A regularized digital filtering technique for the simultaneous reconstruction of a function and its derivatives

In this paper, a new regularized digital filtering technique for the simultaneous approximation of a function and its derivatives is proposed. First, a simple and local method is presented that interpolates the specified sample values exactly. The solution obtained by this method belongs to the space of splines functions, and can be implemented using filter banks. Unfortunately, like most of the methods used to solve interpolation problems using derivatives, it is very sensitive to noise. To overcome this drawback we extend the interpolation method to function approximation by defining a regularized functional, which includes a term forcing the smoothness of the solution. The minimization of this functional is performed by solving a simple linear system of equations or using gradient descent based techniques. Some examples show the improved performance of this technique in noisy environments