Unless I missed it, this seems to skip over an important key step: how is the "foreground" vs "background" identified? There are mask layers, but where do they come from?
"Given a stack of images captured with a fixed focal length at different distances from the scene, our framework allows us to modify the composition of the scene in post-capture"
So I assume it's inferred from the stack of images.
The second video on the page explains they use multiple source photographs taken from different positions with the same camera and lens to produce an approximate depth map of the scene. These multiple views allow for image segmentation based on distance.
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