This paper addresses the problem of reconstructing an image from its nonuniform data and threshold crossings. The problem of reconstructing a two-dimensional signal from its nonuniform data arises in certain medical image problems where either the measurement domain is nonuniform or the measured data are translated to nonuniform samples of the desired image. Reconstruction from threshold crossings has significance in reducing the size of database (image compression) required to store medical images. In this paper, we introduce a deterministic processing via Gram-Schmidt orthogonalization to reconstruct images from their nonuniform data or threshold crossings. This is achieved by first introducing non-orthogonal basis functions in a chosen two-dimensional domain (e.g., for band-limited signal, a possible choice is the two dimensional Fourier domain of the image) that span the signal subspace of the nonuniform data. We then use the Gram-Schmidt procedure to construct a set of orthogonal basis functions that span the linear signal subspace defined by the above-mentioned non-orthogonal basis functions. Next, we project the N-dimensional measurement vector (N is the number of nonuniform data or threshold crossings) into the newly constructed orthogonal basis functions. Finally, the image at any point can be reconstructed by projecting its corresponding basis function on the projection of the measurement vector into the orthogonal basis functions.
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