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Introduction to Principal Component Analysis (PCA)

Prev Tutorial: Support Vector Machines for Non-Linearly Separable Data

Original author Theodore Tsesmelis
Compatibility OpenCV >= 3.0

Goal

In this tutorial you will learn how to:

  • Use the OpenCV class cv::PCA to calculate the orientation of an object.

What is PCA?

Principal Component Analysis (PCA) is a statistical procedure that extracts the most important features of a dataset.

Consider that you have a set of 2D points as it is shown in the figure above. Each dimension corresponds to a feature you are interested in. Here some could argue that the points are set in a random order. However, if you have a better look you will see that there is a linear pattern (indicated by the blue line) which is hard to dismiss. A key point of PCA is the Dimensionality Reduction. Dimensionality Reduction is the process of reducing the number of the dimensions of the given dataset. For example, in the above case it is possible to approximate the set of points to a single line and therefore, reduce the dimensionality of the given points from 2D to 1D.

Moreover, you could also see that the points vary the most along the blue line, more than they vary along the Feature 1 or Feature 2 axes. This means that if you know the position of a point along the blue line you have more information about the point than if you only knew where it was on Feature 1 axis or Feature 2 axis.

Hence, PCA allows us to find the direction along which our data varies the most. In fact, the result of running PCA on the set of points in the diagram consist of 2 vectors called eigenvectors which are the principal components of the data set.

The size of each eigenvector is encoded in the corresponding eigenvalue and indicates how much the data vary along the principal component. The beginning of the eigenvectors is the center of all points in the data set. Applying PCA to N-dimensional data set yields N N-dimensional eigenvectors, N eigenvalues and 1 N-dimensional center point. Enough theory, let’s see how we can put these ideas into code.

How are the eigenvectors and eigenvalues computed?

The goal is to transform a given data set X of dimension p to an alternative data set Y of smaller dimension L. Equivalently, we are seeking to find the matrix Y, where Y is the Karhunen–Loève transform (KLT) of matrix X:

Y=KLT{X}

Organize the data set

Suppose you have data comprising a set of observations of p variables, and you want to reduce the data so that each observation can be described with only L variables, L < p. Suppose further, that the data are arranged as a set of n data vectors x1...xn with each xi representing a single grouped observation of the p variables.

  • Write x1...xn as row vectors, each of which has p columns.
  • Place the row vectors into a single matrix X of dimensions n×p.

Calculate the empirical mean

  • Find the empirical mean along each dimension j=1,...,p.
  • Place the calculated mean values into an empirical mean vector u of dimensions p×1.

    u[j]=1ni=1nX[i,j]

Calculate the deviations from the mean

Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data. Hence, we proceed by centering the data as follows:

  • Subtract the empirical mean vector u from each row of the data matrix X.
  • Store mean-subtracted data in the n×p matrix B.

    B=XhuT

    where h is an n×1 column vector of all 1s:

    h[i]=1,i=1,...,n

Find the covariance matrix

  • Find the p×p empirical covariance matrix C from the outer product of matrix B with itself:

    C=1n1BB

    where * is the conjugate transpose operator. Note that if B consists entirely of real numbers, which is the case in many applications, the "conjugate transpose" is the same as the regular transpose.

Find the eigenvectors and eigenvalues of the covariance matrix

  • Compute the matrix V of eigenvectors which diagonalizes the covariance matrix C:

    V1CV=D

    where D is the diagonal matrix of eigenvalues of C.

  • Matrix D will take the form of an p×p diagonal matrix:

    D[k,l]={λk,k=l0,kl

    here, λj is the j-th eigenvalue of the covariance matrix C

  • Matrix V, also of dimension p x p, contains p column vectors, each of length p, which represent the p eigenvectors of the covariance matrix C.
  • The eigenvalues and eigenvectors are ordered and paired. The j th eigenvalue corresponds to the j th eigenvector.
Note
sources [1], [2] and special thanks to Svetlin Penkov for the original tutorial.

Source Code

  • Downloadable code: Click here
  • Code at glance:
    #include "opencv2/core.hpp"
    #include <iostream>
    using namespace std;
    using namespace cv;
    // Function declarations
    void drawAxis(Mat&, Point, Point, Scalar, const float);
    double getOrientation(const vector<Point> &, Mat&);
    void drawAxis(Mat& img, Point p, Point q, Scalar colour, const float scale = 0.2)
    {
    double angle = atan2( (double) p.y - q.y, (double) p.x - q.x ); // angle in radians
    double hypotenuse = sqrt( (double) (p.y - q.y) * (p.y - q.y) + (p.x - q.x) * (p.x - q.x));
    // Here we lengthen the arrow by a factor of scale
    q.x = (int) (p.x - scale * hypotenuse * cos(angle));
    q.y = (int) (p.y - scale * hypotenuse * sin(angle));
    line(img, p, q, colour, 1, LINE_AA);
    // create the arrow hooks
    p.x = (int) (q.x + 9 * cos(angle + CV_PI / 4));
    p.y = (int) (q.y + 9 * sin(angle + CV_PI / 4));
    line(img, p, q, colour, 1, LINE_AA);
    p.x = (int) (q.x + 9 * cos(angle - CV_PI / 4));
    p.y = (int) (q.y + 9 * sin(angle - CV_PI / 4));
    line(img, p, q, colour, 1, LINE_AA);
    }
    double getOrientation(const vector<Point> &pts, Mat &img)
    {
    //Construct a buffer used by the pca analysis
    int sz = static_cast<int>(pts.size());
    Mat data_pts = Mat(sz, 2, CV_64F);
    for (int i = 0; i < data_pts.rows; i++)
    {
    data_pts.at<double>(i, 0) = pts[i].x;
    data_pts.at<double>(i, 1) = pts[i].y;
    }
    //Perform PCA analysis
    PCA pca_analysis(data_pts, Mat(), PCA::DATA_AS_ROW);
    //Store the center of the object
    Point cntr = Point(static_cast<int>(pca_analysis.mean.at<double>(0, 0)),
    static_cast<int>(pca_analysis.mean.at<double>(0, 1)));
    //Store the eigenvalues and eigenvectors
    vector<Point2d> eigen_vecs(2);
    vector<double> eigen_val(2);
    for (int i = 0; i < 2; i++)
    {
    eigen_vecs[i] = Point2d(pca_analysis.eigenvectors.at<double>(i, 0),
    pca_analysis.eigenvectors.at<double>(i, 1));
    eigen_val[i] = pca_analysis.eigenvalues.at<double>(i);
    }
    // Draw the principal components
    circle(img, cntr, 3, Scalar(255, 0, 255), 2);
    Point p1 = cntr + 0.02 * Point(static_cast<int>(eigen_vecs[0].x * eigen_val[0]), static_cast<int>(eigen_vecs[0].y * eigen_val[0]));
    Point p2 = cntr - 0.02 * Point(static_cast<int>(eigen_vecs[1].x * eigen_val[1]), static_cast<int>(eigen_vecs[1].y * eigen_val[1]));
    drawAxis(img, cntr, p1, Scalar(0, 255, 0), 1);
    drawAxis(img, cntr, p2, Scalar(255, 255, 0), 5);
    double angle = atan2(eigen_vecs[0].y, eigen_vecs[0].x); // orientation in radians
    return angle;
    }
    int main(int argc, char** argv)
    {
    // Load image
    CommandLineParser parser(argc, argv, "{@input | pca_test1.jpg | input image}");
    parser.about( "This program demonstrates how to use OpenCV PCA to extract the orientation of an object.\n" );
    parser.printMessage();
    Mat src = imread( samples::findFile( parser.get<String>("@input") ) );
    // Check if image is loaded successfully
    if(src.empty())
    {
    cout << "Problem loading image!!!" << endl;
    return EXIT_FAILURE;
    }
    imshow("src", src);
    // Convert image to grayscale
    Mat gray;
    cvtColor(src, gray, COLOR_BGR2GRAY);
    // Convert image to binary
    Mat bw;
    threshold(gray, bw, 50, 255, THRESH_BINARY | THRESH_OTSU);
    // Find all the contours in the thresholded image
    vector<vector<Point> > contours;
    findContours(bw, contours, RETR_LIST, CHAIN_APPROX_NONE);
    for (size_t i = 0; i < contours.size(); i++)
    {
    // Calculate the area of each contour
    double area = contourArea(contours[i]);
    // Ignore contours that are too small or too large
    if (area < 1e2 || 1e5 < area) continue;
    // Draw each contour only for visualisation purposes
    drawContours(src, contours, static_cast<int>(i), Scalar(0, 0, 255), 2);
    // Find the orientation of each shape
    getOrientation(contours[i], src);
    }
    imshow("output", src);
    return EXIT_SUCCESS;
    }
    Designed for command line parsing.
    Definition utility.hpp:890
    n-dimensional dense array class
    Definition mat.hpp:829
    _Tp & at(int i0=0)
    Returns a reference to the specified array element.
    bool empty() const
    Returns true if the array has no elements.
    int rows
    the number of rows and columns or (-1, -1) when the matrix has more than 2 dimensions
    Definition mat.hpp:2155
    Principal Component Analysis.
    Definition core.hpp:2498
    _Tp y
    y coordinate of the point
    Definition types.hpp:202
    _Tp x
    x coordinate of the point
    Definition types.hpp:201
    std::string String
    Definition cvstd.hpp:151
    #define CV_64F
    Definition interface.h:79
    #define CV_PI
    Definition cvdef.h:380
    void imshow(const String &winname, InputArray mat)
    Displays an image in the specified window.
    int waitKey(int delay=0)
    Waits for a pressed key.
    CV_EXPORTS_W Mat imread(const String &filename, int flags=IMREAD_COLOR_BGR)
    Loads an image from a file.
    void cvtColor(InputArray src, OutputArray dst, int code, int dstCn=0, AlgorithmHint hint=cv::ALGO_HINT_DEFAULT)
    Converts an image from one color space to another.
    void drawContours(InputOutputArray image, InputArrayOfArrays contours, int contourIdx, const Scalar &color, int thickness=1, int lineType=LINE_8, InputArray hierarchy=noArray(), int maxLevel=INT_MAX, Point offset=Point())
    Draws contours outlines or filled contours.
    double threshold(InputArray src, OutputArray dst, double thresh, double maxval, int type)
    Applies a fixed-level threshold to each array element.
    double contourArea(InputArray contour, bool oriented=false)
    Calculates a contour area.
    void findContours(InputArray image, OutputArrayOfArrays contours, OutputArray hierarchy, int mode, int method, Point offset=Point())
    Finds contours in a binary image.
    int main(int argc, char *argv[])
    Definition highgui_qt.cpp:3
    Definition core.hpp:107
    STL namespace.
Note
Another example using PCA for dimensionality reduction while maintaining an amount of variance can be found at opencv_source_code/samples/cpp/pca.cpp

Explanation

  • Read image and convert it to binary

Here we apply the necessary pre-processing procedures in order to be able to detect the objects of interest.

// Load image
CommandLineParser parser(argc, argv, "{@input | pca_test1.jpg | input image}");
parser.about( "This program demonstrates how to use OpenCV PCA to extract the orientation of an object.\n" );
parser.printMessage();
Mat src = imread( samples::findFile( parser.get<String>("@input") ) );
// Check if image is loaded successfully
if(src.empty())
{
cout << "Problem loading image!!!" << endl;
return EXIT_FAILURE;
}
imshow("src", src);
// Convert image to grayscale
Mat gray;
cvtColor(src, gray, COLOR_BGR2GRAY);
// Convert image to binary
Mat bw;
threshold(gray, bw, 50, 255, THRESH_BINARY | THRESH_OTSU);
  • Extract objects of interest

Then find and filter contours by size and obtain the orientation of the remaining ones.

// Find all the contours in the thresholded image
vector<vector<Point> > contours;
findContours(bw, contours, RETR_LIST, CHAIN_APPROX_NONE);
for (size_t i = 0; i < contours.size(); i++)
{
// Calculate the area of each contour
double area = contourArea(contours[i]);
// Ignore contours that are too small or too large
if (area < 1e2 || 1e5 < area) continue;
// Draw each contour only for visualisation purposes
drawContours(src, contours, static_cast<int>(i), Scalar(0, 0, 255), 2);
// Find the orientation of each shape
getOrientation(contours[i], src);
}
  • Extract orientation

Orientation is extracted by the call of getOrientation() function, which performs all the PCA procedure.

//Construct a buffer used by the pca analysis
int sz = static_cast<int>(pts.size());
Mat data_pts = Mat(sz, 2, CV_64F);
for (int i = 0; i < data_pts.rows; i++)
{
data_pts.at<double>(i, 0) = pts[i].x;
data_pts.at<double>(i, 1) = pts[i].y;
}
//Perform PCA analysis
PCA pca_analysis(data_pts, Mat(), PCA::DATA_AS_ROW);
//Store the center of the object
Point cntr = Point(static_cast<int>(pca_analysis.mean.at<double>(0, 0)),
static_cast<int>(pca_analysis.mean.at<double>(0, 1)));
//Store the eigenvalues and eigenvectors
vector<Point2d> eigen_vecs(2);
vector<double> eigen_val(2);
for (int i = 0; i < 2; i++)
{
eigen_vecs[i] = Point2d(pca_analysis.eigenvectors.at<double>(i, 0),
pca_analysis.eigenvectors.at<double>(i, 1));
eigen_val[i] = pca_analysis.eigenvalues.at<double>(i);
}

First the data need to be arranged in a matrix with size n x 2, where n is the number of data points we have. Then we can perform that PCA analysis. The calculated mean (i.e. center of mass) is stored in the cntr variable and the eigenvectors and eigenvalues are stored in the corresponding std::vector’s.

  • Visualize result

The final result is visualized through the drawAxis() function, where the principal components are drawn in lines, and each eigenvector is multiplied by its eigenvalue and translated to the mean position.

// Draw the principal components
circle(img, cntr, 3, Scalar(255, 0, 255), 2);
Point p1 = cntr + 0.02 * Point(static_cast<int>(eigen_vecs[0].x * eigen_val[0]), static_cast<int>(eigen_vecs[0].y * eigen_val[0]));
Point p2 = cntr - 0.02 * Point(static_cast<int>(eigen_vecs[1].x * eigen_val[1]), static_cast<int>(eigen_vecs[1].y * eigen_val[1]));
drawAxis(img, cntr, p1, Scalar(0, 255, 0), 1);
drawAxis(img, cntr, p2, Scalar(255, 255, 0), 5);
double angle = atan2(eigen_vecs[0].y, eigen_vecs[0].x); // orientation in radians
double angle = atan2( (double) p.y - q.y, (double) p.x - q.x ); // angle in radians
double hypotenuse = sqrt( (double) (p.y - q.y) * (p.y - q.y) + (p.x - q.x) * (p.x - q.x));
// Here we lengthen the arrow by a factor of scale
q.x = (int) (p.x - scale * hypotenuse * cos(angle));
q.y = (int) (p.y - scale * hypotenuse * sin(angle));
line(img, p, q, colour, 1, LINE_AA);
// create the arrow hooks
p.x = (int) (q.x + 9 * cos(angle + CV_PI / 4));
p.y = (int) (q.y + 9 * sin(angle + CV_PI / 4));
line(img, p, q, colour, 1, LINE_AA);
p.x = (int) (q.x + 9 * cos(angle - CV_PI / 4));
p.y = (int) (q.y + 9 * sin(angle - CV_PI / 4));
line(img, p, q, colour, 1, LINE_AA);

Results

The code opens an image, finds the orientation of the detected objects of interest and then visualizes the result by drawing the contours of the detected objects of interest, the center point, and the x-axis, y-axis regarding the extracted orientation.