相敏全息的线性时间算法
Linear time algorithm for phase sensitive holography
摘要：
全息搜索算法，如直接搜索和模拟退火允许高质量的全息图，以长的执行时间为代价。这是由于$O(N_x N_y)$的单次迭代计算成本和所需的$O(N_x N_y)$阶迭代次数，其中$N_x$和$N_y$是图像维度。这给出了$O(N_x^2 N_y^2)$阶的组合性能。在本文中，我们使用了一种新的技术，以减少迭代成本$O(1)$相敏计算机生成的全息图给出了最终的算法性能的$O(N_x N_y)$。我们通过重新制定均方误差度量来实现这一点，以允许从衍射场计算均方误差度量，而不需要正向变换步骤。对于$1024\times 1024$像素测试图像，与传统的直接搜索相比，这给了我们一个$\approx 50,000\times$加速，几乎没有额外的复杂性。当应用于相位调制或幅度调制设备时，所提出的算法在$O(N_x N_y)$时间内收敛于全局最小均方误差。相比之下，大多数现存算法不能保证获得全局最小值，并且那些算法具有至少$O(N_x^2 N_y^2)$的计算复杂度，其中朴素算法是$O((N_xN_y)!)$。
英文摘要：
Holographic search algorithms such as direct search and simulated annealing allow high-quality holograms to be generated at the expense of long execution times. This is due to single iteration computational costs of $O(N_x N_y)$ and number of required iterations of order $O(N_x N_y)$, where $N_x$ and $N_y$ are the image dimensions. This gives a combined performance of order $O(N_x^2 N_y^2)$. In this paper we use a novel technique to reduce the iteration cost down to $O(1)$ for phase-sensitive computer generated holograms giving a final algorithmic performance of $O(N_x N_y)$. We do this by reformulating the mean-squared error metric to allow it to be calculated from the diffraction field rather than requiring a forward transform step. For a $1024\times 1024$ pixel test images this gave us a $\approx 50,000\times$ speed-up when compared with traditional direct search with little additional complexity. When applied to phase-modulating or amplitude-modulating devices the proposed algorithm converges on a global minimum mean squared error in $O(N_x N_y)$ time. By comparison, most extant algorithms do not guarantee a global minimum is obtained and those that do have a computational complexity of at least $O(N_x^2 N_y^2)$ with the naive algorithm being $O((N_xN_y)!)$.
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