Cost-based image patch matching is at the core of various techniques in computer vision, photogrammetry and remote sensing. When the subpixel disparity between the reference patch in the source and target images is required, either the cost function or the target image have to be interpolated. While cost-based interpolation is the easiest to implement, multiple works have shown that image based interpolation can increase the accuracy of the subpixel matching, but usually at the cost of expensive search procedures. This, however, is problematic, especially for very computation intensive applications such as stereo matching or optical flow computation. In this paper, we show that closed form formulae for subpixel disparity computation for the case of one dimensional matching, e.g., in the case of rectified stereo images where the search space is of one dimension, exists when using the standard NCC, SSD and SAD cost functions. We then demonstrate how to generalize the proposed formulae to the case of high dimensional search spaces, which is required for unrectified stereo matching and optical flow extraction. We also compare our results with traditional cost volume interpolation formulae as well as with state-of-the-art cost-based refinement methods, and show that the proposed formulae bring a small improvement over the state-of-the-art cost-based methods in the case of one dimensional search spaces, and a significant improvement when the search space is two dimensional.

翻译：基于代价的图像块匹配是计算机视觉、摄影测量与遥感领域多种技术的核心。当需要源图像与目标图像中参考块之间的子像素视差时，必须对代价函数或目标图像进行插值。虽然基于代价的插值最容易实现，但多项研究表明，基于图像的插值可以提高子像素匹配的精度，但通常以昂贵的搜索过程为代价。然而，这对于计算密集型应用（如立体匹配或光流计算）而言存在问题。本文证明，对于一维匹配情况（例如，搜索空间为一维的校正立体图像），当使用标准NCC、SSD和SAD代价函数时，存在子像素视差计算的闭合形式公式。随后，我们演示如何将所提出的公式推广到高维搜索空间的情况，这对于未校正立体匹配和光流提取是必需的。我们还将结果与传统代价体插值公式以及最先进的基于代价的细化方法进行比较，表明所提出的公式在一维搜索空间情况下对最先进的基于代价的方法略有改进，而在二维搜索空间情况下则显著提升。