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Using the Navier-Cauchy equation for motion estimation in dynamic imaging

1. 

Department of Mathematics, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

2. 

Department of Mathematics, University of Würzburg, Emil-Fischer-Straße 40, 97074 Würzburg, Germany

*Corresponding author: Sandra Warnecke

Received  June 2021 Revised  December 2021 Early access April 2022

Fund Project: The first and second authors are supported by the Deutsche Forschungsgemeinschaft under grant HA 8176/1-1

Tomographic image reconstruction is well understood if the specimen being studied is stationary during data acquisition. However, if this specimen changes its position during the measuring process, standard reconstruction techniques can lead to severe motion artefacts in the computed images. Solving a dynamic reconstruction problem therefore requires to model and incorporate suitable information on the dynamics in the reconstruction step to compensate for the motion.

Many dynamic processes can be described by partial differential equations which thus could serve as additional information for the purpose of motion compensation. In this article, we consider the Navier-Cauchy equation which characterizes small elastic deformations and serves, for instance, as a simplified model for respiratory motion. Our goal is to provide a proof-of-concept that by incorporating the deformation fields provided by this PDE, one can reduce the respective motion artefacts in the reconstructed image. To this end, we solve the Navier-Cauchy equation prior to the image reconstruction step using suitable initial and boundary data. Then, the thus computed deformation fields are incorporated into an analytic dynamic reconstruction method to compute an image of the unknown interior structure. The feasibility is illustrated with numerical examples from computerized tomography.

Citation: Bernadette N. Hahn, Melina-Loren Kienle Garrido, Christian Klingenberg, Sandra Warnecke. Using the Navier-Cauchy equation for motion estimation in dynamic imaging. Inverse Problems and Imaging, doi: 10.3934/ipi.2022018
References:
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S. S. Antman, Nonlinear Problems of Elasticity, Second edition, Applied Mathematical Sciences, 107. Springer, New York, 2005.

[2]

C. BlondelR. VaillantG. Malandain and N. Ayache, 3D tomographic reconstruction of coronary arteries using a precomputed 4D motion field, Physics in Medicine and Biology, 49 (2004), 2197-2208.  doi: 10.1088/0031-9155/49/11/006.

[3]

V. BoutchkoR. RayzN. VandeheyJ. O'NeilT. BudingerP. Nico and W. Moses, Imaging and modeling of flow in porous media using clinical nuclear emission tomography systems and computational fluid dynamics, Journal of Applied Geophysics, 76 (2012), 74-81. 

[4]

M. Burger, H. Dirks, L. Frerking, A. Hauptmann, T. Helin and S. Siltanen, A variational reconstruction method for undersampled dynamic x-ray tomography based on physical motion models, Inverse Problems, 33 (2017), 124008, 24 pp. doi: 10.1088/1361-6420/aa99cf.

[5]

M. BurgerH. Dirks and C.-B. Schönlieb, A variational model for joint motion estimation and image reconstruction, SIAM Journal on Imaging Sciences, 11 (2018), 94-128.  doi: 10.1137/16M1084183.

[6]

C. ChenB. Gris and O. Öktem, A new variational model for joint image reconstruction and motion estimation in spatiotemporal imaging, SIAM J. Imaging Sciences, 12 (2019), 1686-1719.  doi: 10.1137/18M1234047.

[7]

C. P. Chen and W. von Wahl, Das rand-anfangswertproblem für quasilineare wellengleichungen in sobolevräumen niedriger ordnung, J. Reine Angew. Math., 337 (1982), 77-112.  doi: 10.1515/crll.1982.337.77.

[8]

J. Chung and L. Nguyen, Motion estimation and correction in photoacoustic tomographic reconstruction, SIAM J. Imaging Sci., 10 (2017), 216-242.  doi: 10.1137/16M1082901.

[9]

J. Chung, A. K. Saibaba, M. Brown and E. Westman, Efficient generalized golub-kahan based methods for dynamic inverse problems, Inverse Problems, 34 (2018), 024005, 29 pp. doi: 10.1088/1361-6420/aaa0e1.

[10]

P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity, Studies in Mathematics and its Applications, 20. North-Holland Publishing Co., Amsterdam, 1988.

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C. CrawfordK. KingC. Ritchie and J. Godwin, Respiratory compensation in projection imaging using a magnification and displacement model, IEEE Transactions on Medical Imaging, 15 (1996), 327-332.  doi: 10.1109/42.500141.

[12]

L. DesbatS. Roux and P. Grangeat, Compensation of some time dependent deformations in tomography, IEEE Transactions on Medical Imaging, 26 (2007), 261-269.  doi: 10.1109/TMI.2006.889743.

[13]

J. Fitzgerald and P. Danias, Effect of motion on cardiac spect imaging: Recognition and motion correction, Journal of Nuclear Cardiology, 8 (2001), 701-706.  doi: 10.1067/mnc.2001.118694.

[14]

F. GigengackL. RuthottoM. BurgerC. WoltersX. Jiang and K. Schäfers, Motion correction in dual gated cardiac pet using mass-preserving image registration, IEEE Trans. Med. Imag., 31 (2012), 698-712.  doi: 10.1109/TMI.2011.2175402.

[15]

E. GravierY. Yang and M. Jin, Tomographic reconstruction of dynamic cardiac image sequences, IEEE Transactions on Image Processing, 16 (2007), 932-942.  doi: 10.1109/TIP.2006.891328.

[16]

B. Hahn, Reconstruction of dynamic objects with affine deformations in dynamic computerized tomography, J. Inverse Ill-Posed Probl., 22 (2014), 323-339.  doi: 10.1515/jip-2012-0094.

[17]

B. N. Hahn, Efficient algorithms for linear dynamic inverse problems with known motion, Inverse Problems, 30 (2014), 035008, 20 pp. doi: 10.1088/0266-5611/30/3/035008.

[18]

B. N. Hahn, Motion estimation and compensation strategies in dynamic computerized tomography, Sensing and Imaging, 18 (2017), 1-20.  doi: 10.1007/s11220-017-0159-6.

[19]

B. N. Hahn and M.-L. Kienle Garrido, An efficient reconstruction approach for a class of dynamic imaging operators, Inverse Problems, 35 (2019), 094005, 26 pp. doi: 10.1088/1361-6420/ab178b.

[20]

B. N. Hahn, M.-L. Kienle Garrido and E. T. Quinto, Microlocal properties of dynamic Fourier integral operators, Time-Dependent Problems in Imaging and Parameter Identification, (2021), 85–120. doi: 10.1007/978-3-030-57784-1_4.

[21]

B. N. Hahn and E. T. Quinto, Detectable singularities from dynamic radon data, SIAM Journal on Imaging Sciences, 9 (2016), 1195-1225.  doi: 10.1137/16M1057917.

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Classics in Mathematics, Springer, Berlin, 2009.

[23]

T. J. R. HughesT. Kato and J. E. Marsden, Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1977), 273-294.  doi: 10.1007/BF00251584.

[24]

A. A. IsolaA. ZieglerT. KoehlerW. Niessen and M. Grass, Motion-compensated iterative cone-beam CT image reconstruction with adapted blobs as basis functions, Physics in Medicine and Biology, 53 (2008), 6777-6797.  doi: 10.1088/0031-9155/53/23/009.

[25]

J. Kastner, B. Plank and C. Heinzl, Advanced x-ray computed tomography methods: High resolution CT, phase contrast CT, quantitative CT and 4DCT, Digital Industrial Radiology and Computed Tomography (DIR 2015), Ghent, Belgium, (2015).

[26]

A. Katsevich, An accurate approximate algorithm for motion compensation in two-dimensional tomography, Inverse Problems, 26 (2010), 065007, 16 pp. doi: 10.1088/0266-5611/26/6/065007.

[27]

A. Katsevich, A local approach to resolution analysis of image reconstruction in tomography, SIAM J. Appl. Math., 77 (2017), 1706-1732.  doi: 10.1137/17M1112108.

[28]

A. KatsevichM. Silver and A. Zamyatin, Local tomography and the motion estimation problem, SIAM J. Imaging Sci., 4 (2011), 200-219.  doi: 10.1137/100796728.

[29]

S. Kindermann and A. Leitão, On regularization methods for inverse problems of dynamic type, Numer. Funct. Anal. Optim., 27 (2006), 139-160.  doi: 10.1080/01630560600569973.

[30]

V. P. Krishnan and E. T. Quinto, Microlocal analysis in tomography, Handbook of Mathematical Methods in Imaging, Springer, New York, 1, 2, 3 (2015), 847-902.

[31]

D. Le BihanC. PouponA. Amadon and F. Lethimonnier, Artifacts and pitfalls in diffusion mri, Journal of Magnetic Resonance Imaging, 24 (2006), 478-488. 

[32]

J. Liu, X. Zhang, X. Zhang, H. Zhao, Y. Gao, D. Thomas, D. Low and H. Gao, 5D respiratory motion model based image reconstruction algorithm for 4D cone-beam computed tomography, Inverse Problems, 31 (2015), 115007, 21 pp. doi: 10.1088/0266-5611/31/11/115007.

[33]

W. Lu and T. R. Mackie, Tomographic motion detection and correction directly in sinogram space, Phys. Med. Biol., 47 (2002), 1267-1284.  doi: 10.1088/0031-9155/47/8/304.

[34]

D. MankeK. Nehrke and P. Börnert, Novel prospective respiratory motion correction approach for free-breathing coronary mr angiography using a patient-adapted affine motion model, Magnetic Resonance in Medicine, 50 (2003), 122-131.  doi: 10.1002/mrm.10483.

[35]

F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.

[36]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898718324.

[37]

R. OtazoE. Candès and D. Sodickson, Low-rank plus sparse matrix decomposition for accelerated dynamic mri with separation of background and dynamic components, Magnetic Resonance in Medicine, 73 (2015), 1125-1136.  doi: 10.1002/mrm.25240.

[38]

S. Rabieniaharatbar, Invertibility and stability for a generic class of radon transforms with application to dynamic operators, Journal of Inverse and Ill-Posed Problems, 27 (2018), 469-486.  doi: 10.1515/jiip-2018-0014.

[39]

M. ReyesG. MalandainP. KoulibalyM. González-Ballester and J. Darcourt, Model-based respiratory motion compensation for emission tomography image reconstruction, Physics in Medicine and Biology, 52 (2007), 3579-3600.  doi: 10.1088/0031-9155/52/12/016.

[40]

U. Schmitt and A. Louis, Efficient algorithms for the regularization of dynamic inverse problems. I. Theory, Inverse Problems, 18 (2002), 645-658.  doi: 10.1088/0266-5611/18/3/308.

[41]

U. SchmittA. K. LouisC. Wolters and M. Vauhkonen, Efficient algorithms for the regularization of dynamic inverse problems. II. Applications, Inverse Problems, 18 (2002), 659-676.  doi: 10.1088/0266-5611/18/3/309.

[42]

L. A. SheppS. K. Hilal and R. A. Schulz, The tuning fork artifact in computerized tomography, Computer Graphics and Image Processing, 10 (1979), 246-255.  doi: 10.1016/0146-664X(79)90004-2.

[43]

R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Second edition, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511755422.

[44]

F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 2. Fourier Integral Operators, University Series in Mathematics, Plenum Press, New York-London, 1980.

[45]

G. Van EyndhovenJ. Sijbers and J. Batenburg, Combined motion estimation and reconstruction in tomography, Lecture Notes in Computer Science, 7583 (2012), 12-21.  doi: 10.1007/978-3-642-33863-2_2.

[46]

V. Van NieuwenhoveJ. De BeenhouwerT. De SchryverL. Van Hoorebeke and J. Sijbers, Data-driven affine deformation estimation and correction in cone beam computed tomography, IEEE Transactions on Image Processing, 26 (2017), 1441-1451.  doi: 10.1109/TIP.2017.2651370.

[47]

R. Werner, Strahlentherapie Atmungsbewegter Tumoren, Springer Vieweg, Wiesbaden, 2013. doi: 10.1007/978-3-658-01146-8.

[48]

H. Yu and G. Wang, Data consistency based rigid motion artifact reduction in fan-beam CT, IEEE Transactions on Medical Imaging, 26 (2007), 249-260.  doi: 10.1109/TMI.2006.889717.

show all references

References:
[1]

S. S. Antman, Nonlinear Problems of Elasticity, Second edition, Applied Mathematical Sciences, 107. Springer, New York, 2005.

[2]

C. BlondelR. VaillantG. Malandain and N. Ayache, 3D tomographic reconstruction of coronary arteries using a precomputed 4D motion field, Physics in Medicine and Biology, 49 (2004), 2197-2208.  doi: 10.1088/0031-9155/49/11/006.

[3]

V. BoutchkoR. RayzN. VandeheyJ. O'NeilT. BudingerP. Nico and W. Moses, Imaging and modeling of flow in porous media using clinical nuclear emission tomography systems and computational fluid dynamics, Journal of Applied Geophysics, 76 (2012), 74-81. 

[4]

M. Burger, H. Dirks, L. Frerking, A. Hauptmann, T. Helin and S. Siltanen, A variational reconstruction method for undersampled dynamic x-ray tomography based on physical motion models, Inverse Problems, 33 (2017), 124008, 24 pp. doi: 10.1088/1361-6420/aa99cf.

[5]

M. BurgerH. Dirks and C.-B. Schönlieb, A variational model for joint motion estimation and image reconstruction, SIAM Journal on Imaging Sciences, 11 (2018), 94-128.  doi: 10.1137/16M1084183.

[6]

C. ChenB. Gris and O. Öktem, A new variational model for joint image reconstruction and motion estimation in spatiotemporal imaging, SIAM J. Imaging Sciences, 12 (2019), 1686-1719.  doi: 10.1137/18M1234047.

[7]

C. P. Chen and W. von Wahl, Das rand-anfangswertproblem für quasilineare wellengleichungen in sobolevräumen niedriger ordnung, J. Reine Angew. Math., 337 (1982), 77-112.  doi: 10.1515/crll.1982.337.77.

[8]

J. Chung and L. Nguyen, Motion estimation and correction in photoacoustic tomographic reconstruction, SIAM J. Imaging Sci., 10 (2017), 216-242.  doi: 10.1137/16M1082901.

[9]

J. Chung, A. K. Saibaba, M. Brown and E. Westman, Efficient generalized golub-kahan based methods for dynamic inverse problems, Inverse Problems, 34 (2018), 024005, 29 pp. doi: 10.1088/1361-6420/aaa0e1.

[10]

P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity, Studies in Mathematics and its Applications, 20. North-Holland Publishing Co., Amsterdam, 1988.

[11]

C. CrawfordK. KingC. Ritchie and J. Godwin, Respiratory compensation in projection imaging using a magnification and displacement model, IEEE Transactions on Medical Imaging, 15 (1996), 327-332.  doi: 10.1109/42.500141.

[12]

L. DesbatS. Roux and P. Grangeat, Compensation of some time dependent deformations in tomography, IEEE Transactions on Medical Imaging, 26 (2007), 261-269.  doi: 10.1109/TMI.2006.889743.

[13]

J. Fitzgerald and P. Danias, Effect of motion on cardiac spect imaging: Recognition and motion correction, Journal of Nuclear Cardiology, 8 (2001), 701-706.  doi: 10.1067/mnc.2001.118694.

[14]

F. GigengackL. RuthottoM. BurgerC. WoltersX. Jiang and K. Schäfers, Motion correction in dual gated cardiac pet using mass-preserving image registration, IEEE Trans. Med. Imag., 31 (2012), 698-712.  doi: 10.1109/TMI.2011.2175402.

[15]

E. GravierY. Yang and M. Jin, Tomographic reconstruction of dynamic cardiac image sequences, IEEE Transactions on Image Processing, 16 (2007), 932-942.  doi: 10.1109/TIP.2006.891328.

[16]

B. Hahn, Reconstruction of dynamic objects with affine deformations in dynamic computerized tomography, J. Inverse Ill-Posed Probl., 22 (2014), 323-339.  doi: 10.1515/jip-2012-0094.

[17]

B. N. Hahn, Efficient algorithms for linear dynamic inverse problems with known motion, Inverse Problems, 30 (2014), 035008, 20 pp. doi: 10.1088/0266-5611/30/3/035008.

[18]

B. N. Hahn, Motion estimation and compensation strategies in dynamic computerized tomography, Sensing and Imaging, 18 (2017), 1-20.  doi: 10.1007/s11220-017-0159-6.

[19]

B. N. Hahn and M.-L. Kienle Garrido, An efficient reconstruction approach for a class of dynamic imaging operators, Inverse Problems, 35 (2019), 094005, 26 pp. doi: 10.1088/1361-6420/ab178b.

[20]

B. N. Hahn, M.-L. Kienle Garrido and E. T. Quinto, Microlocal properties of dynamic Fourier integral operators, Time-Dependent Problems in Imaging and Parameter Identification, (2021), 85–120. doi: 10.1007/978-3-030-57784-1_4.

[21]

B. N. Hahn and E. T. Quinto, Detectable singularities from dynamic radon data, SIAM Journal on Imaging Sciences, 9 (2016), 1195-1225.  doi: 10.1137/16M1057917.

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Classics in Mathematics, Springer, Berlin, 2009.

[23]

T. J. R. HughesT. Kato and J. E. Marsden, Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1977), 273-294.  doi: 10.1007/BF00251584.

[24]

A. A. IsolaA. ZieglerT. KoehlerW. Niessen and M. Grass, Motion-compensated iterative cone-beam CT image reconstruction with adapted blobs as basis functions, Physics in Medicine and Biology, 53 (2008), 6777-6797.  doi: 10.1088/0031-9155/53/23/009.

[25]

J. Kastner, B. Plank and C. Heinzl, Advanced x-ray computed tomography methods: High resolution CT, phase contrast CT, quantitative CT and 4DCT, Digital Industrial Radiology and Computed Tomography (DIR 2015), Ghent, Belgium, (2015).

[26]

A. Katsevich, An accurate approximate algorithm for motion compensation in two-dimensional tomography, Inverse Problems, 26 (2010), 065007, 16 pp. doi: 10.1088/0266-5611/26/6/065007.

[27]

A. Katsevich, A local approach to resolution analysis of image reconstruction in tomography, SIAM J. Appl. Math., 77 (2017), 1706-1732.  doi: 10.1137/17M1112108.

[28]

A. KatsevichM. Silver and A. Zamyatin, Local tomography and the motion estimation problem, SIAM J. Imaging Sci., 4 (2011), 200-219.  doi: 10.1137/100796728.

[29]

S. Kindermann and A. Leitão, On regularization methods for inverse problems of dynamic type, Numer. Funct. Anal. Optim., 27 (2006), 139-160.  doi: 10.1080/01630560600569973.

[30]

V. P. Krishnan and E. T. Quinto, Microlocal analysis in tomography, Handbook of Mathematical Methods in Imaging, Springer, New York, 1, 2, 3 (2015), 847-902.

[31]

D. Le BihanC. PouponA. Amadon and F. Lethimonnier, Artifacts and pitfalls in diffusion mri, Journal of Magnetic Resonance Imaging, 24 (2006), 478-488. 

[32]

J. Liu, X. Zhang, X. Zhang, H. Zhao, Y. Gao, D. Thomas, D. Low and H. Gao, 5D respiratory motion model based image reconstruction algorithm for 4D cone-beam computed tomography, Inverse Problems, 31 (2015), 115007, 21 pp. doi: 10.1088/0266-5611/31/11/115007.

[33]

W. Lu and T. R. Mackie, Tomographic motion detection and correction directly in sinogram space, Phys. Med. Biol., 47 (2002), 1267-1284.  doi: 10.1088/0031-9155/47/8/304.

[34]

D. MankeK. Nehrke and P. Börnert, Novel prospective respiratory motion correction approach for free-breathing coronary mr angiography using a patient-adapted affine motion model, Magnetic Resonance in Medicine, 50 (2003), 122-131.  doi: 10.1002/mrm.10483.

[35]

F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.

[36]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898718324.

[37]

R. OtazoE. Candès and D. Sodickson, Low-rank plus sparse matrix decomposition for accelerated dynamic mri with separation of background and dynamic components, Magnetic Resonance in Medicine, 73 (2015), 1125-1136.  doi: 10.1002/mrm.25240.

[38]

S. Rabieniaharatbar, Invertibility and stability for a generic class of radon transforms with application to dynamic operators, Journal of Inverse and Ill-Posed Problems, 27 (2018), 469-486.  doi: 10.1515/jiip-2018-0014.

[39]

M. ReyesG. MalandainP. KoulibalyM. González-Ballester and J. Darcourt, Model-based respiratory motion compensation for emission tomography image reconstruction, Physics in Medicine and Biology, 52 (2007), 3579-3600.  doi: 10.1088/0031-9155/52/12/016.

[40]

U. Schmitt and A. Louis, Efficient algorithms for the regularization of dynamic inverse problems. I. Theory, Inverse Problems, 18 (2002), 645-658.  doi: 10.1088/0266-5611/18/3/308.

[41]

U. SchmittA. K. LouisC. Wolters and M. Vauhkonen, Efficient algorithms for the regularization of dynamic inverse problems. II. Applications, Inverse Problems, 18 (2002), 659-676.  doi: 10.1088/0266-5611/18/3/309.

[42]

L. A. SheppS. K. Hilal and R. A. Schulz, The tuning fork artifact in computerized tomography, Computer Graphics and Image Processing, 10 (1979), 246-255.  doi: 10.1016/0146-664X(79)90004-2.

[43]

R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Second edition, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511755422.

[44]

F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 2. Fourier Integral Operators, University Series in Mathematics, Plenum Press, New York-London, 1980.

[45]

G. Van EyndhovenJ. Sijbers and J. Batenburg, Combined motion estimation and reconstruction in tomography, Lecture Notes in Computer Science, 7583 (2012), 12-21.  doi: 10.1007/978-3-642-33863-2_2.

[46]

V. Van NieuwenhoveJ. De BeenhouwerT. De SchryverL. Van Hoorebeke and J. Sijbers, Data-driven affine deformation estimation and correction in cone beam computed tomography, IEEE Transactions on Image Processing, 26 (2017), 1441-1451.  doi: 10.1109/TIP.2017.2651370.

[47]

R. Werner, Strahlentherapie Atmungsbewegter Tumoren, Springer Vieweg, Wiesbaden, 2013. doi: 10.1007/978-3-658-01146-8.

[48]

H. Yu and G. Wang, Data consistency based rigid motion artifact reduction in fan-beam CT, IEEE Transactions on Medical Imaging, 26 (2007), 249-260.  doi: 10.1109/TMI.2006.889717.

Figure 1.  The mapping $ \Phi^{-1}_t $ correlates the state $ f_t $ at time $ t $ to the reference state $ f_0 $ at the initial time
Figure 2.  Initial state $ f_0 $ of a phantom (left) and its singularities (right)
Figure 4.  Illustration of the boundary: The nodes 1 and 2 lie directly on the continuous boundary, and their behaviour is prescribed by the Dirichlet data $ \psi $. For the node 0, the stencil for the update scheme only can be applied with the help of an interpolation since the values of the ghost node are not available. The average of the values of the nodes 1 and 2 are used to create an auxiliary node which corresponds to a slightly 'shifted' boundary
Figure 3.  We illustrate the stencil for our numerical scheme. For the update of the values at node $ x_{i, j} $ from $ t_n\to t_{n+1} $, we have to provide information about the values at the other marked nodes
Figure 5.  Cross-section of the numerical phantom during one cycling breath. The first image corresponds to the reference state, the second and third image correspond to the body after a quarter and after one half of a breathing cycle, respectively. The fourth image illustrates the body after one period when the initial configuration is reached again
Figure 6.  Static and dynamic reconstruction results of the initial state function
Figure 7.  Illustration of the numerical solution of the Navier-Cauchy equation with analytical boundary data. The initial density distribution used for solving the Navier-Cauchy equation is given in the first image. The second, third and fourth image correspond to the configurations after a quarter, after one half and after a full period of the breathing cycle
Figure 8.  Dynamic reconstruction with motion information from solving the PDE with noisy boundary data
Figure 9.  Dynamic reconstruction results with motion information from solving the PDE with only a small number of boundary nodes
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