Lifting for Blind Deconvolution in Random Mask Imaging: Identifiability and Convex Relaxation

In this paper we analyze the blind deconvolution of an image and an unknown blur in a coded imaging system. The measurements consist of subsampled convolution of an unknown blurring kernel with multiple random binary modulations (coded masks) of the image. To perform the deconvolution, we consider a standard lifting of the image and the blurring kernel that transforms the measurements into a set of linear equations of the matrix formed by their outer product. Any rank-one solution to this system of equation provides a valid pair of an image and a blur. We first express the necessary and sufficient conditions for the uniqueness of a rank-one solution under some additional assumptions (uniform subsampling and no limit on the number of coded masks). These conditions are special case of a previously established result regarding identifiability in the matrix completion problem. We also characterize a low-dimensional subspace model for the blur kernel that is sufficient to guarantee identifiability, including the interesting instance of "bandpass"` blur kernels. Next, assuming the bandpass model for the blur kernel, we show that the image and the blur kernel can be found using nuclear norm minimization. Our main results show that recovery is achieved (with high probability) when the number of masks is on the order of $\mu\log^{2}L\,\log\frac{Le}{\mu}\,\log\log\left(N+1\right)$ where $\mu$ is the \emph{coherence} of the blur, $L$ is the dimension of the image, and $N$ is the number of measured samples per mask.