Compensating for measurement errors
If you can reliably determine which samples are on a rising portion and which are on a falling portion, then perhaps you can do a linear regression on each (near)monotonic segment, and solve for the intersections of the computed regression lines.

If you know the non-linearity equation, then you can do a regression against that curve instead of using linear regression, and compute or numerically solve for the curve intersections, if needed.

This regression method might also help with error or robustness estimates, if needed.