• Previous Article
    Constructing continuous stationary covariances as limits of the second-order stochastic difference equations
  • IPI Home
  • This Issue
  • Next Article
    Total variation and wavelet regularization of orientation distribution functions in diffusion MRI
May  2013, 7(2): 585-609. doi: 10.3934/ipi.2013.7.585

Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$

1. 

Department of Mathematics, Tufts University, Medford, MA 02155, United States

2. 

Department of Mathematics, Stockholm University, 10691 Stockholm

Received  December 2011 Revised  September 2012 Published  May 2013

We define a general curvilinear Radon transform in $\mathbb{R}^3$, and we develop its microlocal properties. We prove that singularities can be added (or masked) in any backprojection reconstruction method for this transform. We use the microlocal properties of the transform to develop a local backprojection reconstruction algorithm that decreases the effect of the added singularities and reconstructs the shape of the object. This work was motivated by new models in electron microscope tomography in which the electrons travel over curves such as helices or spirals, and we provide reconstructions for a specific transform motivated by this electron microscope tomography problem.
Citation: Eric Todd Quinto, Hans Rullgård. Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$. Inverse Problems & Imaging, 2013, 7 (2) : 585-609. doi: 10.3934/ipi.2013.7.585
References:
[1]

M. Anastasio, Y. Zou, E. Sudley and X. Pan, Local cone-beam tomography image reconstruction on chords,, Journal of the Optical Society of America A, 24 (2007), 1569.  doi: 10.1364/JOSAA.24.001569.  Google Scholar

[2]

J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms,, Duke Math. J., 55 (1987), 943.  doi: 10.1215/S0012-7094-87-05547-5.  Google Scholar

[3]

J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic on line complexes in three-space,, Trans. Amer. Math. Soc., 335 (1993), 877.  doi: 10.1090/S0002-9947-1993-1080733-8.  Google Scholar

[4]

A. Cormack, The Radon transform on a family of curves in the plane,, Proc. Amer. Math. Soc., 83 (1981), 325.  doi: 10.1090/S0002-9939-1981-0624923-1.  Google Scholar

[5]

A. Cormack, The Radon transform on a family of curves in the plane. II,, Proc. Amer. Math. Soc., 86 (1982), 293.  doi: 10.1090/S0002-9939-1982-0667292-4.  Google Scholar

[6]

N. Dairbekov, G. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field,, Advances in Mathematics, 216 (2007), 535.  doi: 10.1016/j.aim.2007.05.014.  Google Scholar

[7]

M. deHoop, H. Smith, G. Uhlmann and R. van der Hilst, Seismic imaging with the generalized Radon transform: A curvelet transform perspective,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/2/025005.  Google Scholar

[8]

D. Fanelli and O. Öktem, Electron tomography: A short overview with an emphasis on the absorption potential model for the forward problem,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/1/013001.  Google Scholar

[9]

A. Faridani, D. Finch, E. L. Ritman and K. T. Smith, Local tomography II,, SIAM Journal of Applied Mathematics, 57 (1997), 1095.  doi: 10.1137/S0036139995286357.  Google Scholar

[10]

A. Faridani, E. L. Ritman and K. T. Smith, Local tomography,, SIAM Journal of Applied Mathematics, 52 (1992), 459.  doi: 10.1137/0152026.  Google Scholar

[11]

R. Felea, Composition of Fourier integral operators with fold and blowdown singularities,, Comm. Partial Differential Equations, 30 (2005), 1717.  doi: 10.1080/03605300500299968.  Google Scholar

[12]

R. Felea and E. T. Quinto, The microlocal properties of the Local 3-D SPECT operator,, SIAM J. Math. Anal., 43 (2011), 1145.  doi: 10.1137/100807703.  Google Scholar

[13]

D. Finch, I.-R. Lan and G. Uhlmann, Microlocal analysis of the restricted X-ray transform with sources on a curve,, in, 47 (2003), 193.   Google Scholar

[14]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-Ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 81.  doi: 10.1007/s12220-007-9007-6.  Google Scholar

[15]

I. Gelfand, M. Graev and N. Vilenkin, "Generalized Functions,", 5 Academic Press, 5 (1966).   Google Scholar

[16]

I. Gelfand and M. I. Graev, Integral transformations connected with straight line complexes in a complex affine space,, Soviet Math. Doklady, 2 (1961), 809.   Google Scholar

[17]

I. Gelfand and M. I. Graev, Complexes of straight lines in the space $\mathbbC^n$,, Functional Analysis and its Applications, 2 (1968), 219.   Google Scholar

[18]

I. Gelfand, M. I. Graev and Z. Shapiro, A local problem of integral geometry in a space of curves,, Functional Anal. and Appl., 13 (1979), 87.   Google Scholar

[19]

A. Greenleaf, A. Seeger and S. Waigner, Estimates for generalized Radon transforms in three and four dimensions,, Contemporary Mathematics, 251 (2000), 243.  doi: 10.1090/conm/251/03873.  Google Scholar

[20]

A. Greenleaf. and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Mathematical Journal, 58 (1989), 205.  doi: 10.1215/S0012-7094-89-05811-0.  Google Scholar

[21]

A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms,, Ann. Inst. Fourier, 40 (1990), 443.  doi: 10.5802/aif.1220.  Google Scholar

[22]

A. Greenleaf and G. Uhlmann, Microlocal Techniques in Integral Geometry,, Contemporary Math., 113 (1990), 121.  doi: 10.1090/conm/113/1108649.  Google Scholar

[23]

V. Guillemin, "Some Remarks on Integral Geometry,", Technical report, (1975).   Google Scholar

[24]

V. Guillemin and S. Sternberg, "Geometric Asymptotics,", American Mathematical Society, (1977).   Google Scholar

[25]

P. W. Hawkes and E. Kasper, "Principles of Electron Optics. Volume 3. Wave Optics,", Academic Press, (1994).   Google Scholar

[26]

S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds,, Acta Math., 113 (1965), 153.  doi: 10.1007/BF02391776.  Google Scholar

[27]

S. Holman and P. Stefanov, The weighted Doppler transform,, Inverse Probl. Imaging, 4 (2010), 111.  doi: 10.3934/ipi.2010.4.111.  Google Scholar

[28]

L. Hörmander, Fourier integral operators, I,, Acta Mathematica, 127 (1971), 79.  doi: 10.1007/BF02392052.  Google Scholar

[29]

L. Hörmander, "The Analysis of Linear Partial Differential Operators,", I. Springer Verlag, I (1983).   Google Scholar

[30]

A. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224.  doi: 10.1137/S0036139998336043.  Google Scholar

[31]

A. Katsevich, Improved Cone Beam Local Tomography,, Inverse Problems, 22 (2006), 627.  doi: 10.1088/0266-5611/22/2/015.  Google Scholar

[32]

V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging,, Inverse Problems and Imaging, 5 (2011), 659.  doi: 10.3934/ipi.2011.5.659.  Google Scholar

[33]

Á. Kurusa, Support curves of invertible Radon transforms,, Arch. Math. (Basel), 61 (1993), 448.  doi: 10.1007/BF01207544.  Google Scholar

[34]

A. Lawrence, J. Bouwer, G. Perkins and M. Ellisman, Transform-based backprojection for volume reconstruction of large format electron microscope tilt series,, J. Structural Bio., 154 (2006), 144.  doi: 10.1016/j.jsb.2005.12.012.  Google Scholar

[35]

A. Louis and P. Maaş, Contour reconstruction in 3-D X-Ray CT,, IEEE Trans. Medical Imaging, 12 (1993), 764.  doi: 10.1109/42.251129.  Google Scholar

[36]

C. Nolan and M. Cheney, Synthetic Aperture inversion,, Inverse Problems, 18 (2002), 221.  doi: 10.1088/0266-5611/18/1/315.  Google Scholar

[37]

S. Phan, J. Bouwer, J. Lanman, M. Terada and A. Lawrence, Non-linear bundle adjustment for electron tomography,, in, 1 (2009), 604.  doi: 10.1109/CSIE.2009.864.  Google Scholar

[38]

S. Phan and A. Lawrence, Tomography of large format electron microscope tilt series: Image alignment and volume reconstruction,, in, 2 (2008), 176.  doi: 10.1109/CISP.2008.535.  Google Scholar

[39]

D. Popov, The generalized radon transform on the plane, the inverse transform, and the Cavalieri conditions,, Functional Analysis and Its Applications, 35 (2001), 270.  doi: 10.1023/A:1013126507543.  Google Scholar

[40]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331.  doi: 10.1090/S0002-9947-1980-0552261-8.  Google Scholar

[41]

E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, 7 (2008), 321.   Google Scholar

[42]

E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM Journal of Applied Mathematics, 68 (2008), 1282.  doi: 10.1137/07068326X.  Google Scholar

[43]

E. T. Quinto, A. Rieder and T. Schuster, Local Algorithms in Sonar,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/3/035006.  Google Scholar

[44]

L. Reimer, "Transmission Electron Microscopy,", 36 of Springer series in optical sciences. Springer Verlag, 36 (1997).  doi: 10.1007/978-3-662-14824-2.  Google Scholar

[45]

V. Romanov, "Integral Geometry and Inverse Problems for Hyperbolic Equations,", 26 of Springer Tracts in Natural Philosophy. Springer Verlag, 26 (1969).  doi: 10.1007/978-3-642-80781-7.  Google Scholar

[46]

P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds,, Amer. J. Math., 130 (2008), 239.  doi: 10.1353/ajm.2008.0003.  Google Scholar

[47]

E. Vainberg, I. Kazak and V. Kurozaev, Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections,, Soviet Journal of Nondestructive Testing, 17 (1981), 415.   Google Scholar

show all references

References:
[1]

M. Anastasio, Y. Zou, E. Sudley and X. Pan, Local cone-beam tomography image reconstruction on chords,, Journal of the Optical Society of America A, 24 (2007), 1569.  doi: 10.1364/JOSAA.24.001569.  Google Scholar

[2]

J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms,, Duke Math. J., 55 (1987), 943.  doi: 10.1215/S0012-7094-87-05547-5.  Google Scholar

[3]

J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic on line complexes in three-space,, Trans. Amer. Math. Soc., 335 (1993), 877.  doi: 10.1090/S0002-9947-1993-1080733-8.  Google Scholar

[4]

A. Cormack, The Radon transform on a family of curves in the plane,, Proc. Amer. Math. Soc., 83 (1981), 325.  doi: 10.1090/S0002-9939-1981-0624923-1.  Google Scholar

[5]

A. Cormack, The Radon transform on a family of curves in the plane. II,, Proc. Amer. Math. Soc., 86 (1982), 293.  doi: 10.1090/S0002-9939-1982-0667292-4.  Google Scholar

[6]

N. Dairbekov, G. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field,, Advances in Mathematics, 216 (2007), 535.  doi: 10.1016/j.aim.2007.05.014.  Google Scholar

[7]

M. deHoop, H. Smith, G. Uhlmann and R. van der Hilst, Seismic imaging with the generalized Radon transform: A curvelet transform perspective,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/2/025005.  Google Scholar

[8]

D. Fanelli and O. Öktem, Electron tomography: A short overview with an emphasis on the absorption potential model for the forward problem,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/1/013001.  Google Scholar

[9]

A. Faridani, D. Finch, E. L. Ritman and K. T. Smith, Local tomography II,, SIAM Journal of Applied Mathematics, 57 (1997), 1095.  doi: 10.1137/S0036139995286357.  Google Scholar

[10]

A. Faridani, E. L. Ritman and K. T. Smith, Local tomography,, SIAM Journal of Applied Mathematics, 52 (1992), 459.  doi: 10.1137/0152026.  Google Scholar

[11]

R. Felea, Composition of Fourier integral operators with fold and blowdown singularities,, Comm. Partial Differential Equations, 30 (2005), 1717.  doi: 10.1080/03605300500299968.  Google Scholar

[12]

R. Felea and E. T. Quinto, The microlocal properties of the Local 3-D SPECT operator,, SIAM J. Math. Anal., 43 (2011), 1145.  doi: 10.1137/100807703.  Google Scholar

[13]

D. Finch, I.-R. Lan and G. Uhlmann, Microlocal analysis of the restricted X-ray transform with sources on a curve,, in, 47 (2003), 193.   Google Scholar

[14]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-Ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 81.  doi: 10.1007/s12220-007-9007-6.  Google Scholar

[15]

I. Gelfand, M. Graev and N. Vilenkin, "Generalized Functions,", 5 Academic Press, 5 (1966).   Google Scholar

[16]

I. Gelfand and M. I. Graev, Integral transformations connected with straight line complexes in a complex affine space,, Soviet Math. Doklady, 2 (1961), 809.   Google Scholar

[17]

I. Gelfand and M. I. Graev, Complexes of straight lines in the space $\mathbbC^n$,, Functional Analysis and its Applications, 2 (1968), 219.   Google Scholar

[18]

I. Gelfand, M. I. Graev and Z. Shapiro, A local problem of integral geometry in a space of curves,, Functional Anal. and Appl., 13 (1979), 87.   Google Scholar

[19]

A. Greenleaf, A. Seeger and S. Waigner, Estimates for generalized Radon transforms in three and four dimensions,, Contemporary Mathematics, 251 (2000), 243.  doi: 10.1090/conm/251/03873.  Google Scholar

[20]

A. Greenleaf. and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Mathematical Journal, 58 (1989), 205.  doi: 10.1215/S0012-7094-89-05811-0.  Google Scholar

[21]

A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms,, Ann. Inst. Fourier, 40 (1990), 443.  doi: 10.5802/aif.1220.  Google Scholar

[22]

A. Greenleaf and G. Uhlmann, Microlocal Techniques in Integral Geometry,, Contemporary Math., 113 (1990), 121.  doi: 10.1090/conm/113/1108649.  Google Scholar

[23]

V. Guillemin, "Some Remarks on Integral Geometry,", Technical report, (1975).   Google Scholar

[24]

V. Guillemin and S. Sternberg, "Geometric Asymptotics,", American Mathematical Society, (1977).   Google Scholar

[25]

P. W. Hawkes and E. Kasper, "Principles of Electron Optics. Volume 3. Wave Optics,", Academic Press, (1994).   Google Scholar

[26]

S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds,, Acta Math., 113 (1965), 153.  doi: 10.1007/BF02391776.  Google Scholar

[27]

S. Holman and P. Stefanov, The weighted Doppler transform,, Inverse Probl. Imaging, 4 (2010), 111.  doi: 10.3934/ipi.2010.4.111.  Google Scholar

[28]

L. Hörmander, Fourier integral operators, I,, Acta Mathematica, 127 (1971), 79.  doi: 10.1007/BF02392052.  Google Scholar

[29]

L. Hörmander, "The Analysis of Linear Partial Differential Operators,", I. Springer Verlag, I (1983).   Google Scholar

[30]

A. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224.  doi: 10.1137/S0036139998336043.  Google Scholar

[31]

A. Katsevich, Improved Cone Beam Local Tomography,, Inverse Problems, 22 (2006), 627.  doi: 10.1088/0266-5611/22/2/015.  Google Scholar

[32]

V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging,, Inverse Problems and Imaging, 5 (2011), 659.  doi: 10.3934/ipi.2011.5.659.  Google Scholar

[33]

Á. Kurusa, Support curves of invertible Radon transforms,, Arch. Math. (Basel), 61 (1993), 448.  doi: 10.1007/BF01207544.  Google Scholar

[34]

A. Lawrence, J. Bouwer, G. Perkins and M. Ellisman, Transform-based backprojection for volume reconstruction of large format electron microscope tilt series,, J. Structural Bio., 154 (2006), 144.  doi: 10.1016/j.jsb.2005.12.012.  Google Scholar

[35]

A. Louis and P. Maaş, Contour reconstruction in 3-D X-Ray CT,, IEEE Trans. Medical Imaging, 12 (1993), 764.  doi: 10.1109/42.251129.  Google Scholar

[36]

C. Nolan and M. Cheney, Synthetic Aperture inversion,, Inverse Problems, 18 (2002), 221.  doi: 10.1088/0266-5611/18/1/315.  Google Scholar

[37]

S. Phan, J. Bouwer, J. Lanman, M. Terada and A. Lawrence, Non-linear bundle adjustment for electron tomography,, in, 1 (2009), 604.  doi: 10.1109/CSIE.2009.864.  Google Scholar

[38]

S. Phan and A. Lawrence, Tomography of large format electron microscope tilt series: Image alignment and volume reconstruction,, in, 2 (2008), 176.  doi: 10.1109/CISP.2008.535.  Google Scholar

[39]

D. Popov, The generalized radon transform on the plane, the inverse transform, and the Cavalieri conditions,, Functional Analysis and Its Applications, 35 (2001), 270.  doi: 10.1023/A:1013126507543.  Google Scholar

[40]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331.  doi: 10.1090/S0002-9947-1980-0552261-8.  Google Scholar

[41]

E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, 7 (2008), 321.   Google Scholar

[42]

E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM Journal of Applied Mathematics, 68 (2008), 1282.  doi: 10.1137/07068326X.  Google Scholar

[43]

E. T. Quinto, A. Rieder and T. Schuster, Local Algorithms in Sonar,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/3/035006.  Google Scholar

[44]

L. Reimer, "Transmission Electron Microscopy,", 36 of Springer series in optical sciences. Springer Verlag, 36 (1997).  doi: 10.1007/978-3-662-14824-2.  Google Scholar

[45]

V. Romanov, "Integral Geometry and Inverse Problems for Hyperbolic Equations,", 26 of Springer Tracts in Natural Philosophy. Springer Verlag, 26 (1969).  doi: 10.1007/978-3-642-80781-7.  Google Scholar

[46]

P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds,, Amer. J. Math., 130 (2008), 239.  doi: 10.1353/ajm.2008.0003.  Google Scholar

[47]

E. Vainberg, I. Kazak and V. Kurozaev, Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections,, Soviet Journal of Nondestructive Testing, 17 (1981), 415.   Google Scholar

[1]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013

[2]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[3]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1

[4]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649

[5]

C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems & Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457

[6]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems & Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

[7]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029

[8]

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

[9]

Earl Berkson. Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces. Electronic Research Announcements, 2010, 17: 90-103. doi: 10.3934/era.2010.17.90

[10]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027

[11]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721

[12]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579

[13]

Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005

[14]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7

[15]

Michael Music. The nonlinear Fourier transform for two-dimensional subcritical potentials. Inverse Problems & Imaging, 2014, 8 (4) : 1151-1167. doi: 10.3934/ipi.2014.8.1151

[16]

Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339

[17]

Matti Viikinkoski, Mikko Kaasalainen. Shape reconstruction from images: Pixel fields and Fourier transform. Inverse Problems & Imaging, 2014, 8 (3) : 885-900. doi: 10.3934/ipi.2014.8.885

[18]

Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007

[19]

Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems & Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341

[20]

Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic & Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]