A Convex Quadratic Formulation for Numerical Differentiation of Noisy Signals via Total Variation Minimization

In this paper, a new formulation for numerical differentiation of noisy signals based on Total Variation Minimization (TVM) is presented. The main existing approach for this task is to minimize a functional using the Euler-Lagrange equation. This method leads to a nonlinear second-order partial differential equation that must be solved using iterative numerical techniques. Whereas in the method proposed in this paper, instead of using a functional, the cost function is expressed directly in terms of a vector that contains the unknown samples of the numerical derivative. This difference in the definition of cost function leads to a significant simplification in solving the problem. In fact, since the proposed cost function is convex and quadratic, it can be effectively solved with high accuracy using available software.