formalization, proposed by C. Lopez, some of the texton densities can be interpreted as  densities of the positive and negative parts of the image curvature at different scales. In this remarkable experiment, C. Lopez proved that one of the simplest contrast invariant scale spaces beats by far the human discrimination performance.  From left to right and up-down : 
\newline
1-an original preattentively undiscriminable texture pair.  The central  square of the image is made of rotated "10's" and the rest of rotated "S's".  Those patterns are different, but have the same number of bars, angles, etc.
\newline
2-curvature motion applied to the original up to some scale
3-negative part of the curvature at the same scale
4-positive part of the curvature at the same scale
5-multichannel segmentation of the multiimage made of the curvatures
6-negative part of the curvature at scale 0.
As seen in 2, 3, 4 and 5, this nonlinear scale space permits to discriminate easily the undiscriminable textures. 
}
\end{figure}
}

\def\PSKELTROIS{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPSKELTROIS
\epsfbox{\DDIR/3dskel03.ps}
}}
\caption[Three dimensional median filter]{\Pstyle
Three dimensional median filter. The original three dimensional image is made of 20 cuts of a vertebra.
We display on the left column three succesives cuts, and on the next column their level-lines with levels
multiple of 20. The next column shows these three cuts after one iteration
of the median filter done with a 3D ball of radius 2, and the resulting level-lines.
}
\end{figure}
}


\def\PMEDIANDEBRUITE{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=10cm
\epsfbox{\DDIR/MedianDB.ps}
}}
\caption[Median filter based denoising]{\Pstyle
Denoising properties of the median filter.
Left : image altered by a 40\% salt and pepper noise. Right : Three iterations of the median filter on a $3 \times 3$ square mask.
}
\end{figure}
} 


\def\PMEDBIRD{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=13cm
\epsfbox{\DDIR/MED_lacor.ps}
}}
\caption[Smoothing effect of the median filter on level lines]{\Pstyle
Smoothing effect of the median filter on level lines.
Above : original image, then  all of its level lines (boundaries of level sets) with levels multiple of 12, then level lines of level 100. Below : result of two iterations of a median filter on a disk with radius 2, then corresponding level lines (levels multiple of 12), then level lines with level 100.
}
\end{figure}
}


\def\PLIONEL1
{
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\epsfxsize=7cm
\epsfbox{\DDIR/lionel.eps} & & 
\epsfxsize=7cm
\epsfbox{\DDIR/lionel1.eps} \\
\epsfxsize=7cm
\epsfbox{\DDIR/lionel3.eps} & & 
\epsfxsize=7cm
\epsfbox{\DDIR/lionel6.eps} 
\end{tabular} 
\end{center}
\caption[Affine scale space of a ``hand'' curve]{\Pstyle Affine scale space of a ``hand'' curve, performed with the alternate affine erosion-dilation scheme.
(scales 0, 1, 20, 400). Experiment : Lionel Moisan.}
\end{figure}
}

\def\PER_EROSION
{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=\PPMEDCURV
\epsfbox{\DDIR/er_evol.ps}
}}
\caption[Erosion and motion of level lines]{\Pstyle 
The erosion  does a smoothing
independent of the curvature of the level lines. Left~: image of a simple
shape. Right~: difference of this image with its eroded by a ball with radius 6. We see in black the points which
have changed. The width of the difference is constant.
}
\end{figure}
}

\def\PAMSS
{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=11cm
\epsfbox{\DDIR/plancheAMSS.ps}
}}
\caption[Curvature motion finite difference scheme and scale calibration]{\Pstyle \label {curvaturemotiondifferencescheme} Curvature motion finite difference scheme and scale calibration. Image filtered by curvature motion at scales 1, 2, 3, 4, 5. In order to give a sound numerical meaning to the scale, a calibration of the numerical scales (number of iterations) is made in such a way that a disk with radius $t$ shrinks to a point at scale $t$.
}
\end{figure}
}

\def\PMED
{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=11cm
\epsfbox{\DDIR/plancheMED.ps}
}}
\caption[Curvature motion by median filtering]{\Pstyle Iterated median filter with approximately calibrated  scales 1, 1.5, 2, 2.5, 3.
}\end{figure}
}

\def\POSAMSS
{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=11cm
\epsfbox{\DDIR/plancheAMSS.ps}
}}
\caption[Curvature motion finite difference scheme applied on each level set separately (threshold superposition principle)]{\Pstyle Curvature motion finite difference scheme applied on each level set separately, at scales 1, 2, 3, 4, 5. The processed image is then reconstructed by the threshold superposition principle.  In contrast with the same scheme directly applied on the image, this scheme yields a fully contrast invariant smoothing. However, a comparison with  Figure \ref{curvaturemotiondifferencescheme} shows that the resulting images are very close to each  other. This shows that the contrast invariance is almost achieved when applying the finite difference scheme directly on the image. The experiment makes sense if the original image is of good quality, that is relatively smooth and with no strong oscillations. In that case, it can be considered as a distance function to each one of its own level sets.  As we shall see in Figure \ref {figure_amss_extrema}, if the initial image is noisy, the difference between both methods can be huge. }  
\end{figure}
}


\def\PMAXAMSS
{
\begin{figure}
\begin{center}
\epsfxsize=4.5cm
\begin{tabular}{cc}
\epsfbox{\DDIR/AMSS/BRUIT40.ps}
\end{tabular}
\begin{tabular}{cc}
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/AMSS2.ps} &
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/AMSS3.ps} \\
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/ZAMSS2.ps} &
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/ZAMSS3.ps} \\
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/KAMSS2.ps} &
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/KAMSS3.ps} \\
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/OSAMSS2.ps} &
\epsfxsize=4.5cm
\epsfbox{\DDIR/AMSS/OSAMSS3.ps} \\
\end{tabular} 
\end{center}
\caption[Various implementations of  curvature motion on noisy images]{\Pstyle \label{figure_amss_extrema} Various implementations of curvature motion on a noisy image. Middle : scale 2, Right : scale 3. From top to bottom : finite difference scheme (FDS), then FDS applied on the image previously zoomed by a factor 2, FDS applied on the image after its extrema have been "killed" (the reference area is  given by the area of the disk vanishing at the desired scale), FDS applied separately on each level set, with application of the threshold superposition principle. Time computations on a Pentium 200, for achieving scale 3, and per millions of pixels, are respectively from above to below : 23.75 s, 1 mn 32s, 24.1 s, 10 mins. The third scheme offers a good compromise between time computation and results.}
\end{figure}
}




\def\PAMSSAFFINE
{
\begin{figure}
\begin{center}
\epsfxsize=7.8cm
\epsfbox{\DDIR/AMSSAFF/ORIG.ps} 
\epsfxsize=3.8cm
\epsfbox{\DDIR/AMSSAFF/O3.ps}
\end{center}
\caption[Finite difference scheme for AMSS]{\Pstyle \label{figure_amss_affine_char} Finite difference scheme for AMSS. 
Left : Original noisy image of letters, middle : same image without the extrema with area less than 80 pixels (extrema killer).  Right : result of AMSS at calibrated scale 4.
 }
\end{figure}
}


\def\PMAMUN
{\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=12cm
\epsfbox{\DDIR/mam1.ps}
}}
\caption[AMG model (Affine, Morphological, Galilean) used for image sequence denoising]{\Pstyle AMG model (Affine, Morphological, Galilean) used for image sequence ``denoising''. Above : three successive images extracted from a sequence. Second row : resulting images at calibrated scale 100 pixels (scale at which a spatiotemporal sphere of 100 pixels disapears). Third row : Some noise has been added to the original sequence (25\% of the pixels are corrupted). Bottom : resulting images at scale 100.} \end{figure} }

\def\PMAMDEUX
{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=12cm
\epsfbox{\DDIR/mam2.ps}
}}
\caption[AMG  model used for image sequence ``denoising'', synthetic images]{\Pstyle The affine, morphological, galilean (AMG) model used for image sequence restoration (extraction of coherent trajectories). Above : three successive images extracted from a synthetic sequence, made of salt and pepper noise, plus some squares placed at random locations. In addition, a little black square in uniform motion has been added in the whole sequence. Bottom : resulting images at calibrated scale 500p (scale at which a spatio-temporal sphere of 500 pixels  disapears by AMG). Only the little black square remains, as it has a coherent motion.
} \label{syntheticsequencedenoised}
\end{figure}
}


\def\PEXTKILL
{
\begin{figure}[!ht]
\centerline{\hbox{
\epsfxsize=12cm
\epsfbox{\DDIR/ExtremaKill.ps}
}}
\caption[Extrema Killer]{\Pstyle  Extrema Killer : maxima killer followed by minima killer.  The extrema killer removes all connected components of upper and lower level sets with area less than some threshold, here equal to 20 pixels.}
\label{extremakiller}
\end{figure}
}

\def\PEXTBIRDKILL
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=5cm \epsfbox{\DDIR/AREAOPENINGBIRD/bird1z4.ps} &
\epsfxsize=5cm \epsfbox{\DDIR/AREAOPENINGBIRD/K20.ps} \\
\epsfxsize=5cm \epsfbox{\DDIR/AREAOPENINGBIRD/l4bird1z4.ps} &
\epsfxsize=5cm \epsfbox{\DDIR/AREAOPENINGBIRD/l4K20.ps} \\
\end{tabular}
\end{center}
\caption[Extrema Killer]{\Pstyle Extrema Killer : maxima killer followed by minima killer. Above, right: extrema killer of  the above, left image, 20 pixels area. Below : level lines (for levels multiple of 4) of the image before and after the application of the extrema killer. Notice the strong simplification of the topographic map : most digital images have many small oscillations generating extrema.}
\label{extremakiller2}
\end{figure}
}



\def\PDAMIERAMSS
{
\begin{figure}[!ht]
%%\centerline{\hbox{
\begin{center}
\epsfxsize=3.8cm
\epsfbox{\DDIR/DAMIER/damier.ps} 
\epsfxsize=3.8cm
\epsfbox{\DDIR/DAMIER/fdsmcm.ps} 
\epsfxsize=3.8cm
\epsfbox{\DDIR/DAMIER/usc_ext_damier.ps} 
\epsfxsize=3.8cm
\epsfbox{\DDIR/DAMIER/lsc_ext_damier.ps} 
\end{center}
%%}}
\caption[Evans-Spruck extensions and the chessboard dilemna]
{\label{damieretextensions}\Pstyle The chessboard dilemna.  Left: chessboard image. Next: result with the finite difference scheme (FDS, Chapter \ref{chapter22}) of the curvature motion, applied up to a fixed scale. The creation of a new gray level proves that the scheme is not fully contrast invariant. Indeed, by Proposition \ref{morphoimpliquecaracteristique}, a contrast invariant operator does not create new levels. The new observed gray level corresponds to an average  of the existing ones, black and white. The next two images 
are obtained by the Evans-Spruck extension of the curvature motion, first under the assumption that the image is u.s.c. and second under the l.s.c. assumption. Thus, the schemes are in both cases  fully contrast invariant and are extensions of the curvature motion as specified in Theorem \ref {extensionapleinementmorpho}.}
\end{figure}
}


\def\PMODEFINDING
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=7.8cm
\epsfbox{\DDIR/ModeFinding/mode2.ps}&
\epsfxsize=7.8cm
\epsfbox{\DDIR/ModeFinding/Hmode2.ps}
\end{tabular}
\end{center}
\caption[Multiscale histogram modes  by using the linear scale space (heat equation)]{\Pstyle Multiscale histogram modes  by using the linear scale space (heat equation).  Histograms are strongly oscillatory functions, presenting many peaks due to flat image regions and quantization effects.  We call {\it modes}of the histogram all the intervals whose endpoints are two successive local minima. We call central point of the mode the unique maximal point between them.  In order to get a more and more global analysis of the histogram, the heat equation may be applied. We then talk about "modes at scale $t$".  On the left, we display the original histogram (scale 0) and the histogram at scale 100, corresponding to a convolution with the gaussian of standard deviation 50. Only three modes are left.  On the right, we display the so-called "fingerprints" of the histogram, a scale-space representation where scale increases downwards. The white curves display the minima of the smoothed histogram and the black curves the maxima of the histogram.  By a classical property of the heat equation, maxima and minima collapse by pairs, thus yielding the observed "fingerprint" organization of multiscale modes : each mode at scale $t$ can be followed upwards to the fine scales and contains more and more submodes.  Modes are therefore organized in a tree.  No new mode is ever created when the scale increases. This is an instance where the Witkin linear scale space works nicely !
}
\end{figure}
}

\def\PMACKWONE
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW1/biz0.ps}&
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW1/biz15.ps} \\
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW1/biz30.ps}&
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW1/biz60.ps}
\end{tabular}
\end{center}
\label{MACKW1L}
\caption[Curve evolution by the heat equation.]{\Pstyle Curve evolution by the heat equation. The coordinates of the curves are parameterized by the arc length, and then smoothed as real functions of the length, by applying to them the heat equation. From A to D, the coordinates are smoothed with an increasing scale. Each coordinate function therefore is $C^\infty$ ;  the evolving curve can, all the same, generate self-crossings (C) or singularities (D).  }
\end{figure}
}


\def\PMACKWTWO
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW2/biz0.ps}&
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW2/biz15.ps} \\
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW2/biz30.ps}&
\epsfxsize=6cm
\epsfbox{\DDIR/PMACKW2/biz60.ps}
\end{tabular}
\end{center}
\label{MACKW2L}
\caption[Curve evolution by the renormalized heat equation.]{\Pstyle Curve evolution by the `` renormalized heat equation'' (Mackworth-Mokhtarian). At each smoothing step, the coordinates of the curves are reparameterized by the arc length of the smoothed curve. From A to D, the curve is smoothed with an increasing scale. Note that, in contrast with the linear heat equation (Figure \ref{MACKW1L}), the evolving curve shows no singularities and does not cross itself. }
\end{figure}
}


\def\PKVDBAD
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=4cm
\epsfbox{\DDIR/KVD_bad/twocircle.ps} &
\epsfxsize=4cm
\epsfbox{\DDIR/KVD_bad/SM1.ps} \vspace{-1cm} \\
\epsfxsize=4cm
\epsfbox{\DDIR/KVD_bad/SSM2.ps} &
\epsfxsize=4cm
\epsfbox{\DDIR/KVD_bad/SSM3.ps} \vspace{-1cm}
\end{tabular}
\end{center}
\label{KVDBAD}
\caption[Dynamic Shape and non local interactions.]{\Pstyle \label{DS1} Non local interactions in the Dynamic Shape method.  Two close disks (top-left) interact as scale grows and create a qualitatively different and new shape. The change of topology, at the scale where  both  shapes merge into one, entails the appearance of a singularity (a cusp) on  the shape(s) boundaries.
Top right : Dynamic Shape evolution at the small critical scale where the cusp appears. Bottom : further
evolution when scale grows : creation of a new global shape.}
\end{figure}
}


\def\PKVDSTRANGE
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{ccc}
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/strange.ps} &
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/SM.ps} &
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/SM2.ps} \vspace{-1cm}  \\
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/SM3.ps} &
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/SM4.ps} &
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/SM5.ps}  \vspace{-1cm}
\end{tabular}
\end{center}
\label{KVDSTRANGE}
\caption[Dynamic Shape, nonlocal behaviour of shapes.]{\Pstyle \label{DS2}  Nonlocal behaviour of shapes with the Dynamic Shape method.  This image displays the smoothing of two irregular shapes by the Dynamic Shape  method (Koenderink and Van Dorn).  Top left : initial image, made of two irregular shapes. Top to bottom and left to right: smoothing with increasing scales.  Notice how, the convolution being made with
gaussians of increasing variance, the shapes merge  more and more. We do not have 
a separate analyis of both shapes but a ``joint analysis'' of both. This joint analysis depends of course a lot upon the initial distance between both shapes. 
}
\end{figure}
}

\def\PKVDLOCALIZED
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{ccc}
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/strange.ps}&
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/AM1.ps} &
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/AM2.ps} \vspace{-1cm}  \\
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/AM3.ps} &
\epsfxsize=4cm \hspace{-1cm}
\epsfbox{\DDIR/KVDstrange/AM4.ps}&
\epsfxsize=4cm \hspace{-1cm} 
\epsfbox{\DDIR/KVDstrange/AM5.ps}  \vspace{-1cm} 
\end{tabular}
\end{center}
\label{KVDLOCALIZED}
\caption[Localized, iterated version of the Dynamic Shape.]{\Pstyle \label{DS3} The Bence-Merriman-Osher shape smoothing method is a localized and iterated version of the Dynamic Shape
method.  A convolution of the binary image with small-sized gaussians is alternated with mid-level thresholding.  It uses (top, left) as initial data the same shapes as in Figure \ref{KVDSTRANGE}.
 From top to bottom and left to right: smoothing with increasing scales. Notice that the 
shapes keep separate. In fact, their is no interaction between their evolutions. Each
one evolves as it would do alone.
}
\end{figure}
}

%%\def\PKramer
%%{
%%\begin{figure}[!ht]
%%\begin{center}
%%\begin{tabular}{ccc}
%%\epsfxsize=5cm 
%%\epsfbox{\DDIR/Kramer/b4.ps} & 
%%\epsfxsize=5cm
%%\epsfbox{\DDIR/Kramer/blur.ps} &
%% \epsfxsize=5cm
%%\epsfbox{\DDIR/Kramer/deblur.ps}
%%\end{tabular}
%%\end{center}
%%\label{PPKRAMER}
%%\caption[Shock Filter implemented by using a non flat morphogical %%filter.]{\Pstyle Shock Filter implemented by using a non flat morphogical %%filter. From left to right: original image, blurred image using Heat Equation, %%result of the shock filter.}
%%  kramer in out 0.5 2
%%\end{figure}
%%}


\def\POshKramer
{
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=7cm 
\epsfbox{\DDIR/KRAMER2/victor2.ps} & 
\epsfxsize=7cm
\epsfbox{\DDIR/KRAMER2/blur.ps} \\
 \epsfxsize=7cm
\epsfbox{\DDIR/KRAMER2/kram2.ps} &
 \epsfxsize=7cm
\epsfbox{\DDIR/KRAMER2/Osh2.ps} \\
\end{tabular}
\begin{tabular}{ccc}
\epsfxsize=4.5cm 
\epsfbox{\DDIR/KRAMER2/Cvic.ps} & 
\epsfxsize=4.5cm 
\epsfbox{\DDIR/KRAMER2/Ckram2.ps} & 
\epsfxsize=4.5cm 
\epsfbox{\DDIR/KRAMER2/COsh2.ps} 
\end{tabular}
\end{center}
\label{PPOSHKRAMER}
\caption[Shock Filter implemented by using non flat morphogical 
filters.]{\Pstyle Shock Filter implemented by using  non flat morphogical 
filters. Top, left :original image, right: blurred image using Heat Equation,
Middle-left: two iterations of the kramer filter, Middle-right: two iterations 
of the Rudin-Osher filter. The scale parameter is chosen such that the parabola
 passes the range of the image at a distance of 6 pixels.
Down: zoom version of a detail, left: original image, middle: kramer filter, 
right: Rudin-Osher filter. We see a tendancy of this last to smooth shapes 
toward circles. }
%%  kramer in out 0.5 2
\end{figure}
}

%%666666666666666666666666666666666666666666666666666666

%%DESCRIPTIONS DES FIGURES DU LIVRE

%%999999999999999999999999999999999999999999999999999999

\def\MEDIANFILTER{
\begin{figure}
\centerline{\hbox{
\epsfxsize=12cm
\epsfbox{\FDIR/Medianfilter.ps}
}}
\caption[Fig :]{\Pstyle 
When $h$ is small, the parabola $P(h)$ with equation $x+ hy^2 = m$
divides $D(0,1)$ into two connected components. The
median value $m(h)$ of $x+ hy^2$ on $D(0,1)$  simply is the value $m$ for which these
two connected components have equal area.}

{.  \label{figuremedianfilter}}
\end{figure}
}


\def\WMEDIANFIG{
\begin{figure}
\centerline{\hbox{
\epsfxsize=12cm
\epsfbox{\FDIR/Wmedian.eps}
}}
\caption[Fig : Discretization of a median filter]{\Pstyle Discretization of a median filter.  The digital image (above,left) is considered as constant on each square pixel. The support of the illustrated median filter is a square centered at zero. The subfigure above, left, shows the intersections of the nine neighboring pixels with the mask.  The values correspond to the areas of those intersections.
 These values are used as weights to the pixel values around zero. They permit to build the weighted histogram of the neighborhood of  the central pixel. In this histogram (below), the  values 0 and 2 appear once with weight 0.5, the value 1 appears once with weight 0.25, the value 2 does not appear, the value
4 appears once with weight 1 and once with weight 0.25, resulting in a histogram value of 1.25, etc. The obtained median value is 4.

} \label{Wmedianfig}
\end{figure}
}



%\def\MONOGRAPH{
%\begin{figure}
%\centerline{\hbox{
%\epsfxsize=12cm
%\epsfbox{\FDIR/MonoGraph.ps}
%}}
%\caption{\Pstyle A monotone graph G and its inverse graph $G^{-1}$}
%\end{figure}
%}


\def\MEDIANITERE{
\begin{figure}
\centerline{\hbox{
\epsfxsize=6cm
\epsfbox{\FDIR/MedianItere.ps}
}}
\caption[Fig : Illustrating the inequality  (\ref{median1})]{\Pstyle
Illustrating the inequality  (\ref{median1}).
}
\end{figure}
}

%CETTE FIGURE SEMBLE AVOIR DISPARU :

\def\LOCALITY{
\begin{figure}
\centerline{\hbox{
\epsfxsize=13cm
\epsfbox{\FDIR/Locality.ps}
}}
\caption[Fig : Affine invariance and locality]{\Pstyle Affine invariance and locality.}
\end{figure}
}

\def\LOCINCLUS
{
\begin{figure}
\centerline{\hbox{
\epsfxsize=10cm
\epsfbox{\FDIR/local.ps}
}}
\caption[Fig : Local Inclusion Principle]{\Pstyle Local Inclusion Principle (Definition \ref{definitionshapelocalinclusionprinciple}).}
\end{figure}
}

\def\AFFINEITERE
{
\begin{figure}
\centerline{\hbox{
\epsfxsize=6cm
\epsfbox{\FDIR/affi.ps}
}}
\caption[Fig : Illustration of the proof of the inequality (\ref{ iteratedaffine1})]{\Pstyle Illustration of the proof of the inequality (\ref{ iteratedaffine1}).}
\end{figure}
}

\def\PPyramide
{
\begin{figure}
\centerline{\hbox{
\epsfxsize=6cm
\epsfbox{\FDIR/pyramide.ps}
}}
\caption[Fig : The scale-space visual pyramid]{\Pstyle  The scale-space visual pyramid : perception is thought of as a flow of images passing through transition operators $T_{t,s}$. These operators receive an image previously analysed at scale $s$ and deliver an image analysed at a larger scale $t$. The scale $t=0$ corresponds to the original percept.  In this simple model, the perception process is irreversible (no feedback from coarse scales to fine scales).}
\end{figure}

}

\def\Pstencil{
\begin{figure}
\centerline{\hbox{
\epsfxsize=3.5cm
\epsfbox{\FDIR/stencil2.ps}
}}
\caption[Fig : A 3x3 stencil]{A 3x3 stencil}
\label{STENCIL}
\end{figure}
}

\def\PCHORDARCSET{
\begin{figure}
\centerline{\hbox{
\epsfxsize=5cm
\epsfbox{\FDIR/chordarcset.eps}
}}
\caption[Illustration of the Corollary \ref{erosionaffinefacile}]{\Pstyle \label{erosionaffinefacilefigure} Illustration of the Corollary \ref{erosionaffinefacile}.}
\end{figure}
}

\def\PINTRINSICCOORD{
\begin{figure}
\centerline{\hbox{
\epsfxsize=4cm
\epsfbox{\FDIR/intrinsicch.eps}
}}
\caption[Intrinsic coordinates]{\Pstyle Intrinsic coordinates}
\end{figure}
}

\def\PAPPROXTHRESHOLD{
\begin{figure}
\centerline{\hbox{
\epsfxsize=10cm
\epsfbox{\FDIR/ThresholdAp.eps}
}}
\caption[Approximation of the threshold function]{\Pstyle Approximation of the
threshold function $\gamma_\lambda$ from above and from below}
\end{figure}
}

%%�����������������������������������������

%%FIGURES DU CHAPITRE MOVIE SCALE SPACE :

%%�����������������������������������������

\def\Pfigtrois{
\begin{figure}
\centerline{\hbox{\epsfxsize=8cm \epsfbox{\FDIR/mvfigure3.eps} }}
\caption[Fig : The AMG model erodes a circle in acceleration only only on one side.]{\Pstyle The AMG model erodes a circle in acceleration only on one side. Indeed, 
when the curvature and the acceleration have opposite signs, the evolution in scale is zero. (see the AMG equation).}
\end{figure}
}



\def\Pfigquatre{
\begin{figure}
\centerline{\hbox{\epsfxsize=8cm \epsfbox{\FDIR/mvfigure4.eps} }}
\label{saddle}
\caption[Fig : Saddle points of level surfaces are not moved by the AMG model.]{\Pstyle Saddle points of level surfaces remain steady by the AMG model. Indeed, our scale-space can be seen as a motion in scale of gray level-surfaces (isophotes). The level-surfaces that are straight in time  correspond to a uniform translation and are  not changed by the smoothing. Therefore, the two thin cylindric level-surfaces drawn left and right in the  figures above do not move in scale.
Now, by the inclusion principle, two level surfaces can never cross during the evolution
in scale. Since, as displayed in the picture, it is possible to squeeze any surface saddle point  between two such steady cylinders,  it follows that saddle points do not move in scale as well. This property is readable in the scale space equation : at saddle points, the positive part of the product of the curvature and of the acceleration is zero.}
\end{figure}
}

\def\Pfigcinq{\begin{figure}
\centerline{\hbox{\epsfxsize=4.5cm \epsfbox{\FDIR/mvfigure5.eps} }}
\label{traject}
\caption[Fig : The AMG model  removes the rotating trajectories]{\Pstyle The level-surface in gray cannot become straight~: it would have to
cross the white level-surface which is invariant by the scale space. Now, during the smoothing process, the level-surface in gray will be eroded on  its convex part, and will eventually disappear at a fixed scale : it cannot converge to any steady surface since all of them  are straight in time.
Thus, trajectories that are contained in the grey level surface end being removed from the movie.   }
\end{figure}
}


\def\Pflow{
\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\epsfxsize=4.5cm \epsfbox{\FDIR/flow1.ps} &
\epsfxsize=4.5cm \epsfbox{\FDIR/flow2.ps} \\
\end{tabular}
\end{center}
\caption[Fig : Optical flow definition]{\Pstyle According to the optical flow definition, all above drawn velocity vectors are possible, since they allow the moving point to  
remain on the same level surface. One possibility to get rid of this ambiguity is choose as velocity the  direction which does not change the orientation of the
spatial gradient.}
\end{figure}
}


%%\def\Pflaw{\begin{figure}[htb2]
%%\centerline{\hbox{\epsfxsize=4cm \epsfbox{\FDIR/flow2.ps} }} 
%%\caption[Fig : Optical flow definition II]{\Pstyle We choose the direction which does not change the orientation of the
%%spatial gradient.}
%%\end{figure}
%%}

\input{\FDIR/graph.tex}










