Generalized Closed-Form Formulae for Feature-Based Subpixel Alignment in Patch-Based Matching

Patch-based matching is a technique meant to measure the disparity between pixels in a source and target image and is at the core of various methods in computer vision. When the subpixel disparity between the source and target images is required, the cost function or the target image has to be interpolated. While cost-based interpolation is easier to implement, multiple works have shown that image-based interpolation can increase the accuracy of the disparity estimate. In this paper we review closed-form formulae for subpixel disparity computation for one dimensional matching, e.g., rectified stereo matching, for the standard cost functions used in patch-based matching. We then propose new formulae to generalize to high-dimensional search spaces, which is necessary for unrectified stereo matching and optical flow. We also compare the image-based interpolation formulae with traditional cost-based formulae, and show that image-based interpolation brings a significant improvement over the cost-based interpolation methods for two dimensional search spaces, and small improvement in the case of one dimensional search spaces. The zero-mean normalized cross correlation cost function is found to be preferable for subpixel alignment. A new error model, based on very broad assumptions is outlined in the Supplementary Material to demonstrate why these image-based interpolation formulae outperform their cost-based counterparts and why the zero-mean normalized cross correlation function is preferable for subpixel alignement.