# Combining Distance Maps

I have two distance maps that describe distances from blue segment and the red segment. I need to compute a single weight map that describes transition from blue segment to the red segment. The weight map should have values in the 0-1 range where 0 is one segment and 1 is the other one.

There are the distance maps with the respective segments highlighted:

Having $d_{1}$ and $d_{2}$ the distances in the first and second image, I tried the following formula for computing the transition weight map:

$$d=\frac{d_{1}-d_{2}+d_{2max}}{d_{1max}+d_{2max}}$$

where $d_{1max}$, $d_{2max}$ are maximum distances found in the first and second image. The numerator have range $<0, d_{1max}+d_{2max}>$ so the result is in $<0,1>$ range. This resulted in the following map:

However, there are sharp transitions between each segment and the map.

I have also tried multiplicative formula:

$$d=\bar{d_{1}}\cdot (1 - \bar{d_{2}})$$

where $\bar{d_{1}}, \bar{d_{2}}$ are distances normalized to $<0,1>$ range. This resulted in little better transition, but still suffers from the same problem:

Do you have any other ideas how to combine the distance maps to obtain transition map?

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An obvious choice would be $\frac{d_1}{d_1+d_2+\epsilon }$, where $\epsilon$ is a small value to prevent division by zero.

top = ColorNegate[Binarize[ImageApply[StandardDeviation,
Import["http://i.stack.imgur.com/aFNfB.jpg"]]]];
bottom = ColorNegate[Binarize[ImageApply[StandardDeviation,
Import["http://i.stack.imgur.com/Kipvo.jpg"]]]];
tDist = ImageData[DistanceTransform[top]];
bDist = ImageData[DistanceTransform[bottom]];

\[Epsilon] = 10^-10;
Image[tDist/(tDist + bDist + \[Epsilon])] // ImageAdjust


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How simple. Thanks! – Libor Dec 26 '12 at 17:59