# Voronoi-based segmentation on image manifolds

### Description

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].

### Usage

1 |

### Arguments

`x` |
An |

`seeds` |
An |

`mask` |
An optional |

`lambda` |
A numeric value. The regularization parameter used in the
metric, determining the trade-off between the Euclidean distance in the
image plane and the contribution of the gradient of |

### Details

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:

1 |

,
where g is the gradient of image `x`

at point v.

`lambda`

controls the weight of the Euclidean distance term.
When `lambda`

tends to infinity, d tends to the Euclidean
distance. When `lambda`

tends to 0, d tends to the intensity
gradient of the image.

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 `lambda`

can
suffer from artifacts coming from the metric approximation.

### Value

An `Image`

object or an array, containing the labelled objects.

### 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 `propagate`

function.

### Author(s)

The original CellProfiler code is from Anne Carpenter <carpenter@wi.mit.edu>, Thouis Jones <thouis@csail.mit.edu>, In Han Kang <inthek@mit.edu>. Responsible for this implementation: Greg Pau.

### References

[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

### See Also

`bwlabel`

, `watershed`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 | ```
## 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
}
display(x, title='Image')
display(seeds/max(seeds), title='Seeds')
display(segmented, title="Voronoi regions", all=TRUE)
``` |