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Determination of fractional pixel location via an optimal group delay estimation approach

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Abstract

The main problem to be addressed in this paper is on the high required computational powers of the existing fractional pixel location search algorithms. The current solution method is based on the computation of the phase shift of the power spectral densities of two consecutive frames of a video sequence. Since the probability density function of each frame of the video sequence is required for computing its power spectral density but it is unknown in the practical situation, this method is not applied in the practical situation. To address this issue, this paper proposes an optimal group delay estimation method. First, the motion vectors are represented as the sum of the coarse-scale motion vectors and the fine-scale motion vectors. A full integer-pixel location search is employed for finding the coarse-scale motion vectors. Then, the determination of the fine-scale motion vectors is formulated as an optimal motion estimation problem. The rounding operator is further applied to the obtained solution of the optimization problem. If the fine-scale motion vectors are the zero vector, then it is not required to find the fractional pixel location. Since only few motion vectors are located at the half-pixel locations and the quarter-pixel locations, the main contribution and the advantage of our proposed method are on the significant reduction in the required computational power. As a result, our proposed method allows decoding of a high-quality video in the real time.

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Acknowledgements

This paper was supported partly by the National Nature Science Foundation of China (Nos. U1701266, 61372173 and 61671163), the Team Project of the Education Ministry of the Guangdong Province (2017KCXTD011), the Guangdong Higher Education Engineering Technology Research Center for Big Data on Manufacturing Knowledge Patent (No. 501130144), and Hong Kong Innovation and Technology Commission, Enterprise Support Scheme (No. S/E/070/17).

Author information

Authors and Affiliations

  1. Faculty of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, China

    Nili Tian & Bingo Wing-Kuen Ling

  2. Australian Joint Research Centre for Building Information Modelling, School of Built Environment, Curtin University, Perth, WA, 6845, Australia

    Changzhi Wu & Xiangyu Wang

  3. Department of Mathematics and Statistics, Curtin University, Perth, WA, 6002, Australia

    Kok-Lay Teo

Authors
  1. Nili Tian
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  2. Bingo Wing-Kuen Ling
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  3. Changzhi Wu
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  4. Xiangyu Wang
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  5. Kok-Lay Teo
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Correspondence to Bingo Wing-Kuen Ling.

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Appendix

Appendix

$$ {\hat{\mathbf{B}}}_{k} \equiv \left[ {B_{k} \left( {p_{0,k} ,q_{0,k} } \right), \ldots ,B_{k} \left( {p_{0,k} ,q_{0,k} + N - 1} \right), \ldots ,B_{k} \left( {p_{0,k} + N - 1,q_{0,k} } \right), \ldots ,B_{k} \left( {p_{0,k} + N - 1,q_{0,k} + N - 1} \right)} \right]^{\text{T}} $$
(4)
$$ {\hat{\mathbf{B}}}_{k + 1} \equiv \left[ {B_{k + 1} \left( {0,0} \right), \ldots ,B_{k + 1} \left( {0,N - 1} \right), \ldots ,B_{k + 1} \left( {N - 1,0} \right), \ldots ,B_{k + 1} \left( {N - 1,N - 1} \right)} \right]^{\text{T}} $$
(5)
$$ {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) \equiv \left[ {\cos \left( {0\omega_{1} + 0\omega_{2} } \right), \ldots ,\cos \left( {0\omega_{1} + \left( {N - 1} \right)\omega_{2} } \right), \ldots ,\cos \left( {\left( {N - 1} \right)\omega_{1} + 0\omega_{2} } \right), \ldots ,\cos \left( {\left( {N - 1} \right)\omega_{1} + \left( {N - 1} \right)\omega_{2} } \right)} \right]^{\text{T}} $$
(6)
$$ {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) \equiv \left[ {\sin \left( {0\omega_{1} + 0\omega_{2} } \right), \ldots ,\sin \left( {0\omega_{1} + \left( {N - 1} \right)\omega_{2} } \right), \ldots ,\sin \left( {\left( {N - 1} \right)\omega_{1} + 0\omega_{2} } \right), \ldots ,\sin \left( {\left( {N - 1} \right)\omega_{1} + \left( {N - 1} \right)\omega_{2} } \right)} \right]^{\text{T}} $$
(7)
$$ \begin{aligned} - \frac{\text{d}}{{{\text{d}}\omega_{1} }}\angle H_{k} \left( {\omega_{1} ,\omega_{2} } \right) & = - \frac{\text{d}}{{{\text{d}}\omega_{1} }}\tan^{ - 1} \frac{{{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)}}{{{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)}} \\ & = - \frac{{\left( \begin{aligned} \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)\frac{\text{d}}{{{\text{d}}\omega_{1} }}\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ - \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)\frac{\text{d}}{{{\text{d}}\omega_{1} }}\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ \end{aligned} \right)}}{{\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} + \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} }} \\ \end{aligned} $$
(9)
$$ {\tilde{\mathbf{B}}}_{k} \equiv \left[ {0B_{k} \left( {p_{0,k} ,q_{0,k} } \right), \ldots ,0B_{k} \left( {p_{0,k} ,q_{0,k} + N - 1} \right), \ldots ,\left( {N - 1} \right)B_{k} \left( {p_{0,k} + N - 1,q_{0,k} } \right), \ldots ,\left( {N - 1} \right)B_{k} \left( {p_{0,k} + N - 1,q_{0,k} + N - 1} \right)} \right]^{\text{T}} $$
(10)
$$ {\tilde{\mathbf{B}}}_{k + 1} \equiv \left[ {0B_{k + 1} \left( {0,0} \right), \ldots ,0B_{k + 1} \left( {0,N - 1} \right), \ldots ,\left( {N - 1} \right)B_{k + 1} \left( {N - 1,0} \right), \ldots ,\left( {N - 1} \right)B_{k + 1} \left( {N - 1,N - 1} \right)} \right]^{\text{T}} $$
(11)
$$ \begin{aligned} & - \frac{\text{d}}{{{\text{d}}\omega_{1} }}\angle H_{k} \left( {\omega_{1} ,\omega_{2} } \right) \\ & \quad = - \frac{{\left( \begin{aligned} \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ \left( \begin{aligned} - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\tilde{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\tilde{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ - {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\tilde{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\tilde{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ \end{aligned} \right) \hfill \\ - \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ \left( \begin{aligned} - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\tilde{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\tilde{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ + {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\tilde{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\tilde{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ \end{aligned} \right) \hfill \\ \end{aligned} \right)}}{{\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} + \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{T} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{T} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} }} \\ \end{aligned} $$
(12)
$$ - \frac{\text{d}}{{{\text{d}}\omega_{1} }}\angle H_{k} \left( {\omega_{1} ,\omega_{2} } \right) = - \frac{{\left( \begin{aligned} \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)\left( {{\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\mathbf{\dddot B}}_{k} - {\mathbf{\ddot{B}}}_{k} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\mathbf{\dddot B}}_{k} - {\mathbf{\ddot{B}}}_{k} } \right){\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ - {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\dot{\mathbf{B}}}_{k} - {\dot{\mathbf{B}}}_{k}^{\text{T}} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right){\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\mathbf{\ddot{B}}}_{k} - {\mathbf{\ddot{B}}}_{k}^{\text{T}} + {\mathbf{\dddot B}}_{k}^{\text{T}} - {\mathbf{\dddot B}}_{k} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ \end{aligned} \right)}}{{\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\dot{\mathbf{B}}}_{k} - {\dot{\mathbf{B}}}_{k}^{\text{T}} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} + \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} }} $$
(13)
$$ \begin{aligned} - \frac{\text{d}}{{{\text{d}}\omega_{2} }}\angle H_{k} \left( {\omega_{1} ,\omega_{2} } \right) & = - \frac{\text{d}}{{{\text{d}}\omega_{2} }}\tan^{ - 1} \frac{{{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)}}{{{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)}} \\ & \quad = - \frac{{\left( \begin{aligned} \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)\frac{\text{d}}{{{\text{d}}\omega_{2} }}\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ - \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)\frac{\text{d}}{{{\text{d}}\omega_{2} }}\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ \end{aligned} \right)}}{{\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} + \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{{^{\text{T}} }} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{{^{\text{T}} }} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} }} \\ \end{aligned} $$
(14)
$$ {\bar{\mathbf{B}}}_{k} \equiv \left[ {0B_{k} \left( {p_{0,k} ,q_{0,k} } \right), \ldots ,\left( {N - 1} \right)B_{k} \left( {p_{0,k} ,q_{0,k} + N - 1} \right), \ldots ,0B_{k} \left( {p_{0,k} + N - 1,q_{0,k} } \right), \ldots ,\left( {N - 1} \right)B_{k} \left( {p_{0,k} + N - 1,q_{0,k} + N - 1} \right)} \right]^{\text{T}} $$
(15)
$$ {\bar{\mathbf{B}}}_{k + 1} \equiv \left[ {0B_{k + 1} \left( {0,0} \right), \ldots ,\left( {N - 1} \right)B_{k + 1} \left( {0,N - 1} \right), \ldots ,0B_{k + 1} \left( {N - 1,0} \right), \ldots ,\left( {N - 1} \right)B_{k + 1} \left( {N - 1,N - 1} \right)} \right]^{\text{T}} $$
(16)
$$ \begin{aligned} & - \frac{\text{d}}{{{\text{d}}\omega_{2} }}\angle H_{k} \left( {\omega_{1} ,\omega_{2} } \right) \\ & \quad = - \frac{{\left( \begin{aligned} \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ \left( \begin{aligned} - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\bar{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\bar{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ - {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\bar{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\bar{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ \end{aligned} \right) \hfill \\ - \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{T} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ \left( \begin{aligned} - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\bar{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\bar{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ + {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\bar{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\bar{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ \end{aligned} \right) \hfill \\ \end{aligned} \right)}}{{\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) - {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} + \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{\text{T}} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{T} \left( {\omega_{1} ,\omega_{2} } \right){\hat{\mathbf{B}}}_{k + 1} {\hat{\mathbf{B}}}_{k}^{T} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} }} \\ \end{aligned} $$
(17)
$$ - \frac{\text{d}}{{{\text{d}}\omega_{2} }}\angle H_{k} \left( {\omega_{1} ,\omega_{2} } \right) = - \frac{{\left( \begin{aligned} \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)\left( {{\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\mathbf{B}}_{k}^{{{\prime \prime \prime }}} - {\mathbf{B}}_{k}^{{\prime \prime }} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\mathbf{B}}_{k}^{{{\prime \prime \prime }}} - {\mathbf{B}}_{k}^{{\prime \prime }} } \right){\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right)} \right) \hfill \\ - {\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\dot{\mathbf{B}}}_{k} - {\dot{\mathbf{B}}}_{k}^{\text{T}} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right){\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\mathbf{B}}_{k}^{{\prime \prime }} - {\mathbf{B}}_{k}^{{{\prime \prime {\rm T}}}} + {\mathbf{B}}_{k}^{{{\prime \prime \prime {\rm T}}}} - {\mathbf{B}}_{k}^{{{\prime \prime \prime }}} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right) \hfill \\ \end{aligned} \right)}}{{\left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right)\left( {{\dot{\mathbf{B}}}_{k} - {\dot{\mathbf{B}}}_{k}^{\text{T}} } \right){\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} + \left( {{\mathbf{c}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{c}}\left( {\omega_{1} ,\omega_{2} } \right) + {\mathbf{s}}^{\text{T}} \left( {\omega_{1} ,\omega_{2} } \right){\dot{\mathbf{B}}}_{k} {\mathbf{s}}\left( {\omega_{1} ,\omega_{2} } \right)} \right)^{2} }} $$
(18)

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Tian, N., Ling, B.WK., Wu, C. et al. Determination of fractional pixel location via an optimal group delay estimation approach. SIViP 13, 1477–1485 (2019). https://doi.org/10.1007/s11760-019-01497-2

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