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:

![](images/Harris_Detector_Original_Image.jpg)

The detected corners are surrounded by a small black circle

![](images/Harris_Detector_Result.jpg)
