Phase Imaging: A Compressive Sensing Approach
Abstract
Since Wolfgang Pauli posed the question in 1933, whether the probability densities Ψ(r)² (realspace image) and Ψ(q)² (reciprocal space image) uniquely determine the wave function Ψ(r) [1], the so called Pauli Problem sparked numerous methods in all fields of microscopy [2, 3]. Reconstructing the complete wave function Ψ(r) = a(r)eiφ(r) with the amplitude a(r) and the phase φ(r) from the recorded intensity enables the possibility to directly study the electric and magnetic properties of the sample through the phase. In transmission electron microscopy (TEM), electron holography is by far the most established method for phase reconstruction [4]. Requiring a high stability of the microscope, next to the installation of a biprism in the TEM, holography cannot be applied to any microscope straightforwardly. Recently, a phase retrieval approach was proposed using conventional TEM electron diffractive imaging (EDI). Using the SAD aperture as reciprocalspace constraint, a localized sample structure can be reconstructed from its diffraction pattern and a realspace image using the hybrid inputoutput algorithm [5]. We present an alternative approach using compressive phaseretrieval [6]. Our approach does not require a realspace image. Instead, random complimentary pairs of checkerboard masks are cut into a 200 nm Pt foil covering a conventional TEMmore »
 Authors:
 Publication Date:
 Research Org.:
 Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1379440
 Report Number(s):
 PNNLSA124072
Journal ID: ISSN 14319276; applab
 DOE Contract Number:
 AC0576RL01830
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Microscopy and Microanalysis; Journal Volume: 23; Journal Issue: S1
 Country of Publication:
 United States
 Language:
 English
Citation Formats
Schneider, Sebastian, Stevens, Andrew, Browning, Nigel D., Pohl, Darius, Nielsch, Kornelius, and Rellinghaus, Bernd. Phase Imaging: A Compressive Sensing Approach. United States: N. p., 2017.
Web. doi:10.1017/S1431927617001155.
Schneider, Sebastian, Stevens, Andrew, Browning, Nigel D., Pohl, Darius, Nielsch, Kornelius, & Rellinghaus, Bernd. Phase Imaging: A Compressive Sensing Approach. United States. doi:10.1017/S1431927617001155.
Schneider, Sebastian, Stevens, Andrew, Browning, Nigel D., Pohl, Darius, Nielsch, Kornelius, and Rellinghaus, Bernd. 2017.
"Phase Imaging: A Compressive Sensing Approach". United States.
doi:10.1017/S1431927617001155.
@article{osti_1379440,
title = {Phase Imaging: A Compressive Sensing Approach},
author = {Schneider, Sebastian and Stevens, Andrew and Browning, Nigel D. and Pohl, Darius and Nielsch, Kornelius and Rellinghaus, Bernd},
abstractNote = {Since Wolfgang Pauli posed the question in 1933, whether the probability densities Ψ(r)² (realspace image) and Ψ(q)² (reciprocal space image) uniquely determine the wave function Ψ(r) [1], the so called Pauli Problem sparked numerous methods in all fields of microscopy [2, 3]. Reconstructing the complete wave function Ψ(r) = a(r)eiφ(r) with the amplitude a(r) and the phase φ(r) from the recorded intensity enables the possibility to directly study the electric and magnetic properties of the sample through the phase. In transmission electron microscopy (TEM), electron holography is by far the most established method for phase reconstruction [4]. Requiring a high stability of the microscope, next to the installation of a biprism in the TEM, holography cannot be applied to any microscope straightforwardly. Recently, a phase retrieval approach was proposed using conventional TEM electron diffractive imaging (EDI). Using the SAD aperture as reciprocalspace constraint, a localized sample structure can be reconstructed from its diffraction pattern and a realspace image using the hybrid inputoutput algorithm [5]. We present an alternative approach using compressive phaseretrieval [6]. Our approach does not require a realspace image. Instead, random complimentary pairs of checkerboard masks are cut into a 200 nm Pt foil covering a conventional TEM aperture (cf. Figure 1). Used as SAD aperture, subsequently diffraction patterns are recorded from the same sample area. Hereby every mask blocks different parts of gold particles on a carbon support (cf. Figure 2). The compressive sensing problem has the following formulation. First, we note that the complexvalued reciprocalspace wavefunction is the Fourier transform of the (also complexvalued) realspace wavefunction, Ψ(q) = F[Ψ(r)], and subsequently the diffraction pattern image is given by Ψ(q)2 = F[Ψ(r)]2. We want to find Ψ(r) given a few differently coded diffraction pattern measurements yn = F[HnΨ(r)]2, where the matrices Hn encode the mask structure of the aperture. This is a nonlinear inverse problem, but has been shown to be solvable even in the underdetermined case [6]. Since each diffraction pattern yn contains diffraction information from selected regions of the same sample, the differences in each pattern contain local phase information, which can be combined to form a full estimate of the realspace wavefunction[7]. References: [1] W. Pauli in “Die allgemeinen Prinzipien der Wellenmechanik“, ed. H Geiger and W Scheel, (Julius Springer, Berlin). [2] A. Tonomura, Rev. Mod. Phys. 59 (1987), p. 639. [3] J. Miao et al, Nature 400 (1999), p. 342. [4] H. Lichte et al, Annu. Rev. Mater. Res. 37 (2007), p. 539. [5] J. Yamasaki et al, Appl. Phys. Lett. 101 (2012), 234105. [6] P Schniter and S Rangan. Signal Proc., IEEE Trans. on. 64(4), (2015), pp. 1043. [7] Supported by the Chemical Imaging, Signature Discovery, and Analytics in Motion initiatives at PNNL. PNNL is operated by Battelle Memorial Inst. for the US DOE; contract DEAC0576RL01830.},
doi = {10.1017/S1431927617001155},
journal = {Microscopy and Microanalysis},
number = S1,
volume = 23,
place = {United States},
year = 2017,
month = 7
}

Tomographic imaging modalities sample subjects with a discrete, finite set of measurements, while the underlying object function is continuous. Because of this, inversion of the imaging model, even under ideal conditions, necessarily entails approximation. The error incurred by this approximation can be important when there is rapid variation in the object function or when the objects of interest are small. In this work, we investigate this issue with the Fourier transform (FT), which can be taken as the imaging model for magnetic resonance imaging (MRl) or some forms of wave imaging. Compressive sensing has been successful for inverting this datamore »

The Potential for Bayesian Compressive Sensing to Significantly Reduce Electron Dose in High Resolution STEM Images
The use of high resolution imaging methods in the scanning transmission electron microscope (STEM) is limited in many cases by the sensitivity of the sample to the beam and the onset of electron beam damage (for example in the study of organic systems, in tomography and during insitu experiments). To demonstrate that alternative strategies for image acquisition can help alleviate this beam damage issue, here we apply compressive sensing via Bayesian dictionary learning to high resolution STEM images. These experiments successively reduce the number of pixels in the image (thereby reducing the overall dose while maintaining the high resolution information)more » 
Compressive Sensing Based Machine Learning Strategy For Characterizing The Flow Around A Cylinder With Limited Pressure Measurements
Compressive sensing is used to determine the flow characteristics around a cylinder (Reynolds number and pressure/flow field) from a sparse number of pressure measurements on the cylinder. Using a supervised machine learning strategy, library elements encoding the dimensionally reduced dynamics are computed for various Reynolds numbers. Convex L1 optimization is then used with a limited number of pressure measurements on the cylinder to reconstruct, or decode, the full pressure field and the resulting flow field around the cylinder. Aside from the highly turbulent regime (large Reynolds number) where only the Reynolds number can be identified, accurate reconstruction of the pressuremore »