# American Institute of Mathematical Sciences

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  March 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386
##### References:
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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). [2] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Proceedings of Fifth International Conference on Computer Vision, (1995), 694-699. [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. [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. [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. [6] I. M. Gelfand, R. A. Silverman and et al., Calculus of Variations, Courier Corporation, 2000. [7] A. Handlovičová, K. 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. [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. [9] J. Jost, Postmodern Analysis, Third edition, Universitext, Springer-Verlag, Berlin, 2005. [10] V. Klement, T. 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. [11] F. Maes, A. Collignon, D. Vandermeulen, G. Marchal and P. Suetens, Multimodality image registration by maximisation of mutual information, IEEE Transactions on Medical Imaging, 16 (1997), 187-198. [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. [13] T. Makela, P. Clarysse, O. Sipila, N. Pauna, Q. C. Pham, T. 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. [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. [15] D. P. Peng, B. Merriman, S. Osher, H. 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. [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. [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. [18] R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9986-3. [19] D. Rueckert, L. I. Sonoda, C. Hayes, D. L. Hill, M. 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. [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. [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. [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.
Diagram of the proposed method
Volume elements and the central points
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$^{dist}$: $N_1 = 200, N_2 = 200, \alpha = 1.0, \beta = 3.25, \gamma = 2.5$
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$^{dist}$: $N_1 = 200, N_2 = 200, \alpha = 1.5, \beta = 3.75, \gamma = 3.25$
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
The results of OF$^{dist}$ registration of object marked by green line in 6a, and global MIM registration of the whole scene
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: $K_{S_1} = K_{S_2} = K_{T} = 9.0\cdot 10^{-6}$
Results of registration of $S_1,S_2$ with target image $T$. Parameters of OF$^{dist}$: $N_1 = 256, N_2 = 218, \alpha = 1.25, \beta = 3.5, \gamma = 3.0$
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
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
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
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
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
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