Measuring the average thickness of traces in an image

Question

Here's the problem: I have a number of binary images composed by traces of different thickness. Below there are two images to illustrate the problem:

First Image - size: 711 x 643 px

Second Image - size: 930 x 951 px

What I need is to measure the average thickness (in pixels) of the traces in the images. In fact, the average thickness of traces in an image is a somewhat subjective measure. So, what I need is a measure that have some correlation with the radius of the trace, as indicated in the figure below:

Notes

• Since the measure doesn't need to be very precise, I am willing to trade precision for speed. In other words, speed is an important factor to the solution of this problem.

• There might be intersections in the traces.

• The trace thickness might not be constant, but an average measure is OK (even the maximum trace thickness is acceptable).

• The trace will always be much longer than it is wide.

Answer 1

I'd suggest this algorithm:

1. Apply a distance transformation to the image, so that all background pixels are set to 0, all foreground pixels are set to the distance from the background
2. Find the local maxima in the distance transformed image. These are points in the middle of the lines. Put their pixel values (i.e. distances from the background) image into a list
3. Calculate the median or average of that list

Answer 2

I was impressed by @nikie's answer, and gave it a try ...

I simplified the algorithm for just getting the maximum value, not the mean, so evading the local maxima detection algorithm. I think this is enough if the stroke is well-behaved (although for self intersecting lines it may not be accurate).

The program in Mathematica is:

m = Import["http://imgur.com/3Zs7m.png"]   (* Get image from web*)
s = Abs[ImageData[m] - 1];                 (* Invert colors to detect background *)
k = DistanceTransform[Image[s]]            (* White Pxs converted to distance to black*)
k // ImageAdjust                           (* Show the image *)
Max[ImageData[k]]                          (* Get the max stroke width *)

The generated result is

The numerical value (28.46 px X 2) fits pretty well my measurement of 56 px (Although your value is 100px :* )

Edit - Implemented the full algorithm

Well ... sort of ... instead of searching the local maxima, finding the fixed point of the distance transformation. Almost, but not quite completely unlike the same thing :)

m = Import["http://imgur.com/3Zs7m.png"];   (*Get image from web*)
s = Abs[ImageData[m] - 1];         (*Invert colors to detect background*)
k = DistanceTransform[Image[s]];   (*White Pxs converted to distance to black*)
Print["Distance to Background*"]
k // ImageAdjust                   (*Show the image*)
Print["Local Maxima"]
weights = 
    Binarize[FixedPoint[ImageAdjust@DistanceTransform[Image[#], .4] &,s]]  
Print["Stroke Width =", 
     2 Mean[Select[Flatten[ImageData[k]] Flatten[ImageData[weights]], # != 0 &]]]

As you may see, the result is very similar to the previous one, obtained with the simplified algorithm.