Voronoi-based segmentation on image manifolds

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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

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propagate(x, seeds, mask=NULL, lambda=1e-4)

Arguments

x

An Image object or an array, containing the image to segment.

seeds

An Image object or an array, containing the seeding objects of the already identified regions.

mask

An optional Image object or an array, containing the binary image mask of the regions that can be segmented. If missing, the whole image is segmented.

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 x. See details.

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:

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d^2 = ( (t(dv)*g)^2 + lambda*t(dv)*dv )/(lambda + 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

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  ## 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)