Droplet Location

To extract the most benefit from our confocal microscopy images means figuring out the sizes and locations of all the droplets in the image. It is then possible to determine detailed structural information at a fixed time, for example, or track individual droplet motion over time to see how the emulsion ages.

If all the droplets were the same size, this would be relatively simple. Firstly the size need only be measured once, which is easy to do, so the issue of determining sizes for each droplet in an image goes away. Secondly, a brightness peak in a region of a size equalling the known droplet size is a good estimate of droplet location. This approach relies heavily on having a single characteristic length scale, the droplet size, in the problem.

For polydisperse emulsions the droplets are all different sizes, however, so there is no single characteristic length scale. Assuming a large length scale, appropriate to the biggest droplets, means that clusters of smaller droplets are undifferentiated, while using a small length scale means that single large features are interpreted as clusters of smaller ones. Using simple extensions of the monodisperse “masking” approach also offers no clear and simple route to measuring droplet size. Closely packed droplets having a range of sizes are especially hard to resolve.

We have devised an algorithm based on the Euclidean Distance Map (EDM) that addresses this problem and allows us to locate droplets and estimate their sizes.

For a binary image, in which regions of interest are bright pixels against a black background, calculate the (Euclidean) distance of each bright pixel from the nearest background pixel. Pixels near the centre of a bright region will have large distance values, and those at the edges will have the value one. Create a new image whose pixels contain these distance values. This new image is the EDM.

A large feature in the original microscope image becomes a large bright region after some threshold has been applied to generate the corresponding binary image. The centre of this bright region will be a long way from the background and will be a local maximum in the EDM. The values of local maxima in the EDM are therefore estimates of feature sizes, and the positions of those local maxima are estimates of feature positions.

emulsion image

EDM

skeleton image

reconstituted image

To illustrate the idea, here is a part of an emulsion image.

The emulsion has a volume fraction of 30% and is flocculated using 0.9% PEO. The image here is a quarter of a single confocal slice (from a stack of 64 slices) and measures 53.65 µm square. The average droplet diameter is 6 µm. The binary image is created by applying a brightness threshold cut-off. The corresponding EDM is shown below:

Large droplets give rise to bright regions in the EDM. Different sized droplets in the original image become regions having different maximum brightness in the EDM, with the location of those maxima matching the centres of the droplets in the original image.

As an aside, we note that the EDM shows bridges linking droplets, caused by the fact that the brightness between droplets that are touching does not fall to zero. This bridging effect can be enhanced by applying a morphological transformation called thinning. The output of thinning is the skeleton shown below. This suggests a way of analysing the droplet network in terms of coordination number and stress chains, for example.

The EDM approach has a number of strengths. It is relatively insensitive to noise in the original image. Also, droplets that are partially outside the viewing area can be correctly located provided their centres are within the viewing box. In addition the technique is efficient in the sense that it exploits the prior knowledge that the features to be located are spheres. It works particularly well in three dimensions.

Our implementation, which takes account of the fact that original image voxels are rectangular prisms, applies thresholding and iterative feature location and removal. Below is an example of a reconstituted image based on features located in the image above. The reconstructed version looks less populated partly because in real microscope images droplets are “smeared” in the direction of the optical axis.