Measuring Particle Sizes and Process Control – Controlling Size Process using Bhattacharyya Distance

Keox Technologies occasionally encounters a request in measuring particle size distribution.  That is, when making measurements on powders, minerals, rice, beans, grains, and even contaminant particles, a specific dimension number isn’t given but rather a distribution (or terminology used more in the Imaging world, a histogram).


The challenging part is how to determine if one distribution is similar to the control distribution or if it’s significantly different?  How does Process Engineering determine if incoming raw materials are within specification or that the fabrication process is out-of-control?


There are many different metrics of comparing – ranging from simple to complex, from heavily implied assumptions to multi-dimensional models.  One metric that Keox Technologies like to use for comparing distributions is the Bhattacharyya Distance.  The Bhattacharyya Distance is a useful measure as it is “self-consistent”, unbiased, and applicable to any distribution.


Using a grounded coffee project as an example, Keox Technologies needed to determine if the grind size is consistent.  As connoisseur coffee drinkers attest to – size of the grind need to be appropriate for the brew technique used!


To satisfy our habitual coffee drinkers around the world, here are two samples to be compared:


Figure 1 – Sample A and Sample B


Image processing steps are used to determine the size of each grind and a distribution is constructed:


Figure 2 – Size Distributions of Sample A and Sample B


Using the defined Bhattacharyya definition for discrete probability distributions:


Figure 3 – Bhattacharyya Distance for Discrete Distributions


The example above yielded a Bhattacharyya Distance (DB) of 0.292726.


The determination if this number is acceptable or not is established by additional process experiments, quality control specifications, etc.  Nonetheless, Bhattacharyya Distance would be a trustworthy metric for Quality to rely on.