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Harris corner detector {#tutorial_harris_detector}
======================
Goal
----
In this tutorial you will learn:
- What features are and why they are important
- Use the function @ref 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 \f$I\f$. We are going to sweep a window \f$w(x,y)\f$ (with displacements \f$u\f$
in the x direction and \f$v\f$ in the right direction) \f$I\f$ and will calculate the variation of
intensity.
\f[E(u,v) = \sum _{x,y} w(x,y)[ I(x+u,y+v) - I(x,y)]^{2}\f]
where:
- \f$w(x,y)\f$ is the window at position \f$(x,y)\f$
- \f$I(x,y)\f$ is the intensity at \f$(x,y)\f$
- \f$I(x+u,y+v)\f$ is the intensity at the moved window \f$(x+u,y+v)\f$
- 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:
\f[\sum _{x,y}[ I(x+u,y+v) - I(x,y)]^{2}\f]
- Using *Taylor expansion*:
\f[E(u,v) \approx \sum _{x,y}[ I(x,y) + u I_{x} + vI_{y} - I(x,y)]^{2}\f]
- Expanding the equation and cancelling properly:
\f[E(u,v) \approx \sum _{x,y} u^{2}I_{x}^{2} + 2uvI_{x}I_{y} + v^{2}I_{y}^{2}\f]
- Which can be expressed in a matrix form as:
\f[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}\f]
- Let's denote:
\f[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}\f]
- So, our equation now is:
\f[E(u,v) \approx \begin{bmatrix}
u & v
\end{bmatrix}
M
\begin{bmatrix}
u \\
v
\end{bmatrix}\f]
- A score is calculated for each window, to determine if it can possibly contain a corner:
\f[R = det(M) - k(trace(M))^{2}\f]
where:
- det(M) = \f$\lambda_{1}\lambda_{2}\f$
- trace(M) = \f$\lambda_{1}+\lambda_{2}\f$
a window with a score \f$R\f$ greater than a certain value is considered a "corner"
Code
----
This tutorial code's is shown lines below. You can also download it from
[here](https://github.com/opencv/opencv/tree/master/samples/cpp/tutorial_code/TrackingMotion/cornerHarris_Demo.cpp)
@include samples/cpp/tutorial_code/TrackingMotion/cornerHarris_Demo.cpp
Explanation
-----------
Result
------
The original image:
The detected corners are surrounded by a small black circle
|
