Variational methods in computational microscopy
Many Microscopy techniques have been developed to explore atomic structures of material and biological specimens at a nano-scale. Coherent Diffractive Imaging (CDI) and Atomic Electron Tomography (AET) are the most powerful among these techniques. Mathematics, especially variational optimization, also follows and supports microscopy, helping to solve various image processing problems such as deblurring and denoising. In our latest research, variational methods can help obtain super-resolution ptychography images, marking a substantial improvement in computational microscopy and bypassing the resolution limit. While sub-pixel shifting and structured probes are two crucial keys to the super-resolution problem, total variation and l1 regularization play a significant role in reconstruction. In another research project, we explore the magnetic vector fields created in the vacancy between atoms of a magnetized material. Our analysis shows that the vector tomography needs in-plane rotation and constraint support to guarantee reconstruction. The sparsity inducing l1 minimization is the right tool to generate highly accurate support. Our variational methods have successfully solved a wide range of image processing problems. We will continue to utilize these techniques again in our upcoming research.