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Abstract

Direct ptychography enables the retrieval of information encoded in the phase of an electron wave passing through a thin sample by deconvolving the interference effects of a converged probe with known aberrations. Under the weak phase object approximation, this permits the optimal transfer of information using non-iterative techniques. However, the achievable resolution of the technique is traditionally limited by the probe step size -- setting stringent Nyquist sampling requirements. At the same time, parallax imaging has emerged as a dose-efficient phase-retrieval technique which relaxes sampling requirements and enables scan-upsampling. Here, we formulate parallax imaging as a quadratic approximation to part of the direct ptychography kernel and use this insight to enable upsampling in direct ptychography. We validate our analytical results numerically using simulated and experimental reconstructions.