# ‘Weighted’ distance transform

A distance transform (as done by DistanceTransform[]) on a scattering of background pixels effectively creates a landscape of basins with conical morphology, each one starting from the same depth (zero) and with the same slope (one). The ridges between the basins are therefore equidistant between the minima, which is why a watershed transform after a distance transform approximates a Voroni diagram.

What I want to do is create a ‘basin landscape’ with the same slope in each basin but from minima of different depth, so that a ridge between two minima would be closer to the shallower minimum.

Does anyone now if there is an established, fast algorithm to do something like this? An obvious but slow approach would be to do a separate distance transform for each background pixel to create independent basins, translate those up and down the z axis as appropriate, and then find the global minimum of each pixel over the set of basins. This would scale linearly with the number of background pixels though, which in my application at least would be a problem.

Here is some code that make an example dataset and does the basic distance transform, ignoring the depths.

``locations = {{10, 157}, {26, 129}, {37, 22}, {42, 190}, {44,      158}, {58, 118}, {91, 169}, {99, 152}}; depths = {0, 40, 20, 30, 10, 25, 15, 5}; image = Image[ReplacePart[Table[1, {100}, {200}], locations -> 0]]; distances = ImageData[DistanceTransform[image]]; ListPlot3D[Reverse[distances], BoxRatios -> Full, Boxed -> False] Image[WatershedComponents[Image[distances]]] ``

And here is my method:

``separateBasins = Table[    image =      Image[ReplacePart[       Table[1, {100}, {200}], {locations[[l]]} -> 0]];    distances = ImageData[DistanceTransform[image]] + depths[[l]], {l,      1, Length[locations]}]; newBasins = Map[Min, Transpose[separateBasins, {3, 1, 2}], {2}]; ListPlot3D[Reverse[newBasins], BoxRatios -> Full, Boxed -> False] Image[WatershedComponents[Image[newBasins]]] ``

which works but unsurprisingly is almost 10 times as long (eight distance transforms plus the finding of the minima). Anyone know if there is something more efficient, before I dig in to the various low-level distance transform algorithms?