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Machine Vision and Applications

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01.02.2021 | Original Paper

Numerical joint invariant level set formulation with unique image segmentation result

verfasst von: Reza Aghayan

Erschienen in: Machine Vision and Applications | Ausgabe 1/2021

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Abstract

The level set method is one of the most widely used and powerful techniques in image science such as image/motion segmentation, object tracking, etc. This paper brings up an unstudied issue with discretized level set algorithms about ‘the non-uniqueness’ of segmentation results which is different from the problem of ‘the existence’ of a result. Our solution is to numerically approximate the level set formulation based on suitable combination of some visual joint invariants, leading to the unique segmentation results, therefore unique visual joint invariant numerical signatures—independent of contour initialization and what visual group is applied. To figure out ‘the cause’ of resulting unique segmentations in this scheme, we utilize the level set algorithm to introduce three energy features—called fingerprints, flows, and stem charts. Our experimental results indicate that curvature-based energies can be classified in terms of these characteristics—depending merely on the nature of each energy. Besides, the energies generated by the current discretization are ‘positive,’ while the visual joint invariant curvature-based energies sketch charts with ‘negative’ values.

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As far as I know, unlike the problem of the existence of a solution, it has not appeared in the literature.

Even if the uniqueness exists in the continuum.

A computed tomography technique by which arterial and venous vessels are visualized through the body.

The number of points in the original curve that map to a single generic point in the resulting signature.

A ‘joint invariant’ of the action of a group G refers to an algebraic function that depends on several points of E, having the property that its value is unchanged under simultaneous action of the group elements on the point configuration—in other words, a joint invariant is a real-valued function \(\mathrm {J}: \mathrm {E}^{\times \mathrm {k}}\longrightarrow {\mathbb {R}}\) so that for each \(\mathrm {g}\in \mathrm {G}\) and any mesh \(\lbrace \mathrm {x}_{1}, \ldots , \mathrm {x}_{\mathrm{k}}\rbrace \subset \mathrm {E}\): \(\mathrm {J}(\mathrm {g}\cdot \mathrm {x}_{1},\ldots ,\mathrm {g}\cdot \mathrm {x}_{\mathrm{k}}) = \mathrm {J}(\mathrm {x}_{1}, \ldots , \mathrm {x}_{\mathrm {k}})\).

In visual applications, the visual group \({{\mathbb {V}}}\) is either the Euclidean, affine, similarity, or projective group.

In fact, our scheme replaces the level set curvature \(\kappa \) by the joint curvature \({\kappa ^{\vartriangle }_{S{\mathbb {E}}}}\) of the level-curve in each iteration.

Since \(\kappa _{\mathrm {S}{\mathbb {I}}}=\kappa _{\mathrm {S}{\mathbb {E}},\mathrm {s}}/\kappa _{\mathrm {S} {\mathbb {E}}}^{2}\), showing the same result in the similarity case seems trivial, therefore we investigated the discussed independence where the LSF (7) is discretized by the affine joint invariants with a completely different ‘nature’ and ‘lots of computational complexity,’ compared to the other ones.

Hence, from now on, there is no need to be mentioned the name of the visual group used in our approach, and the resulting level set algorithm will be denoted only by “JILS.”

According to [35], this flexibility is very helpful, for instance to consider equally and unequally spaced meshes, resulting closer approximations to DISCs, and to apply the m-mean signature and the m-difference techniques to minimize the effects of noise and indeterminacy in the resulting signatures. Here, it helps to generate a wide variety of the curvature-based \({\mathbb {V}}\)JIEs.

In our study, \( 0 \leqslant \alpha , \beta \leqslant 1 \).

In general, based on our observations, each of these features is able to classify curvature-based energies.

Pollefeys, M., Nist\(\acute{{\rm r}}\), D., Frahm, J.-M., et al.: Detailed real-time urban 3D reconstruction from video. Int. J. Comput. Vis. 78, 143–167 (2008)

Kolev, K., Klodt, M., Brox, T., Esedoglu, S., Cremers, D.: Continuous global optimization in multiview 3D reconstruction. Int. J. Comput. Vis. 84(1), 80–96 (2009)CrossRef

Linda, G.S., George, C.S.: Computer Vision. Prentice-Hall, New York (2001)

Pal, N.R., Pal, S.K.: A review on image segmentation techniques. Pattern Recognit. 26, 1277–1294 (1993)CrossRef

Lee, L.K., Liew, S.C., Thong, W.J.: A review of image segmentation methodologies in med-ical image, In: Advanced computer and communication engineering technology, pp. 1069–1080. Springer, New York (2015)

Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetCrossRef

Saito, A., Nawano, S., Shimizu, A.: Joint optimization of segmentation and shape prior from level-set-based statistical shape model, and its application to the automated segmentation of abdominal organs. Med. Image Anal. 28, 46–65 (2016)CrossRef

Wu, Q., Gan, Y., Lin, B., Zhang, Q., Chang, H.: An active contour model based on fused texture features for image segmentation. Neurocomputing 151, 1133–1141 (2015)CrossRef

Chang, H., Chen, Z., Huang, Q., Shi, J., Li, X.: Graph-based learning for segmentation of 3D ultrasound images. Neurocomputing 151, 632–644 (2015)CrossRef

10.

Abdelsamea, M.M., Gnecco, G., Gaber, M.M.: An efficient self-organizing active contour model for image segmentation. Neurocomputing 149, 820–835 (2015)CrossRef

11.

Gibou, F., Fedkiw, R., Osher, S.: A review of level-set methods and some recent applications. J. Comput. Phys. 353, 82–109 (2018)MathSciNetCrossRef

12.

Wan, M., Gu, G., Sun, J., et al.: A level set method for infrared image segmentation using global and local information. Remote Sens. 10, 1039 (2018). https://doi.org/10.3390/rs10071039CrossRef

13.

Li, C.M., Xu, C.Y., Gui, C.F., Fox, M.D.: Level set evolution without re-initialization: a new variational formulation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, San Diego, pp. 430–436 (2005)

14.

Paragios, N., Deriche, R.: Geodesic active contours and level sets for detection and tracking of moving objects. IEEE Transaction on Pattern Analysis and Machine Intelligence 22, 1–15 (2000)CrossRef

15.

Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22, 61–79 (1997)CrossRef

16.

Ning, J., Zhang, L., Zhang, D., Wu, C.: Interactive image segmentation by maximal simi-larity based region merging. Pattern Recogn. 43, 445–456 (2010)CrossRef

17.

Zhang, K., Song, H., Zhang, L.: Active contours driven by local image fitting energy. Pattern Recogn. 43, 1199–1206 (2010)CrossRef

18.

Wan, M., Gu, G., Qian, W., Ren, K., Chen, Q.: Hybrid active contour model based on edge gradients and regional multi-features for infrared image segmentation. Optik 140, 833–842 (2017)CrossRef

19.

Çataloluk, H., Çelebi, F.H.: A novel hybrid model for two-phase image segmentation: GSA based Chan–Vese algorithm. Eng. Appl. Artif. Intell. 73, 22–30 (2018)CrossRef

20.

Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10, 266–277 (2001)CrossRef

21.

Mumford, D., Shah, J.: Optimal approximation by piecewise smooth function and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)MathSciNetCrossRef

22.

Tsai, A., Yezzi, A., Willsky, A.: Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. Image Process. 10, 1169–1186 (2001)CrossRef

23.

Vese, L., Chan, T.: A multiphase level set framework for image segmentation using the Mumford–Shah model. Int. J. Comput. Vis. 50, 271–293 (2002)CrossRef

24.

Zhang, K., Zhang, L., Song, H., Zhou, W.: Active contours with selective local or global segmentation: a new formulation and level set method. Image Vis. Comput. 28, 668–676 (2010)CrossRef

25.

Li, C.M., Kao, C.Y., Gore, J.C., Ding, Z.H.: Minimization of region-scalable fitting energy for image segmentation. IEEE Trans. Image Process. 17, 1940–1949 (2008)MathSciNetCrossRef

26.

Lankton, S., Tannenbaum, A.: Localizing region-based active contours. IEEE Trans. Image Process. 17, 2029–2039 (2008)MathSciNetCrossRef

27.

Wang, X., Huang, D., Xu, H.: An efficient local Chan–Vese model for image segmentation. Pattern Recogn. 43, 603–618 (2010)CrossRef

28.

Wang, L., Li, C., Sun, Q., Xia, D., Kao, C.-Y.: Active contours driven by local and global intensity fitting energy with application to brain MR image segmentation. Comput. Med. Imaging Graph. 33, 520–531 (2009)CrossRef

29.

Cao, J., Wu, X.: A novel level set method for image segmentation by combining local and global information. J. Mod. Opt. 64, 2399–2412 (2017)CrossRef

30.

Olver, P.J.: Classical Invariant Theory. Cambridge University Press, New York (1999)CrossRef

31.

Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Comput. Vis. 26, 107–135 (1998)CrossRef

32.

Cartan, É.: La méthode du repére mobile, la théorie des groupes continus et les espaces général -isés, Exposés de Géométrie, no. 5, Paris, Hermann et cie (1935)

33.

Pauwels, E., Moons, T., Van Gool, L.J., Kempenaers, P., Oosterlinck, A.: Foundations of semi-differential invariants. Int. J. Comput. Vis. 14, 25–48 (1995)CrossRef

34.

Boutin, M.: Numerically invariant signature curves. Int. J. Comput. Vis. 40(3), 235–248 (2000) CrossRef

35.

Aghayan, R., Ellis, T., Dehmeshki, J.: Planar numerical signature theory applied to object recognition. J. Math. Imaging Vis. 48(3), 583–605 (2014)MathSciNetCrossRef

36.

Aghayan, R.: Orientation-invariant numerically invariant joint signatures in curve analysis. Int. J. Comput. Math. 3(1), 13–30 (2018)MathSciNet

37.

Aghayan, R.: Signature-inverse theorem in mesh group-planes \(-\) The new formulation. In: Proceedings of the 49th Annual Iranian Mathematics Conference—Computer Science Section, Tehran, IRAN, August 23–26, pp. 2310–2332 (2018)

38.

Aghayan, R.: Visual groups and their structural equations. In: Proceedings of the 49th Annual Iranian Mathematics Conference—Geometry Section, Tehran, IRAN, August 23–26, pp. 21–35 (2018)

39.

Aghayan, R.: Signature-inverse Theorem in Mesh group-planes (2020). arXiv:2006.03759

40.

Aghayan, R.: Generating visual invariants applied to curve analysis, revised (2020)

Titel: Numerical joint invariant level set formulation with unique image segmentation result
verfasst von: Reza Aghayan
Publikationsdatum: 01.02.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Machine Vision and Applications / Ausgabe 1/2021
Print ISSN: 0932-8092
Elektronische ISSN: 1432-1769
DOI: https://doi.org/10.1007/s00138-020-01134-w

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