The goals of a co-variant detector were discussed in Covariant detectors fundamentals. This page introduces a few general principles that are at the basis of most covariant detection algorithms. Consider an input image $$ and a two dimensional continuous and invertible warp $w$. The *warped image* $w[]$ is defined to be
\[ w[] = w^{-1}, \]
or, equivalently,
\[ w[](x,y) = (w^{-1}(x,y)), (x,y)^2. \]
Note that, while $w$ pushes pixels forward, from the original to the transformed image domain, defining the transformed image $'$ requires inverting the warp and composing $$ with $w^{-1}$.
The goal a covariant detector is to extract the same local features irregardless of image transformations. The detector is said to be covariant or equivariant with a class of warps $w{W}$ if, when the feature $R$ is detected in image $$, then the transformed feature $w[R]$ is detected in the transformed image $w[]$.
The net effect is that a covariant feature detector appears to “track” image transformations; however, it is important to note that a detector *is not a tracker* because it processes images individually rather than jointly as part of a sequence.
An intuitive way to construct a covariant feature detector is to extract features in correspondence of images structures that are easily identifiable even after a transformation. Example of specific structures include dots, corners, and blobs. These will be generically indicated as **corners** in the followup.
A covariant detector faces two challenges. First, corners have, in practice, an infinite variety of individual appearances and the detector must be able to capture them to be of general applicability. Second, the way corners are identified and detected must remain stable under transformations of the image. These two problems are addressed in Local maxima of a cornerness measure and Covariant detection by normalization respectively.
One way to decide whether an image region $R$ contains a corner is to compare the local appearance to a model or template of the corner; the result of this comparisons produces a *cornerness score* at that location. This page describe general theoretical properties of the cornerness and the detection process. Concrete examples of cornerness are given in Cornerness measures.
A **cornerness measure** associate a score to all possible feature locations in an image $$. As described in Feature geometry and feature frames, the location or, more in general, pose $u$ of a feature $R$ is the warp $w$ that maps the canonical feature frame $R_0$ to $R$:
\[ R = u[R_0]. \]
The goal of a cornerness measure is to associate a score $F(u;)$ to all possible feature poses $u$ and use this score to extract a finite number of co-variant features from any image.
Given the cornerness of each candidate feature, the detector must extract a finite number of them. However, the cornerness of features with nearly identical pose must be similar (otherwise the cornerness measure would be unstable). As such, simply thresholding $F(w;)$ would detect an infinite number of nearly identical features rather than a finite number.
The solution is to detect features in correspondence of the local maxima of the score measure:
\[ w_1,,w_n