As an example, we want to measure $f(t)$, but it's hard to get it directly. What we can do is to measure $f(t)-f(t-\tau)$. How to use the information we have to find $f(t)$?
The only method I know is, when $\tau$ is small enough, $$f(t)-f(t-\tau) \approx \tau f'(t)$$ then integrate it. But I still feel this is not good enough.
Motivation
We need to measure the instant frequency of a tuneable laser source (or the instant phase $\phi(t))$, and through swept-wavelength interferometer we get the beat signal $\cos(\phi(t)-\phi(t-\tau))$, where $\tau$ is the delay of two arms, so I'm struggling with dealing this issue. If any solutions here, please feel welcome to show your wisdom.
The transfer function of the measurement device is
$$ H_M (s) = 1 - e^{- \tau s} $$
Note that if the delay $\tau$ is "sufficiently small", then $H_M (s) \approx \tau s$, which is a scaled differentiator.
One would like to find a "compensator" that undoes the measurement. This compensator could have the following transfer function
$$ H_C (s) = \frac{1}{H_M (s)} = \frac{1}{1 - e^{- \tau s}} $$
Again, note that if the delay $\tau$ is "sufficiently small", then $H_C (s) \approx \frac{1}{\tau s}$, which is a scaled integrator.
The aforementioned compensator's could be implemented by the following
$$ y (t) - y (t - \tau) = u (t) $$
where $u$ is the compensator's input and $y$ is the compensator's output. Do you have access to a delay line? How accurate is your measurement of $\tau$?