Harris corner detector

Goal

In this tutorial you will learn:

  • What features are and why they are important
  • Use the function 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

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    . We are going to sweep a window

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    (with displacements

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    in the x direction and

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    in the right direction)

    System Message: WARNING/2 (I)

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

    and will calculate the variation of intensity.

    System Message: WARNING/2 (E(u,v) = \sum _{x,y} w(x,y)[ I(x+u,y+v) - I(x,y)]^{2})

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    where:

    • System Message: WARNING/2 (w(x,y))

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      is the window at position

      System Message: WARNING/2 ((x,y))

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    • System Message: WARNING/2 (I(x,y))

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      is the intensity at

      System Message: WARNING/2 ((x,y))

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    • System Message: WARNING/2 (I(x+u,y+v))

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      is the intensity at the moved window

      System Message: WARNING/2 ((x+u,y+v))

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  • 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:

    System Message: WARNING/2 (\sum _{x,y}[ I(x+u,y+v) - I(x,y)]^{2})

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

  • Using Taylor expansion:

    System Message: WARNING/2 (E(u,v) \approx \sum _{x,y}[ I(x,y) + u I_{x} + vI_{y} - I(x,y)]^{2})

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

  • Expanding the equation and cancelling properly:

    System Message: WARNING/2 (E(u,v) \approx \sum _{x,y} u^{2}I_{x}^{2} + 2uvI_{x}I_{y} + v^{2}I_{y}^{2})

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

  • Which can be expressed in a matrix form as:

    System Message: WARNING/2 (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})

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

  • Let’s denote:

    System Message: WARNING/2 (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})

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

  • So, our equation now is:

    System Message: WARNING/2 (E(u,v) \approx \begin{bmatrix} u & v \end{bmatrix} M \begin{bmatrix} u \\ v \end{bmatrix})

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

    System Message: WARNING/2 (R = det(M) - k(trace(M))^{2})

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

    where:

    • det(M) =

      System Message: WARNING/2 (\lambda_{1}\lambda_{2})

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    • trace(M) =

      System Message: WARNING/2 (\lambda_{1}+\lambda_{2})

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    a window with a score

    System Message: WARNING/2 (R)

    latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014-OpenBSD_Ports) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. (/usr/local/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/share/texmf-dist/tex/latex/base/size12.clo)) (/usr/local/share/texmf-dist/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.165 \endinput ^^M No pages of output. Transcript written on math.log.

    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

#include "opencv2/highgui/highgui.hpp"
#include "opencv2/imgproc/imgproc.hpp"
#include <iostream>
#include <stdio.h>
#include <stdlib.h>

using namespace cv;
using namespace std;

/// Global variables
Mat src, src_gray;
int thresh = 200;
int max_thresh = 255;

char* source_window = "Source image";
char* corners_window = "Corners detected";

/// Function header
void cornerHarris_demo( int, void* );

/** @function main */
int main( int argc, char** argv )
{
  /// Load source image and convert it to gray
  src = imread( argv[1], 1 );
  cvtColor( src, src_gray, CV_BGR2GRAY );

  /// Create a window and a trackbar
  namedWindow( source_window, CV_WINDOW_AUTOSIZE );
  createTrackbar( "Threshold: ", source_window, &thresh, max_thresh, cornerHarris_demo );
  imshow( source_window, src );

  cornerHarris_demo( 0, 0 );

  waitKey(0);
  return(0);
}

/** @function cornerHarris_demo */
void cornerHarris_demo( int, void* )
{

  Mat dst, dst_norm, dst_norm_scaled;
  dst = Mat::zeros( src.size(), CV_32FC1 );

  /// Detector parameters
  int blockSize = 2;
  int apertureSize = 3;
  double k = 0.04;

  /// Detecting corners
  cornerHarris( src_gray, dst, blockSize, apertureSize, k, BORDER_DEFAULT );

  /// Normalizing
  normalize( dst, dst_norm, 0, 255, NORM_MINMAX, CV_32FC1, Mat() );
  convertScaleAbs( dst_norm, dst_norm_scaled );

  /// Drawing a circle around corners
  for( int j = 0; j < dst_norm.rows ; j++ )
     { for( int i = 0; i < dst_norm.cols; i++ )
          {
            if( (int) dst_norm.at<float>(j,i) > thresh )
              {
               circle( dst_norm_scaled, Point( i, j ), 5,  Scalar(0), 2, 8, 0 );
              }
          }
     }
  /// Showing the result
  namedWindow( corners_window, CV_WINDOW_AUTOSIZE );
  imshow( corners_window, dst_norm_scaled );
}

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