medpy.graphcut.energy_label.boundary_stawiaski

medpy.graphcut.energy_label.boundary_stawiaski(graph, label_image, gradient_image)

Boundary term based on the sum of border voxel pairs differences.

An implementation of the boundary term in [1], suitable to be used with the graph_from_labels function.

Determines for each two supplied regions the voxels forming their border assuming \(ndim*2\)-connectedness (e.g. \(3*2=6\) for 3D). From the gradient magnitude values of each end-point voxel the border-voxel pairs, the highest one is selected and passed to a strictly positive and decreasing function \(g(x)\), which is defined as:

\[g(x) = \left(\frac{1}{1+|x|}\right)^k\]

,where \(k=2\). The final weight \(w_{i,j}\) between two regions \(r_i\) and \(r_j\) is then determined by the sum of all these neighbour values:

\[w_{i,j} = \sum_{e_{m,n}\in F_{(r_i,r_j)}}g(\max(|I(m)|,|I(n)|))\]

, where \(F_{(r_i,r_j)}\) is the set of border voxel-pairs \(e_{m,n}\) between the regions \(r_i\) and \(r_j\) and \(|I(p)|\) the absolute of the gradient magnitude at the voxel \(p\)

This boundary_function works as an edge indicator in the original image. In simpler words the weight (and therefore the energy) is obtained by summing the local contrast along the boundaries between two regions.

Parameters:

graphGCGraph

The graph to add the weights to.

label_imagendarray

The label image. Must contain consecutively labelled regions starting from index 1.

gradient_imagendarray

The gradient image.

Notes

This function requires the gradient magnitude image of the original image to be passed along. That means that graph_from_labels has to be called with boundary_term_args set to the gradient image. This can be obtained e.g. with generic_gradient_magnitude and prewitt from scipy.ndimage.

This function is tested on 2D and 3D images and theoretically works for all dimensionalities.

References

[1]

Stawiaski J., Decenciere E., Bidlaut F. “Interactive Liver Tumor Segmentation Using Graph-cuts and watershed” MICCAI 2008 participation