This is a cool idea. I think it is pretty straightforward, although I don't have any experience with image processing. So the basic idea, is you have data of stress vs strain, and you know that Hooke's law describes the behaviour during the linear portion of the curve (before plastic deformation occurs).
So, get a fit for the linear portion. Since it is linear and there is no force at rest, it should literally just be y = mx. Then process your images to get displacement (x), and plug it into your model and it should output the stress.
I'm sure you could try to create a model to include the nonlinear aspects too. But it would be very complicated to do from first principles. You would probably be better off just doing a spline. But the plastic region of your data will have a much higher variance than the elastic, so this could make it tricky. Hope you have a lot of data.
So, no need to interpolate, do it properly and for your data. You will need a different fit for each material though, because they will all have their different properties such as Young's modulus and futility, yield strength, strain-hardening etc. And I'm not sure where you will apply this too, because real-life material behaviour can vary from a lab setting. Materials in real applications can have defects, degradation, different climate, etc.
