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Numerical joint invariant level set formulation with unique image segmentation result

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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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Notes

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

  2. Even if the uniqueness exists in the continuum.

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

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

  5. 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 \(\lbrace \mathrm {x}_{1}, \ldots , \mathrm {x}_{\mathrm{k}}\rbrace \subset \mathrm {E}\): \(\mathrm {J}(\mathrm {g}\cdot \mathrm {x}_{1},\ldots ,\mathrm {g}\cdot \mathrm {x}_{\mathrm{k}}) = \mathrm {J}(\mathrm {x}_{1}, \ldots , \mathrm {x}_{\mathrm {k}})\).

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

  7. 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.

  8. 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.

  9. 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.”

  10. 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.

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

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

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Aghayan, R. Numerical joint invariant level set formulation with unique image segmentation result. Machine Vision and Applications 32, 33 (2021). https://doi.org/10.1007/s00138-020-01134-w

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