OpenCV  4.7.0-dev Open Source Computer Vision
Harris corner detector

Next Tutorial: Shi-Tomasi corner detector

Original author Ana Huamán
Compatibility OpenCV >= 3.0

Goal

In this tutorial you will learn:

• What features are and why they are important
• Use the function cv::cornerHarris to detect corners using the Harris-Stephens method.

Theory

What is a feature?

• In computer vision, usually we need to find matching points between different frames of an environment. Why? If we know how two images relate to each other, we can use both images to extract information of them.
• When we say matching points we are referring, in a general sense, to characteristics in the scene that we can recognize easily. We call these characteristics features.
• So, what characteristics should a feature have?
• It must be uniquely recognizable

Types of Image Features

To mention a few:

• Edges
• Corners (also known as interest points)
• Blobs (also known as regions of interest )

In this tutorial we will study the corner features, specifically.

Why is a corner so special?

• Because, since it is the intersection of two edges, it represents a point in which the directions of these two edges change. Hence, the gradient of the image (in both directions) have a high variation, which can be used to detect it.

How does it work?

• Let's look for corners. Since corners represents a variation in the gradient in the image, we will look for this "variation".
• Consider a grayscale image $$I$$. We are going to sweep a window $$w(x,y)$$ (with displacements $$u$$ in the x direction and $$v$$ in the y direction) $$I$$ and will calculate the variation of intensity.

$E(u,v) = \sum _{x,y} w(x,y)[ I(x+u,y+v) - I(x,y)]^{2}$

where:

• $$w(x,y)$$ is the window at position $$(x,y)$$
• $$I(x,y)$$ is the intensity at $$(x,y)$$
• $$I(x+u,y+v)$$ is the intensity at the moved window $$(x+u,y+v)$$
• Since we are looking for windows with corners, we are looking for windows with a large variation in intensity. Hence, we have to maximize the equation above, specifically the term:

$\sum _{x,y}[ I(x+u,y+v) - I(x,y)]^{2}$

• Using Taylor expansion:

$E(u,v) \approx \sum _{x,y}[ I(x,y) + u I_{x} + vI_{y} - I(x,y)]^{2}$

• Expanding the equation and cancelling properly:

$E(u,v) \approx \sum _{x,y} u^{2}I_{x}^{2} + 2uvI_{x}I_{y} + v^{2}I_{y}^{2}$

• Which can be expressed in a matrix form as:

$E(u,v) \approx \begin{bmatrix} u & v \end{bmatrix} \left ( \displaystyle \sum_{x,y} w(x,y) \begin{bmatrix} I_x^{2} & I_{x}I_{y} \\ I_xI_{y} & I_{y}^{2} \end{bmatrix} \right ) \begin{bmatrix} u \\ v \end{bmatrix}$

• Let's denote:

$M = \displaystyle \sum_{x,y} w(x,y) \begin{bmatrix} I_x^{2} & I_{x}I_{y} \\ I_xI_{y} & I_{y}^{2} \end{bmatrix}$

• So, our equation now is:

$E(u,v) \approx \begin{bmatrix} u & v \end{bmatrix} M \begin{bmatrix} u \\ v \end{bmatrix}$

• A score is calculated for each window, to determine if it can possibly contain a corner:

$R = det(M) - k(trace(M))^{2}$

where:

• det(M) = $$\lambda_{1}\lambda_{2}$$
• trace(M) = $$\lambda_{1}+\lambda_{2}$$

a window with a score $$R$$ greater than a certain value is considered a "corner"

Result

The original image:

The detected corners are surrounded by a small black circle