Descriptor Matching and Association

To conclude this chapter, it is finally necessary to say a few words about descriptor matching.

Let $I_1$ and $I_2$ be two images to be analyzed, and let $\mathbf {p}_1$ and $\mathbf {p}_2$ be two points, possibly keypoints, identified in the first and second images, respectively. To determine whether these two image points represent the same point, normally observed from different viewpoints and therefore affected by affine transformations (translations, scale changes, and rotations), homographies, and possibly changes in illumination, it is necessary to define some form of metric $d(\mathbf{p}_1,\mathbf{p}_2)$ for performing the comparison. A specific metric can be defined for each descriptor. In general, the most widely used metrics are L1 (Manhattan, SAD) and L2 (Euclidean, SSD).

Since more than one point will certainly be extracted from each image, a search must be performed, and each point in the first image is associated only with the point in the second image having the minimum distance according to the selected metric:

$$\hat{\mathbf{p}_{2} } = \argmin_i d(\mathbf{p}_1, \mathbf{p}_{2,i} )$$ (8.1)

Usually, to reduce the number of incorrect matches, an association is accepted only if the metric is below a given threshold and the ratio between the best and second-best matches is below a second, uniqueness threshold.

Finally, after finding $\mathbf {p}_2$, the best match for point $\mathbf {p}_1$ in the second image, it can be verified that $\mathbf {p}_2$ does not have better matches in the first image.