Fourier Transform Holography (FTH) and Coherent Diffraction Imaging (CDI) experimental results: Top:FTH experimental results. (a) Captured hologram, the red rectangle denotes the missing data area of 41×41 pixels. (b) IFT reconstruction from the complete hologram. (c) IFT reconstruction from incomplete hologram. (d) CS reconstruction from incomplete hologram. Bottom: CDI experimental results. (a) Captured diffraction pattern with red rectangle showing the removed part. (b) HIO reconstruction after 2000 iterations with complete data (c) HIO reconstruction after 2000 iterations with missing data (d) HIO+CS Reconstruction (2000 HIO iterations total).
Project Description
In both lensless Fourier transform holography (FTH) and coherent diffraction imaging (CDI), a beamstop is used to block strong intensities which exceed the limited dynamic range of the sensor, causing a loss in low-frequency information, making high quality reconstructions difficult or even impossible. In this paper, we show that an image can be recovered from high-frequencies alone, thereby overcoming the beamstop problem in both FTH and CDI. The only requirement is that the object is sparse in a known basis, a common property of most natural and manmade signals. The reconstruction method relies on compressed sensing (CS) techniques, which ensure signal recovery from incomplete measurements. Specifically, in FTH, we perform compressed sensing (CS) reconstruction of captured holograms and show that this method is applicable not only to standard FTH, but also multiple or extended reference FTH. For CDI, we propose a new phase retrieval procedure, which combines Fienup’s hybrid input-output (HIO) method and CS. Both numerical simulations and proof-of-principle experiments are shown to demonstrate the effectiveness and robustness of the proposed CS-based reconstructions in dealing with missing data in both FTH and CDI.
Publications
"A Compressed Sensing Approach to Solving the Dynamic Range Problem in Fourier Transform Holography"
K. He and O. Cossairt
Imaging and Applied Optics 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper CW2F.3.
[PDF]
"High Dynamic Range Coherent Imaging Using Compressed Sensing"
K. He, M. Sharma, O. Cossairt
Optics Express, November 2015.
[PDF]
Images
Beam stop issue in coherent imaging:
Beam stop in lensless imaging, causing central low-frequencies information lost in the diffraction pattern.
Image reconstruction from high-frequencies alone:
(a) Object, (b) The Fourier spectrum of the object, the black rectangle mimics the beam stop, (c) convention reconstruction by taking the inverse Fourier transform of the incomplete measurement, (d) CS reconstruction.
Image reconstruction from a single hologram with missing central data in multiple references FTH:
(a) Sample mask contains an object and 2 references size 512×512. (b) Simulated hologram with 41×41 pixel region missing in the center. (c) Inverse FT of (b) by setting unmeasured region to zero. (d) CS reconstruction from (b).
Image reconstruction from a single hologram with missing central data in HERALDO:
(a) The sample mask contains an object and a thin slit reference. (b) Simulated processed hologram with 41×41 pixels region missing in the center. (c) Inverse FT of (b) by setting unmeasured region to zero. (d) CS reconstruction.
Missing data size and noise impact on reconstruction quality:
Left: Reconstructed images with different missing area sizes for 35db noise, using IFT and CS algorithms.
Right: The SNR of reconstructed images is plotted against missing area sizes using IFT and CS methods. Plots for different noise levels are shown in different colors.
FTH experimental setup and results:
Top: FTH experimental setup. L1: collimator, O: object mask: containing an object and a point reference, L2 : Fourier transforming lens.
Bottom: FTH real experiment. (a) Captured hologram, the red rectangle denotes the missing data area of 41×41 pixels. (b) IFT reconstruction from the complete hologram. (c) IFT reconstruction from incomplete hologram. (d) CS reconstruction from incomplete hologram.
Reconstruction of an oversampled diffraction pattern with missing central data:
(a) An original 128×128 pixels image is embedded in a 512×512 pixels array. (b) The oversampled diffraction intensity pattern. The data at the central 31×31 pixels are removed. (c) HIO reconstruction after 2000 iterations (d) reconstruction after 2000 iterations using proposed HIO+CS algorithm.
Reconstruction error VS iteration number; Proposed HIO+CS algorithm
Left: Reconstruction error vs. iteration number of the missing data region (black) and high frequency region (blue) for conventional HIO. Reconstruction error of the missing data region (red) for the proposed HIO+CS algorithm.
Right: The block diagram of the proposed HIO+CS method.
Missing data size and noise impact on reconstruction quality:
Left: reconstructed images with different missing area sizes using HIO and HIO+CS algorithms (Only central 200×200 pixels are shown out of 512×512).
Right: The SNR of reconstructed images is plotted against missing area sizes using HIO and HIO+CS algorithms. Plots for different noise levels are shown in different colors.
CDI experimental setup and results:
Top: CDI experimental setup. L1: collimator, S: aperture stop, O: object, L2 : Fourier transforming lens.
Bottom: CDI experiment. (a) Captured diffraction pattern with red rectangle showing the removed part. (b) HIO reconstruction after 2000 iterations with complete data (c) HIO reconstruction after 2000 iterations with missing data (d) HIO+CS Reconstruction (2000 HIO iterations total).
Acknowledgements
This work was supported in part by NSF CAREER grant IIS-1453192, ONR grant 1(GG010550)//N00014-14-1-0741, and a Northwestern University McCormick Catalyst grant.
