Bayesian Estimation for Continuous-Time Sparse Stochastic Processes
This work addresses signal recovery challenges in fields like communications or imaging, but it is incremental as it builds on existing spline theory and compares to known regularization techniques.
The paper tackles the problem of denoising and interpolating continuous-time sparse stochastic processes from limited noisy or noiseless samples, deriving Bayesian estimators and showing that regularization methods can perform close to the minimum mean-square error estimator under certain conditions.
We consider continuous-time sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal in-between (interpolation problem). By relying on tools from the theory of splines, we derive the joint a priori distribution of the samples and show how this probability density function can be factorized. The factorization enables us to tractably implement the maximum a posteriori and minimum mean-square error (MMSE) criteria as two statistical approaches for estimating the unknowns. We compare the derived statistical methods with well-known techniques for the recovery of sparse signals, such as the $\ell_1$ norm and Log ($\ell_1$-$\ell_0$ relaxation) regularization methods. The simulation results show that, under certain conditions, the performance of the regularization techniques can be very close to that of the MMSE estimator.