• Previous Article
    Smoothing Newton method for $ \ell^0 $-$ \ell^2 $ regularized linear inverse problem
  • IPI Home
  • This Issue
  • Next Article
    On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging
February  2022, 16(1): 119-152. doi: 10.3934/ipi.2021043

A mathematical perspective on radar interferometry

Department of Mathematics, North Carolina State University, Campus Box 8205, Raleigh, NC 27695, USA

* Corresponding author: Mikhail Gilman

Received  July 2020 Revised  April 2021 Published  February 2022 Early access  July 2021

Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two or more radar images of the same target taken at slightly different imaging conditions and/or different times. Its goal is to derive additional information about the target, such as elevation. While this kind of task requires centimeter-level accuracy, the interaction of radar signals with the target, as well as the lack of precision in antenna position and other disturbances, generate ambiguities in the image phase that are orders of magnitude larger than the effect of interest.

Yet the common exposition of radar interferometry in the literature often skips such topics. This may lead to unrealistic requirements for the accuracy of determining the parameters of imaging geometry, unachievable precision of image co-registration, etc. To address these deficiencies, in the current work we analyze the problem of interferometric height reconstruction and provide a careful and detailed account of all the assumptions and requirements to the imaging geometry and data processing needed for a successful extraction of height information from the radar data. We employ two most popular scattering models for radar targets: an isolated point scatterer and delta-correlated extended scatterer, and highlight the similarities and differences between them.

Citation: Mikhail Gilman, Semyon Tsynkov. A mathematical perspective on radar interferometry. Inverse Problems and Imaging, 2022, 16 (1) : 119-152. doi: 10.3934/ipi.2021043
References:
[1]

R. Bamler and P. Hartl, Synthetic aperture radar interferometry, Inverse Problems, 14 (1998), R1–R54. doi: 10.1088/0266-5611/14/4/001.

[2] F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces, Translated and Edited by Carol B. Vesecky and John F. Vesecky. International Series in Natural Philosophy, Pergamon Press, Oxford-New York, 1979. 
[3] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, New York, 1963. 
[4]

F. BovengaD. DerauwF. M. RanaC. BarbierA. ReficeN. Veneziani and R. Vitulli, Multi-chromatic analysis of SAR images for coherent target detection, Remote Sensing, 6 (2014), 8822-8843. 

[5]

G. Brigot, M. Simard, E. Colin-Koeniguer and A. Boulch, Retrieval of forest vertical structure from PolInSAR data by machine learning using LIDAR-derived features, Remote Sensing, 11 (2019), 381.

[6]

M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312.  doi: 10.1137/S0036144500368859.

[7]

M. Cheney and B. Borden, Fundamentals of radar imaging, CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 79 (2009).

[8] S. Cloude, Polarisation: Applications in Remote Sensing, Oxford University Press, 2010. 
[9]

M. CrosettoO. MonserratM. Cuevas-GonzálezN. Devanthéry and B. Crippa, Persistent scatterer interferometry: A review, ISPRS Journal of Photogrammetry and Remote Sensing, 115 (2016), 78-89. 

[10]

I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data. Algorithms and Implementation, Artech House, Boston, 2005.

[11]

L. J. Cutrona, Synthetic Aperture Radar, 2$^{nd}$ edition, M. Skolnik, editor, Radar Handbook, 21, McGraw-Hill, New-York, 1990.

[12]

D. DerauwA. Orban and C. Barbier, Wide band SAR sub-band splitting and inter-band coherence measurements, Remote Sensing Letters, 1 (2010), 133-140. 

[13]

M. EinederC. MinetP. SteigenbergerX. Cong and T. Fritz, Imaging geodesy — toward centimeter-level ranging accuracy with TerraSAR-X, IEEE Transactions on Geoscience and Remote Sensing, 49 (2011), 661-671. 

[14]

A. FerrettiC. Prati and F. Rocca, Permanent scatterers in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 8-20. 

[15]

H. ForooshJ. B. Zerubia and M. Berthod, Extension of phase correlation to subpixel registration, IEEE Transactions on Image Processing, 11 (2002), 188-200. 

[16] G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing, Electronic Engineering Systems Series. CRC Press, Boca Raton, FL, 1999. 
[17]

G. Franceschetti and D. Riccio, Scattering, Natural Surfaces, and Fractals, Elsevier, 2006.

[18]

F. GatelliA. Monti GuamieriF. ParizziP. PasqualiC. Prati and F. Rocca, The wavenumber shift in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 32 (1994), 855-865. 

[19]

M. Gilman, E. Smith and S. Tsynkov, Transionospheric Synthetic Aperture Imaging, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham, Switzerland, 2017.

[20]

M. Gilman and S. Tsynkov, A mathematical model for SAR imaging beyond the first Born approximation, SIAM J. Imaging Sci., 8 (2015), 186-225.  doi: 10.1137/140973025.

[21]

R. M. GoldsteinH. A. Zebker and C. L. Werner, Satellite radar interferometry: Two-dimensional phase unwrapping, Radio science, 23 (1988), 713-720. 

[22]

J. W. Goodman, Statistical properties of laser speckle patterns, in Laser Speckle and Related Phenomena, (1984), 9–75.

[23]

M. Guizar-SicairosS. T. Thurman and J. R. Fienup, Efficient subpixel image registration algorithms, Opt. Lett., 33 (2008), 156-158. 

[24]

R. F. Hanssen, Radar Interferometry: Data Interpretation and Error Analysis Remote Sensing and Digital Image Processing, Kluwer Academic Publishers, New York, 2001.

[25]

E. W. Hoen and H. A. Zebker, Penetration depths inferred from interferometric volume decorrelation observed over the Greenland ice sheet, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 2571-2583. 

[26]

J. A. JacksonB. D. Rigling and R. L. Moses, Canonical scattering feature models for 3D and bistatic SAR, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 525-541. 

[27]

C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach, Springer, 1996.

[28]

D. Just and R. Bamler, Phase statistics of interferograms with applications to synthetic aperture radar, Applied Optics, 33 (1994), 4361-4368. 

[29] J.-S. Lee and E. Pottier, Polarimetric Radar Imaging from Basics to Applications, Optical Science and Engineering. CRC Press, Boca Raton, 2009. 
[30]

G. D. MartinoA. IodiceD. Riccio and G. Ruello, Equivalent number of scatterers for SAR speckle modeling, IEEE Transactions on Geoscience and Remote Sensing, 52 (2014), 2555-2564. 

[31] D. Massonnet and J. -Claude Souyris, Imaging with Synthetic Aperture Radar, Engineering Sciences: Electrical Engineering. EFPL Press. Distributed by CRC Press, Lausanne, Switzerland, 2008. 
[32]

A. MoreiraP. Prats-IraolaM. YounisG. KriegerI. Hajnsek and K. P. Papathanassiou, A tutorial on synthetic aperture radar, IEEE Geoscience and Remote Sensing Magazine, 1 (2013), 6-43. 

[33]

C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images, Artech House Remote Sensing Library. Artech House, Boston, 1998.

[34]

P. A. RosenS. HensleyI. R. JoughinF. K. LiS. N. MadsenE. Rodriguez and R. M. Goldstein, Synthetic aperture radar interferometry, Proceedings of the IEEE, 88 (2000), 333-382. 

[35]

H. S. StoneM. OrchardE. Chang and S. Martucci, A fast direct Fourier-based algorithm for subpixel registration of images, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 2235-2243. 

[36]

Q. Tian and M. N. Huhns, Algorithms for subpixel registration, Computer Vision, Graphics, and Image Processing, 35 (1986), 220-233. 

[37]

A. Voronovich, Wave Scattering from Rough Surfaces, Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1999.

[38]

M. WermuthA. HauschildO. Montenbruck and R. Kahle, TerraSAR-Xprecise orbit determination with real-time GPS ephemerides, Advances in Space Research, 50 (2012), 549-559. 

[39]

B. Yazıcı, I. Son and H. Cagri Yanik, Doppler synthetic aperture radar interferometry: A novel SAR interferometry for height mapping using ultra-narrowband waveforms, Inverse Problems, 34 (2018), 055003. doi: 10.1088/1361-6420/aab24c.

[40]

B. YonelI. Son and B. Yazici, Exact multistatic interferometric imaging via generalized ıirtinger flow, IEEE Transactions on Computational Imaging, 6 (2020), 711-726. 

[41]

Y. T. YoonM. EinederN. Yague-Martinez and O. Montenbruck, TerraSAR-Xprecise trajectory estimation and quality assessment, IEEE Transactions on Geoscience and Remote Sensing, 47 (2009), 1859-1868. 

[42]

H. A. Zebker and J. Villasenor, Decorrelation in interferometric radar echoes, IEEE Transactions on Geoscience and Remote Sensing, 30 (1992), 950-959. 

show all references

References:
[1]

R. Bamler and P. Hartl, Synthetic aperture radar interferometry, Inverse Problems, 14 (1998), R1–R54. doi: 10.1088/0266-5611/14/4/001.

[2] F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces, Translated and Edited by Carol B. Vesecky and John F. Vesecky. International Series in Natural Philosophy, Pergamon Press, Oxford-New York, 1979. 
[3] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, New York, 1963. 
[4]

F. BovengaD. DerauwF. M. RanaC. BarbierA. ReficeN. Veneziani and R. Vitulli, Multi-chromatic analysis of SAR images for coherent target detection, Remote Sensing, 6 (2014), 8822-8843. 

[5]

G. Brigot, M. Simard, E. Colin-Koeniguer and A. Boulch, Retrieval of forest vertical structure from PolInSAR data by machine learning using LIDAR-derived features, Remote Sensing, 11 (2019), 381.

[6]

M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312.  doi: 10.1137/S0036144500368859.

[7]

M. Cheney and B. Borden, Fundamentals of radar imaging, CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 79 (2009).

[8] S. Cloude, Polarisation: Applications in Remote Sensing, Oxford University Press, 2010. 
[9]

M. CrosettoO. MonserratM. Cuevas-GonzálezN. Devanthéry and B. Crippa, Persistent scatterer interferometry: A review, ISPRS Journal of Photogrammetry and Remote Sensing, 115 (2016), 78-89. 

[10]

I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data. Algorithms and Implementation, Artech House, Boston, 2005.

[11]

L. J. Cutrona, Synthetic Aperture Radar, 2$^{nd}$ edition, M. Skolnik, editor, Radar Handbook, 21, McGraw-Hill, New-York, 1990.

[12]

D. DerauwA. Orban and C. Barbier, Wide band SAR sub-band splitting and inter-band coherence measurements, Remote Sensing Letters, 1 (2010), 133-140. 

[13]

M. EinederC. MinetP. SteigenbergerX. Cong and T. Fritz, Imaging geodesy — toward centimeter-level ranging accuracy with TerraSAR-X, IEEE Transactions on Geoscience and Remote Sensing, 49 (2011), 661-671. 

[14]

A. FerrettiC. Prati and F. Rocca, Permanent scatterers in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 8-20. 

[15]

H. ForooshJ. B. Zerubia and M. Berthod, Extension of phase correlation to subpixel registration, IEEE Transactions on Image Processing, 11 (2002), 188-200. 

[16] G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing, Electronic Engineering Systems Series. CRC Press, Boca Raton, FL, 1999. 
[17]

G. Franceschetti and D. Riccio, Scattering, Natural Surfaces, and Fractals, Elsevier, 2006.

[18]

F. GatelliA. Monti GuamieriF. ParizziP. PasqualiC. Prati and F. Rocca, The wavenumber shift in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 32 (1994), 855-865. 

[19]

M. Gilman, E. Smith and S. Tsynkov, Transionospheric Synthetic Aperture Imaging, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham, Switzerland, 2017.

[20]

M. Gilman and S. Tsynkov, A mathematical model for SAR imaging beyond the first Born approximation, SIAM J. Imaging Sci., 8 (2015), 186-225.  doi: 10.1137/140973025.

[21]

R. M. GoldsteinH. A. Zebker and C. L. Werner, Satellite radar interferometry: Two-dimensional phase unwrapping, Radio science, 23 (1988), 713-720. 

[22]

J. W. Goodman, Statistical properties of laser speckle patterns, in Laser Speckle and Related Phenomena, (1984), 9–75.

[23]

M. Guizar-SicairosS. T. Thurman and J. R. Fienup, Efficient subpixel image registration algorithms, Opt. Lett., 33 (2008), 156-158. 

[24]

R. F. Hanssen, Radar Interferometry: Data Interpretation and Error Analysis Remote Sensing and Digital Image Processing, Kluwer Academic Publishers, New York, 2001.

[25]

E. W. Hoen and H. A. Zebker, Penetration depths inferred from interferometric volume decorrelation observed over the Greenland ice sheet, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 2571-2583. 

[26]

J. A. JacksonB. D. Rigling and R. L. Moses, Canonical scattering feature models for 3D and bistatic SAR, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 525-541. 

[27]

C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach, Springer, 1996.

[28]

D. Just and R. Bamler, Phase statistics of interferograms with applications to synthetic aperture radar, Applied Optics, 33 (1994), 4361-4368. 

[29] J.-S. Lee and E. Pottier, Polarimetric Radar Imaging from Basics to Applications, Optical Science and Engineering. CRC Press, Boca Raton, 2009. 
[30]

G. D. MartinoA. IodiceD. Riccio and G. Ruello, Equivalent number of scatterers for SAR speckle modeling, IEEE Transactions on Geoscience and Remote Sensing, 52 (2014), 2555-2564. 

[31] D. Massonnet and J. -Claude Souyris, Imaging with Synthetic Aperture Radar, Engineering Sciences: Electrical Engineering. EFPL Press. Distributed by CRC Press, Lausanne, Switzerland, 2008. 
[32]

A. MoreiraP. Prats-IraolaM. YounisG. KriegerI. Hajnsek and K. P. Papathanassiou, A tutorial on synthetic aperture radar, IEEE Geoscience and Remote Sensing Magazine, 1 (2013), 6-43. 

[33]

C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images, Artech House Remote Sensing Library. Artech House, Boston, 1998.

[34]

P. A. RosenS. HensleyI. R. JoughinF. K. LiS. N. MadsenE. Rodriguez and R. M. Goldstein, Synthetic aperture radar interferometry, Proceedings of the IEEE, 88 (2000), 333-382. 

[35]

H. S. StoneM. OrchardE. Chang and S. Martucci, A fast direct Fourier-based algorithm for subpixel registration of images, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 2235-2243. 

[36]

Q. Tian and M. N. Huhns, Algorithms for subpixel registration, Computer Vision, Graphics, and Image Processing, 35 (1986), 220-233. 

[37]

A. Voronovich, Wave Scattering from Rough Surfaces, Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1999.

[38]

M. WermuthA. HauschildO. Montenbruck and R. Kahle, TerraSAR-Xprecise orbit determination with real-time GPS ephemerides, Advances in Space Research, 50 (2012), 549-559. 

[39]

B. Yazıcı, I. Son and H. Cagri Yanik, Doppler synthetic aperture radar interferometry: A novel SAR interferometry for height mapping using ultra-narrowband waveforms, Inverse Problems, 34 (2018), 055003. doi: 10.1088/1361-6420/aab24c.

[40]

B. YonelI. Son and B. Yazici, Exact multistatic interferometric imaging via generalized ıirtinger flow, IEEE Transactions on Computational Imaging, 6 (2020), 711-726. 

[41]

Y. T. YoonM. EinederN. Yague-Martinez and O. Montenbruck, TerraSAR-Xprecise trajectory estimation and quality assessment, IEEE Transactions on Geoscience and Remote Sensing, 47 (2009), 1859-1868. 

[42]

H. A. Zebker and J. Villasenor, Decorrelation in interferometric radar echoes, IEEE Transactions on Geoscience and Remote Sensing, 30 (1992), 950-959. 

Figure 1.  Traditional presentation of geometry for radar interferometry in the vertical cross-track plane. Points $ {\mathit{\boldsymbol{z}}} $ and $ {\mathit{\boldsymbol{T}}} $ are on the same circle centered at $ {\mathit{\boldsymbol{x}}}^ {(0)} $ (dashed line)
Figure 2.  Calculation of the flat Earth phase in cross-track radar interferometry. The points $ {\mathit{\boldsymbol{z}}}' $ and $ {\mathit{\boldsymbol{T}}} $ have zero elevation, and both circles are centered at $ {\mathit{\boldsymbol{x}}}^ {(0)} $
Figure 3.  The annulus in the vertical cross-track plane due to the main lobe of the $ \mathop{\mathrm{{sinc}}}\nolimits $ in $ (49) $. It is centered at $ {\mathit{\boldsymbol{x}}} $ and has central radius $ R_ {\mathit{\boldsymbol{z}}} $. Its thickness $ 2\Delta_ {\rm{R}} $ is defined by the system bandwidth $ B $, see formula (50). Vertical localization of radar targets can be performed using either interferometry or external information about the elevation
Figure 4.  Radar interferometry with two coequal antennas. All coordinates are specified in the slant reference frame $ (u,v) $. To illustrate formula (88), note that $ l_1+l_2 = z_ {vb}-z_ {va} $
Figure 5.  Interferometry of an extended vertically stratified scatterer. In the analysis of (106), $ V(z'_u) = \tau \mathop{\mathrm{{sinc}}}\nolimits (Bz'_u/c) $ is considered as a function of $ z_u = y_u+z'_u $ (cf. Fig. 3)
[1]

Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems and Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024

[2]

Venkateswaran P. Krishnan, Eric Todd Quinto. Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Problems and Imaging, 2011, 5 (3) : 659-674. doi: 10.3934/ipi.2011.5.659

[3]

Keji Liu. Scattering by impenetrable scatterer in a stratified ocean waveguide. Inverse Problems and Imaging, 2019, 13 (6) : 1349-1365. doi: 10.3934/ipi.2019059

[4]

Kaitlyn Muller. The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging. Inverse Problems and Imaging, 2016, 10 (2) : 549-561. doi: 10.3934/ipi.2016011

[5]

Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of Doppler synthetic aperture radar. Inverse Problems and Imaging, 2019, 13 (6) : 1283-1307. doi: 10.3934/ipi.2019056

[6]

Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems and Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030

[7]

T. Varslo, C E Yarman, M. Cheney, B Yazıcı. A variational approach to waveform design for synthetic-aperture imaging. Inverse Problems and Imaging, 2007, 1 (3) : 577-592. doi: 10.3934/ipi.2007.1.577

[8]

Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems and Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77

[9]

Guanghui Hu, Andrea Mantile, Mourad Sini, Tao Yin. Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles. Inverse Problems and Imaging, 2020, 14 (6) : 1025-1056. doi: 10.3934/ipi.2020054

[10]

Seonho Park, Maciej Rysz, Kaitlin L. Fair, Panos M. Pardalos. Synthetic-Aperture Radar image based positioning in GPS-denied environments using Deep Cosine Similarity Neural Networks. Inverse Problems and Imaging, 2021, 15 (4) : 763-785. doi: 10.3934/ipi.2021013

[11]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027

[12]

Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems and Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064

[13]

Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems and Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033

[14]

Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems and Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231

[15]

Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022025

[16]

Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems and Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009

[17]

Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems and Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026

[18]

Roland Griesmaier. Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Problems and Imaging, 2009, 3 (3) : 389-403. doi: 10.3934/ipi.2009.3.389

[19]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems and Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[20]

Bruno Sixou, Valentina Davidoiu, Max Langer, Francoise Peyrin. Absorption and phase retrieval with Tikhonov and joint sparsity regularizations. Inverse Problems and Imaging, 2013, 7 (1) : 267-282. doi: 10.3934/ipi.2013.7.267

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (302)
  • HTML views (327)
  • Cited by (0)

Other articles
by authors

[Back to Top]