\HeaderA{propagate}{Voronoi-based segmentation on image manifolds}{propagate}
\begin{Description}\relax
Find boundaries between adjacent regions in an image, where seeds
have been already identified in the individual regions to be
segmented. The method finds the Voronoi region of each seed on
a manifold with a metric controlled by local image properties.
The method is motivated by the problem of finding the borders of
cells in microscopy images, given a labelling of the nuclei
in the images.

Algorithm and implementation are from Jones et al. [1].
\end{Description}
\begin{Usage}
\begin{verbatim}
  propagate(x, seeds, mask=NULL, lambda=1e-4, ext, seed.centers)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] An \code{Image} object or an array, containing the image to segment.
\item[\code{seeds}] An \code{Image} object or an array, containing the seeding objects of
the already identified regions.
\item[\code{mask}] An optional \code{Image} object or an array, containing
the binary image mask of the regions that can be segmented. If missing, 
the whole image is segmented.
\item[\code{lambda}] A numeric value. The regularisation parameter used in the
metric, determining the trade-off between the Euclidian distance in the
image plane and the contribution of the gradient of \code{x}. See details.
\item[\code{ext}] Deprecated.
\item[\code{seed.centers}] Deprecated.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The method operates by computing a discretized approximation of the
Voronoi regions for given seed points on a Riemann manifold with a
metric controlled by local image features.

Under this metric, the infinitesimal distance d between points
v and v+dv is defined by: 
\begin{alltt}d\textasciicircum{}2 = ( (t(dv)*g)\textasciicircum{}2 + lambda*t(dv)*dv )/(lambda + 1)\end{alltt},
where g is the gradient of image \code{x} at point v.

\code{lambda} controls the weight of the Euclidian distance term. 
When \code{lambda} tends to infinity, d tends to the Euclidian
distance. When \code{lambda} tends to 0, d tends to the intensity
gradient of the image.

To avoid to rely too much on single noisy pixels, the gradient is
computed on a neighborhood of 3x3 pixels.

Segmentation of the Voronoi regions in the vicinity of flat areas
(having a null gradient) with small values of \code{lambda} can
suffer from artefacts coming from the metric approximation.
\end{Details}
\begin{Value}
An \code{Image} object or an array, containing the labelled objects.
\end{Value}
\begin{Section}{License}
The implementation is based on CellProfiler C++ source code [2, 3].
An LGPL license was granted by Thouis Jones to use this part of CellProfiler's code for the \code{propagate} function.
\end{Section}
\begin{Author}\relax
Original CellProfiler code: Anne Carpenter \textless{}carpenter@wi.mit.edu\textgreater{},
Thouis Jones \textless{}thouis@csail.mit.edu\textgreater{}, In Han Kang \textless{}inthek@mit.edu\textgreater{}.

Port for this package: Oleg Sklyar, Gregoire Pau, Wolfgang Huber
\end{Author}
\begin{References}\relax
[1] T. Jones, A. Carpenter and P. Golland,
"Voronoi-Based Segmentation of Cells on Image Manifolds",
CVBIA05 (535-543), 2005

[2] A. Carpenter, T.R. Jones, M.R. Lamprecht, C. Clarke, I.H. Kang,
O. Friman, D. Guertin, J.H. Chang, R.A. Lindquist, J. Moffat,
P. Golland and D.M. Sabatini, "CellProfiler: image analysis software
for identifying and quantifying cell phenotypes", Genome Biology 2006, 7:R100

[3] CellProfiler: http://www.cellprofiler.org
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{bwlabel}{bwlabel}}, \code{\LinkA{watershed}{watershed}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## a paraboloid mountain in a plane
n = 400
x = (n/4)^2 - matrix(
       (rep(1:n, times=n) - n/2)^2 + (rep(1:n, each=n) - n/2)^2,
       nrow=n, ncol=n)
x = normalize(x)

## 4 seeds
seeds = array(0, dim=c(n,n))
seeds[51:55, 301:305] = 1
seeds[301:305, 101:105] = 2
seeds[201:205, 141:145] = 3
seeds[331:335, 351:355] = 4

lambda = 10^seq(-8, -1, by=1)
segmented = Image(dim=c(dim(x), length(lambda)))

for(i in seq(along=lambda)) {
  prop = propagate(x, seeds, lambda=lambda[i])
  prop = prop/max(prop)
  segmented[,,i] = prop
}

if(interactive()){
  display(x, title='Image')
  display(seeds/max(seeds), title='Seeds')
  display(segmented, title="Voronoi regions")
}

\end{ExampleCode}
\end{Examples}

