Abstract

In compressive digital holography, we reconstruct sparse object wavefields from undersampled holograms by solving an 1-minimization problem. Applying wavelet transformations to the object wavefields produces the necessary sparse representations, but prior work clings to transformations with too few vanishing moments. We put several wavelet transformations belonging to different wavelet families to the test by evaluating their sparsifying properties, the number of hologram samples that are required to reconstruct the sparse wavefields perfectly, and the robustness of the reconstructions to additive noise and sparsity defects. In particular, we recommend the CDF 9/7 and 17/11 wavelet transformations, as well as their reverse counter-parts, because they yield sufficiently sparse representations for most accustomed wavefields in combination with robust reconstructions. These and other recommendations are procured from simulations and are validated using biased, noisy holograms.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2017 (3)

2016 (3)

T. Egami, R. Horisaki, L. Tian, and J. Tanida, “Relaxation of mask design for single-shot phase imaging with a coded aperture,” Appl. Opt. 55(8), 1830–1837 (2016).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

2015 (1)

2014 (3)

2013 (5)

P. Clemente, V. Durán, E. Tajahuerce, P. Andrés, V. Climent, and J. Lancis, “Compressive holography with a single-pixel detector,” Opt. Lett. 38(14), 2524–2527 (2013).
[Crossref] [PubMed]

H. Monajemi, S. Jafarpour, M. Gavish, Stat 330/CME 362 Collaboration D. L. Donoho, “Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices,” Proc. Natl. Acad. Sci. U.S.A. 110(4), 1181–1186 (2013).
[Crossref] [PubMed]

Y. Rivenson, A. Stern, and B. Javidi, “Improved depth resolution by single-exposure in-line compressive holography,” Appl. Opt. 52(1), A223–A231 (2013).
[Crossref] [PubMed]

Y. Rivenson, A. Stern, and B. Javidi, “Overview of compressive sensing techniques applied in holography [Invited],” Appl. Opt. 52(1), A423–A432 (2013).
[Crossref] [PubMed]

M.A. Herman, J. Tidman, D. Hewitt, T. Weston, and L. McMackin, “A higher-speed compressive sensing camera through multi-diode design,” Proc. SPIE 8717, 871706 (2013).
[Crossref]

2012 (1)

2011 (6)

M. Liebling, “Fresnelab: sparse representations of digital holograms,” Proc. SPIE 8138, 81380I (2011).
[Crossref]

Y. Rivenson and A. Stern, “Conditions for practicing compressive Fresnel holography,” Opt. Lett. 36(17), 3365–3367 (2011).
[Crossref] [PubMed]

Y. Rivenson, A. Stern, and J. Rosen, “Compressive multiple view projection incoherent holography,” Opt. Express 19(7), 6109–6118 (2011).
[Crossref] [PubMed]

S. Lim, D. L. Marks, and D. J. Brady, “Sampling and processing for compressive holography [Invited],” Appl. Opt. 50(34), H75–H86 (2011).
[Crossref] [PubMed]

S. Becker, J. Bobin, and E. J. Candès, “NESTA: A fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4(1), 1–39 (2011).
[Crossref]

E. J. Candès and Y. Plan, “A Probabilistic and RIPless Theory of Compressed Sensing,” IEEE Trans. Inf. Theory 57(11), 7235–7254 (2011).
[Crossref]

2010 (4)

2009 (3)

2008 (1)

E. J. Candes and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

2007 (3)

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and Depth from a Conventional Camera with a Coded Aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2007).
[Crossref]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

2003 (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

2002 (1)

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9), R85–R101 (2002).
[Crossref]

1995 (1)

D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet Shrinkage: Asymptopia?” J. R. Stat. Soc Series B Stat. Methodol. (Methodological) 57(2), 301–369 (1995).

1993 (1)

M. Unser, A. Aldroubi, and M. Eden, “A family of polynomial spline wavelet transforms,” Signal Processing 30(2), 141–162 (1993).
[Crossref]

Adcock, B.

B. Adcock, A. C. Hansen, C. Poon, and B. Roman are preparing a manuscript to be called “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing.”

Aino, M.

Aldroubi, A.

M. Unser, A. Aldroubi, and M. Eden, “A family of polynomial spline wavelet transforms,” Signal Processing 30(2), 141–162 (1993).
[Crossref]

Allier, C.

Andrés, P.

Angelini, E.

Aronova, M. A.

M. D. Guay, W. Czaja, M. A. Aronova, and R. D. Leapman, “Compressed Sensing Electron Tomography for Determining Biological Structure,” Sci. Rep. 6, 27614 (2016).
[Crossref] [PubMed]

Atlan, M.

Balber, S.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2007).
[Crossref]

Becker, S.

S. Becker, J. Bobin, and E. J. Candès, “NESTA: A fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4(1), 1–39 (2011).
[Crossref]

Belzer, B.

J. D. Villasenor, B. Belzer, and J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4(8), 1053–1060(2010).
[Crossref]

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

Bobin, J.

S. Becker, J. Bobin, and E. J. Candès, “NESTA: A fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4(1), 1–39 (2011).
[Crossref]

Brady, D. J.

Candes, E. J.

E. J. Candes and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Candès, E. J.

S. Becker, J. Bobin, and E. J. Candès, “NESTA: A fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4(1), 1–39 (2011).
[Crossref]

E. J. Candès and Y. Plan, “A Probabilistic and RIPless Theory of Compressed Sensing,” IEEE Trans. Inf. Theory 57(11), 7235–7254 (2011).
[Crossref]

Chen, C.

C. Chen and J. Huang, “The benefit of tree sparsity in accelerated {MRI},” Med. Image Anal. 18(6), 834–842 (2014).
[Crossref] [PubMed]

Chen, H.

Choi, K.

Clemente, P.

Climent, V.

Cossairt, O.

Czaja, W.

M. D. Guay, W. Czaja, M. A. Aronova, and R. D. Leapman, “Compressed Sensing Electron Tomography for Determining Biological Structure,” Sci. Rep. 6, 27614 (2016).
[Crossref] [PubMed]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2007).
[Crossref]

De Pierro, A. R.

M. V. W. Zibetti, E. S. Helou, E. X. Migueles, and A. R. De Pierro, “Accelerating the over-relaxed iterative shrinkage-thresholding algorithms with fast and exact line search for high resolution tomographic image reconstruction,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2305–2308.

Denis, L.

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Donoho, D. L.

H. Monajemi, S. Jafarpour, M. Gavish, Stat 330/CME 362 Collaboration D. L. Donoho, “Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices,” Proc. Natl. Acad. Sci. U.S.A. 110(4), 1181–1186 (2013).
[Crossref] [PubMed]

D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet Shrinkage: Asymptopia?” J. R. Stat. Soc Series B Stat. Methodol. (Methodological) 57(2), 301–369 (1995).

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2007).
[Crossref]

Durán, V.

Durand, F.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and Depth from a Conventional Camera with a Coded Aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Eden, M.

M. Unser, A. Aldroubi, and M. Eden, “A family of polynomial spline wavelet transforms,” Signal Processing 30(2), 141–162 (1993).
[Crossref]

Egami, R.

Egami, T.

Fergus, R.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and Depth from a Conventional Camera with a Coded Aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Foucart, S.

S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing (Birkhäuser, 2013).
[Crossref]

Fournel, T.

Fournier, C.

Freeman, W. T.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and Depth from a Conventional Camera with a Coded Aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Friedlander, M. P.

E. van den Berg and M. P. Friedlander, “Probing the Pareto Frontier for Basis Pursuit Solutions,” SIAM J. Sci. Comput. 31(2), 890–912 (2009).
[Crossref]

Gavish, M.

H. Monajemi, S. Jafarpour, M. Gavish, Stat 330/CME 362 Collaboration D. L. Donoho, “Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices,” Proc. Natl. Acad. Sci. U.S.A. 110(4), 1181–1186 (2013).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2004).

Guay, M. D.

M. D. Guay, W. Czaja, M. A. Aronova, and R. D. Leapman, “Compressed Sensing Electron Tomography for Determining Biological Structure,” Sci. Rep. 6, 27614 (2016).
[Crossref] [PubMed]

Hahn, J.

Hansen, A. C.

B. Adcock, A. C. Hansen, C. Poon, and B. Roman are preparing a manuscript to be called “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing.”

He, K.

Helou, E. S.

M. V. W. Zibetti, E. S. Helou, E. X. Migueles, and A. R. De Pierro, “Accelerating the over-relaxed iterative shrinkage-thresholding algorithms with fast and exact line search for high resolution tomographic image reconstruction,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2305–2308.

Herman, M.A.

M.A. Herman, J. Tidman, D. Hewitt, T. Weston, and L. McMackin, “A higher-speed compressive sensing camera through multi-diode design,” Proc. SPIE 8717, 871706 (2013).
[Crossref]

Hewitt, D.

M.A. Herman, J. Tidman, D. Hewitt, T. Weston, and L. McMackin, “A higher-speed compressive sensing camera through multi-diode design,” Proc. SPIE 8717, 871706 (2013).
[Crossref]

Horisaki, R.

Horstmeyer, R.

Huang, J.

C. Chen and J. Huang, “The benefit of tree sparsity in accelerated {MRI},” Med. Image Anal. 18(6), 834–842 (2014).
[Crossref] [PubMed]

Jafarpour, S.

H. Monajemi, S. Jafarpour, M. Gavish, Stat 330/CME 362 Collaboration D. L. Donoho, “Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices,” Proc. Natl. Acad. Sci. U.S.A. 110(4), 1181–1186 (2013).
[Crossref] [PubMed]

Javidi, B.

Johnstone, I. M.

D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet Shrinkage: Asymptopia?” J. R. Stat. Soc Series B Stat. Methodol. (Methodological) 57(2), 301–369 (1995).

Jolivet, F.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9), R85–R101 (2002).
[Crossref]

Katsaggelos, A.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2007).
[Crossref]

Kerkyacharian, G.

D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet Shrinkage: Asymptopia?” J. R. Stat. Soc Series B Stat. Methodol. (Methodological) 57(2), 301–369 (1995).

Lancis, J.

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2007).
[Crossref]

Leapman, R. D.

M. D. Guay, W. Czaja, M. A. Aronova, and R. D. Leapman, “Compressed Sensing Electron Tomography for Determining Biological Structure,” Sci. Rep. 6, 27614 (2016).
[Crossref] [PubMed]

Levin, A.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and Depth from a Conventional Camera with a Coded Aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Liao, J.

J. D. Villasenor, B. Belzer, and J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4(8), 1053–1060(2010).
[Crossref]

Liebling, M.

M. Liebling, “Fresnelab: sparse representations of digital holograms,” Proc. SPIE 8138, 81380I (2011).
[Crossref]

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

Lim, S.

Llull, P.

Lorenz, D.

Lustig, M.

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J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2004).

D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way (Academic, 2008).

B. Adcock, A. C. Hansen, C. Poon, and B. Roman are preparing a manuscript to be called “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing.”

K. Rao and P.C. Yip, The Transform and Data Compression Handbook (CRC, 2000).
[Crossref]

W. B. Pennebaker and J. L. Mitchell, JPEG: Still Image Data Compression Standard (Van Nostrand Reinhold, 1993).

M. V. W. Zibetti, E. S. Helou, E. X. Migueles, and A. R. De Pierro, “Accelerating the over-relaxed iterative shrinkage-thresholding algorithms with fast and exact line search for high resolution tomographic image reconstruction,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2305–2308.

T.-C. Poon, Digital Holography and Three-Dimensional Display (SpringerUS, 2006).
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Supplementary Material (3)

NameDescription
» Code 1       This MATLAB code generates the figures in the manuscript entitled "Studies on the sparsifying operator in compressive digital holography".
» Data File 1       Underlying values of the phase transition diagrams and undersampling bounds for compressive digital holography with each of eleven wavelet transformations as sparsifying operator.
» Data File 2       PSNR and SNR values for the reconstructed computer-generated object wavefields as a function of the undersampling factor and the specific wavelet transformation.

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Figures (12)

Fig. 1
Fig. 1 The coherence between the angular spectrum method and three one-dimensional wavelet bases decreases as z increases. The signals are decomposed at 9 scales, N = 512, d = 1, λ = 633 nm, and Δx = 3.1 μm.
Fig. 2
Fig. 2 Phase transition diagram for CDH with sparsity in a basis of Haar wavelets. All signals are decomposed at 9 scales, d = 1, N = 512, λ = 633 nm, Δx = 9.3 μm, and z = Nx)2/λ. The blue and orange isolines are the phase transition bounds for respectively the dual B-spline 3 and B-spline 3 wavelet transformation. See Data File 1 for underlying values.
Fig. 3
Fig. 3 The box plots present 1000 restricted isometry constants for CDH with several wavelet transformations as sparsifying operator. The values for the (dual) B-spline wavelet transformations are farthest from 0 and indicate that these transformations are the least robust to additive noise and sparsity defects. Test conditions: d = 1, N = 512, S ∈ {32, 64}, and M ∈ {128, 256, 512}.
Fig. 4
Fig. 4 The computer-generated object wavefields include (a) a transmissive USAF 1951 resolution chart (phase profile is flat) and [(b), (c)] a reflective lens-shaped object with a sag of 6 μm. They are sampled on a 1024 × 1024 square grid with a side length of 9.5 mm and illuminated by a plane wave of wavelength λ = 633 nm.
Fig. 5
Fig. 5 Reconstruction error for the computer-generated object wavefields of the USAF resolution chart [(a), (b)] and the lens-shaped reflective surface [(c), (d)] as a function of the undersampling factor. The wavelet transformations in Table 2 are deployed as sparsifying operator. See Data File 2 for underlying values.
Fig. 6
Fig. 6 Iteration count as a function of the undersampling factor δ for (a) the USAF resolution chart, (b) the lens-shaped relfective surface, (○) the Haar wavelet transformation, and (□) the Battle-Lemarié 3 wavelet transformation. The reconstruction error is visualized by the color of the markers.
Fig. 7
Fig. 7 Amplitude of the computer-generated USAF resolution chart’s reconstruction for CDH (δ = 0.05) with the (a) Haar, (b) reverse CDF 9/7 and (c) reverse CDF 17/11 wavelet transformations as sparsifying operator.
Fig. 8
Fig. 8 Amplitude (top row) and phase (bottom row) of the upper left quarter of the computer-generated lens-shaped surface’s reconstruction for CDH (δ = 0.05) with the (a) bl 3, (b) CDF 9/7, (c) CDFr 9/7, (d) CDF 17/11, and (e) CDFr 17/11 wavelet transformations as sparsifying operator.
Fig. 9
Fig. 9 The reconstructions of the cantilevers degrade from the tips (rapid phase variations) towards the support (slower phase variations) as the undersampling factor decreases. The corresponding holograms were captured at a distance of 4.48 cm from the cantilevers.
Fig. 10
Fig. 10 Reconstructed wavefields of a USAF resolution chart (amplitude; first column), a leaf (amplitude; second column) (phase; third column), and an angle grid (amplitude; fourth column). The holograms are undersampled 10 times (δ = 0.1). The reference is shown in row (a), and the sparsifying operators for rows (b), (c), and (d) are respectively the Haar, Daubechies 16, and Battle-Lemarié 3 wavelet transformations.
Fig. 11
Fig. 11 Reconstructed wavefields of a USAF resolution chart (amplitude; first column), a leaf (amplitude; second column) (phase; third column), and an angle grid (amplitude; fourth column). The holograms are undersampled 10 times (δ = 0.1). The reference is shown in row (a), and the sparsifying operators for rows (b), (c), and (d) are respectively the reverse CDF 9/7, CDF 17/11, and B-spline 3 wavelet transformations.
Fig. 12
Fig. 12 Reconstructed wavefields of a USAF resolution chart (amplitude; first column), a leaf (amplitude; second column) (phase; third column), and an angle grid (amplitude; fourth column). The holograms are undersampled 10 times (δ = 0.1). The sparsifying operator for row (a) is the discrete cosine transform and for row (b) total variation minimization is used as prior information.

Tables (3)

Tables Icon

Table 1 Properties of 11 wavelet transformations

Tables Icon

Table 2 Sparsity of the computer-generated object wavefields in eleven wavelet basesa

Tables Icon

Table 3 Sparsity of the experimental object wavefields in eleven wavelet domainsa

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

g T = T g 0 = T Φ f 0 = T Φ Ψ α 0 ,
μ ( Φ , Ψ ) = N d i , j max | ( Φ Ψ ) i , j | 2 ,
α ^ = min α α 1 subject to g T T Φ Ψ α 2 ε ,
c ( 1 δ S ) α 0 2 2 A T α 0 2 2 c ( 1 + δ S ) α 0 2 2 .
g 0 = Φ f 0 = 1 { { f 0 } H ( f x , f y ) } ,
H ( f x , f y ) = { exp [ j 2 π z λ 1 ( λ f x ) 2 ( λ f y ) 2 ] if f x 2 + f y 2 1 / λ 0 if f x 2 + f y 2 > 1 / λ
π ^ ( δ , ρ ) = C ( δ , ρ ) R ,
δ max | ρ Haar * ( δ ) ρ wavelet * ( δ ) | ,
c ( 1 δ S ) α 0 H α 0 α 0 H A T H A T α 0 c ( 1 + δ S ) α 0 H α 0 ,
SER ( f 0 , f ^ ) = 10 log 10 [ Energy ( f 0 ) Energy ( f 0 f ^ ) ] ,
α ^ = min α TV ( α ) subject to g T T Φ Ψ α 2 ε .

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