Abstract
We define a new linearity measure for a wide class of objects consisting of a set of of curves, in both \(2D\) and \(3D\). After initially observing closed curves, which can be represented in a parametric form, we extended the method to connected compound curves—i.e. to connected configurations of a number of curves representable in a parametric form. In all cases, the measured linearities range over the interval \((0,1],\) and do not change under translation, rotation and scaling transformations of the considered curve. We prove that the linearity is equal to \(1\) if and only if the measured curve consists of two straight line overlapping segments. The new linearity measure is theoretically well founded and all related statements are supported with rigorous mathematical proofs. The behavior and applicability of the new linearity measure are explained and illustrated by a number of experiments.
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Notes
Compactness, eccentricity, fractal dimension, roundness, Hu’s moment invariants, affine moment invariants [8], circularity [12] and ellipticity [19, 20]. Several other global shape descriptors were considered, such as elongatedness, rectangularity, and convexity, but were not found to improve classification accuracy.
We have post-processed the original vessel tracings by applying thresholding and thinning.
Some of the images contain small portions of disconnected curves in addition to the main single compound connected curve. Since such curves do not satisfy the requirements of \(\mathcal{L}_{comp}(\mathcal{C})\) then it is possible that \(\mathcal{L}_{comp}(C) > 1\). However, in practice this did not occur for any of the retinal images (the largest computed value is 0.289). In any case, \(\mathcal{L}_{comp}(C) > 1\) can still be used as a shape descriptor that is invariant with respect to similarity transformations.
It is difficult to compare classification accuracy for the ARIA data set against other methods as the majority of the literature using this data set concentrates on vessel segmentation rather than disease classification. Hijazi et al. [13] did perform classification, but only considered two classes (86 AMD and 56 normal images), achieving 75 % accuracy. Moreover, they treated the blood vessels as noise and eliminated them, analyzing instead image intensity histograms. In order to more directly compare our results with theirs, we removed the diabetic cases to create a two class problem, achieving 80.6 % accuracy. Since traced centerlines were not provided for the full image database, we also tested the full set of source images for the two classes (92 AMD and 60 normals), and extracted blood vessel centerlines using a general purpose ridge detector [23]. Although the quality of the vessel detection was not high it captured some important structural information from the images, and the classification accuracy obtained using \(\mathcal{L}_{comp}(\mathcal{C})\) was 82.2 %.
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Acknowledgments
J. Pantović and J. Žunić are also with the Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade. This work is partially supported by the Serbian Ministry of Science and Technology/projects OI174026/OI174008. Initial results of this paper were presented in [21]. We would like to thank the following for providing data used in this paper: Zicheng Liu (MSR Action3D Dataset), Andreu-García Gabriela (chicken pieces), St Paul’s Eye Unit, Royal Liverpool University Hospital (ARIA Retinal Image Archive).
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Rosin, P.L., Pantović, J. & Žunić, J. Measuring Linearity of Connected Configurations of a Finite Number of \(2D\) and \(3D\) Curves. J Math Imaging Vis 53, 1–11 (2015). https://doi.org/10.1007/s10851-014-0542-z
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DOI: https://doi.org/10.1007/s10851-014-0542-z