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doc/tutorials/features2d/trackingmotion/harris_detector/harris_detector.markdown

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+Harris corner detector {#tutorial_harris_detector}
+======================
+
+@tableofcontents
+
+@next_tutorial{tutorial_good_features_to_track}
+
+|    |    |
+| -: | :- |
+| 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 @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 y 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
+----
+
+@add_toggle_cpp
+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
+@end_toggle
+
+@add_toggle_java
+This tutorial code's is shown lines below. You can also download it from
+[here](https://github.com/opencv/opencv/tree/master/samples/java/tutorial_code/TrackingMotion/harris_detector/CornerHarrisDemo.java)
+@include samples/java/tutorial_code/TrackingMotion/harris_detector/CornerHarrisDemo.java
+@end_toggle
+
+@add_toggle_python
+This tutorial code's is shown lines below. You can also download it from
+[here](https://github.com/opencv/opencv/tree/master/samples/python/tutorial_code/TrackingMotion/harris_detector/cornerHarris_Demo.py)
+@include samples/python/tutorial_code/TrackingMotion/harris_detector/cornerHarris_Demo.py
+@end_toggle
+
+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)