Superresolution theory and techniques seek to recover signals from samples in the presence of blur and noise. Discrete image registration can be an approach to fuse information from different sets of samples of the same signal. Quantization errors in the spatial domain are inherent to digital images. We consider superresolution and discrete image registration for one-dimensional spatially-limited piecewise constant functions which are subject to blur which is Gaussian or a mixture of Gaussians as well as to round-off errors. We describe a signal-dependent measurement matrix which captures both types of effects. For this setting we show that the difficulties in determining the discontinuity points from two sets of samples even in the absence of other types of noise. If the samples are also subject to statistical noise, then it is necessary to align and segment the data sequences to make the most effective inferences about the amplitudes and discontinuity points. Under some conditions on the blur, the noise, and the distance between discontinuity points, we prove that we can correctly align and determine the first samples following each discontinuity point in two data sequences with an approach based on dynamic programming.

Sampling and quantization are standard practices in signal and image processing, but a theoretical understanding of their impact is incomplete. We consider discrete image registration when the underlying function is a one-dimensional spatially-limited piecewise constant function. For ideal noiseless sampling the number of samples from each region of the support of the function generally depends on the placement of the sampling grid. Therefore, if the samples of the function are noisy, then image registration requires alignment and segmentation of the data sequences. One popular strategy for aligning images is selecting the maximum from cross-correlation template matching. To motivate more robust and accurate approaches which also address segmentation, we provide an example of a one-dimensional spatially-limited piecewise constant function for which the cross-correlation technique can perform poorly on noisy samples. While earlier approaches to improve the method involve normalization, our example suggests a novel strategy in our setting. Difference sequences, thresholding, and dynamic programming are well-known techniques in image processing. We prove that they are tools to correctly align and segment noisy data sequences under some conditions on the noise. We also address some of the potential difficulties that could arise in a more general case.