Reconstruction of 2-D bandlimited discrete signals from nonuniform samples

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Abstract
A reconstruction method is developed for nonuniformly sampled bandlimited 1-D and 2-D discrete signals. In several applications we need to interpolate over regularly spaced points given irregularly sampled values of the signal. By exploiting the aliasing relationship in the DFT (sampled frequency) domain rather than in the continuous Fourier transform domain, we can exactly reconstruct the discrete signal subject to the harmonically limited constraint. This approach is especially attractive in the 2-D case since a novel closed-form interpolation formula is obtained. First, a transform matrix is obtained which relates the uniformly spaced frequency samples to the specified nonuniformly spaced sample values of the signal. This is done in the 1-D as well as in the 2-D case. The required uniformly spaced interpolation is then obtained by forming the inverse DFT. Unlike in the 1-D case, in the 2-D case the transform matrix may not exist even when all the sample points are distinct. Necessary conditions for the existence of the transform matrix have been investigated for the 2-D case. An efficient algorithm to compute the transform matrix has been developed by exploiting the block structure of the matrix characterising the system of equations in the 2-D case.

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