March  2021, 14(3): 1145-1160. doi: 10.3934/dcdss.2020386

Signed-distance function based non-rigid registration of image series with varying image intensity

a. 

Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13,120 00 Prague, Czech Republic

b. 

Department of Radiology, Institute for clinical and experimental medicine, Vídeňská 1958/9, Praha 4,140 21, Czech Republic

c. 

Inria, France

d. 

LMS, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, France

e. 

School of Biomedical Engineering & Imaging Sciences, St Thomas' Hospital, King's College London, UK

* Corresponding author: katerina.skardova@fjfi.cvut.cz

Received  January 2019 Revised  February 2020 Published  June 2020

In this paper we propose a method for locally adjusted optical flow-based registration of multimodal images, which uses the segmentation of object of interest and its representation by the signed-distance function (OF$ ^{dist} $ method). We deal with non-rigid registration of the image series acquired by the Modiffied Look-Locker Inversion Recovery (MOLLI) magnetic resonance imaging sequence, which is used for a pixel-wise estimation of $ T_1 $ relaxation time. The spatial registration of the images within the series is necessary to compensate the patient's imperfect breath-holding. The evolution of intensities and a large variation of image contrast within the MOLLI image series, together with the myocardium of left ventricle (the object of interest) typically not being the most distinct object in the scene, makes the registration challenging. The paper describes all components of the proposed OF$ ^{dist} $ method and their implementation. The method is then compared to the performance of a standard mutual information maximization-based registration method, applied either to the original image (MIM) or to the signed-distance function (MIM$ ^{dist} $). Several experiments with synthetic and real MOLLI images are carried out. On synthetic image with a single object, MIM performed the best, while OF$ ^{dist} $ and MIM$ ^{dist} $ provided better results on synthetic images with more than one object and on real images. When applied to signed-distance function of two objects of interest, MIM$ ^{dist} $ provided a larger registration error, but more homogeneously distributed, compared to OF$ ^{dist} $. For the real MOLLI image series with left ventricle pre-segmented using a level-set method, the proposed OF$ ^{dist} $ registration performed the best, as is demonstrated visually and by measuring the increase of mutual information in the object of interest and its neighborhood.

Citation: Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386
References:
[1]

J. F. Aujol and G. Aubert, Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi Equations, INRIA Res. Rep, 4507 (2002).   Google Scholar

[2]

V. CasellesR. Kimmel and G. Sapiro, Geodesic active contours, Proceedings of Fifth International Conference on Computer Vision, (1995), 694-699.   Google Scholar

[3]

S. Chen and A. Rahman, Contrast enhancement using recursive mean-separate histogram equalisation for scalable brightness preservation, IEEE Transactions on Consumer Electronics, 49 (2003), 1301-1309.   Google Scholar

[4]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. Ⅰ, Journal of Differential Geometry, 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.  Google Scholar

[5]

M. A. Fischler and R. A. Elschlager, The representation and matching of pictorial structures, IEEE Transactions on computers, C-22 (1973), 67-92.  doi: 10.1109/T-C.1973.223602.  Google Scholar

[6]

I. M. Gelfand, R. A. Silverman and et al., Calculus of Variations, Courier Corporation, 2000. Google Scholar

[7]

A. HandlovičováK. Mikula and F. Sgallari, Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution, Numerische Mathematik, 93 (2003), 675-695.  doi: 10.1007/s002110100374.  Google Scholar

[8]

B. K. P. Horn and B. G. Schunck, Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.  doi: 10.1016/0004-3702(81)90024-2.  Google Scholar

[9]

J. Jost, Postmodern Analysis, Third edition, Universitext, Springer-Verlag, Berlin, 2005.  Google Scholar

[10]

V. KlementT. Oberhuber and D. Ševčovič, Application of the level-set model with constraints in image segmentation, Numerical Mathematics: Theory, Methods and Applications, 9 (2016), 147-168.  doi: 10.4208/nmtma.2015.m1418.  Google Scholar

[11]

F. MaesA. CollignonD. VandermeulenG. Marchal and P. Suetens, Multimodality image registration by maximisation of mutual information, IEEE Transactions on Medical Imaging, 16 (1997), 187-198.   Google Scholar

[12]

J. A. Maintz and M. A. Viergever, A survey of medical image registration, Medical Image Analysis, 2 (1998), 1-36.  doi: 10.1016/S1361-8415(01)80026-8.  Google Scholar

[13]

T. MakelaP. ClarysseO. SipilaN. PaunaQ. C. PhamT. Katila and I. E. Magnin, A review of cardiac image registration methods, IEEE Transactions on Medical Imaging, 21 (2002), 1011-1021.  doi: 10.1109/TMI.2002.804441.  Google Scholar

[14]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[15]

D. P. PengB. MerrimanS. OsherH. K. Zhao and M. Kang, A PDE-based fast local level set method, Journal of Computational Physics, 155 (1999), 410-438.  doi: 10.1006/jcph.1999.6345.  Google Scholar

[16]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[17]

G. Peyré, M. Péchaud, R. Keriven, L. D. Cohen and et al., Geodesic methods in computer vision and graphics, Foundations and Trends® in Computer Graphics and Vision, 5 (2010), 197–397. Google Scholar

[18]

R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9986-3.  Google Scholar

[19]

D. RueckertL. I. SonodaC. HayesD. L. HillM. O. Leach and D. J. Hawkes, Nonrigid registration using free-form deformations: Application to breast MR images, IEEE Transactions on Medical Imaging, 18 (1999), 712-721.  doi: 10.1109/42.796284.  Google Scholar

[20]

J. A. Sethian, Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws, Journal of Differential Geometry, 31 (1990), 131-161.  doi: 10.4310/jdg/1214444092.  Google Scholar

[21]

J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Monographs on Applied and Computational Mathematics, 3. Cambridge University Press, Cambridge, 1996.  Google Scholar

[22]

T. W. Tang and A. C. Chung, Non-rigid image registration using graph-cuts, International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), Springer, (2007), 916-924.   Google Scholar

show all references

References:
[1]

J. F. Aujol and G. Aubert, Signed distance functions and viscosity solutions of discontinuous Hamilton-Jacobi Equations, INRIA Res. Rep, 4507 (2002).   Google Scholar

[2]

V. CasellesR. Kimmel and G. Sapiro, Geodesic active contours, Proceedings of Fifth International Conference on Computer Vision, (1995), 694-699.   Google Scholar

[3]

S. Chen and A. Rahman, Contrast enhancement using recursive mean-separate histogram equalisation for scalable brightness preservation, IEEE Transactions on Consumer Electronics, 49 (2003), 1301-1309.   Google Scholar

[4]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. Ⅰ, Journal of Differential Geometry, 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.  Google Scholar

[5]

M. A. Fischler and R. A. Elschlager, The representation and matching of pictorial structures, IEEE Transactions on computers, C-22 (1973), 67-92.  doi: 10.1109/T-C.1973.223602.  Google Scholar

[6]

I. M. Gelfand, R. A. Silverman and et al., Calculus of Variations, Courier Corporation, 2000. Google Scholar

[7]

A. HandlovičováK. Mikula and F. Sgallari, Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution, Numerische Mathematik, 93 (2003), 675-695.  doi: 10.1007/s002110100374.  Google Scholar

[8]

B. K. P. Horn and B. G. Schunck, Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.  doi: 10.1016/0004-3702(81)90024-2.  Google Scholar

[9]

J. Jost, Postmodern Analysis, Third edition, Universitext, Springer-Verlag, Berlin, 2005.  Google Scholar

[10]

V. KlementT. Oberhuber and D. Ševčovič, Application of the level-set model with constraints in image segmentation, Numerical Mathematics: Theory, Methods and Applications, 9 (2016), 147-168.  doi: 10.4208/nmtma.2015.m1418.  Google Scholar

[11]

F. MaesA. CollignonD. VandermeulenG. Marchal and P. Suetens, Multimodality image registration by maximisation of mutual information, IEEE Transactions on Medical Imaging, 16 (1997), 187-198.   Google Scholar

[12]

J. A. Maintz and M. A. Viergever, A survey of medical image registration, Medical Image Analysis, 2 (1998), 1-36.  doi: 10.1016/S1361-8415(01)80026-8.  Google Scholar

[13]

T. MakelaP. ClarysseO. SipilaN. PaunaQ. C. PhamT. Katila and I. E. Magnin, A review of cardiac image registration methods, IEEE Transactions on Medical Imaging, 21 (2002), 1011-1021.  doi: 10.1109/TMI.2002.804441.  Google Scholar

[14]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[15]

D. P. PengB. MerrimanS. OsherH. K. Zhao and M. Kang, A PDE-based fast local level set method, Journal of Computational Physics, 155 (1999), 410-438.  doi: 10.1006/jcph.1999.6345.  Google Scholar

[16]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[17]

G. Peyré, M. Péchaud, R. Keriven, L. D. Cohen and et al., Geodesic methods in computer vision and graphics, Foundations and Trends® in Computer Graphics and Vision, 5 (2010), 197–397. Google Scholar

[18]

R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9986-3.  Google Scholar

[19]

D. RueckertL. I. SonodaC. HayesD. L. HillM. O. Leach and D. J. Hawkes, Nonrigid registration using free-form deformations: Application to breast MR images, IEEE Transactions on Medical Imaging, 18 (1999), 712-721.  doi: 10.1109/42.796284.  Google Scholar

[20]

J. A. Sethian, Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws, Journal of Differential Geometry, 31 (1990), 131-161.  doi: 10.4310/jdg/1214444092.  Google Scholar

[21]

J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Monographs on Applied and Computational Mathematics, 3. Cambridge University Press, Cambridge, 1996.  Google Scholar

[22]

T. W. Tang and A. C. Chung, Non-rigid image registration using graph-cuts, International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), Springer, (2007), 916-924.   Google Scholar

Figure 1.  Diagram of the proposed method
Figure 2.  Volume elements and the central points
Figure 3.  Source image (A), target image (B), the absolute value of their difference before registration (C) and after registration by OF$ ^{dist} $, MIM and MIM$ ^{dist} $ (D-F). Parameters of OF$ ^{dist} $: $ N_1 = 200, N_2 = 200, \alpha = 1.0, \beta = 3.25, \gamma = 2.5 $
Figure 4.  Source image (A), target image (B) and the absolute value of their difference before (C) and after registration by OF, MIM and MIM$ ^{dist} $ (D-F). Parameters of OF$ ^{dist} $: $ N_1 = 200, N_2 = 200, \alpha = 1.5, \beta = 3.75, \gamma = 3.25 $
Figure 5.  Absolute value of the difference between the target and source signed-distance function before and after registration by OF$ ^{dist} $ and MIM$ ^{dist} $. The signed-distance functions are computed on 10-pixel-wide neighborhood of the edges of the object and set to constant outside the neighborhood
Figure 6.  The results of OF$ ^{dist} $ registration of object marked by green line in 6a, and global MIM registration of the whole scene
Figure 7.  Images from the MOLLI sequence with segmented myocardium. Parameters for the outer edge detection: $ K_{S_1} = 1.3\cdot 10^{-6} $, $ K_{S_2} = 1.3\cdot 10^{-6} $, $ K_{T} = 2.3\cdot 10^{-6} $. Parameters for the inner edge detection: $ K_{S_1} = K_{S_2} = K_{T} = 9.0\cdot 10^{-6} $
Figure 8.  Results of registration of $ S_1,S_2 $ with target image $ T $. Parameters of OF$ ^{dist} $: $ N_1 = 256, N_2 = 218, \alpha = 1.25, \beta = 3.5, \gamma = 3.0 $
Table 1.  Norms of difference between the target 3b image and source image 3a before and after registration by OF$ ^{dist} $, MIM and MIM$ ^{dist} $
$ \| T - S \|_2 $ $ \| T - S_{OF^{dist}} \|_2 $ $ \| T - S_{MIM} \|_2 $ $ \| T - S_{MIM^{dist}} \|_2 $
9462.47 3130.83 2173.93 2607.66
$ \| T - S \|_2 $ $ \| T - S_{OF^{dist}} \|_2 $ $ \| T - S_{MIM} \|_2 $ $ \| T - S_{MIM^{dist}} \|_2 $
9462.47 3130.83 2173.93 2607.66
Table 2.  Norms of difference between the target image in Figure 4b image and source image in Figure 4a before and after registration by OF$ ^{dist} $, MIM and MIM$ ^{dist} $
$ \| T - S \|_2 $ $ \| T - S_{OF^{dist}} \|_2 $ $ \| T - S_{MIM} \|_2 $ $ \| T - S_{MIM^{dist}} \|_2 $
6654.44 2199.07 4121.95 3602.27
$ \| T - S \|_2 $ $ \| T - S_{OF^{dist}} \|_2 $ $ \| T - S_{MIM} \|_2 $ $ \| T - S_{MIM^{dist}} \|_2 $
6654.44 2199.07 4121.95 3602.27
Table 3.  Norms of difference between target and source signed-distance function before and after registration by OF$ ^{dist} $ and MIM$ ^{dist} $. The source and target objects can be seen in Figure 4a and 4b, respectively
$ \| \phi_T - \phi_S \|_2 $ $ \| \phi_T - \phi_{S,{OF^{dist}}} \|_2 $ $ \| \phi_T - \phi_{S,{MIM^{dist}}} \|_2 $
0.849915 0.239367 0.2950385
$ \| \phi_T - \phi_S \|_2 $ $ \| \phi_T - \phi_{S,{OF^{dist}}} \|_2 $ $ \| \phi_T - \phi_{S,{MIM^{dist}}} \|_2 $
0.849915 0.239367 0.2950385
Table 4.  Norms of difference between the target image 4b and source image 4a before registration, after registration of one object by OF$ ^{dist} $, and after global registration by MIM
$ \| T - S \|_2 $ $ \| T - S_{1,OF^{dist}} \|_2 $ $ \| T - S_{MIM} \|_2 $
6654.44 5062.1 4121.95
$ \| T - S \|_2 $ $ \| T - S_{1,OF^{dist}} \|_2 $ $ \| T - S_{MIM} \|_2 $
6654.44 5062.1 4121.95
Table 5.  MI target image and source images from Figure 8 before and after registration by OF$ ^{dist} $ and MIM and MIM$ ^{dist} $. The MI was computed only in the surroundings of the segmented objects
$ i $ $ MI(T,S_i) $ $ MI(T,S_{i,OF^{dist}}) $ $ MI(T,S_{i,MIM}) $ $ MI(T,S_{i,MIM^{dist}}) $
1 1.1556 1.2184 1.2273 1.2170
2 1.1012 1.2034 1.2052 1.1964
$ i $ $ MI(T,S_i) $ $ MI(T,S_{i,OF^{dist}}) $ $ MI(T,S_{i,MIM}) $ $ MI(T,S_{i,MIM^{dist}}) $
1 1.1556 1.2184 1.2273 1.2170
2 1.1012 1.2034 1.2052 1.1964
[1]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017

[2]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321

[3]

Israa Mohammed Khudher, Yahya Ismail Ibrahim, Suhaib Abduljabbar Altamir. Individual biometrics pattern based artificial image analysis techniques. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2020056

[4]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013

[5]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[6]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[7]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[8]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709

[9]

Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341

[10]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[11]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[12]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018

[13]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[14]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[15]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[16]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (69)
  • HTML views (269)
  • Cited by (0)

[Back to Top]