Detecting curves from binary image?

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I am trying to detect curves with a pre-determined shaped from a dataset. The input image will be similar to the following picture:

I want to find a line through the left-most and right-most points, as roughly indicated in the following picture:

So far, I can imagine two ways to get this result:

First find a way to extract the left-most points and right-most points and using a 3rd/5th degree polynomial or a hyperbolic sine fit through these points.
Using something like a hough transform but applying it for curved lines.

So far, I have not found a good way to get the results I need, also caused by the fact that the data set is quite large [8890x1312] (above images are stretched in the x direction), making it extremely slow to loop over each pixel.

Do you guys have any ideas to tackle this problem?