Descriptor Comparison and Association

As a conclusion to this chapter, it is essential to say a few words about the topic of comparison.

Let $I_1$ and $I_2$ be two images to be analyzed, and let $\mathbf {p}_1$ and $\mathbf {p}_2$ be two characteristic points identified in the first and second images, respectively. To determine whether these two image points represent the same point, typically observed from different viewpoints and thus affected by affine transformations (translations, scaling changes, rotations), homographies, and possibly changes in brightness, it is necessary to define some form of metric $d(\mathbf{p}_1,\mathbf{p}_2)$ to perform this comparison.

Associated with each descriptor, a specific metric can be defined. In general, the most commonly used metrics are L1 (Manhattan, SAD) and L2 (Euclidean, SSD).

Since the points extracted from the two images will certainly be more than one, a scanning process must be performed, and each point from the first image will be associated only with that point from the second image which has the minimum distance according to the selected metric:

$$\hat{\mathbf{p}_{2} } = \argmin_i d(\mathbf{p}_1, \mathbf{p}_{2,i} )$$ (7.1)

Typically, to reduce the number of incorrect matches, the association is confirmed only if the metric is below a specified threshold and the ratio between the best match and the second-best match is below a second uniqueness threshold.

Finally, after finding $\mathbf {p}_2$, the best association of point $\mathbf {p}_1$ on the second image, it can be verified that $\mathbf {p}_2$ does not have better associations on the first image.