Irregular periodic sampling of images and their derivatives

Given a band-limited signal, we consider the sampling of the signal and its first K derivatives in a periodic manner. The mathematical concept of frames is utilized in the analysis of the sequence of sampling functions. It is shown that the frame operator of this sequence can be expressed as matrix-valued function multiplying a vector-valued function. An important property of this matrix is that the maximum and minimum eigenvalues are equal (in some sense) to the upper and lower frame bounds. We present a way for finding the dual frame and, thereby, a way for reconstructing the signal from its samples. Using the matrix approach we also show that if no sampling of the signal itself is involved, the sampling scheme can not be stabilized by oversampling. The formalism can be extended to two dimensions to permit representation of images.