Signal Reconstruction in Multi-Windows Spline-Spaces Using the Dual System

The letter presents a non-massive parallel procedure to compute the biorthogonal dual system used for signal reconstruction in the case of spline-type spaces with multiple generators. The basis of this algorithm are the properties of the projection operator and the invertibility of the Gramian in the case of a Riesz basis. We use a parallel approach in both time and frequency for the computation of the dual system obtained by translation and sampling of a finitely number of atoms. Since there are many applications in signal and image processing where the spline-type spaces (also known as shift-invariant spaces) play a central role, fast computing methods are needed, especially in the multi-windows case, where the computations are expensive from the execution time and from memory storage point of view. We test the implementation on car crash data.