Harris corner detector

A corner is a point whose local neighborhood stands in two dominant and different edge directions. In other words, a corner can be interpreted as the junction of two edges, where an edge is a sudden change in image brightness.[3] Corners are the important features in the image, and they are generally termed as interest points which are invariant to translation, rotation and illumination. Although corners are only a small percentage of the image, they contain the most important features in restoring image information, and they can be used to minimize the amount of processed data for motion tracking, image stitching, building 2D mosaics, stereo vision, image representation and other related computer vision areas.

In order to capture the corners from the image, researchers have proposed many different corner detectors including the Kanade-Lucas-Tomasi (KLT) operator and the Harris operator which are most simple, efficient and reliable for use in corner detection. These two popular methodologies are both closely associated with and based on the local structure matrix. Compared to the Kanade-Lucas-Tomasi corner detector, the Harris corner detector provides good repeatability under changing illumination and rotation, and therefore, it is more often used in stereo matching and image database retrieval. Although there still exists drawbacks and limitations, the Harris corner detector is still an important and fundamental technique for many computer vision applications.

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 ${\displaystyle (x,y)\in W}$(window) and shifting it by ${\displaystyle (\Delta x,\Delta y)}$. The sum of squared differences (SSD) between these two patches, denoted ${\displaystyle f}$, is given by:

${\displaystyle f(\Delta x,\Delta y)={\underset {(x_{k},y_{k})\in W}{\sum }}\left(I(x_{k},y_{k})-I(x_{k}+\Delta x,y_{k}+\Delta y)\right)^{2}}$

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

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

This produces the approximation

${\displaystyle f(\Delta x,\Delta y)\approx {\underset {(x,y)\in W}{\sum }}\left(I_{x}(x,y)\Delta x+I_{y}(x,y)\Delta y\right)^{2},}$

which can be written in matrix form:

${\displaystyle f(\Delta x,\Delta y)\approx {\begin{pmatrix}\Delta x&\Delta y\end{pmatrix}}M{\begin{pmatrix}\Delta x\\\Delta y\end{pmatrix}},}$

where M is the structure tensor,

${\displaystyle M={\underset {(x,y)\in W}{\sum }}{\begin{bmatrix}I_{x}^{2}&I_{x}I_{y}\\I_{x}I_{y}&I_{y}^{2}\end{bmatrix}}={\begin{bmatrix}{\underset {(x,y)\in W}{\sum }}I_{x}^{2}&{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}\\{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}&{\underset {(x,y)\in W}{\sum }}I_{y}^{2}\end{bmatrix}}}$

Commonly, Harris corner detector algorithm can be divided into five steps.

1. Color to grayscale
2. Spatial derivative calculation
3. Structure tensor setup
4. Harris response calculation
5. Non-maximum suppression

1. Harris-Laplace Corner Detector[9]
2. Differential Morphological Decomposition Based Corner Detector[10]
3. Multi-scale Bilateral Structure Tensor Based Corner Detector[11]

1. Image Alignment, Stitching and Registration[12]
2. 2D Mosaics Creation[13]
3. 3D Scene Modeling and Reconstruction[14]
4. Motion Detection[15]
5. Object Recognition[16]
6. Image Indexing and Content-based Retrieval[17]
7. Video Tracking[18]

• Structure tensor
• Harris affine region detector
• Corner detection
• Feature detection (computer vision)
• Computer vision
• List of computer vision topics

• "Learn OpenCV by Examples : Harris Corner Detection"
• "Harris Corner Detection - OpenCV Documentation"
• "Harris Corner Detection - OpenCV-Python Tutorials". Archived from the original on 2021-05-07.
• Online Implementation of the Harris Corner Detector - IPOL