We introduce a framework on a general order pyramid transform for variational multiresolution registration. The traditional pyramid transform shrinks the image to an image of the quarter size. If the result of the pyramid transform is expressed in the same landscape with the original images, the result of the transform yields a low-resolution image. The pyramid transform is achieved by subsampling after linear smoothing. Furthermore, the dual operation of the pyramid transform is achieved by linear interpolation after upsampling. The rational-order pyramid transform is decomposed into subsampling by linear interpolation and the traditional pyramid transform. By controlling ration between subsampling for linear interpolation and subsampling in the pyramid transform, we construct the rational-order pyramid transform.
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