Let's say I have an image like this one, and have masked it so all yellow pixels are 1 and all black pixels are 0. I would like to (1) fit the largest possible circles containing solely yellow pixels inside the yellow blobs; multiple circles may fit in irregular blobs, (2) from all such circles, identify which has the biggest radius, and (3) find the location of its center. What image processing techniques can I use for this? I am familiar with some morphological operations (like these), but can't get to where I want to go yet.
do a distance transform. you'll see why that's a good idea: for every pixel you get the shortest distance to a border. that's exactly the radius of an inscribed circle.
from this, just find the pixel with the largest value.
if you're curious, throw a "non-maximum suppression" on it. that is a kind of "morphological" kernel operation where you set a pixel to 0 if any neighbor has a strictly larger value.
First, the question you mentioned in the comments contains good ideas for sure.
Secondly, an approach using morphological operations is to implement an iterative algorithm that performs successive Morphological Erosions using a disk as the structural element. The idea is to begin with a large radius disk (half of the smallest image dimension is sufficient) and to stop when the returned image contains at least one white pixel (assuming that 1 is white). All of them are the centers of the biggest circles that can be contained in your "yellow zone".
This approach may be quite slow as it tests all the possibilities. Optimisation is certainly possible (dichotomy method maybe).
Hope it will help.