Abstract

Infrared scattering scanning near-field optical microscopy (IR s-SNOM) provides for spectroscopic imaging with nanometer spatial resolution, yet full spatio-spectral imaging is constrained by long measurement times. Here, we demonstrate the application of compressed sensing algorithms to achieve hyperspectral FTIR-based nano-imaging at an order of magnitude faster imaging speed to achieve the same spectral content compared to conventional approaches. At the example of the spectroscopy of a single vibrational resonance, we discuss the relationship of prior knowledge of sparseness of the employed Fourier base functions and sub-sampling. Compressed sensing nano-FTIR spectroscopy promises both rapid and sensitive chemical nano-imaging which is highly relevant in academic and industrial settings for fundamental and applied nano- and bio-materials research.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

G. F. Mohsin, F.-J. Schmitt, C. Kanzler, J. Dirk Epping, S. Flemig, and A. Hornemann, “Structural characterization of melanoidin formed from d-glucose and l-alanine at different temperatures applying FTIR, NMR, EPR, and MALDI-ToF-MS,” Food Chem. 245, 761–767 (2018).
[Crossref] [PubMed]

R. O. Freitas, C. Deneke, F. C. B. Maia, H. G. Medeiros, T. Moreno, P. Dumas, Y. Petroff, and H. Westfahl, “Low-aberration beamline optics for synchrotron infrared nanospectroscopy,” Opt. Express 26(9), 11238–11249 (2018).
[Crossref] [PubMed]

2017 (4)

L. Ran, Y. Zhang, W. Wei, and Q. Zhang, “A hyperspectral image classification framework with spatial pixel pair features,” Sensors (Basel) 17(10), 2421 (2017).
[Crossref] [PubMed]

A. Hornemann, D. Sinning, S. Cortes, L. Campino, P. Emmer, K. Kuhls, G. Ulm, M. Frohme, and B. Beckhoff, “A pilot study on fingerprinting Leishmania species from the Old World using Fourier transform infrared spectroscopy,” Anal. Bioanal. Chem. 409(29), 6907–6923 (2017).
[Crossref] [PubMed]

I. Amenabar, S. Poly, M. Goikoetxea, W. Nuansing, P. Lasch, and R. Hillenbrand, “Hyperspectral infrared nanoimaging of organic samples based on Fourier transform infrared nanospectroscopy,” Nat. Commun. 8, 14402 (2017).
[Crossref] [PubMed]

J. J. Humston, I. Bhattacharya, M. Jacob, and C. M. Cheatum, “Compressively sampled two-dimensional infrared spectroscopy that preserves line shape information,” J. Phys. Chem. A 121(16), 3088–3093 (2017).
[Crossref] [PubMed]

2016 (4)

I. August, Y. Oiknine, M. AbuLeil, I. Abdulhalim, and A. Stern, “Miniature compressive ultra-spectral imaging system utilizing a single liquid crystal phase retarder,” Sci. Rep. 6(1), 23524 (2016).
[Crossref] [PubMed]

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

Y. Mao, Y. Wang, J. Zhou, and H. Jia, “An infrared image super-resolution reconstruction method based on compressive sensing,” Infrared Phys. Technol. 76, 735–739 (2016).
[Crossref]

B. Pollard, F. C. B. Maia, M. B. Raschke, and R. O. Freitas, “Infrared Vibrational Nanospectroscopy by Self-Referenced Interferometry,” Nano Lett. 16(1), 55–61 (2016).
[Crossref] [PubMed]

2015 (5)

B. T. O’Callahan, W. E. Lewis, S. Möbius, J. C. Stanley, E. A. Muller, and M. B. Raschke, “Broadband infrared vibrational nano-spectroscopy using thermal blackbody radiation,” Opt. Express 23(25), 32063–32074 (2015).
[Crossref] [PubMed]

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
[Crossref] [PubMed]

A. S. Stern and J. C. Hoch, “A new approach to compressed sensing for NMR,” Magn. Reson. Chem. 53(11), 908–912 (2015).
[Crossref] [PubMed]

S. Mastel, A. A. Govyadinov, T. V. A. G. de Oliveira, I. Amenabar, and R. Hillenbrand, “Nanoscale-resolved chemical identification of thin organic films using infrared near-field spectroscopy and standard Fourier transform infrared references,” Appl. Phys. Lett. 106(2), 23113 (2015).
[Crossref]

2014 (3)

X. Lin, G. Wetzstein, Y. Liu, and Q. Dai, “Dual-coded compressive hyperspectral imaging,” Opt. Lett. 39(7), 2044–2047 (2014).
[Crossref] [PubMed]

X. Sui, Q. Chen, G. Gu, and X. Shen, “Infrared super-resolution imaging based on compressed sensing,” Infrared Phys. Technol. 63, 119–124 (2014).
[Crossref]

H. A. Bechtel, E. A. Muller, R. L. Olmon, M. C. Martin, and M. B. Raschke, “Ultrabroadband infrared nanospectroscopic imaging,” Proc. Natl. Acad. Sci. U.S.A. 111(20), 7191–7196 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (4)

J. N. Sanders, S. K. Saikin, S. Mostame, X. Andrade, J. R. Widom, A. H. Marcus, and A. Aspuru-Guzik, “Compressed sensing for multidimensional spectroscopy experiments,” J. Phys. Chem. Lett. 3(18), 2697–2702 (2012).
[Crossref] [PubMed]

F. Huth, A. Govyadinov, S. Amarie, W. Nuansing, F. Keilmann, and R. Hillenbrand, “Nano-FTIR absorption spectroscopy of molecular fingerprints at 20 nm spatial resolution,” Nano Lett. 12(8), 3973–3978 (2012).
[Crossref] [PubMed]

A. C. Jones and M. B. Raschke, “Thermal Infrared Near-Field Spectroscopy,” Nano Lett. 12(3), 1475–1481 (2012).
[Crossref] [PubMed]

A. Gottwald, R. Klein, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “Current capabilities at the Metrology Light Source,” Metrologia 49(2), S146–S151 (2012).
[Crossref]

2011 (1)

D. J. Holland, M. J. Bostock, L. F. Gladden, and D. Nietlispach, “Fast multidimensional NMR spectroscopy using compressed sensing,” Angew. Chem. Int. Ed. Engl. 50(29), 6548–6551 (2011).
[Crossref] [PubMed]

2009 (2)

A. J. Huber, A. Ziegler, T. Köck, and R. Hillenbrand, “Infrared nanoscopy of strained semiconductors,” Nat. Nanotechnol. 4(3), 153–157 (2009).
[Crossref] [PubMed]

A. Cohen, W. Dahmen, and R. DeVore, “Compressed sensing and best k-terms approximation,” J. Am. Math. Soc. 22(1), 211–231 (2009).
[Crossref]

2008 (1)

A. Hartschuh, “Tip-enhanced near-field optical microscopy,” Angew. Chem. Int. Ed. Engl. 47(43), 8178–8191 (2008).
[Crossref] [PubMed]

2007 (1)

E. J. Candés and T. Tao, “The Dantzig selector: statistical estimation when p is much larger than n,” Ann. Stat. 35(6), 2313–2351 (2007).
[Crossref]

2006 (4)

E. J. Candés, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

E. J. Candés and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 51(4), 1289–1306 (2006).
[Crossref]

2005 (2)

E. J. Candés and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[Crossref]

D. C. Fernandez, R. Bhargava, S. M. Hewitt, and I. W. Levin, “Infrared spectroscopic imaging for histopathologic recognition,” Nat. Biotechnol. 23(4), 469–474 (2005).
[Crossref] [PubMed]

2000 (2)

B. Knoll and F. Keilmann, “Infrared conductivity mapping for nanoelectronics,” Appl. Phys. Lett. 77(24), 3980–3982 (2000).
[Crossref]

R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85(14), 3029–3032 (2000).
[Crossref] [PubMed]

1998 (2)

G. Downey, “Food and food ingredient authentication by mid-infrared spectroscopy and chemometrics,” TrAC Trends Analyt. Chem. 17(7), 418–424 (1998).
[Crossref]

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20(1), 33–61 (1998).
[Crossref]

1993 (2)

S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41(12), 3397–3415 (1993).
[Crossref]

P. Hansen and D. O’Leary, “The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems,” SIAM J. Sci. Comput. 14(6), 1487–1503 (1993).
[Crossref]

1991 (1)

D. Helm, H. Labischinski, G. Schallehn, and D. Naumann, “Classification and identification of bacteria by Fourier-transform infrared spectroscopy,” J. Gen. Microbiol. 137(1), 69–79 (1991).
[Crossref] [PubMed]

Abdulhalim, I.

I. August, Y. Oiknine, M. AbuLeil, I. Abdulhalim, and A. Stern, “Miniature compressive ultra-spectral imaging system utilizing a single liquid crystal phase retarder,” Sci. Rep. 6(1), 23524 (2016).
[Crossref] [PubMed]

AbuLeil, M.

I. August, Y. Oiknine, M. AbuLeil, I. Abdulhalim, and A. Stern, “Miniature compressive ultra-spectral imaging system utilizing a single liquid crystal phase retarder,” Sci. Rep. 6(1), 23524 (2016).
[Crossref] [PubMed]

Ahlers, F. J.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
[Crossref] [PubMed]

Amarie, S.

F. Huth, A. Govyadinov, S. Amarie, W. Nuansing, F. Keilmann, and R. Hillenbrand, “Nano-FTIR absorption spectroscopy of molecular fingerprints at 20 nm spatial resolution,” Nano Lett. 12(8), 3973–3978 (2012).
[Crossref] [PubMed]

Amenabar, I.

I. Amenabar, S. Poly, M. Goikoetxea, W. Nuansing, P. Lasch, and R. Hillenbrand, “Hyperspectral infrared nanoimaging of organic samples based on Fourier transform infrared nanospectroscopy,” Nat. Commun. 8, 14402 (2017).
[Crossref] [PubMed]

S. Mastel, A. A. Govyadinov, T. V. A. G. de Oliveira, I. Amenabar, and R. Hillenbrand, “Nanoscale-resolved chemical identification of thin organic films using infrared near-field spectroscopy and standard Fourier transform infrared references,” Appl. Phys. Lett. 106(2), 23113 (2015).
[Crossref]

Andrade, X.

J. N. Sanders, S. K. Saikin, S. Mostame, X. Andrade, J. R. Widom, A. H. Marcus, and A. Aspuru-Guzik, “Compressed sensing for multidimensional spectroscopy experiments,” J. Phys. Chem. Lett. 3(18), 2697–2702 (2012).
[Crossref] [PubMed]

Aprojanz, J.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

Aspuru-Guzik, A.

J. N. Sanders, S. K. Saikin, S. Mostame, X. Andrade, J. R. Widom, A. H. Marcus, and A. Aspuru-Guzik, “Compressed sensing for multidimensional spectroscopy experiments,” J. Phys. Chem. Lett. 3(18), 2697–2702 (2012).
[Crossref] [PubMed]

August, I.

I. August, Y. Oiknine, M. AbuLeil, I. Abdulhalim, and A. Stern, “Miniature compressive ultra-spectral imaging system utilizing a single liquid crystal phase retarder,” Sci. Rep. 6(1), 23524 (2016).
[Crossref] [PubMed]

August, Y.

Baringhaus, J.

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H. A. Bechtel, E. A. Muller, R. L. Olmon, M. C. Martin, and M. B. Raschke, “Ultrabroadband infrared nanospectroscopic imaging,” Proc. Natl. Acad. Sci. U.S.A. 111(20), 7191–7196 (2014).
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Pakdehi, D. M.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

Patoka, P.

Petroff, Y.

Pierz, K.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
[Crossref] [PubMed]

Pollard, B.

B. Pollard, F. C. B. Maia, M. B. Raschke, and R. O. Freitas, “Infrared Vibrational Nanospectroscopy by Self-Referenced Interferometry,” Nano Lett. 16(1), 55–61 (2016).
[Crossref] [PubMed]

Poly, S.

I. Amenabar, S. Poly, M. Goikoetxea, W. Nuansing, P. Lasch, and R. Hillenbrand, “Hyperspectral infrared nanoimaging of organic samples based on Fourier transform infrared nanospectroscopy,” Nat. Commun. 8, 14402 (2017).
[Crossref] [PubMed]

Ran, L.

L. Ran, Y. Zhang, W. Wei, and Q. Zhang, “A hyperspectral image classification framework with spatial pixel pair features,” Sensors (Basel) 17(10), 2421 (2017).
[Crossref] [PubMed]

Raschke, M. B.

B. Pollard, F. C. B. Maia, M. B. Raschke, and R. O. Freitas, “Infrared Vibrational Nanospectroscopy by Self-Referenced Interferometry,” Nano Lett. 16(1), 55–61 (2016).
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B. T. O’Callahan, W. E. Lewis, S. Möbius, J. C. Stanley, E. A. Muller, and M. B. Raschke, “Broadband infrared vibrational nano-spectroscopy using thermal blackbody radiation,” Opt. Express 23(25), 32063–32074 (2015).
[Crossref] [PubMed]

H. A. Bechtel, E. A. Muller, R. L. Olmon, M. C. Martin, and M. B. Raschke, “Ultrabroadband infrared nanospectroscopic imaging,” Proc. Natl. Acad. Sci. U.S.A. 111(20), 7191–7196 (2014).
[Crossref] [PubMed]

A. C. Jones and M. B. Raschke, “Thermal Infrared Near-Field Spectroscopy,” Nano Lett. 12(3), 1475–1481 (2012).
[Crossref] [PubMed]

Richter, M.

A. Gottwald, R. Klein, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “Current capabilities at the Metrology Light Source,” Metrologia 49(2), S146–S151 (2012).
[Crossref]

Rivenson, Y.

Romberg, J.

E. J. Candés, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Rühl, E.

Saikin, S. K.

J. N. Sanders, S. K. Saikin, S. Mostame, X. Andrade, J. R. Widom, A. H. Marcus, and A. Aspuru-Guzik, “Compressed sensing for multidimensional spectroscopy experiments,” J. Phys. Chem. Lett. 3(18), 2697–2702 (2012).
[Crossref] [PubMed]

Sanders, J. N.

J. N. Sanders, S. K. Saikin, S. Mostame, X. Andrade, J. R. Widom, A. H. Marcus, and A. Aspuru-Guzik, “Compressed sensing for multidimensional spectroscopy experiments,” J. Phys. Chem. Lett. 3(18), 2697–2702 (2012).
[Crossref] [PubMed]

Saunders, M.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20(1), 33–61 (1998).
[Crossref]

Schallehn, G.

D. Helm, H. Labischinski, G. Schallehn, and D. Naumann, “Classification and identification of bacteria by Fourier-transform infrared spectroscopy,” J. Gen. Microbiol. 137(1), 69–79 (1991).
[Crossref] [PubMed]

Schmitt, F.-J.

G. F. Mohsin, F.-J. Schmitt, C. Kanzler, J. Dirk Epping, S. Flemig, and A. Hornemann, “Structural characterization of melanoidin formed from d-glucose and l-alanine at different temperatures applying FTIR, NMR, EPR, and MALDI-ToF-MS,” Food Chem. 245, 761–767 (2018).
[Crossref] [PubMed]

Scholze, F.

A. Gottwald, R. Klein, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “Current capabilities at the Metrology Light Source,” Metrologia 49(2), S146–S151 (2012).
[Crossref]

Schumacher, H. W.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
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Seyller, T.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
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Shen, X.

X. Sui, Q. Chen, G. Gu, and X. Shen, “Infrared super-resolution imaging based on compressed sensing,” Infrared Phys. Technol. 63, 119–124 (2014).
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Sinning, D.

A. Hornemann, D. Sinning, S. Cortes, L. Campino, P. Emmer, K. Kuhls, G. Ulm, M. Frohme, and B. Beckhoff, “A pilot study on fingerprinting Leishmania species from the Old World using Fourier transform infrared spectroscopy,” Anal. Bioanal. Chem. 409(29), 6907–6923 (2017).
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Stanley, J. C.

Stern, A.

I. August, Y. Oiknine, M. AbuLeil, I. Abdulhalim, and A. Stern, “Miniature compressive ultra-spectral imaging system utilizing a single liquid crystal phase retarder,” Sci. Rep. 6(1), 23524 (2016).
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Y. August, C. Vachman, Y. Rivenson, and A. Stern, “Compressive hyperspectral imaging by random separable projections in both the spatial and the spectral domains,” Appl. Opt. 52(10), D46–D54 (2013).
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Stern, A. S.

A. S. Stern and J. C. Hoch, “A new approach to compressed sensing for NMR,” Magn. Reson. Chem. 53(11), 908–912 (2015).
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Stosch, R.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
[Crossref] [PubMed]

Sui, X.

X. Sui, Q. Chen, G. Gu, and X. Shen, “Infrared super-resolution imaging based on compressed sensing,” Infrared Phys. Technol. 63, 119–124 (2014).
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Tao, T.

E. J. Candés and T. Tao, “The Dantzig selector: statistical estimation when p is much larger than n,” Ann. Stat. 35(6), 2313–2351 (2007).
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E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

E. J. Candés and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

E. J. Candés, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[Crossref]

E. J. Candés and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[Crossref]

Tegenkamp, C.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
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A. Gottwald, R. Klein, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “Current capabilities at the Metrology Light Source,” Metrologia 49(2), S146–S151 (2012).
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A. Hornemann, D. Sinning, S. Cortes, L. Campino, P. Emmer, K. Kuhls, G. Ulm, M. Frohme, and B. Beckhoff, “A pilot study on fingerprinting Leishmania species from the Old World using Fourier transform infrared spectroscopy,” Anal. Bioanal. Chem. 409(29), 6907–6923 (2017).
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A. Gottwald, R. Klein, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “Current capabilities at the Metrology Light Source,” Metrologia 49(2), S146–S151 (2012).
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Vachman, C.

Wang, Y.

Y. Mao, Y. Wang, J. Zhou, and H. Jia, “An infrared image super-resolution reconstruction method based on compressive sensing,” Infrared Phys. Technol. 76, 735–739 (2016).
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Wei, W.

L. Ran, Y. Zhang, W. Wei, and Q. Zhang, “A hyperspectral image classification framework with spatial pixel pair features,” Sensors (Basel) 17(10), 2421 (2017).
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Westfahl, H.

Wetzstein, G.

Widom, J. R.

J. N. Sanders, S. K. Saikin, S. Mostame, X. Andrade, J. R. Widom, A. H. Marcus, and A. Aspuru-Guzik, “Compressed sensing for multidimensional spectroscopy experiments,” J. Phys. Chem. Lett. 3(18), 2697–2702 (2012).
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Wundrack, S.

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
[Crossref] [PubMed]

Zhang, Q.

L. Ran, Y. Zhang, W. Wei, and Q. Zhang, “A hyperspectral image classification framework with spatial pixel pair features,” Sensors (Basel) 17(10), 2421 (2017).
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Zhang, Y.

L. Ran, Y. Zhang, W. Wei, and Q. Zhang, “A hyperspectral image classification framework with spatial pixel pair features,” Sensors (Basel) 17(10), 2421 (2017).
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Zhang, Z.

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
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S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41(12), 3397–3415 (1993).
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Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
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Zhou, J.

Y. Mao, Y. Wang, J. Zhou, and H. Jia, “An infrared image super-resolution reconstruction method based on compressive sensing,” Infrared Phys. Technol. 76, 735–739 (2016).
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A. J. Huber, A. Ziegler, T. Köck, and R. Hillenbrand, “Infrared nanoscopy of strained semiconductors,” Nat. Nanotechnol. 4(3), 153–157 (2009).
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2D Mater. (1)

M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, J. Lidzba, T. Seyller, F. Hohls, F. J. Ahlers, and H. W. Schumacher, “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Mater. 3(4), 41002 (2016).
[Crossref]

Anal. Bioanal. Chem. (1)

A. Hornemann, D. Sinning, S. Cortes, L. Campino, P. Emmer, K. Kuhls, G. Ulm, M. Frohme, and B. Beckhoff, “A pilot study on fingerprinting Leishmania species from the Old World using Fourier transform infrared spectroscopy,” Anal. Bioanal. Chem. 409(29), 6907–6923 (2017).
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E. J. Candés and T. Tao, “The Dantzig selector: statistical estimation when p is much larger than n,” Ann. Stat. 35(6), 2313–2351 (2007).
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Commun. Pure Appl. Math. (1)

E. J. Candés, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
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Food Chem. (1)

G. F. Mohsin, F.-J. Schmitt, C. Kanzler, J. Dirk Epping, S. Flemig, and A. Hornemann, “Structural characterization of melanoidin formed from d-glucose and l-alanine at different temperatures applying FTIR, NMR, EPR, and MALDI-ToF-MS,” Food Chem. 245, 761–767 (2018).
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IEEE Trans. Inf. Theory (4)

E. J. Candés and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[Crossref]

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
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D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 51(4), 1289–1306 (2006).
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E. J. Candés and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
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IEEE Trans. Signal Process. (1)

S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41(12), 3397–3415 (1993).
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X. Sui, Q. Chen, G. Gu, and X. Shen, “Infrared super-resolution imaging based on compressed sensing,” Infrared Phys. Technol. 63, 119–124 (2014).
[Crossref]

Y. Mao, Y. Wang, J. Zhou, and H. Jia, “An infrared image super-resolution reconstruction method based on compressive sensing,” Infrared Phys. Technol. 76, 735–739 (2016).
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D. Helm, H. Labischinski, G. Schallehn, and D. Naumann, “Classification and identification of bacteria by Fourier-transform infrared spectroscopy,” J. Gen. Microbiol. 137(1), 69–79 (1991).
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J. J. Humston, I. Bhattacharya, M. Jacob, and C. M. Cheatum, “Compressively sampled two-dimensional infrared spectroscopy that preserves line shape information,” J. Phys. Chem. A 121(16), 3088–3093 (2017).
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J. N. Sanders, S. K. Saikin, S. Mostame, X. Andrade, J. R. Widom, A. H. Marcus, and A. Aspuru-Guzik, “Compressed sensing for multidimensional spectroscopy experiments,” J. Phys. Chem. Lett. 3(18), 2697–2702 (2012).
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J. Phys. Condens. Matter (1)

M. Kruskopf, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, C.-C. Kalmbach, A. Müller, J. Baringhaus, C. Tegenkamp, F. J. Ahlers, and H. W. Schumacher, “Epitaxial graphene on SiC: modification of structural and electron transport properties by substrate pretreatment,” J. Phys. Condens. Matter 27(18), 185303 (2015).
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Magn. Reson. Chem. (1)

A. S. Stern and J. C. Hoch, “A new approach to compressed sensing for NMR,” Magn. Reson. Chem. 53(11), 908–912 (2015).
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Metrologia (1)

A. Gottwald, R. Klein, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “Current capabilities at the Metrology Light Source,” Metrologia 49(2), S146–S151 (2012).
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F. Huth, A. Govyadinov, S. Amarie, W. Nuansing, F. Keilmann, and R. Hillenbrand, “Nano-FTIR absorption spectroscopy of molecular fingerprints at 20 nm spatial resolution,” Nano Lett. 12(8), 3973–3978 (2012).
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A. C. Jones and M. B. Raschke, “Thermal Infrared Near-Field Spectroscopy,” Nano Lett. 12(3), 1475–1481 (2012).
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B. Pollard, F. C. B. Maia, M. B. Raschke, and R. O. Freitas, “Infrared Vibrational Nanospectroscopy by Self-Referenced Interferometry,” Nano Lett. 16(1), 55–61 (2016).
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I. August, Y. Oiknine, M. AbuLeil, I. Abdulhalim, and A. Stern, “Miniature compressive ultra-spectral imaging system utilizing a single liquid crystal phase retarder,” Sci. Rep. 6(1), 23524 (2016).
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L. Ran, Y. Zhang, W. Wei, and Q. Zhang, “A hyperspectral image classification framework with spatial pixel pair features,” Sensors (Basel) 17(10), 2421 (2017).
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Figures (4)

Fig. 1
Fig. 1 Experimental setup. The spectral irradiance, E e,  ν ˜ , of the IR radiation emitted by the electron storage ring MLS is shown in (a). The features around 2000 cm−1 originate from the diamond window separating the beamline from the storage ring. The radiation is coupled into an AFM based s-SNOM with a Michelson interferometer (b). The AFM is operated in tapping mode. The detector signal is demodulated at the nth harmonic of the fundamental cantilever frequency Ω to extract the near-field signal, and an interferogram is recorded (c). Fourier transform (FT) yields the spectrum (d), which is spatially mapped using 41 x 44 pixels over an 8 μm by 8 μm scan area of a nanostructured SiC surface. The topography of the indent is shown in the inset.
Fig. 2
Fig. 2 Histogram of indices of relevant basis functions obtained when subsequently applying compressed sensing to the measurements of each position of the considered array.
Fig. 3
Fig. 3 Example interferogram (a) of measured (blue line), randomly sampled (black dots) and reconstructed (red) data from the randomly sampled points. The inset shows a zoom into a 100 μm long stretch (highlighted in gray) of the optical path difference within the center burst. The corresponding amplitude of the Fourier transform (FT) is shown in (b), comparing the FT based on measured (blue) and reconstructed (red) interferogram data. The black curve shows the FT from the sub-sampled and interpolated data points. The inset plots both amplitude (solid line) and phase (dashed line) of the FT within the spectral region of the phonon peak, showing excellent agreement in both variables between the measured data (blue) and those reconstructed from the subsampled data (red).
Fig. 4
Fig. 4 Spectral mapping of the 6H-SiC surface covered by mono- and bi-layer graphene around a mechanically created indent, shown as a topographical image in (a). Cracks develop as lines from the lower-left to the upper right corner of the image. The lines going from the upper-left to the lower-right corner correspond to terraces on the SiC surface. Close to the terrace edges bi-layer graphene has formed, leading to the dark stripes in the AFM phase image shown in (b). A 41 x 44 array of spectra has been recorded over the area shown in (a). The analysis of peak heights of the SiC-phonon resonance in the spectral region around 900 cm−1 results in a 2D map shown in (c) with excellent agreement between measured and reconstructed data. Similarly, the peak position has been analyzed in (d), where the reconstructed data reproduce the main features of the measured data.

Equations (3)

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S( kΔ ν ˜ )= n=0 N x 1 I( nΔx ) e 2πin k N x ,
y(t)= i=1 K z i ϕ π(i) (t)
min z z 0 subject to y=Az,

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