Exact image deconvolution from multiple FIR blurs

We address the problem of restoring an image from its noisy convolutions with two or more blur functions (channels). Deconvolution from multiple blurs is, in general, better conditioned than from a single blur, and can be performed without regularization for moderate noise levels. We characterize the problem of missing data at the image boundaries, and show that perfect reconstruction is impossible (even in the no-noise case) almost surely unless there are at least three channels. Conversely, when there are at least three channels, we show that perfect reconstruction is not only possible almost surely in the absence of noise, but also that it can be accomplished by finite impulse response (FIR) filtering. Such FIR reconstruction is vastly more efficient computationally than the least-squares solution, and is suitable for low noise levels. Even in the high-noise case, the estimates obtained by FIR filtering provide useful starting points for iterative least-squares algorithms. We present results on the minimum possible sizes of such deconvolver filters. We derive expressions for the mean-square errors in the FIR reconstructions, and show that performance comparable to that of the least-squares reconstruction may be obtained with relatively small deconvolver filters. Finally, we demonstrate the FIR reconstruction on synthetic and real data.