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Quaternion Harmonic moments and extreme learning machine for color object recognition

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Abstract

The quaternary orthogonal moments have been widely used as color image descriptors owe to their remarkable color and shape information encapsulation capability. Their computation, however, depends on finding the optimal value of a unit pure quaternion parameter, which is done empirically and with no warranty of optimality. We propose a 2D color object recognition method that relies on the quaternion-valued parameter-free disc-harmonic moment invariants (QHMs) fed into the quaternion extreme learning machine (QELM). The role of this latter is to maintain the correlation between the four parts, real and imaginary, of the quaternary descriptor coefficients. Several datasets are used for recognition experiments. We draw the conclusion that: (1) our quaternion-valued QHMs invariants outperform other quaternary moments, (2) the quaternion-valued moment invariants give results better than the modulus-based moment invariants and (3) the QELM yields results better than the state-of-the-art classifiers.

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Authors and Affiliations

  1. Laboratory of Informatics and Modeling, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, P.B. 1796, Fez, 30003, Morocco

    Nisrine Dad, Noureddine En-nahnahi & Said El Alaoui Ouatik

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  1. Nisrine Dad
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  2. Noureddine En-nahnahi
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  3. Said El Alaoui Ouatik
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Correspondence to Nisrine Dad.

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Appendix: Operations on quaternions

Appendix: Operations on quaternions

A quaternion [23] is a generalization of a complex number. It consists of one real part and three imaginary parts as q = qa + qbi + qcj + qdk.

The imaginary units i, j and k follow the combinations:i2 = j2 = k2 = − 1, ij = k, jk = i, ki = j, ji = −k, kj = −i, ik = −j.

The conjugate and the modulus of a quaternion number q are respectively defined by

$$ q^{\ast}=q^{a}-q^{b}i-q^{c}j-q^{d}k, $$
(A1)
$$ |q|=\sqrt{(q^{a})^{2}+(q^{b})^{2}+(q^{c})^{2}+(q^{d})^{2}}. $$
(A2)

The dot product of two quaternions q1 and q2 is defined as:

$$ q_{1}\bullet q_{2}= {q_{1}^{a}} {q_{2}^{a}} + {q_{1}^{b}} {q_{2}^{b}} i+{q_{1}^{c}} {q_{2}^{c}} j+{q_{1}^{d}} {q_{2}^{d}} k, $$
(A3)

and the cross product as:

$$ \begin{array}{@{}rcl@{}} q_{1}\otimes q_{2}&=& ({q_{1}^{a}} {q_{2}^{a}}-{q_{1}^{b}} {q_{2}^{b}}-{q_{1}^{c}} {q_{2}^{c}}-{q_{1}^{d}} {q_{2}^{d}}) + ({q_{1}^{a}} {q_{2}^{b}}+{q_{1}^{b}} {q_{2}^{a}}+{q_{1}^{c}} {q_{2}^{d}}-{q_{1}^{d}} {q_{2}^{c}})i\\ &&+ ({q_{1}^{a}} {q_{2}^{c}}-{q_{1}^{b}} {q_{2}^{d}}+{q_{1}^{c}} {q_{2}^{a}}+{q_{1}^{d}} {q_{2}^{b}})j + ({q_{1}^{a}} {q_{2}^{d}}+{q_{1}^{b}} {q_{2}^{c}}-{q_{1}^{c}} {q_{2}^{b}}+{q_{1}^{d}} {q_{2}^{a}})k. \end{array} $$
(A4)

The quaternions space, or also called Hamilton space, is denoted \(\mathbb {H}\).

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Dad, N., En-nahnahi, N. & El Alaoui Ouatik, S. Quaternion Harmonic moments and extreme learning machine for color object recognition. Multimed Tools Appl 78, 20935–20959 (2019). https://doi.org/10.1007/s11042-019-7381-2

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  • DOI: https://doi.org/10.1007/s11042-019-7381-2

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