On Friday, I presented the pattern unmixing problem. Today, I’ll discuss how we solved it.
Coelho, Luis Pedro, Tao Peng, and Robert F. Murphy. “Quantifying the Distribution of Probes Between Subcellular Locations Using Unsupervised Pattern Unmixing.” Bioinformatics 26.12 (2010): i7–i12. DOI: 10.1093/bioinformatics/btq220
The first step is to extract objects. For this, we use a combination of global & local thresholding: this means that a pixel is on if it is both above a global threshold which identifies the cells from the background and a local threshold (which identifies subcellular objects ).
We then group the objects found using k-means clustering. Here is what we obtain for a lysosomal picture (different colours mean different clusters) .
and the equivalent for the mitochondrial image:
You will see that the mitochondrial image has many green things and less dark purple objects, but both mitochondrial and lysosomal images have all of the groups. Now (and this is an important point): we do not attempt to classify each individual object, only to estimate the mixture.
Of course, if we had the identity of each object, the mixture would be trivially estimated. But we do not need to identify each object. In fact, to attempt to do so would be a gross violation of Vapnik’s Dictum (which says do not solve, as an intermediate step, a harder problem than the one you are trying to solve). It is easier to just estimate the mixtures .
In this formulation it might not even matter much that some of the objects we detect correspond to multiple biological objects!
How do we solve the mixture problem? Latent Dirichlet allocation or basis pursuit. The details are in the paper, but I will jump to the punchline.
We tested the method using a dataset where we had manipulated the cell tagging so we know the ground truth (but the algorithm, naturally, does not see it). On the graph below, the x-axis is the (hidden) truth and the y-axis is the automated estimate. In green, the perfect diagonal; and each dot represents one condition:
I will note that each individual dot in the above plot represents several images from each condition. On a single image (or single cell) level the prediction is not so accurate. Only by aggregating a large number of objects can the model predict well.
This also points out why it may be very difficult for humans to perform this task (nobody has tried to do it, actually).
|||A global threshold did not appear to be sufficient for this because there is a lot of in-cell background light (auto-fluorescence and auto-focus light).|
|||For this picture, I used 5 clusters to get 5 different colours. The real process used a larger number, obtained by minimising BIC.|
|||Sure, we can then reverse engineer and obtain a probability distribution for each individual object, but that is not the goal.|
- Old Work: Unsupervised Subcellular Pattern Unmixing (metarabbit.wordpress.com)