余弦变换的二元相位恢复：基于局部曲率最小化
Binary Phase Retrieval of Cosine Transforms via Local Curvature Minimization
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- University of Texas at Austin（德克萨斯大学奥斯汀分校）
- Purdue University（普渡大学）
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中文总结 AI 辅助
本文针对余弦变换的二元相位恢复问题，提出通过最小化重建变换的曲率来求解，实验表明该方法可稳定重建图像，多尺度结构相似性指数达0.95，且求解时间近似线性增长。
中文摘要 AI 辅助
余弦变换因其计算简便和高能量紧凑性，在图像和视频压缩中得到了频繁使用。这引起了光学界对通过光学方式计算余弦变换以用于压缩的兴趣。当在傅里叶平面成像以捕获余弦变换时，会遇到相位恢复问题：光场既有幅度又有相位，但相机只能捕获场的幅度。余弦变换将相位恢复问题限制在二元域中，而非单位圆上，但一般的相位恢复解决方案并未利用这一点。在这项工作中，我们证明了余弦变换相位恢复中的符号错误会导致变换在错误处的Hessian矩阵幅度显著增加。基于此，我们开发了一种算法，通过最小化重建余弦变换的曲率来解决这个二元相位恢复问题。我们通过计算实验证明，给定图像的余弦变换幅度，我们能够一致地重建原始图像，其多尺度结构相似性指数达到0.95。我们表明，求解时间随求解像素数量近似线性增长。
英文摘要
Cosine transforms see frequent use in image and video compression due to their ease of computation and high energy compaction. This has sparked interest within the optics community in computing cosine transforms optically for compression. When imaging in the Fourier plane to capture a cosine transform, one encounters a phase retrieval problem: optical fields have both magnitude and phase, but cameras only capture the magnitude of the field. Cosine transforms restrict the phase retrieval problem to a binary domain as opposed to the unit circle, but general phase retrieval solutions do not leverage this. In this work, we demonstrate that sign errors in phase retrieval for cosine transforms result in a significant increase in the magnitude of the Hessian of the transform at the error. Motivated by this, we develop an algorithm to solve this binary phase retrieval problem by minimizing the curvature of the reconstructed cosine transform. We demonstrate via computational experiments that given the magnitude of the cosine transform of an image, we can consistently reconstruct the original image within a multiscale structural similarity index of 0.95. We show that the solve time grows approximately linearly with the number of pixels solved.
