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Advanced leaf image retrieval via Multidimensional Embedding Sequence Similarity (MESS) method

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Abstract

A novel method for shape analysis and similarity measurement is introduced based on a time series matching approach. It applies to shapes represented through one-dimensional signals and has as objectives to utilize efficiently the provided information and to optimize the shape matching process. The new technique is tested on boundaries from leaf images, after their conversion into 1D sequences using either the Centroid Contour Distance (CCD) or the Angle code (AC) measurements. In the core of the new method lies the ‘time delay’-based transformation of a given 1D sequence to an ensemble of vectors embedded in a multivariate phase space. The resulting point set is considered as representative of the leaf identity. Inter-leaf comparisons are carried out in a pairwise fashion by employing the multidimensional, Wald–Wolfowitz, statistical test for the ‘two-sample problem’, which implicitly performs shape matching and similarity quantification. The comparative experimentation shows that the complexity of our method is moderate, while the leaf retrieval performance, compared to that achieved by standard matching procedures usually employed with the CCD and AC representations, is greatly improved.

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Authors and Affiliations

  1. Department of Computer Engineering and Informatics, University of Patras, Patras, Greece

    F. Fotopoulou

  2. Artificial Intelligence and Information Analysis Lab, Department of Informatics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    N. Laskaris

  3. Electronics Lab., Department of Physics, University of Patras, Patras, Greece

    G. Economou & S. Fotopoulos

Authors
  1. F. Fotopoulou
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  2. N. Laskaris
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  3. G. Economou
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  4. S. Fotopoulos
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Corresponding author

Correspondence to S. Fotopoulos.

Appendix A. The multivariate Wald–Wolfowitz test

Appendix A. The multivariate Wald–Wolfowitz test

The Wald–Wolfowitz multivariate statistical test [22] assesses the commonality between two different sets of multivariate observations.

The output of the test can be expressed as the probability that two point-samples are coming from the same distribution. Its great advantage is that it is model-free and this stems from the graph–theoretic origin of the test, which is actually based on the concept of MST graph [29, 30]. The MST is a spanning tree containing exactly (N − 1) edges, for which the sum of edge weights is minimum. In WW-test, the graph is built over points in Rd: a single node corresponds to every given point, the weight associated with every possible edge is the corresponding interpoint Euclidean distance. WW-test can be used to test the hypothesis H 0, whether any two given multidimensional point samples {Xi}i = 1:m and {Yi}i = 1:n are coming from the same multivariate distribution. A great advantage is that no a priori assumption about the distribution of points in the two samples is a prerequisite [31].

In the first step, the sample identity of each point is not taken into account and the MST of the overall sample is constructed.

Then, based on the sample identities of the points, a test statistic R is computed. R is the total number of runs, while a run is defined as a consecutive sequence of identical sample identities. Rejection of H 0 is for small values of R. The null distribution of this statistic has been derived, based on combinatorial analysis. It has been shown that the quantity

$$ W = \frac{R - E[R]}{{\sqrt {{\text{Var}}[R]} }} $$

approaches (asymptotically) the standard normal distribution, while the mean E[R] and variance Var[R/C] of R depend on the sizes m and n of the two point-samples and can be computed using the following analytical expressions:

$$ E[R] = \frac{2mn}{N} + 1 $$
$$ {\text{Var}}[R/C] = \frac{2mn}{N(N - 1)} \times \left\{ {\frac{2mn - N}{N} \,+\, \frac{C - N + 2}{(N - 2)(N - 3)}[N(N - 1) - 4mn + 2]} \right\} $$

where N = m + n, C is the number of edge pairs sharing a common node defined as \( C = \frac{1}{2}\sum\nolimits_{i - 1}^{N} {d{}_{i}(d_{i} - 1)} \) and d i is the degree of the ith node.

The above analysis enables the computation of the significance level (and p value) for the acceptance of the hypothesis H 0.

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Fotopoulou, F., Laskaris, N., Economou, G. et al. Advanced leaf image retrieval via Multidimensional Embedding Sequence Similarity (MESS) method. Pattern Anal Applic 16, 381–392 (2013). https://doi.org/10.1007/s10044-011-0254-6

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