Corner detection is an approach used within computer vision systems to extract certain kinds of features and infer the contents of an image. Corner detection is frequently used in motion detection, image registration, video tracking, image mosaicing, panorama stitching, 3D reconstruction and object recognition. Corner detection overlaps with the topic of interest point detection.

Formalization

A corner can be defined as the intersection of two edges. A corner can also be defined as a point for which there are two dominant and different edge directions in a local neighbourhood of the point.

An interest point is a point in an image which has a well-defined position and can be robustly detected. This means that an interest point can be a corner but it can also be, for example, an isolated point of local intensity maximum or minimum, line endings, or a point on a curve where the curvature is locally maximal.

In practice, most so-called corner detection methods detect interest points in general, and in fact, the term "corner" and "interest point" are used more or less interchangeably through the literature.^[1] As a consequence, if only corners are to be detected it is necessary to do a local analysis of detected interest points to determine which of these are real corners. Examples of edge detection that can be used with post-processing to detect corners are the Kirsch operator and the Frei-Chen masking set.^[2]

"Corner", "interest point" and "feature" are used interchangeably in literature, confusing the issue. Specifically, there are several blob detectors that can be referred to as "interest point operators", but which are sometimes erroneously referred to as "corner detectors". Moreover, there exists a notion of ridge detection to capture the presence of elongated objects.

Corner detectors are not usually very robust and often require large redundancies introduced to prevent the effect of individual errors from dominating the recognition task.

One determination of the quality of a corner detector is its ability to detect the same corner in multiple similar images, under conditions of different lighting, translation, rotation and other transforms.

A simple approach to corner detection in images is using correlation, but this gets very computationally expensive and suboptimal. An alternative approach used frequently is based on a method proposed by Harris and Stephens (below), which in turn is an improvement of a method by Moravec.

Moravec corner detection algorithm

This is one of the earliest corner detection algorithms and defines a corner to be a point with low self-similarity.^[3] The algorithm tests each pixel in the image to see if a corner is present, by considering how similar a patch centered on the pixel is to nearby, largely overlapping patches. The similarity is measured by taking the sum of squared differences (SSD) between the corresponding pixels of two patches. A lower number indicates more similarity.

If the pixel is in a region of uniform intensity, then the nearby patches will look similar. If the pixel is on an edge, then nearby patches in a direction perpendicular to the edge will look quite different, but nearby patches in a direction parallel to the edge will result in only a small change. If the pixel is on a feature with variation in all directions, then none of the nearby patches will look similar.

The corner strength is defined as the smallest SSD between the patch and its neighbours (horizontal, vertical and on the two diagonals). The reason is that if this number is high, then the variation along all shifts is either equal to it or larger than it, so capturing that all nearby patches look different.

If the corner strength number is computed for all locations, that it is locally maximal for one location indicates that a feature of interest is present in it.

As pointed out by Moravec, one of the main problems with this operator is that it is not isotropic: if an edge is present that is not in the direction of the neighbours (horizontal, vertical, or diagonal), then the smallest SSD will be large and the edge will be incorrectly chosen as an interest point.^[4]

The Harris & Stephens / Shi–Tomasi corner detection algorithms

Harris and Stephens^[5] improved upon Moravec's corner detector by considering the differential of the corner score with respect to direction directly, instead of using shifted patches. (This corner score is often referred to as autocorrelation, since the term is used in the paper in which this detector is described. However, the mathematics in the paper clearly indicate that the sum of squared differences is used.)

Without loss of generality, we will assume a grayscale 2-dimensional image is used. Let this image be given by ${\displaystyle I}$. Consider taking an image patch over the area ${\displaystyle (u,v)}$ and shifting it by ${\displaystyle (x,y)}$. The weighted sum of squared differences (SSD) between these two patches, denoted ${\displaystyle S}$, is given by:

${\displaystyle S(x,y)=\sum _{u}\sum _{v}w(u,v)\,\left(I(u+x,v+y)-I(u,v)\right)^{2}}$

${\displaystyle I(u+x,v+y)}$ can be approximated by a Taylor expansion. Let ${\displaystyle I_{x}}$ and ${\displaystyle I_{y}}$ be the partial derivatives of ${\displaystyle I}$, such that

${\displaystyle I(u+x,v+y)\approx I(u,v)+I_{x}(u,v)x+I_{y}(u,v)y}$

This produces the approximation

${\displaystyle S(x,y)\approx \sum _{u}\sum _{v}w(u,v)\,\left(I_{x}(u,v)x+I_{y}(u,v)y\right)^{2},}$

which can be written in matrix form:

${\displaystyle S(x,y)\approx {\begin{bmatrix}x&y\end{bmatrix}}A{\begin{bmatrix}x\\y\end{bmatrix}},}$

where A is the structure tensor,

${\displaystyle A=\sum _{u}\sum _{v}w(u,v){\begin{bmatrix}I_{x}(u,v)^{2}&I_{x}(u,v)I_{y}(u,v)\\I_{x}(u,v)I_{y}(u,v)&I_{y}(u,v)^{2}\end{bmatrix}}={\begin{bmatrix}\langle I_{x}^{2}\rangle &\langle I_{x}I_{y}\rangle \\\langle I_{x}I_{y}\rangle &\langle I_{y}^{2}\rangle \end{bmatrix}}}$

In words, we find the covariance of the partial derivative of the image intensity ${\displaystyle I}$ with respect to the ${\displaystyle x}$ and ${\displaystyle y}$ axes.

Angle brackets denote averaging (i.e. summation over ${\displaystyle (u,v)}$). ${\displaystyle w(u,v)}$Zdroj:https://en.wikipedia.org?pojem=Corner_detection
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