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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. Bae, J. 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. Burger, J. 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. Chen, J. L. Shi, M. 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. Chumchob, K. 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. Hodneland, A. Lundervold, J. 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. Hu, Y. Xie, L. 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. Ibrahim, K. 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. Loeckx, P. Slagmolen, F. Maes, D. Vandermeulen and P. Suetens, Nonrigid image registration using conditional mutual information, IEEE Transactions on Medical Imaging, 29 (2010), 19-29. Google Scholar [20] F. Maes, A. Collignon, D. Vandermeulen, G. 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. Papafitsoros, C. 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. Pluim, J. 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. Pöschl, J. 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. Rohlfing, C. R. Maurer, D. 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. Sotiras, C. 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. Tai, J. 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. Wu, J. 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. Yilun, Y. Junfeng, Y. 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. Zhang, K. 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. Zhang, K. 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. ZHU, X.-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. Zhu, X.-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. Bae, J. 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. Burger, J. 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. Chen, J. L. Shi, M. 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. Chumchob, K. 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. Hodneland, A. Lundervold, J. 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. Hu, Y. Xie, L. 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. Ibrahim, K. 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. Loeckx, P. Slagmolen, F. Maes, D. Vandermeulen and P. Suetens, Nonrigid image registration using conditional mutual information, IEEE Transactions on Medical Imaging, 29 (2010), 19-29. Google Scholar [20] F. Maes, A. Collignon, D. Vandermeulen, G. 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. Papafitsoros, C. 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. Pluim, J. 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. Pöschl, J. 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. Rohlfing, C. R. Maurer, D. 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. Sotiras, C. 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. Tai, J. 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. Wu, J. 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. Yilun, Y. Junfeng, Y. 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. Zhang, K. 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. Zhang, K. 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. ZHU, X.-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. Zhu, X.-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
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 $\Omega$
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 $|X| +|Y|$
Example of a multilevel representation of images
Example 1: Comparison of three different models. Clearly only Our Model works while NGF, MI fail completely
Example 2: Comparison of different models to register T-1 and T2-MRI images. New Model performs the best
Example 3: Registration of a second pair of MRI images (T1 and T2). New Model performs the best
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.
Example 4: High-b- and Low-b-value Diffusion-weighted MRIs (of $256\times 256$) using different models. New Model performs the best
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
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
Left: Log scale plot of the distance Dm versus ALM iteration numbers for examples 2-5
Example 6: Registering a PET image to an MRI vimage. New model performs better than others in this example
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
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
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
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