Rendition: Reclaiming what a black box takes away

The premise of our work is deceptively familiar: A black box $f(\cdot)$ has altered an image $\mathbf{x} \rightarrow f(\mathbf{x})$. Recover the image $\mathbf{x}$. This black box might be any number of simple or complicated things: a linear or non-linear filter, some app on your phone, etc. The latter is a good canonical example for the problem we address: Given only "the app" and an image produced by the app, find the image that was fed to the app. You can run the given image (or any other image) through the app as many times as you like, but you can not look inside the (code for the) app to see how it works. At first blush, the problem sounds a lot like a standard inverse problem, but it is not in the following sense: While we have access to the black box $f(\cdot)$ and can run any image through it and observe the output, we do not know how the block box alters the image. Therefore we have no explicit form or model of $f(\cdot)$. Nor are we necessarily interested in the internal workings of the black box. We are simply happy to reverse its effect on a particular image, to whatever extent possible. This is what we call the "rendition" (rather than restoration) problem, as it does not fit the mold of an inverse problem (blind or otherwise). We describe general conditions under which rendition is possible, and provide a remarkably simple algorithm that works for both contractive and expansive black box operators. The principal and novel take-away message from our work is this surprising fact: One simple algorithm can reliably undo a wide class of (not too violent) image distortions. A higher quality pdf of this paper is available at http://www.milanfar.org