• Previous Article
    Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography
  • IPI Home
  • This Issue
  • Next Article
    Incorporating structural prior information and sparsity into EIT using parallel level sets
April  2019, 13(2): 309-335. doi: 10.3934/ipi.2019016

An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration

Department of Mathematical Sciences and EPSRC Liverpool Centre for Mathematics in Healthcare, The University of Liverpool, Liverpool L69 7ZL, UK

* Corresponding author: Ke Chen  http://www.liverpool.ac.uk/~cmchenke

Received  March 2018 Revised  October 2018 Published  January 2019

Fund Project: Both authors are supported by the UK EPSRC grant EP/N014499/1

In this work we propose a variational model for multi-modal image registration. It minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. We first present a theoretical analysis of the proposed model. Then, to solve the model numerically, we use an augmented Lagrangian method (ALM) to reformulate it to a few more amenable subproblems (each giving rise to an Euler-Lagrange equation that is discretized by finite difference methods) and solve iteratively the main linear systems by the fast Fourier transform; a multilevel technique is employed to speed up the initialisation and avoid likely local minima of the underlying functional. Finally we show the convergence of the ALM solver and give numerical results of the new approach. Comparisons with some existing methods are presented to illustrate its effectiveness and advantages.

Citation: Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems & Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016
References:
[1]

E. BaeJ. Shi and X.-C. Tai, Graph cuts for curvature based image denoising, IEEE Transactions on Image Processing, 20 (2011), 1199-1210.  doi: 10.1109/TIP.2010.2090533.  Google Scholar

[2]

M. BurgerJ. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration, SIAM Journal on Scientific Computing, 35 (2013), 132-148.  doi: 10.1137/110835955.  Google Scholar

[3]

Y. M. ChenJ. L. ShiM. Rao and J. S. Lee, Deformable multi-modal image registration by maximizing renyi's statistical dependence measure, Inverse Problems and Imaging, 9 (2015), 79-103.  doi: 10.3934/ipi.2015.9.79.  Google Scholar

[4]

N. Chumchob, Vectorial total variation-based regularization for variational image registration, IEEE Transactions on Image Processing, 22 (2013), 4551-4559.  doi: 10.1109/TIP.2013.2274749.  Google Scholar

[5]

N. Chumchob and K. Chen, Improved variational image registration model and a fast algorithm for its numerical approximation, Numerical Methods for Partial Differential Equations, 28 (2012), 1966-1995.  doi: 10.1002/num.20710.  Google Scholar

[6]

N. ChumchobK. Chen and C. Brito-Loeza, A fourth-order variational image registration model and its fast multigrid algorithm, Multiscale Modeling & Simulation, 9 (2011), 89-128.  doi: 10.1137/100788239.  Google Scholar

[7]

M. Droske and W. Ring, A mumford-shah level-set approach for geometric image registration, SIAM journal on Applied Mathematics, 66 (2006), 2127-2148.  doi: 10.1137/050630209.  Google Scholar

[8]

J. Feydy, B. Charlier, F. V. Vialard and G. Peyre, Optimal transport for diffeomorphic registration, International Conference on Medical Image Computing and Computer-Assisted Intervention, MICCAI 2017: Medical Image Computing and Computer Assisted Intervention - MICCAI, (2017), 291-299, https://arXiv.org/abs/1706.05218v1. doi: 10.1007/978-3-319-66182-7_34.  Google Scholar

[9]

B. Fischer and J. Modersitzki, Fast diffusion registration, Contemp. Math., 313 (2002), 117-129.  doi: 10.1090/conm/313/05372.  Google Scholar

[10]

B. Fischer and J. Modersitzki, Curvature based image registration, Journal of Mathematical Imaging and Vision, 18 (2003), 81-85.  doi: 10.1023/A:1021897212261.  Google Scholar

[11]

B. Fischer and J. Modersitzki, Ill-posed medicine - an introduction to image registration, Inverse Problems, 24 (2008), 034008, 16 pp. doi: 10.1088/0266-5611/24/3/034008.  Google Scholar

[12]

E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration, Inverse Problems, 20 (2004), 1621-1638.  doi: 10.1088/0266-5611/20/5/018.  Google Scholar

[13]

E. Haber and J. Modersitzki, Image registration with guaranteed displacement regularity, International Journal of Computer Vision, 71 (2007), 361-372.  doi: 10.1007/s11263-006-8984-4.  Google Scholar

[14]

S. Henn, A multigrid method for a fourth-order diffusion equation with application to image processing, SIAM Journal on Scientific Computing, 27 (2005), 831-849.  doi: 10.1137/040611124.  Google Scholar

[15]

E. HodnelandA. LundervoldJ. Rørvik and A. Z. Munthe-Kaas, Normalized gradient fields for nonlinear motion correction of dce-mri time series, Computerized Medical Imaging and Graphics, 38 (2014), 202-210.   Google Scholar

[16]

W. HuY. XieL. Li and W. Zhang, A total variation based nonrigid image registration by combining parametric and non-parametric transformation models, Neurocomputing, 144 (2014), 222-237.  doi: 10.1016/j.neucom.2014.05.031.  Google Scholar

[17]

M. IbrahimK. Chen and C. Brito-Loeza, A novel variational model for image registration using gaussian curvature, Geometry, Imaging and Computing, 1 (2014), 417-446.  doi: 10.4310/GIC.2014.v1.n4.a2.  Google Scholar

[18]

L. König and J. Rühaak, A fast and accurate parallel algorithm for non-linear image registration using normalized gradient fields, in Biomedical Imaging (ISBI), 2014 IEEE 11th International Symposium on, IEEE, 2014,580-583. Google Scholar

[19]

D. LoeckxP. SlagmolenF. MaesD. Vandermeulen and P. Suetens, Nonrigid image registration using conditional mutual information, IEEE Transactions on Medical Imaging, 29 (2010), 19-29.   Google Scholar

[20]

F. MaesA. CollignonD. VandermeulenG. Marchal and P. Suetens, Multimodality image registration by maximization of mutual information, IEEE Transactions on Tedical Imaging, 16 (1997), 187-198.  doi: 10.1109/42.563664.  Google Scholar

[21]

A. Mang and G. Biros, An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration, SIAM Journal on Imaging Sciences, 8 (2015), 1030-1069.  doi: 10.1137/140984002.  Google Scholar

[22]

A. Mang and G. Biros, Constrained $h^1$-regularization schemes for diffeomorphic image registration, SIAM Journal on Imaging Sciences, 9 (2016), 1154-1194.  doi: 10.1137/15M1010919.  Google Scholar

[23]

J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, SIAM, 2009. doi: 10.1137/1.9780898718843.  Google Scholar

[24]

F. P. Oliveira and J. M. R. Tavares, Medical image registration: A review, Computer Methods in Biomechanics and Biomedical Engineering, 17 (2014), 73-93.  doi: 10.1080/10255842.2012.670855.  Google Scholar

[25]

K. PapafitsorosC. B. Schoenlieb and B. Sengul, Combined first and second order total variation inpainting using split bregman, Image Processing On Line, 3 (2013), 112-136.  doi: 10.5201/ipol.2013.40.  Google Scholar

[26]

J. P. PluimJ. A. Maintz and M. A. Viergever, Mutual-information-based registration of medical images: A survey, IEEE Transactions on Medical Imaging, 22 (2003), 986-1004.  doi: 10.1109/TMI.2003.815867.  Google Scholar

[27]

C. PöschlJ. Modersitzki and O. Scherzer, A variational setting for volume constrained image registration, Inverse Problems and Imaging, 4 (2010), 505-522.  doi: 10.3934/ipi.2010.4.505.  Google Scholar

[28]

T. RohlfingC. R. MaurerD. A. Bluemke and M. A. Jacobs, Volume-preserving nonrigid registration of mr breast images using free-form deformation with an incompressibility constraint, IEEE transactions on medical imaging, 22 (2003), 730-741.  doi: 10.1109/TMI.2003.814791.  Google Scholar

[29]

G. Roland and L. T. Patrick, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, 1989. doi: 10.1137/1.9781611970838.  Google Scholar

[30]

J. Rühaak, L. König, M. Hallmann, N. Papenberg, S. Heldmann, H. Schumacher and B. Fischer, A fully parallel algorithm for multimodal image registration using normalized gradient fields, in Biomedical Imaging (ISBI), 2013 IEEE 10th International Symposium on, IEEE, 2013,572-575. Google Scholar

[31]

A. SotirasC. Davatzikos and N. Paragios, Deformable medical image registration: A survey, IEEE Transactions on Medical Imaging, 32 (2013), 1153-1190.  doi: 10.1109/TMI.2013.2265603.  Google Scholar

[32]

X.-C. TaiJ. Hahn and G. J. Chung, A fast algorithm for Euler's elastica model using augmented lagrangian method, SIAM Journal on Imaging Sciences, 4 (2011), 313-344.  doi: 10.1137/100803730.  Google Scholar

[33]

P. Viola and W. M. Wells Ⅲ, Alignment by maximization of mutual information, International Journal of Computer Vision, 24 (1997), 137-154.   Google Scholar

[34]

C. Wu and X. C. Tai, Augmented lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[35]

C. WuJ. Zhang and X.-C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2011), 237-261.  doi: 10.3934/ipi.2011.5.237.  Google Scholar

[36]

C. Xing and P. Qiu, Intensity-based image registration by nonparametric local smoothing, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2081-2092.   Google Scholar

[37]

M. Yashtini and S. H. Kang, A fast relaxed normal two split method and an effective weighted TV approach for E1uler's elastica image inpainting, SIAM Journal on Imaging Sciences, 9 (2016), 1552-1581.  doi: 10.1137/16M1063757.  Google Scholar

[38]

W. YilunY. JunfengY. Wotao and Z. Yin, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272.  doi: 10.1137/080724265.  Google Scholar

[39]

J. Zhang and K. Chen, Variational image registration by a total fractional-order variation model, Journal of Computational Physics, 293 (2015), 442-461.  doi: 10.1016/j.jcp.2015.02.021.  Google Scholar

[40]

J. ZhangK. Chen and B. Yu, An improved discontinuity-preserving image registration model and its fast algorithm, Applied Mathematical Modelling, 40 (2016), 10740-10759.  doi: 10.1016/j.apm.2016.08.009.  Google Scholar

[41]

J. ZhangK. Chen and B. Yu, A novel high-order functional based image registration model with inequality constraint, Mathematics with Applications, 72 (2016), 2887-2899.  doi: 10.1016/j.camwa.2016.10.018.  Google Scholar

[42]

X. Zhou, Weak lower semicontinuity of a functional with any order, Journal of Mathematical Analysis and Applications, 221 (1998), 217-237.  doi: 10.1006/jmaa.1997.5881.  Google Scholar

[43]

W. ZHUX.-C. TAI and T. CHAN, Augmented lagrangian method for a mean curvature based image denoising model, Imaging, 7 (2013), 1409-1432.  doi: 10.3934/ipi.2013.7.1409.  Google Scholar

[44]

W. ZhuX.-C. Tai and T. Chan, Image segmentation using euler's elastica as the regularization, Journal of Scientific Computing, 57 (2013), 414-438.  doi: 10.1007/s10915-013-9710-3.  Google Scholar

show all references

References:
[1]

E. BaeJ. Shi and X.-C. Tai, Graph cuts for curvature based image denoising, IEEE Transactions on Image Processing, 20 (2011), 1199-1210.  doi: 10.1109/TIP.2010.2090533.  Google Scholar

[2]

M. BurgerJ. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration, SIAM Journal on Scientific Computing, 35 (2013), 132-148.  doi: 10.1137/110835955.  Google Scholar

[3]

Y. M. ChenJ. L. ShiM. Rao and J. S. Lee, Deformable multi-modal image registration by maximizing renyi's statistical dependence measure, Inverse Problems and Imaging, 9 (2015), 79-103.  doi: 10.3934/ipi.2015.9.79.  Google Scholar

[4]

N. Chumchob, Vectorial total variation-based regularization for variational image registration, IEEE Transactions on Image Processing, 22 (2013), 4551-4559.  doi: 10.1109/TIP.2013.2274749.  Google Scholar

[5]

N. Chumchob and K. Chen, Improved variational image registration model and a fast algorithm for its numerical approximation, Numerical Methods for Partial Differential Equations, 28 (2012), 1966-1995.  doi: 10.1002/num.20710.  Google Scholar

[6]

N. ChumchobK. Chen and C. Brito-Loeza, A fourth-order variational image registration model and its fast multigrid algorithm, Multiscale Modeling & Simulation, 9 (2011), 89-128.  doi: 10.1137/100788239.  Google Scholar

[7]

M. Droske and W. Ring, A mumford-shah level-set approach for geometric image registration, SIAM journal on Applied Mathematics, 66 (2006), 2127-2148.  doi: 10.1137/050630209.  Google Scholar

[8]

J. Feydy, B. Charlier, F. V. Vialard and G. Peyre, Optimal transport for diffeomorphic registration, International Conference on Medical Image Computing and Computer-Assisted Intervention, MICCAI 2017: Medical Image Computing and Computer Assisted Intervention - MICCAI, (2017), 291-299, https://arXiv.org/abs/1706.05218v1. doi: 10.1007/978-3-319-66182-7_34.  Google Scholar

[9]

B. Fischer and J. Modersitzki, Fast diffusion registration, Contemp. Math., 313 (2002), 117-129.  doi: 10.1090/conm/313/05372.  Google Scholar

[10]

B. Fischer and J. Modersitzki, Curvature based image registration, Journal of Mathematical Imaging and Vision, 18 (2003), 81-85.  doi: 10.1023/A:1021897212261.  Google Scholar

[11]

B. Fischer and J. Modersitzki, Ill-posed medicine - an introduction to image registration, Inverse Problems, 24 (2008), 034008, 16 pp. doi: 10.1088/0266-5611/24/3/034008.  Google Scholar

[12]

E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration, Inverse Problems, 20 (2004), 1621-1638.  doi: 10.1088/0266-5611/20/5/018.  Google Scholar

[13]

E. Haber and J. Modersitzki, Image registration with guaranteed displacement regularity, International Journal of Computer Vision, 71 (2007), 361-372.  doi: 10.1007/s11263-006-8984-4.  Google Scholar

[14]

S. Henn, A multigrid method for a fourth-order diffusion equation with application to image processing, SIAM Journal on Scientific Computing, 27 (2005), 831-849.  doi: 10.1137/040611124.  Google Scholar

[15]

E. HodnelandA. LundervoldJ. Rørvik and A. Z. Munthe-Kaas, Normalized gradient fields for nonlinear motion correction of dce-mri time series, Computerized Medical Imaging and Graphics, 38 (2014), 202-210.   Google Scholar

[16]

W. HuY. XieL. Li and W. Zhang, A total variation based nonrigid image registration by combining parametric and non-parametric transformation models, Neurocomputing, 144 (2014), 222-237.  doi: 10.1016/j.neucom.2014.05.031.  Google Scholar

[17]

M. IbrahimK. Chen and C. Brito-Loeza, A novel variational model for image registration using gaussian curvature, Geometry, Imaging and Computing, 1 (2014), 417-446.  doi: 10.4310/GIC.2014.v1.n4.a2.  Google Scholar

[18]

L. König and J. Rühaak, A fast and accurate parallel algorithm for non-linear image registration using normalized gradient fields, in Biomedical Imaging (ISBI), 2014 IEEE 11th International Symposium on, IEEE, 2014,580-583. Google Scholar

[19]

D. LoeckxP. SlagmolenF. MaesD. Vandermeulen and P. Suetens, Nonrigid image registration using conditional mutual information, IEEE Transactions on Medical Imaging, 29 (2010), 19-29.   Google Scholar

[20]

F. MaesA. CollignonD. VandermeulenG. Marchal and P. Suetens, Multimodality image registration by maximization of mutual information, IEEE Transactions on Tedical Imaging, 16 (1997), 187-198.  doi: 10.1109/42.563664.  Google Scholar

[21]

A. Mang and G. Biros, An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration, SIAM Journal on Imaging Sciences, 8 (2015), 1030-1069.  doi: 10.1137/140984002.  Google Scholar

[22]

A. Mang and G. Biros, Constrained $h^1$-regularization schemes for diffeomorphic image registration, SIAM Journal on Imaging Sciences, 9 (2016), 1154-1194.  doi: 10.1137/15M1010919.  Google Scholar

[23]

J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, SIAM, 2009. doi: 10.1137/1.9780898718843.  Google Scholar

[24]

F. P. Oliveira and J. M. R. Tavares, Medical image registration: A review, Computer Methods in Biomechanics and Biomedical Engineering, 17 (2014), 73-93.  doi: 10.1080/10255842.2012.670855.  Google Scholar

[25]

K. PapafitsorosC. B. Schoenlieb and B. Sengul, Combined first and second order total variation inpainting using split bregman, Image Processing On Line, 3 (2013), 112-136.  doi: 10.5201/ipol.2013.40.  Google Scholar

[26]

J. P. PluimJ. A. Maintz and M. A. Viergever, Mutual-information-based registration of medical images: A survey, IEEE Transactions on Medical Imaging, 22 (2003), 986-1004.  doi: 10.1109/TMI.2003.815867.  Google Scholar

[27]

C. PöschlJ. Modersitzki and O. Scherzer, A variational setting for volume constrained image registration, Inverse Problems and Imaging, 4 (2010), 505-522.  doi: 10.3934/ipi.2010.4.505.  Google Scholar

[28]

T. RohlfingC. R. MaurerD. A. Bluemke and M. A. Jacobs, Volume-preserving nonrigid registration of mr breast images using free-form deformation with an incompressibility constraint, IEEE transactions on medical imaging, 22 (2003), 730-741.  doi: 10.1109/TMI.2003.814791.  Google Scholar

[29]

G. Roland and L. T. Patrick, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, 1989. doi: 10.1137/1.9781611970838.  Google Scholar

[30]

J. Rühaak, L. König, M. Hallmann, N. Papenberg, S. Heldmann, H. Schumacher and B. Fischer, A fully parallel algorithm for multimodal image registration using normalized gradient fields, in Biomedical Imaging (ISBI), 2013 IEEE 10th International Symposium on, IEEE, 2013,572-575. Google Scholar

[31]

A. SotirasC. Davatzikos and N. Paragios, Deformable medical image registration: A survey, IEEE Transactions on Medical Imaging, 32 (2013), 1153-1190.  doi: 10.1109/TMI.2013.2265603.  Google Scholar

[32]

X.-C. TaiJ. Hahn and G. J. Chung, A fast algorithm for Euler's elastica model using augmented lagrangian method, SIAM Journal on Imaging Sciences, 4 (2011), 313-344.  doi: 10.1137/100803730.  Google Scholar

[33]

P. Viola and W. M. Wells Ⅲ, Alignment by maximization of mutual information, International Journal of Computer Vision, 24 (1997), 137-154.   Google Scholar

[34]

C. Wu and X. C. Tai, Augmented lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[35]

C. WuJ. Zhang and X.-C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2011), 237-261.  doi: 10.3934/ipi.2011.5.237.  Google Scholar

[36]

C. Xing and P. Qiu, Intensity-based image registration by nonparametric local smoothing, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2081-2092.   Google Scholar

[37]

M. Yashtini and S. H. Kang, A fast relaxed normal two split method and an effective weighted TV approach for E1uler's elastica image inpainting, SIAM Journal on Imaging Sciences, 9 (2016), 1552-1581.  doi: 10.1137/16M1063757.  Google Scholar

[38]

W. YilunY. JunfengY. Wotao and Z. Yin, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272.  doi: 10.1137/080724265.  Google Scholar

[39]

J. Zhang and K. Chen, Variational image registration by a total fractional-order variation model, Journal of Computational Physics, 293 (2015), 442-461.  doi: 10.1016/j.jcp.2015.02.021.  Google Scholar

[40]

J. ZhangK. Chen and B. Yu, An improved discontinuity-preserving image registration model and its fast algorithm, Applied Mathematical Modelling, 40 (2016), 10740-10759.  doi: 10.1016/j.apm.2016.08.009.  Google Scholar

[41]

J. ZhangK. Chen and B. Yu, A novel high-order functional based image registration model with inequality constraint, Mathematics with Applications, 72 (2016), 2887-2899.  doi: 10.1016/j.camwa.2016.10.018.  Google Scholar

[42]

X. Zhou, Weak lower semicontinuity of a functional with any order, Journal of Mathematical Analysis and Applications, 221 (1998), 217-237.  doi: 10.1006/jmaa.1997.5881.  Google Scholar

[43]

W. ZHUX.-C. TAI and T. CHAN, Augmented lagrangian method for a mean curvature based image denoising model, Imaging, 7 (2013), 1409-1432.  doi: 10.3934/ipi.2013.7.1409.  Google Scholar

[44]

W. ZhuX.-C. Tai and T. Chan, Image segmentation using euler's elastica as the regularization, Journal of Scientific Computing, 57 (2013), 414-438.  doi: 10.1007/s10915-013-9710-3.  Google Scholar

Figure 1.  Example of Reference and Template images where $ \nabla_n T\cdot \nabla_n R = 0 $ (or one of $ \nabla_n T, \ \nabla_n R $ is zero) a.e in $ \Omega $
Figure 2.  Three examples of the triangle inequality for triangles with sides $ X $, $ Y $ and $ Z $. The left example shows a case where $ |Z| $ is much less than the sum $ |X| +|Y| $ of the other two sides, and the right example shows a case where $ |Z| $ is only slightly less than $ |X| +|Y| $
Figure 3.  Example of a multilevel representation of images
Figure 4.  Example 1: Comparison of three different models. Clearly only Our Model works while NGF, MI fail completely
Figure 5.  Example 2: Comparison of different models to register T-1 and T2-MRI images. New Model performs the best
Figure 6.  Example 3: Registration of a second pair of MRI images (T1 and T2). New Model performs the best
Figure 7.  Comparison of $ 3 $ different models to register the MRI images fin Fig. 6. Example 3 zoomed in the red squares (see Fig. 6): From left to right; Zooms in the reference $ R $ and the registered $ T(\mathbf u) $ using New model, NGF and MI, respectively.
Figure 8.  Example 4: High-b- and Low-b-value Diffusion-weighted MRIs (of $ 256\times 256 $) using different models. New Model performs the best
Figure 9.  Example 5: a pair of MRI images of higher resolution $ 512\times 512 $ by $ 3 $ different models. New Model and MI perform identically, both better than NGF
Figure 10.  Left: Log scale plot of the residual errors for $ \mathbf u $ versus ALM iteration numbers for examples 2-5. Right: Plot of the error $ S_{er} $ values versus ALM iteration numbers for examples 2-5
Figure 11.  Left: Log scale plot of the distance Dm versus ALM iteration numbers for examples 2-5
Figure 12.  Example 6: Registering a PET image to an MRI vimage. New model performs better than others in this example
Table 1.  Run time comparison for all models for the pair of MRI images in Fig. 6
Resolution
$64 \times 64$ $128 \times 128$ $256 \times 256$ $512 \times 512$
Time (s) for New Model 29.836 49.931 117.342 272.578
Time (s) for MI Model 14.794 21.437 48.881 76.398
Time (s) for NGF Model 22.003 42.845 100.961 264.388
Resolution
$64 \times 64$ $128 \times 128$ $256 \times 256$ $512 \times 512$
Time (s) for New Model 29.836 49.931 117.342 272.578
Time (s) for MI Model 14.794 21.437 48.881 76.398
Time (s) for NGF Model 22.003 42.845 100.961 264.388
Table 2.  Registration results of the different models for processing Examples 1-6. The errors are computed using formula (38), (40) and (39). Here, #N is the ratio of the number of pixels where $\nabla_n T\cdot \nabla_n R \neq 0$ over the total number of pixels, whereas #G is the ratio of number pixels where GF(T, R)+TM(T, R) ≠ 0 over the total number of pixels.
Compared Models NGF New Model
#G #N GFer NGFer MIer GFer NGFer MIer
Ex 1 0.2% .02% 0.540 0.964 0.446 0.032 0.932 0.993
Ex 2 49% 24% 0.636 0.640 1.170 0.247 0.756 1.206
Ex 3 49% 23% 0.336 0.491 1.265 0.238 0.389 1.290
Ex 4 49% 20% 0.901 0.856 1.150 0.674 0.800 1.184
Ex 5 43% 37% 0.741 0.656 1.163 0.454 0.623 1.178
Ex 6 48% 23% 0.952 0.957 1.187 0.801 0.920 1.341
Compared Models MI
#G #N GFer NGFer MIer Note:
Ex 1 0.2% .02% 0.370 0.97 0.381 for GFer and NGFer, the smaller the better.
Ex 2 49% 24% 0.490 0.879 1.193
Ex 3 49% 23% 0.463 0.579 1.265 But for MIer, The larger the better.
Ex 4 49% 20% 0.765 0.849 1.154
Ex 5 43% 37% 0.454 0.631 1.163
Ex 6 48% 23% 0.836 0.970 1.254
Compared Models NGF New Model
#G #N GFer NGFer MIer GFer NGFer MIer
Ex 1 0.2% .02% 0.540 0.964 0.446 0.032 0.932 0.993
Ex 2 49% 24% 0.636 0.640 1.170 0.247 0.756 1.206
Ex 3 49% 23% 0.336 0.491 1.265 0.238 0.389 1.290
Ex 4 49% 20% 0.901 0.856 1.150 0.674 0.800 1.184
Ex 5 43% 37% 0.741 0.656 1.163 0.454 0.623 1.178
Ex 6 48% 23% 0.952 0.957 1.187 0.801 0.920 1.341
Compared Models MI
#G #N GFer NGFer MIer Note:
Ex 1 0.2% .02% 0.370 0.97 0.381 for GFer and NGFer, the smaller the better.
Ex 2 49% 24% 0.490 0.879 1.193
Ex 3 49% 23% 0.463 0.579 1.265 But for MIer, The larger the better.
Ex 4 49% 20% 0.765 0.849 1.154
Ex 5 43% 37% 0.454 0.631 1.163
Ex 6 48% 23% 0.836 0.970 1.254
Table 3.  Registration results for $ \frac{\alpha_1}{\lambda} $-dependence tests of New Model for processing Example 3. The relative errors are computed using the normalized gradient fitting formula (38). In all cases, we set $ \alpha = 0.01\alpha_1 $
$\frac{\alpha_1}{\lambda}$ 0.1 0.05 0.025 0.017 0.0125 0.01 0.0075 0.005
Error 0.238 0.237 0.237 0.236 0.237 0.237 0.238 0.24
$\frac{\alpha_1}{\lambda}$ 0.1 0.05 0.025 0.017 0.0125 0.01 0.0075 0.005
Error 0.238 0.237 0.237 0.236 0.237 0.237 0.238 0.24
[1]

Xueyong Wang, Yiju Wang, Gang Wang. An accelerated augmented Lagrangian method for multi-criteria optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 1-9. doi: 10.3934/jimo.2018136

[2]

Chunrong Chen, T. C. Edwin Cheng, Shengji Li, Xiaoqi Yang. Nonlinear augmented Lagrangian for nonconvex multiobjective optimization. Journal of Industrial & Management Optimization, 2011, 7 (1) : 157-174. doi: 10.3934/jimo.2011.7.157

[3]

Chunrong Chen. A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 495-508. doi: 10.3934/naco.2011.1.495

[4]

Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409

[5]

Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687

[6]

Guillaume Bal, Jiaming Chen, Anthony B. Davis. Reconstruction of cloud geometry from high-resolution multi-angle images. Inverse Problems & Imaging, 2018, 12 (2) : 261-280. doi: 10.3934/ipi.2018011

[7]

Xiantao Xiao, Liwei Zhang, Jianzhong Zhang. On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 319-339. doi: 10.3934/jimo.2009.5.319

[8]

Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45

[9]

Yuan Shen, Wenxing Zhang, Bingsheng He. Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints. Journal of Industrial & Management Optimization, 2014, 10 (3) : 743-759. doi: 10.3934/jimo.2014.10.743

[10]

Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165

[11]

Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4193-4223. doi: 10.3934/dcds.2015.35.4193

[12]

Wei Zhu. A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method. Inverse Problems & Imaging, 2017, 11 (6) : 975-996. doi: 10.3934/ipi.2017045

[13]

Egil Bae, Xue-Cheng Tai, Wei Zhu. Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours. Inverse Problems & Imaging, 2017, 11 (1) : 1-23. doi: 10.3934/ipi.2017001

[14]

Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012

[15]

Rentsen Enkhbat, M. V. Barkova, A. S. Strekalovsky. Solving Malfatti's high dimensional problem by global optimization. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 153-160. doi: 10.3934/naco.2016005

[16]

Qian Liu, Xinmin Yang, Heung Wing Joseph Lee. On saddle points of a class of augmented lagrangian functions. Journal of Industrial & Management Optimization, 2007, 3 (4) : 693-700. doi: 10.3934/jimo.2007.3.693

[17]

Yuri B. Suris. Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms. Journal of Geometric Mechanics, 2013, 5 (3) : 365-379. doi: 10.3934/jgm.2013.5.365

[18]

Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673

[19]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039

[20]

Michel Cristofol, Jimmy Garnier, François Hamel, Lionel Roques. Uniqueness from pointwise observations in a multi-parameter inverse problem. Communications on Pure & Applied Analysis, 2012, 11 (1) : 173-188. doi: 10.3934/cpaa.2012.11.173

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (99)
  • HTML views (248)
  • Cited by (0)

Other articles
by authors

[Back to Top]