CAI: Deconvoluting a Radiation Field using a Webcam and Robust PCA (part 3)

In a previous installment, we decomposed a video featuring a webcam bombarded with a proton beam with the Robust PCA (using GoDec or any of the recent Advanced Matrix Factorization solvers). We noted in that installement that the de-convolution of that image sequence provided:

• the detector shot noise (noisy component) 
• the visible range image (the low rank image)
• a proxy for an X-ray photon flux irradiating the webcam/detector (the sparse component).

In a scene without the radiation beam, one expects a simple decomposition to take place for a still scene:

• A still image that instantiates itself as the rank-1 solution (provided the scene is still)
• Some shot noise that is instantiated as a gaussian distribution around the values of the rank-1 solution. This noise distribution is obviously symmetric around zero and denotes the fluctuation of the series of frames in the video around the low rank solution.
• No sparse component.

Because of the additive nature of the proton beam on the image frames, when the proton beam is opened on the webcam, one would expect:

• The same description for the low rank and the noisy frames.
• The sparse component, however, has a distribution of events that is positive. It makes sense, physically, the photon htting the webcam only adds current to the CMOS pixels.

Let us note that there is really nothing in the algorithm that constrains the distribution of elements in the sparse component to be positive. Cable and I decided to look into this a little further.and found out that the flickering clock at the bottom of the video sequence induced a non positive sparse component. By removing that clock from the frames, the sparse component became a lot closer to our expectation, i.e. a series of positive elements. Since the CMOS responds as a function as the incident photon energy, the webcam acts as a  multichannel analyzer or a X-ray spectrometer. The figure provided below features the histograms of the elements in the sparse component of the decomposition. Different histograms represent different values of k (see GoDec ). This k parameter allows one to proactively constrain this distribution to be positive. This result is simply terrific, the algorithm provides a physically obvious solution (a distribution of positive elements in the sparse decomposition) even though it is clearly not constrained into doing so.

Instead of playing with a parameter (k) that has no obvious physical meaning, it might be a good avenue of research to develop Robust PCA solvers for which there is an explicit constraint on the sparse component. In particular, when it comes to video processing, a robust PCA might benefit from a positive or negative only constraint on the sparse component.. 

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