Vector field reconstruction from sparse samples by triple-Laplacian
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Published
Sep 21, 2024
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Abstract
We present a mathematical model to reconstruct vector fields from given sparse samples inside the domain. We applied the presented model to reconstruct the velocity vector field driving macrophages toward the wound during wound healing. In this application, the sparse samples are the sparsely distributed velocity vectors given on macrophage trajectories. The method consists of solving a minimization problem, which leads to applying the Laplace equation with suitable boundary conditions to the two components of the vectors and the vector lengths. The values given by sparse samples are the prescribed Dirichlet conditions inside the domain, and we impose zero Neumann boundary conditions on the domain boundary. Solving the Laplace equation, we obtain a smooth vector field in the whole domain. We prove the existence and uniqueness of a weak solution for the considered partial differential equation with mixed boundary conditions, present its numerical solution, and show numerical results.
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Lupi, G., & Mikula, K.
(2024).
Vector field reconstruction from sparse samples by triple-Laplacian.
Proceedings Of The Conference Algoritmy, , 85 -98.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2158/1030
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References
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[2] Azimzadeh, P., Forsyth, P.A.: Weakly chained matrices, policy iteration, and impulse control. SIAM Journal on Numerical Analysis 54(3) (2016)
[3] Banchs, R.: Natural quartic splines. Research progress report no. 7 on Time Harmonic Field Electric Logging, University of Texas at Austin (1996), https://rbanchs.com/publications/reports.html
[4] Belhachmi, Z., Bucur, D., Burgeth, B., Weickert, J.: How to choose interpolation data in images. SIAM Journal on Applied Mathematics 70(1) (2009)
[5] Fisher, M., Schröder, P., Desbrun, M., Hoppe, H.: Design of tangent vector fields. ACM Transactions on Graphics (TOG) 26(3) (2007)
[6] Gaël, G., Benoı̂t, J., et al.: Eigen v3. http://eigen.tuxfamily.org (2010)
[7] Krutitskii, P.: The mixed harmonic problem in a bounded cracked domain with dirichlet condition on cracks. Journal of Differential Equations 198(2) (2004)
[8] Lage, M., Petronetto, F., Paiva, A., Lopes, H., Lewiner, T., Tavares, G.: Vector field reconstruction from sparse samples with applications. 19th Brazilian Symposium on Computer Graphics and Image Processing (2006)
[9] Lupi, G., Mikula, K., Park, S.A.: Macrophages trajectories smoothing by evolving curves. Tatra Mountains Mathematical Publications 86(1) (2023)
[10] Mikula, K., Ambroz, M., Mokošová, R.: What was the river ister in the time of strabo? A mathematical approach. Tatra Mountains Mathematical Publications 80(3) (2021)
[11] Mussa-Ivaldi, F.: From basis functions to basis fields: vector field approximation from sparse data. Biological Cybernetics 67 (1992)
[12] Nečas, J.: Direct methods in the theory of elliptic equations. Springer Berlin, Heidelberg (2012)
[13] Park, S.A., Sipka, T., Krivá, Z., Nguyen-Chi, M., Lutfalla, G., Mikula, K.: Segmentation-based tracking of macrophages in 2d+time microscopy movies inside a living animal. Computers in Biology and Medicine 153 (2022)
[14] Rektorys, K.: Variational Methods in Mathematics, Science and Engineering. Springer Netherlands (2012)
[15] Schönlieb, C.B.: Partial Differential Equation Methods for Image Inpainting. Cambridge University Press (2015)
[16] Sipka, T., Peroceschi, R., Groß, M., Ellett, F., Pescia, C., Gonzalez, C., Lutfalla, G., Nguyen-Chi, M.: Damage-induced calcium signaling and reactive oxygen species mediate macrophage activation in zebrafish. Frontiers in Immunology 12 (2021)