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Phase-residue Removal Based on Sparse Modeling in Electron Holography

Electron holography is a quantitative phase-detection technique to visualize electrostatic and magnetic fields in and around nano- to micro-meter-size materials. By applying this technique to observe catalytic nanoparticles, weak electrostatic fields around a Pt nanoparticle on a TiO 2 substrate have been visualized as being locally fluctuated [1]. The precision of phase detection can be improved by increasing electron doses and enhancing visibilities of hologram fringes. The precision of ~2  /1000 rad is achieved by averaging a vast number of phase images. The phase image in electron holography is reconstructed from corresponding electron holograms. The wrapped phase image, in which the range is restricted within (−π, +π], is commonly obtained by inverse trigonometric computation. The true phase image is computed using sophisticated phase unwrapping algorithms (for example [2]). However, when the phase image includes phase residues (or phase singularity points), the discontinuity of 2  rad at the residues arises and the true phase values become ambiguous. Because the discontinuity does not disappear by averaging many phase images, residue removal is indispensable for quantitative holography analysis. We used a sparse modelling technique [3, 4] for removing the residues in a phase image. Phase residues were first detected automatically in a wrapped phase image. A corresponding area including a residue was then cropped and repaired by patching a few predefined images on the based of sparse modeling. Finally, residue detection and repair were repeated until all residues were removed. We tested residue removal on a real image, which was an atomic-resolution phase image at a Pt/TiO 2 boundary observed using a 1.2 MV holography electron microscope