Relationship Between Data Smoothing and the Regularization of Inverse Problems

We investigate the practice of regularization (also termed damping) in inverse problems, meaning the use of prior information to supplement observations, in order to suppress instabilities in the solution caused by noisy and incomplete data. Our focus is on forms of regularization that create smooth solutions, for smoothness is often considered a desirable—or at least acceptable—attribute of inverse theory solutions (and especially tomographic images). We consider the general inverse problem, in its continuum limit. By deconstruction into the part controlled by the regularization and the part controlled by the data kernel, we show the general solution depends on a smoothed version of the back-projected data as well as a smoothed version of the generalized inverse. Crucially, the smoothing function that controls both is the solution to the simple data smoothing problem. We then consider how the choice of regularization shapes the smoothing function, in particular exploring the dichotomy between expressing prior information either as a constraint equation (such as a spatial derivative of the solution being small) or as a covariance matrix (such as spatial correlation falling off at a specified rate). By analyzing the data smoothing problem in its continuum limit, we derive analytic solutions for different choices of regularization. We consider four separate cases: (1) the first derivative of the solution is close to zero, (2) the prior covariance is a two-sided declining exponential, (3) the second derivative of the solution is close to zero, and (4) the solution is close to its localized average. First-derivative regularization is put forward as having several attractive properties and few, if any, drawbacks.