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Cost-based image patch matching is at the core of various techniques incomputer vision, photogrammetry and remote sensing. When the subpixel disparitybetween the reference patch in the source and target images is required, eitherthe cost function or the target image have to be interpolated. While cost-basedinterpolation is the easiest to implement, multiple works have shown that imagebased interpolation can increase the accuracy of the subpixel matching, butusually at the cost of expensive search procedures. This, however, isproblematic, especially for very computation intensive applications such asstereo matching or optical flow computation. In this paper, we show that closedform formulae for subpixel disparity computation for the case of onedimensional matching, e.g., in the case of rectified stereo images where thesearch space is of one dimension, exists when using the standard NCC, SSD andSAD cost functions. We then demonstrate how to generalize the proposed formulaeto the case of high dimensional search spaces, which is required forunrectified stereo matching and optical flow extraction. We also compare ourresults with traditional cost volume interpolation formulae as well as withstate-of-the-art cost-based refinement methods, and show that the proposedformulae bring a small improvement over the state-of-the-art cost-based methodsin the case of one dimensional search spaces, and a significant improvementwhen the search space is two dimensional.

Generalized Closed-form Formulae for Feature-based Subpixel Alignment in Patch-based Matching