If I understood right, you need a "raster" background image to display and a gemetric shape that describes that background's silhouette for physical simulations issues.
Looking at your example image I can see a bezier profile so you may store the parameters you used to generate it, along with your image; is up to you to store the rasterized version or use the bezier parameters to generate the displaying image (cpu vs storage trade-off).
The interesting case comes if your background shape does not come from a bezier curve or from an equivalent parametrization: and hand-made artwork for example.
I such cases you may find this paper useful. Here is shown how to get a bezier parametrization from an hand-drawn sketch.
They basically illustrate the process with the following steps:
1 - apply a logarithmic transformation to make it easier to extract the edge.
2 - erosion - dilatation is used to smoth the image
3 - laplacian is applied to find out the edges
once they get the edges, as set of points laying on the shape edge, they start to construct a set of (bezier) curves each of which produces a better edge approximation.
4 - the set of edge points are grouped and for each group a regression line segment is computed
5 - using the intersection of the line segments in the step 4 a set of bezier 2nd grade curves are computed
6 - a method to generate 5th degree Bezier Curve is given: the curve is derived from that obtained in the previous step by letting the derivate and curvature discontinuities be smoothed.
All these steps are obviously computationally relevant and are not intended to be done at runtime.
This should not be a problem since you do not do dynamic backgrounds, if you do so you should have some parametrization and the problem solved in (moved to) an earlier phase of the project.