Fast Quaternion Log-Polar Radial Harmonic Fourier Moments for Color Image Zero-Watermarking

Abstract

As a copyright protection technology, zero-watermarking has received widespread attention in recent years. The zero-watermarking algorithm based on moments has been extensively studied due to its better comprehensive performance and has made great progress. However, three issues of the moments-based zero-watermarking methods should be addressed: first, most of them ignore the analysis and experiment on discriminability, which leads to the high false positive ratio; second, direct computation of the moments from their definition is inefficient, numerically unstable and inaccurate, which severely affects the performances of these moments-based methods; third, most of the algorithms are only suitable for grayscale images, which leads to the limitation of the algorithms in practical applications. In this paper, we present a new color image zero-watermarking using fast quaternion log-polar radial harmonic Fourier moments. This algorithm firstly introduces log-polar coordinates to make traditional RHFMs scale-invariant, secondly improves their speed and accuracy, then introduces the concept of quaternion to make it suitable for color images, and finally applies it to the zero-watermark technology. Theoretical analysis and experimental results show that compared to the existing algorithms, the BER value of the proposed algorithm is 0 under some attacks, which can prove its effectiveness and achieved a good trade-off between discriminability and robustness, and has certain superiority in terms of security, capacity and speed.

Appendices

Appendix A

$$\begin{aligned} \varphi_{n,m}^{{lp^{\prime}}} & = \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g^{\prime}(\rho^{\prime},\theta^{\prime})e^{ - jn\pi \rho } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta } } \\ & \quad + \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g^{\prime}(\rho^{\prime},\theta^{\prime})e^{jn\pi \rho } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta } } \\ & = \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho + \log_{\lambda } (s),\theta + \phi )e^{ - jn\pi \rho } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta } } \\ & \quad + \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho + \log_{\lambda } (s),\theta + \phi )e^{jn\pi \rho } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta } } \\ & = \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )e^{{ - jn\pi (\rho - \log_{\lambda } (s))}} e^{ - jm(\theta - \phi )} {\text{d}}\rho {\text{d}}\theta } } \\ & \quad + \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )e^{{jn\pi (\rho - \log_{\lambda } (s))}} e^{ - jm(\theta - \phi )} {\text{d}}\rho {\text{d}}\theta } } \\ & = e^{{jn\pi \cdot \log_{\lambda } (s) + jm\phi }} \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )} } e^{ - jn\pi \rho } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta \\ & \quad + e^{{ - jn\pi \cdot \log_{\lambda } (s) + jm\phi }} \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )e^{jn\pi \rho } } } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta \\ \end{aligned}$$

(A-1)

Take the absolute value of both sides of this equation, we have

$$\begin{aligned} \left| {\phi_{n,m}^{{lp^{\prime}}} } \right| &= \left| {e^{{jn\pi \cdot \log_{\lambda } (s) + jm\phi }} \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )e^{ - jn\pi \rho } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta }}}\right. \\ &\quad \left.{ {{+ e^{{ - jn\pi \cdot \log_{\lambda } (s) + jm\phi }} \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )e^{jn\pi \rho } e^{ - jm\theta } {\text{d}}\rho {\text{d}}\theta } } } } } \right| \end{aligned}$$

(A-2)

And, consequently

$$\begin{aligned}\left| {\phi_{n,m}^{{lp^{\prime}}} } \right| &= \left| {\int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )e^{ - jn\pi \rho } e^{ - jm\theta } d\rho d\theta } } } \right| \\ &\quad + \left| {\int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(\rho ,\theta )e^{jn\pi \rho } e^{ - jm\theta } d\rho d\theta } } } \right| = \left| {\phi_{n,m}^{lp} } \right| \end{aligned}$$

(A-3)

Appendix B

Take N as an example when N is even

$$\begin{aligned}\phi_{2k,m}& = \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(r,\theta )} } e^{ - j2k\pi r} e^{ - jm\theta } {\text{d}}r{\text{d}}\theta \\ &\quad + \int\limits_{0}^{2\pi } {\int\limits_{ - \infty }^{0} {g(r,\theta )} } e^{j2k\pi r} e^{ - jm\theta } \end{aligned}$$

(B-1)

To achieve discretization, the radius $r$ and angle $\theta$ are divided into $M$ parts, and then

$$r_{u} = \frac{u}{M}\quad \theta_{v} = \frac{2\pi v}{M}$$

(B-2)

The discrete form of RHFM is

$$\begin{aligned} \phi_{2k,m} & = \frac{1}{{M^{2} }}\sum\limits_{u = 0}^{M - 1} {\sum\limits_{v = 0}^{M - 1} {g(r_{u} ,\theta_{v} )} }\\&\quad\times \exp \left( { - j\frac{2\pi }{M}( - k)u} \right)\exp \left( { - jm\frac{2\pi }{M}v} \right) \\ & \quad + \frac{1}{{M^{2} }}\sum\limits_{u = 0}^{M - 1} {\sum\limits_{v = 0}^{M - 1} {g(r_{u} ,\theta_{v} )}}\\&\quad\times \exp \left( { - j\frac{2\pi }{M}ku} \right)\exp \left( { - jm\frac{2\pi }{M}v} \right). \\ \end{aligned}$$

(B-3)

According to the 2D-DFT

$$\hat{f}(\xi_{x} ,\xi_{y} ) = \sum\limits_{{i_{1} = 0}}^{N - 1} {\sum\limits_{{i_{2} = 0}}^{N - 1} {f[i_{1} ,i_{2} ]} } \exp ( - j(i_{1} \xi_{x} + i_{2} \xi_{y} ))$$

(B-4)

where

$$\left\{ {\xi_{x} = \frac{{2\pi k_{1} }}{N},\xi_{y} = \frac{{2\pi k_{2} }}{N}} \right\}$$

(B5)

From this we can obtain

$$\phi_{2k,m} = \hat{f}( - \xi_{x} ,\xi_{y} ) + \hat{f}(\xi_{x} ,\xi_{y} ).$$

(B-6)

When N is 0 and when N is odd, the proof is similar to the above.

Appendix C

The sampling point PP is substituted into the formula of 2D discrete RHFMs, we can obtain,

$$\begin{aligned} \phi_{2k,m} &= \frac{1}{2\pi }\sum\limits_{0}^{2\pi } \sum\limits_{0}^{1} g[i_{1} ,i_{2} ]e^{{j\left( {i_{1} \frac{2\pi pq}{{N^{2} }} - i_{2} \frac{\pi q}{N}} \right)}} \\ &\quad+ \frac{1}{2\pi }\sum\limits_{0}^{2\pi } \sum\limits_{0}^{1} g[i_{1} ,i_{2} ]e^{{ - j\left( {i_{1} \frac{2\pi pq}{{N^{2} }} + i_{2} \frac{\pi q}{N}} \right)}} \end{aligned}$$

(C-1)

I'm going to deform the above equation

$$\begin{aligned} \phi_{2k,m}& = \frac{1}{2\pi }\sum\limits_{{i_{1} = 0}}^{N - 1} e^{{ji_{1} \frac{2\pi pq}{{N^{2}}} }} \sum\limits_{{i_{2} = 0}}^{N - 1} g[i_{1} ,i_{2} ]e^{{ - ji_{2} \frac{\pi q}{N} } } \\ &\quad + \frac{1}{2\pi }\sum\limits_{{i_{1} = 0}}^{N - 1} e^{{ - ji_{1} \frac{2\pi pq}{{N^{2} }} }} \sum\limits_{{i_{2} = 0}}^{N - 1} g[i_{1} ,i_{2} ]e^{{ - ji_{2} \frac{\pi q}{N}}} . \end{aligned}$$

(C-2)

According to the above formula, when N is even, the RHFMs are obtained by adding the Fourier transform of two very similar forms, so let

$$\phi_{1}^{BV} = \frac{1}{2\pi }\sum\limits_{{i_{1} = 0}}^{N - 1} {e^{{ - ji_{1} \frac{2\pi pq}{{N^{2} }}}} } \sum\limits_{{i_{2} = 0}}^{N - 1} {g[i_{1} ,i_{2} ]e^{{ - ji_{2} \frac{\pi q}{N}}} }$$

(C-3)

So let's think about the sum inside

$$\phi_{1}^{BV} [i_{1} ,q] = \frac{1}{2\pi }\sum\limits_{{i_{2} = 0}}^{N - 1} {g[i_{1} ,i_{2} ]e^{{ - ji_{2} \frac{\pi q}{N}}} }$$

(C-4)

It can be seen that $\phi_{1}^{BV}$ is the one-dimensional Fourier transform of $f$ in the direction of $i_{2}$. In order to adjust the range of variable $q$, we carry out zero filling of $f$ in the direction of $i_{2}$

$$g_{{Zi_{2} }} [i_{1} ,i_{2} ] = \left\{ {\begin{array}{*{20}l} {g[i_{1} ,i_{2} ]} \hfill & {0 \le i_{1} < N} \hfill \\ 0 \hfill & {N \le i_{2} < 2N} \hfill \\ \end{array} } \right.$$

(C-5)

Here

$$\begin{aligned} \phi_{1}^{BV} [i_{1} ,q] &= \frac{1}{2\pi }\sum\limits_{{i_{2} = 0}}^{N - 1} {g[i_{1} ,i_{2} ]e^{{ - ji_{2} \frac{\pi q}{N}}} }\\ &= \frac{1}{2\pi }\sum\limits_{{i_{2} = 0}}^{{{2}N - 1}} {g_{{Zi_{2} }} [i_{1} ,i_{2} ]e^{{ - ji_{2} \frac{{{2}\pi q}}{{{2}N}}}} } \end{aligned}$$

(C-6)

Set $q^{\prime} = q + N$, we're going to get

$$\begin{aligned} \phi_{1}^{BV} [i_{1} ,q^{\prime}] &= \frac{1}{2\pi }\sum\limits_{{i_{2} = 0}}^{{{2}N - 1}} {g_{{Zi_{2} }} [i_{1} ,i_{2} ]e^{{ - ji_{2} \frac{{{2}\pi (q^{\prime} - N)}}{{{2}N}}}} }\\& = \frac{1}{2\pi }\sum\limits_{{i_{2} = 0}}^{{{2}N - 1}} {g_{{Zi_{2} }} [i_{1} ,i_{2} ] \cdot ( - 1)^{{i_{2} }} \cdot e^{{ - ji_{2} \frac{{{2}\pi q^{\prime}}}{{{2}N}}}} } \end{aligned}$$

(C-7)

Now let's think about the sum on the outside

$$\phi_{1}^{BV} [p,q] = \sum\limits_{{i_{1} = 0}}^{N - 1} {\phi_{1}^{BV} [i_{1} ,q]e^{{ - ji_{1} \frac{2\pi pq}{{N^{2} }}}} } = \sum\limits_{{i_{1} = 0}}^{N - 1} {\phi_{1}^{BV} [i_{1} ,q]e^{{ - ji_{1} \frac{2\pi q}{N} \cdot \frac{p}{N}}} }$$

(C-8)

Set $\alpha = \frac{p}{N}$, we can obtain

$$\phi_{1}^{BV} [p,q] = \sum\limits_{{i_{1} = 0}}^{N - 1} {\phi_{1}^{BV} [i_{1} ,q]e^{{ - j\frac{{2\pi qi_{1} }}{N} \cdot \alpha }} }$$

(C-9)

From the identity relation ${2}i_{1} q = q^{2} + i_{1}^{2} - (q - i_{1} )^{2}$, we can get by deforming the above equation

$$\phi_{1}^{BV} [p,q] = \sum\limits_{{i_{1} = 0}}^{N - 1} {\phi_{1}^{BV} [i_{1} ,q]e^{{ - \frac{j\pi \alpha }{N}(q^{2} + i_{1}^{2} - (q - i_{1} )^{2} )}} } = e^{{ - \frac{j\pi \alpha }{N}q^{2} }} \sum\limits_{{i_{1} = 0}}^{N - 1} {\phi_{1}^{BV} [i_{1} ,q]e^{{ - \frac{j\pi \alpha }{N}i_{1}^{2} }} } e^{{\frac{j\pi \alpha }{N}(q - i_{1} )^{2} }}$$

(C-10)

So if $s[i_{1} ] = e^{{ - \frac{j\pi \alpha }{N}i_{1}^{2} }}$, $\phi_{1}^{BV} [p,q]$ can be represented as

$$\phi_{1}^{BV} [p,q] = s[q]\sum\limits_{{i_{1} = 0}}^{N - 1} {\phi_{1}^{BV} [i_{1} ,q]s[i_{1} ]} s^{ * } [q - i_{1} ]$$

(C-11)

Thus, the complete formula of RHFMs when N is even can be obtained as follows:

$$\begin{aligned} \varphi_{2k,m} = s[q]\sum\limits_{{i_{1} = 0}}^{N - 1} {F_{{}}^{BV} [i_{1} ,q] \cdot s_{1} [i_{1} ] \cdot s^{*} [q - i_{1} ] }\\ \quad+ s[q]\sum\limits_{{i_{1} = 0}}^{N - 1} {F_{{}}^{BV} [i_{1} ,q] \cdot s_{2} [i_{1} ] \cdot s^{*} [q - i_{1} ]} \end{aligned}$$

(C-12)

Here, $s_{1} [i_{1} ] = \exp \left( { - \frac{j\pi \alpha }{N} \cdot i_{1}^{2} } \right)$, $s_{2} [i_{1} ] = \exp \left( {\frac{j\pi \alpha }{N} \cdot i_{1}^{2} } \right)$.

Appendix D

Take $n$ as an example when $n$ is even:

$$\begin{aligned} \hat{\phi }_{2k,m}^{L} & = \frac{1}{2}\int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {T_{2k}^{*} } } (\rho )\exp ( - \mu m(\theta - \varphi )){\text{d}}\rho {\text{d}}\theta \\ & = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)} } \exp ( - \mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta \\ & \quad + \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)} } \exp (\mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta \\ & = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho + \log_{\lambda } (s),\theta + \varphi ,z)} } \exp ( - \mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta \\ & \quad + \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho + \log_{\lambda } (s),\theta + \phi ,z)} } \exp (\mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta . \\ & = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)} } \exp ( - \mu 2k\pi (\rho - \log_{\lambda } (s)))\exp ( - \mu m(\theta - \varphi ))d\rho d\theta \\ & \quad + \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)} } \exp (\mu 2k\pi (\rho - \log_{\lambda } (s)))\exp ( - \mu m(\theta - \varphi )){\text{d}}\rho {\text{d}}\theta \\ & = \exp (\mu 2k\pi \cdot \log_{\lambda } (s) + \mu m\varphi )\\ &\quad \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)\exp ( - \mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta } } \\ & \quad + \exp ( - \mu 2k\pi \cdot \log_{\lambda } (s) + \mu m\varphi )\\ &\quad \times\int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)} } \exp (\mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta . \\ \end{aligned}$$

Thus, we can obtain

$$\begin{aligned} \left| {\hat{\phi }_{n,m}^{L} } \right| & = \left| {\exp (\mu 2k\pi \cdot \log_{\lambda } (s) + \mu m\varphi )}\right.\\& \quad \left.{\int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)} } \exp ( - \mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta } \right| \\ & \quad + \left| {\exp ( - \mu 2k\pi \cdot \log_{\lambda } (s) + \mu m\varphi )}\right. \\&\quad \left.{\int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {g(\rho ,\theta ,z)} } \exp (\mu 2k\pi \rho )\exp ( - \mu m\theta ){\text{d}}\rho {\text{d}}\theta .} \right| \\ & = \left| {\phi_{n,m}^{L} } \right|. \\ \end{aligned}$$

(D-1)

Therefore, we can say that the quaternion log-polar RHFMs have rotation and scaling invariance.

In the same way, we can obtain the $\left| {\phi_{n,m}^{R} } \right| = \left| {\phi_{n,m}^{R} } \right|.$

Appendix E

To calculate the quaternion RHFMs in log-polar domain, it’s necessary to determine the relationship between the log-polar RHFMs and quaternion log-polar RHFMs. After determining the relationship, it’s easy to obtain FQLPRHFMs according to the FLPRHFMs. The proof process is as follows

$$\begin{aligned} \phi_{n,m}^{R} & = \frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g\left( {\rho ,\theta } \right)T_{n}^{{}} \left( \rho \right)\exp \left( { - \mu m\theta } \right)} } {\text{d}}\rho {\text{d}}\theta \\ & = \frac{1}{2\pi }\int_{0}^{2\pi } \int_{0}^{1} \left[ {g_{R} \left( {\rho ,\theta } \right)i + g_{G} \left( {\rho ,\theta } \right)j + g_{B} \left( {\rho ,\theta } \right)k} \right]\\ &\quad T_{n}^{{}} \left( \rho \right)\exp \left( { - \mu m\theta } \right) {\text{d}}\rho {\text{d}}\theta \\ & = i\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{R} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)\exp \left( { - \mu m\theta } \right)} {\text{d}}\rho {\text{d}}\theta \\ & \quad + j\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{G} \left( {\rho ,\theta } \right)} T_{n}^{{}} \left( \rho \right)\exp \left( { - \mu m\theta } \right)} {\text{d}}\rho {\text{d}}\theta \\ & \quad + k\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{B} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)\exp \left( { - \mu m\theta } \right)} {\text{d}}\rho {\text{d}}\theta \\ & = i\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{R} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)} \left( {\cos m\theta - \mu \sin m\theta } \right){\text{d}}\rho {\text{d}}\theta \\ & \quad + j\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {f_{G} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)\left( {\cos m\theta - \mu \sin m\theta } \right)} {\text{d}}\rho {\text{d}}\theta \\ & \quad + k\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {f_{B} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)\left( {\cos m\theta - \mu \sin m\theta } \right)} {\text{d}}\rho {\text{d}}\theta \\ & = i\left[ {\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{R} \left( {\rho ,\theta } \right)} T_{n}^{{}} \left( \rho \right)} \cos \left( {m\theta } \right){\text{d}}\rho {\text{d}}\theta } \right. \\ & \left. \quad - \mu \frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{R} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)} \sin \left( {m\theta } \right){\text{d}}\rho {\text{d}}\theta \right] \\ & \quad + j\left[ {\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{G} \left( {\rho ,\theta } \right)} T_{n}^{{}} \left( \rho \right)} \cos \left( {m\theta } \right){\text{d}}\rho {\text{d}}\theta } \right. \\ & \left. \quad - \mu \frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{G} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)} \sin \left( {m\theta } \right){\text{d}}\rho {\text{d}}\theta \right] \\ & \quad + k\left[ {\frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{B} \left( {\rho ,\theta } \right)} T_{n} \left( \rho \right)\cos \left( {m\theta } \right)} {\text{d}}\rho {\text{d}}\theta } \right. \\ & \quad \left. { - \mu \frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{1} {g_{B} \left( {\rho ,\theta } \right)} T_{n}^{{}} \left( \rho \right)} \sin \left( {m\theta } \right){\text{d}}\rho {\text{d}}\theta } \right] \\ & = i\left[ {{\text{real}}\left( {\phi_{n,m} \left( {g_{R} } \right)} \right) - \frac{1}{\sqrt 3 }(i + j + k) \cdot {\text{imag}}\left( {\phi_{n,m} \left( {g_{R} } \right)} \right)} \right] \\ & \quad + j\left[ {{\text{real}}\left( {\phi_{n,m} \left( {g_{G} } \right)} \right) - \frac{1}{\sqrt 3 }(i + j + k) \cdot {\text{imag}}\left( {\phi_{n,m} \left( {g_{G} } \right)} \right)} \right] \\ & \quad + k\left[ {{\text{real}}\left( {\phi_{n,m} \left( {g_{B} } \right)} \right) - \frac{1}{\sqrt 3 }(i + j + k) \cdot {\text{imag}}\left( {\phi_{n,m} \left( {g_{B} } \right)} \right)} \right] \\ & = A_{n,m}^{R} + iB_{n,m}^{R} + jC_{n,m}^{R} + kD_{n,m}^{R} \\ \end{aligned}$$

(E-1)