Every month there seems to be another attempt to bypass the laws of optics using software - this is called Computational Imaging. We might as well keep one thread for an overall discussion of this topic.

Here is a brief, incomplete inventory from my own perspective:

Lytro: Attempt to capture the "light field", i.e. not just a 2-D capture of an image but the direction of all light beams entering a lens. This can be used in post to calculate depth in an image, synthesize defocus, detect edges etc. Naturally the amount of data becomes massive, so Lytro is a bit limited WRT resolution. Lytro has had limited commercial success.

Pelican Imaging: A different approach to capturing image depth. Disclaimer - I used to work for Pelican. An array of cameras (4x4) capture simultaneous images with a slight offset. Parallax calculations then detect the distance for every pixel, resulting in a depth map. This depth map can then be used for synthetic defocus, edge detection, gesture recognition etc. Pelican has scaled back its engineering effort and laid off most of its engineering and R&D staff due to lack of funds.

Kinect: Two cameras capture a scene lit by IR light, for gesture recognition. Microsoft seems to focus less on Kinect these days.

Google's Project Tango: Capture from multiple lenses (and multiple exposures) is used for cloud-based computational imaging to create a vectorized 3-D model of an object. Google's focus here is naturally more on the cloud part but Google is shipping a capture device in the form of a tablet with multiple lenses.

Light: Multiple lenses of various focal lengths in one camera shoot simultaneously, computational imaging magic is used to up-rez an image as if it was a high resolution zoom lens. Not yet on the market.

Hmm this is all very Silicon Valley-centric... All these companies except for Kinect are based less than 10 miles from my home. I know there are others, lots going on at MIT of course, some in Stockholm, of course many other projects around the world.

Feel free to add to the list - I'll edit this original post to reflect added information. (Can I edit and OP indefinitely?)