May  2015, 9(2): 395-413. doi: 10.3934/ipi.2015.9.395

Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization

1. 

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

Received  April 2014 Revised  October 2014 Published  March 2015

One challenge for modern imaging methods is the investigation of objects which change during the data acquisition. This occurs in non-destructive testing as well as in medical applications, e.g. on account of patient or organ movements. Due to the object's deformations, the respective imaging modality is described by a dynamic inverse problem. In this paper, a classification scheme for linear dynamic problems depending on the object's motion is provided. Based on this scheme, we study the class in detail where the dynamic problem is still related to the operator in the static case, and where we call the deformations moderate. We proof important properties of the dynamic operator, derive a singular value decomposition and develop suitable regularization methods. The application of these methods to specific problems is illustrated at two examples including dynamic computerized tomography.
Citation: Bernadette N. Hahn. Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization. Inverse Problems & Imaging, 2015, 9 (2) : 395-413. doi: 10.3934/ipi.2015.9.395
References:
[1]

D. Atkinson, D. L. Hill, P. N. Stoyle, P. E. Summers, S. Clare, R. Bowtell and S. F. Keevil, Automatic compensation of motion artefacts in MRI,, Magn. Reson. Med., 41 (1999), 163.  doi: 10.1002/(SICI)1522-2594(199901)41:1<163::AID-MRM23>3.3.CO;2-0.  Google Scholar

[2]

L. Desbat, S. Roux and P. Grangeat, Compensation of some time dependent deformations in tomography,, IEEE Trans. Med. Imag., 26 (2007), 261.  doi: 10.1109/TMI.2006.889743.  Google Scholar

[3]

H. W. Engl and C. W. Groetsch, Inverse and Ill-Posed Problems,, Academic Press, (1986).   Google Scholar

[4]

J. Fitzgerald and P.G. Danias, Effect of motion on cardiac SPECT imaging: Recognition and motion correction,, J. Nucl. Cardiol., 8 (2001), 701.  doi: 10.1067/mnc.2001.118694.  Google Scholar

[5]

F. Gigengack, L. Ruthotto, M. Burger, C. H. Wolters, X. Jiang and K. P. Schäfers, Motion correction in dual gated cardiac PET using mass-preserving image registration,, IEEE Trans. Med. Imag., 31 (2012), 698.  doi: 10.1109/TMI.2011.2175402.  Google Scholar

[6]

G. H. Glover and J. M. Pauly, Projection Reconstruction Techniques for reduction of motion effects in MRI,, Mag. Reson. Med., 28 (1992), 275.  doi: 10.1002/mrm.1910280209.  Google Scholar

[7]

B. Hahn, Reconstruction of dynamic objects with affine deformations in dynamic computerized tomography,, J. Inverse Ill-Posed Probl., 22 (2014), 323.  doi: 10.1515/jip-2012-0094.  Google Scholar

[8]

B. N. Hahn, Efficient algorithms for linear dynamic inverse problems with known motion,, Inverse Problems, 30 (2014).  doi: 10.1088/0266-5611/30/3/035008.  Google Scholar

[9]

B. Hofmann, Regularization for Applied Inverse and Ill-posed Problems,, Teubner, (1986).  doi: 10.1007/978-3-322-93034-7.  Google Scholar

[10]

A. Katsevich, An accurate approximate algorithm for motion compensation in two-dimensional tomography,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/6/065007.  Google Scholar

[11]

A. Katsevich, M. Silver and A. Zamayatin, Local tomography and the motion estimation problem,, SIAM J. Imaging Sci., 4 (2011), 200.  doi: 10.1137/100796728.  Google Scholar

[12]

S. Kindermann and A. Leitão, On regularization methods for inverse problems of dynamic type,, Numer. Func. Anal. Opt., 27 (2006), 139.  doi: 10.1080/01630560600569973.  Google Scholar

[13]

D. Le Bihan, C. Poupon, A. Amadon and F. Lethimonnier, Artifacts and pitfalls in diffusion MRI,, JMRI - J. Magn. Reson. Im., 24 (2006), 478.   Google Scholar

[14]

A. K. Louis, Inverse und Schlecht Gestellte Probleme,, Teubner, (1989).  doi: 10.1007/978-3-322-84808-6.  Google Scholar

[15]

A. K. Louis, Diffusion reconstruction from very noisy tomographic data,, Inverse Probl. Imagine, 4 (2010), 675.  doi: 10.3934/ipi.2010.4.675.  Google Scholar

[16]

W. Lu and T. R. Mackie, Tomographic motion detection and correction directly in sinogram space,, Phys. Med. Biol., 47 (2002), 1267.  doi: 10.1088/0031-9155/47/8/304.  Google Scholar

[17]

S. J. McQuaid and B. F. Hutton, Sources of attenuation-correction artefacts in cardiac PET/CT and SPECT/CT,, Eur. J. Nucl. Med. Mol. Imaging, 35 (2008), 1117.  doi: 10.1007/s00259-008-0718-0.  Google Scholar

[18]

J. L. Müller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications,, SIAM, (2012).   Google Scholar

[19]

F. Natterer, The Mathematics of Computerized Tomography,, John Wiley & Sons, (1986).   Google Scholar

[20]

F. Qiao, T. Pan, J. W. Clark, O. R. Mawlawi, A motion-incorporated reconstruction method for gated PET studies,, Phys. Med. Biol., 51 (2006), 3769.  doi: 10.1088/0031-9155/51/15/012.  Google Scholar

[21]

U. Schmitt and A. K. Louis, Efficient algorithms for the regularization of dynamic inverse problems: I. Theory,, Inverse Problems, 18 (2002), 645.  doi: 10.1088/0266-5611/18/3/308.  Google Scholar

[22]

U. Schmitt, A. K. Louis, C. Wolters and M. Vauhkonen, Efficient algorithms for the regularization of dynamic inverse problems: II. Applications,, Inverse Problems, 18 (2002), 659.  doi: 10.1088/0266-5611/18/3/309.  Google Scholar

[23]

A. Shankaranarayanan, M. Wendt, J. S. Lewin and J. L. Duerk, Two-step navigatorless correction algorithm for radial k-space MRI acquisitions,, Magn. Reson. Med., 45 (2001), 277.   Google Scholar

[24]

L. Shepp, S. Hilal and R. Schulz, The tuning fork artifact in computerized tomography,, Comput. Graph. Image Process., 10 (1979), 246.  doi: 10.1016/0146-664X(79)90004-2.  Google Scholar

show all references

References:
[1]

D. Atkinson, D. L. Hill, P. N. Stoyle, P. E. Summers, S. Clare, R. Bowtell and S. F. Keevil, Automatic compensation of motion artefacts in MRI,, Magn. Reson. Med., 41 (1999), 163.  doi: 10.1002/(SICI)1522-2594(199901)41:1<163::AID-MRM23>3.3.CO;2-0.  Google Scholar

[2]

L. Desbat, S. Roux and P. Grangeat, Compensation of some time dependent deformations in tomography,, IEEE Trans. Med. Imag., 26 (2007), 261.  doi: 10.1109/TMI.2006.889743.  Google Scholar

[3]

H. W. Engl and C. W. Groetsch, Inverse and Ill-Posed Problems,, Academic Press, (1986).   Google Scholar

[4]

J. Fitzgerald and P.G. Danias, Effect of motion on cardiac SPECT imaging: Recognition and motion correction,, J. Nucl. Cardiol., 8 (2001), 701.  doi: 10.1067/mnc.2001.118694.  Google Scholar

[5]

F. Gigengack, L. Ruthotto, M. Burger, C. H. Wolters, X. Jiang and K. P. Schäfers, Motion correction in dual gated cardiac PET using mass-preserving image registration,, IEEE Trans. Med. Imag., 31 (2012), 698.  doi: 10.1109/TMI.2011.2175402.  Google Scholar

[6]

G. H. Glover and J. M. Pauly, Projection Reconstruction Techniques for reduction of motion effects in MRI,, Mag. Reson. Med., 28 (1992), 275.  doi: 10.1002/mrm.1910280209.  Google Scholar

[7]

B. Hahn, Reconstruction of dynamic objects with affine deformations in dynamic computerized tomography,, J. Inverse Ill-Posed Probl., 22 (2014), 323.  doi: 10.1515/jip-2012-0094.  Google Scholar

[8]

B. N. Hahn, Efficient algorithms for linear dynamic inverse problems with known motion,, Inverse Problems, 30 (2014).  doi: 10.1088/0266-5611/30/3/035008.  Google Scholar

[9]

B. Hofmann, Regularization for Applied Inverse and Ill-posed Problems,, Teubner, (1986).  doi: 10.1007/978-3-322-93034-7.  Google Scholar

[10]

A. Katsevich, An accurate approximate algorithm for motion compensation in two-dimensional tomography,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/6/065007.  Google Scholar

[11]

A. Katsevich, M. Silver and A. Zamayatin, Local tomography and the motion estimation problem,, SIAM J. Imaging Sci., 4 (2011), 200.  doi: 10.1137/100796728.  Google Scholar

[12]

S. Kindermann and A. Leitão, On regularization methods for inverse problems of dynamic type,, Numer. Func. Anal. Opt., 27 (2006), 139.  doi: 10.1080/01630560600569973.  Google Scholar

[13]

D. Le Bihan, C. Poupon, A. Amadon and F. Lethimonnier, Artifacts and pitfalls in diffusion MRI,, JMRI - J. Magn. Reson. Im., 24 (2006), 478.   Google Scholar

[14]

A. K. Louis, Inverse und Schlecht Gestellte Probleme,, Teubner, (1989).  doi: 10.1007/978-3-322-84808-6.  Google Scholar

[15]

A. K. Louis, Diffusion reconstruction from very noisy tomographic data,, Inverse Probl. Imagine, 4 (2010), 675.  doi: 10.3934/ipi.2010.4.675.  Google Scholar

[16]

W. Lu and T. R. Mackie, Tomographic motion detection and correction directly in sinogram space,, Phys. Med. Biol., 47 (2002), 1267.  doi: 10.1088/0031-9155/47/8/304.  Google Scholar

[17]

S. J. McQuaid and B. F. Hutton, Sources of attenuation-correction artefacts in cardiac PET/CT and SPECT/CT,, Eur. J. Nucl. Med. Mol. Imaging, 35 (2008), 1117.  doi: 10.1007/s00259-008-0718-0.  Google Scholar

[18]

J. L. Müller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications,, SIAM, (2012).   Google Scholar

[19]

F. Natterer, The Mathematics of Computerized Tomography,, John Wiley & Sons, (1986).   Google Scholar

[20]

F. Qiao, T. Pan, J. W. Clark, O. R. Mawlawi, A motion-incorporated reconstruction method for gated PET studies,, Phys. Med. Biol., 51 (2006), 3769.  doi: 10.1088/0031-9155/51/15/012.  Google Scholar

[21]

U. Schmitt and A. K. Louis, Efficient algorithms for the regularization of dynamic inverse problems: I. Theory,, Inverse Problems, 18 (2002), 645.  doi: 10.1088/0266-5611/18/3/308.  Google Scholar

[22]

U. Schmitt, A. K. Louis, C. Wolters and M. Vauhkonen, Efficient algorithms for the regularization of dynamic inverse problems: II. Applications,, Inverse Problems, 18 (2002), 659.  doi: 10.1088/0266-5611/18/3/309.  Google Scholar

[23]

A. Shankaranarayanan, M. Wendt, J. S. Lewin and J. L. Duerk, Two-step navigatorless correction algorithm for radial k-space MRI acquisitions,, Magn. Reson. Med., 45 (2001), 277.   Google Scholar

[24]

L. Shepp, S. Hilal and R. Schulz, The tuning fork artifact in computerized tomography,, Comput. Graph. Image Process., 10 (1979), 246.  doi: 10.1016/0146-664X(79)90004-2.  Google Scholar

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