
Hi All,

Sorry I have been out of touch...  I was down in NY this weekend
helping my parents out with some unforeseen difficulties.

Regarding Ramin's draft, which looks great.

*) There is at least one previous reference to silhouette
   reconstruction using smoothness, by Ponce.  His reconstruction
   technique is entirely analytic (no voxels at all).  Each silhouette
   is a constraint on the tangents of a smooth analytic surface.  In
   the end the reconstuction approach is very, very hairy.  He also 
   mumbles a bit about using these models in various ways...  All 
   of the objects he reconstructs are very smooth.  
   (I will write to him to ask him what he thinks.  Perhaps he can get
   us some additional references.)

*) I too think we are doing something a little funny with
   probabilities here.

  If each voxel gives rise to independent errors, we have $\Pr(O|f) =
  \prod_{v\in\calV} \Pr(O|f_v)$.  We will assume that a voxel $v$ only
  affects the observed intensities at those pixels that intersect it.
  Formally, let us define the set of intensity
  differences $O(v) = \suchthat{\Delta(p)}{p \in \calP, v\in V(p)}$.  Our
  assumption means that 
  \begin{equation}
  \Pr(O|f) = \prod_{v \in \calV} \Pr(O(v)|f_v).
  \label{eq:likelihood}
  \end{equation}

These steps are often done in reverse order.  First you argue that
given the true state of the world, $f$, the observations are
independent (usually because of noise... but that is not quite right
here because the differences are due to the color of the object and
backgroud).

  \Pr(O|f) = \prod_{v \in \calV} \Pr(O(v)|f)

Then you argue that the observation of a voxel $v$ is not dependent on
the other voxels (but that is not quite right here either because of
occlusion).

  \Pr(O|f) = \prod_{v \in \calV} \Pr(O(v)|f_v)

In the end you get the answer you want with the awkward step above...
but there are still problems.


