Accurate quaternion fractional-order pseudo-Jacobi–Fourier moments

Abstract

Pseudo-Jacobi–Fourier moments (PJFMs) are a set of orthogonal moments which have been successfully applied in the fields of image processing and pattern recognition. In this paper, we present a new set of quaternion fractional-order orthogonal moments for color images, named accurate quaternion fractional-order pseudo-Jacobi–Fourier moments (AQFPJFMs). We initially propose a fast and accurate algorithm for the PJFMs computation of an image using new recursive approach and polar pixel tiling scheme. Then, we define a new set of orthogonal moments, named accurate fractional-order pseudo-Jacobi–Fourier moments, which is characterized by the generic nature and time–frequency analysis capability. We finally extend the gray-level fractional-order PJFMs to color images and present the quaternion fractional-order pseudo-Jacobi–Fourier moments. In addition, we develop a new color image representation for enhancing simultaneously the discriminability and robustness, called mixed low-order moments feature. We conduct extensive experiments to evaluate the performance of the proposed AQFPJFMs, in which the encouraging results demonstrate the efficacy and superiority of the proposed scheme.

Appendices

Appendices

1.1 Appendix A

By Eq. (4), the weighting function is:$w(r) = (1 - r) \cdot r^{2}$.

Here replace the independent variable $r$ of the radial basis function with the new variable $r = r^{\alpha } ,r \in [0,1]$, and the new weight function can be obtained as:

$$ w^{\prime}(r^{\alpha } ) = (1 - r^{\alpha } ) \cdot \alpha r^{\alpha - 1} \cdot r^{2\alpha } . $$

We define a new radial basis functions:

$$ \begin{aligned} J^{\prime}_{n} (r^{\alpha } ) & = \left[ {\frac{{w^{\prime}(r)}}{{rb_{n} }}} \right]^{\frac{1}{2}} P_{n} (r^{\alpha } ) = \left[ {\frac{{(1 - r^{\alpha } ) \cdot \alpha r^{\alpha - 1} \cdot r^{2\alpha } }}{{rb_{n} }}} \right]^{\frac{1}{2}} P_{n} (r^{\alpha } ) \\ & = \left[ {\frac{{(1 - r^{\alpha } ) \cdot \alpha r^{3\alpha - 2} }}{{b_{n} }}} \right]^{\frac{1}{2}} P_{n} (r^{\alpha } ) \\ & = ( - 1)^{n} \left[ {\frac{(2n + 4)}{{(n + 3)(n + 1)}}(1 - r^{\alpha } )\alpha r^{3\alpha - 2} } \right]^{\frac{1}{2}} \times P_{n} (\alpha ,r). \\ \end{aligned} $$

The proof is complete.

1.2 Appendix B

Taking the conjugate of $H_{nm\alpha }^{L}$, there are:

$$ \begin{aligned} \left( {H_{nm\alpha }^{L} } \right)^{ * } & = \left( {\int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }J_{n} (r,\alpha )\exp ( - \mu m\theta )f(r,\theta ,z)} } r{\text{d}}r{\text{d}}\theta } \right)^{ * } \\ & = \int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }(J_{n} (r,\alpha )\exp ( - \mu m\theta )f(r,\theta ,z)} } )^{ * } r{\text{d}}r{\text{d}}\theta \\ & = \int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }(J_{n} (r,\alpha ))^{*} f(r,\theta ,z)(\exp ( - \mu m\theta )} } )^{ * } r{\text{d}}r{\text{d}}\theta \\ & = - \int_{0}^{1} {\int_{0}^{2\pi } {J_{n} (r,\alpha )f(r,\theta ,z)\frac{1}{2\pi }\exp ( - \mu m\theta )} } r{\text{d}}r{\text{d}}\theta \\ & = - H_{( - n)( - m)\alpha }^{R} . \\ \end{aligned} $$

Therefore, $H_{nm\alpha }^{L} = - (H_{( - n)( - m)\alpha }^{R} )^{ * }$.

1.3 Appendix C

$$ \begin{aligned} H_{nm\alpha }^{L} & = \int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }J_{n} (r,\alpha )\exp ( - \mu m\theta )f(r,\theta ,z)} } r{\text{d}}r{\text{d}}\theta \\ & = \int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }J_{n} (r,\alpha )(f_{R} (r,\theta ){\varvec{i}} + f_{G} (r,\theta ){\varvec{j}} + f_{B} (r,\theta ){\varvec{k}})} } r{\text{d}}r{\text{d}}\theta \\ & = \int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }J_{n} (r,\alpha )\exp ( - \mu m\theta )f_{R} (r,\theta )} } r{\text{d}}r{\text{d}}\theta \cdot {\varvec{i}} \\ & \quad + \int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }J_{n} (r,\alpha )\exp ( - \mu m\theta )f_{G} (r,\theta )} } r{\text{d}}r{\text{d}}\theta \cdot {\varvec{j}} \\ & \quad + \int_{0}^{1} {\int_{0}^{2\pi } {\frac{1}{2\pi }J_{n} (r,\alpha )\exp ( - \mu m\theta )f_{B} (r,\theta )} } r{\text{d}}r{\text{d}}\theta \cdot {\varvec{k}} \\ & = [{\text{Re}} (H_{nm\alpha } (f_{R} )) + {\varvec{\mu}}{\text{Im}} (H_{nm\alpha } (f_{R} ))] \cdot {\varvec{i}} \\ & \quad + [{\text{Re}} (H_{nm\alpha } (f_{G} )) + {\varvec{\mu}}{\text{Im}} (H_{nm\alpha } (f_{G} ))] \cdot {\varvec{j}} \\ & \quad + [{\text{Re}} (H_{nm\alpha } (f_{B} )) + {\varvec{\mu}}{\text{Im}} (H_{nm\alpha } (f_{B} ))] \cdot {\varvec{k}} \\ & = A_{nm\alpha }^{L} + B_{nm\alpha }^{L} {\varvec{i}} + C_{nm\alpha }^{L} {\varvec{j}} + D_{nm\alpha }^{L} {\varvec{k}}\user2{.} \\ \end{aligned} $$