How to compare partial data to full data

Hi, I have some data of the output of a process that is obtained experimentally using camera. This data is position-dependent, you can think of it as f(x), where x is a one dimensional variable of position. Due to limited field of view of the camera, only part of the whole region of interest is able to be captured, hence f(x) is partial. I also have a mathematical model that I can solve forwardly to obtain the whole simulated output, let's call it g(y).

Assume that my model is accurate enough, then the partial output data I measured should match my simulated output after some unknown shift, i.e. if I shift f(x) along g(y), for some c where y=x+c, then f and g should match.

My thought is to compute some similarity measure between f and g, so I think I can use MATLAB 'xcorr' to find the cross-correlation between them. To test this, I made up a smooth function g, assign a continuous portion of g as f (by doing this I know the shift), then used [corr,lags]=xcorr(g,f). Interestingly the peak of 'corr' does not occur at the known shift, I figured that's probably because f is only part of g. If I shift the entire g as f instead of assigning part of g as f, then the peak occurs at the right place.

My questions are:

1. Is there any modifications to my approach to make it work?

2. Generally for matching partial data to whole data, what are some approaches/methods and their corresponding MATLAB functions?

Thank you very much!