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  \noindent
  cs294-2, Grouping and Recognition (Prof. Jitendra Malik) \hfill #3\\
  Lecture \##1 (#2) \hfill {\bf DRAFT} Notes by #4\\[-.05in]
  \mbox{}\hrulefill \mbox{}\\}

\begin{document}

% MODIFY THE FOLLOWING TO INCLUDE THE LECTURE #, TOPIC, DATE, and YOUR NAME
\lecturetitle{2}{Introduction}{Jan 25, 1999}{Alex Berg}

% INCLUDE AN OUTLINE OF THE CLASS TOPICS
\begin{itemize} 
  \item Grouping (continuation from last lecture)

  \item Figure / Ground  or Depth Ordering

  \item History

\end{itemize}

% THE NOTES WILL GO HERE

{\bf In the last lecture} we had three related tasks, grouping, figure
ground discrimination, and recognition.

\section{Grouping (continued)}

  The idea is to use various cues to partition an image into groups.
  One motivation here is to organize the overwhelming data of an image
  into coherent simple groups.  We attempt to motivate these cues in
  terms of image statistics.

\begin{description}
\item[Similarity] 

  Various features determine the similarity of two areas: brightness,
  texture, motion, and stereo disparity or depth.  These were covered in
  the previous lecture.

\item[Proximity] 
  Grouping by proximity is based on the idea that things which project
  to nearby points on the image are usually close to each other in the
  world.  One could probably consider this another form of similarity.

\item[Contour Continuation] 

  Here we concentrate on the contour or boundary of a group.  In
  particular the following images motivate the concept of boundaries
  which are not apparent from changes in image intensity.  In fact the
  apparent regions formed by these contours cold not be separated by
  any of the similarity clues above.  These apparent boundaries are
  called {\em subjective contours}, {\em illusory contours}, {\em
  imagined boundaries}, or {\em gradient less edges}.  Figures
  \ref{fig:kanisza1}, \ref{fig:kanisza2}, and \ref{fig:kanisza3} show
  examples of subjective contours and are taken from a paper by Kanisza.

\begin{figure}
\centerline{\psfig{file=kanisza2.eps,height=2in}}
\caption{Illusory Contours}
\label{fig:kanisza1}
\end{figure}

\begin{figure}
\centerline{\psfig{file=kanisza1.eps,height=2in}}
\caption{Illusory Contours}
\label{fig:kanisza2}
\end{figure}

\begin{figure}
\centerline{\psfig{file=kanisza3.eps,height=2in}}
\caption{Terminators and Illusory Contours}
\label{fig:kanisza3}
\end{figure}



  We want to make up a story in terms of image statistics to explain
  this phenomenon.  Basically one can imagine the images are the
  result of a white object occluding the dark disks and lines.
  Examining the features, we might argue that it is unlikely the
  apparent boundaries would line up so smoothly by chance, and
  therefore we ``see'' an occluding object that may or may not in
  actuality be present.  Later on we will look at other cues for
  grouping and for figure/ground determination which might also help
  explain this phenomenon.


\item[Closure]
  
  The physical world consists of objects which are usually seen in
  images as closed contours.  It therefore makes sense for there to be
  a preference for perceiving contours which close a region.

\item[Symmetry and Parallelism]
  
  Many shapes in the physical world are somewhat axisymmetric, most
  notably surfaces of revolution.  As a result their outlines have a
  bilateral symmetry.  Furthermore it is unlikely that there is such
  symmetry by chance.  So we associate symmetric contours as in
  figure~\ref{fig:symmetry1} and tend to associate the area between
  them as a region.

\begin{figure}
\centerline{\psfig{file=symmetry1.eps,height=2in}}
\caption{Symmetry}
\label{fig:symmetry1}
\end{figure}

\item[Familiar Configuration]

  Faces are a good example.  Dark regions for the eyes and mouth, and
  a shadow for the nose or nostril.  We see faces everywhere, even the
  moon.

\end{description}


  Before venturing onto figure/ground discrimination we take another
  look at the relationship between contours and regions.  Many
  segmentation or grouping algorithms use one or the other, but in the
  end the two are more or less equivalent.  Finding the contours that
  bound regions, or finding the regions inside contours, are really
  the same problem.

  The reason we are interested in regions and contours in images is
  that interpretations about this {\em proximal} information (at the
  image plane) provides information about {\em distal}
  features (out in the world).  In particular regions in the image
  correspond to surfaces of objects in the real world.  This points
  out one relationship between grouping and figure/ground
  determination.



\section{Figure / Ground}

  A single image contains a great deal of information about depth
  ordering.  One way to see this is to have yourself blindfolded,
  dragged to a foreign place, and when you are un-blindfolded
  carefully open one eye.  You will probably be able to say a great
  deal... about the depth ordering of the objects you see, even
  without opening the other eye.  We will focus on information which
  helps to determine depth ordering.


\begin{description}

\item[T-Junctions] Actually other junctions can give you information
about figure/ground relationships, but, as shown in figure~\ref{fig:junctions},
t-junctions often indicate which object is in front.  At the same time
the snake shows how this may not be the case.  There the
t-junctions result from texture on the figure, not occlusion.

\begin{figure}
\centerline{\psfig{file=figures2.eps,height=3in}}
\caption{Various junctions}
\label{fig:junctions}
\end{figure}


\item[Increasing Texture Gradient] For textured objects, at an
apparent boundary which does not result from occlusion the texture
gets denser as it is foreshortened.  This can give information about
which object is not occluded and therefore closer.  These same boundaries can result in t-junctions as shown before, and if the background color of the object matches that of the surface behind it, we get terminators.

\item[Convexity] What makes a closed contour look like the outline of
a small object instead of a small hole in a very large object?
Digression about some ancient idea that the night sky was in fact a
very large dark sheet with small holes through to a bright place,
causing people to see stars.  In any event people seem to prefer
explanations with fewer holes.  We can operationalize this preference
by locally assuming that the convex side of a curve is the foreground.
Going back to the illusory contours example we see that this effect
might be helping the illusion somewhat.

\item[Symmetry] By the same argument we used for grouping symmetry can
be helpful. I included the following two examples in
figure~\ref{fig:symmetry2} for fun.  In one the symmetry and convexity
clues agree, and in the other they do not.

\begin{figure}
\centerline{\psfig{file=symmetry2.eps,height=2in}}
\caption{Convexity vs. Symmetry}
\label{fig:symmetry2}
\end{figure}

\item[Occlusion] Occlusion information from stereo can indicate which
object is in front.  Sequence of images of an object in motion can also give depth ordering information.


\end{description}

  
\section{History}

\begin{quote}
``I stand at the window and see a house, trees, sky. Theoretically I
might say there were 327 brightnesses and nuances of color. Do I have
`327'? No. I have sky, house, and trees.''  {\em Max Wertheimer}
\end{quote}

Max Wertheimer worked on grouping around 1923 in Germany.  He launched
the gestalt movement whose motto was, ``The whole is different than
the sum of the parts.''  Wolfgang Kohler, Kurt Kofka, and Hanz Wallach
(aperture problem 1935) were also members of the gestalt movement.
World War II caused them to emigrate from Germany to the United
States.  Their work was less mainstream there and as a result the
gestalt movement ended.  Recently there has been renewed interest in
their work.

Egon Brunswick worked on relating probabilistic inference to the
gestalt principles (probabilistic functionalism) around 1947- mid
1950's.  His work suffered from poor background in statistics and the
lack of computers to do the large amount of computation necessary to
analyze the statistics of images.  He did manage to convince graduate
students to analyze a few images to verify that proximity was in fact a
good cue for grouping.

Edgar Rubin worked on the figure ground distinction in 1915 including the figure vase illusion.


{\bf Next lecture} will look at grouping from similarity of
brightness.  Given image intensity as a function $I(x,y)$ the goal is
to divide it into piecewise smooth patches.  These patches and the
boundaries between them are the result.  Three methods will be
examined, maximum likelihood, Bayesian inference, and spectral graph
theory.

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