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Parameter selection in a Mumford-Shah geometrical model for the detection of thin structures. (English) Zbl 1381.94043

Summary: We present a variational model to perform the segmentation of thin structures in MRI images (namely codimension 1 objects). It is based on the classical Mumford-Shah functional and we have added geometrical priors as constraints. We precisely describe the structure model (that we call tubes). We give existence, uniqueness and regularity results for the solution to the optimization problem. The keypoint is the fact that 2D/3D problems are equivalent to 1D ones. This gives hints to perform an automatic parameter tuning for numerical purpose.

MSC:

94A13 Detection theory in information and communication theory
49J10 Existence theories for free problems in two or more independent variables
49N60 Regularity of solutions in optimal control
49Q10 Optimization of shapes other than minimal surfaces
92C30 Physiology (general)
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[1] Adams, D.R.: Capacity and the obstacle problem. Appl. Math. Optim. 8(1), 39-57 (1982) · Zbl 0503.35039 · doi:10.1007/BF01447750
[2] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs Oxford University Press, London (2000) · Zbl 0957.49001
[3] Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via gamma-convergence. Commun. Pure Appl. Math. XLIII, 999-1036 (1990) · Zbl 0722.49020 · doi:10.1002/cpa.3160430805
[4] An, J.-H.; Bigeleisen, P.; Damelin, S., Identification of nerves in ultrasound scans using modified Mumford-Shah model and prior information, WCECS 2011, San Francisco, October 19-21, 2011
[5] Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. SIAM, Philadelphia (2005) · Zbl 1095.49001
[6] Aubert, G., Aujol, J.-F., Blanc-Féraud, L.: Detecting codimension-two objects in an image with Ginsburg-Landau models. Int. J. Comput. Vis. 65(1/2), 29-42 (2005) · Zbl 1287.94008 · doi:10.1007/s11263-005-3847-y
[7] Aubert, G., Blanc-Féraud, L., Graziani, D.: Analysis of a new variational model to restore point-like and curve-like singularities in imaging. Appl. Math. Optim. 67(1), 73-96 (2013) · Zbl 1259.93021 · doi:10.1007/s00245-012-9181-1
[8] Aubert, G., Graziani, D.: Variational approximation for detecting point-like target problems. ESAIM Control Optim. Calc. Var. 17(4), 909-930 (2011) · Zbl 1238.49024 · doi:10.1051/cocv/2010029
[9] Aubert, G., Graziani, D.: A relaxation result for an inhomogeneous functional preserving point-like and curve-like singularities in image processing. Int. J. Math. 23(11), 1250120 (2012), 14 pp. · Zbl 1258.49011 · doi:10.1142/S0129167X12501200
[10] Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing, Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147. Springer, Berlin (2006) · Zbl 1110.35001
[11] Chen, L.Q., Shen, J.: Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108(2-3), 147-158 (1998) · Zbl 1017.65533 · doi:10.1016/S0010-4655(97)00115-X
[12] David, G.: Mumford-Shah minimizers on thin plates. Calc. Var. Partial Differ. Equ. 27(2), 203-232 (2006) · Zbl 1103.49022 · doi:10.1007/s00526-006-0018-0
[13] Francfort, G.A., Le, N.Q., Serfaty, S.: Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case. ESAIM Control Optim. Calc. Var. 15(3), 576-598 (2009) · Zbl 1168.49041 · doi:10.1051/cocv:2008041
[14] Graziani, D., Blanc-Féraud, L., Aubert, G.: A formal Γ-convergence approach for the detection of points in 2-D images. SIAM J. Imaging Sci. 3(3), 578-594 (2010) · Zbl 1195.92046 · doi:10.1137/080741604
[15] Kirbas, C., Quek, F.: A review of vessel extraction techniques and algorithms. J. ACM Comput. Surv. 36(2), 81-121 (2004) · doi:10.1145/1031120.1031121
[16] Klann, E., Ramlau, R., Ring, W.: A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Probl. Imaging 5(1), 137-166 (2011) · Zbl 1213.94015 · doi:10.3934/ipi.2011.5.137
[17] Lesage, D., Angelini, E.D., Bloch, I., Funka-Lea, G.: A review of 3d vessel lumen segmentation techniques: models, features and extraction schemes. Med. Image Anal. 13, 819-845 (2009) · doi:10.1016/j.media.2009.07.011
[18] Mahmoodi, S.: Edge detection filter based on Mumford-Shah Green function. SIAM J. Imaging Sci. 5(1), 343-365 (2012) · Zbl 1250.65039 · doi:10.1137/100811349
[19] Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123-142 (1987) · Zbl 0616.76004 · doi:10.1007/BF00251230
[20] Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. XLVII(5), 577-685 (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[21] Péchaud, M., Peyré, G., Keriven, R.: Extraction of vessels networks over an orientation domain (2008) · Zbl 0503.35039
[22] Ramlau, R., Ring, W.: Regularization of ill-posed Mumford-Shah models with perimeter penalization. Inverse Problems 26(11), 115001 (2010), 25 pp. · Zbl 1226.47105 · doi:10.1088/0266-5611/26/11/115001
[23] Adams, R., John, F.: Sobolev Spaces. Pure and Applied Mathematics Series, vol. 140. Academic Press, San Diego (2003) · Zbl 1098.46001
[24] Rouchdy, Y.; Cohen, L., Image segmentation by geodesic voting. Application to the extraction of tree structures from confocal microscope images, 1-5 (2008) · doi:10.1109/ICPR.2008.4761763
[25] Vicente, D.: Approximation result for a Mumford-Shah functional adapted to a segmentation problem (2014). https://hal.archives-ouvertes.fr/hal-01087205 · Zbl 0691.49036
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