  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.
