This article by Robert Laganière, author of OpenCV Computer Vision Application Programming Cookbook Second Edition, includes that images are generally produced using a digital camera, which captures a scene by projecting light going through its lens onto an image sensor. The fact that an image is formed by the projection of a 3D scene onto a 2D plane implies the existence of important relationships between a scene and its image and between different images of the same scene. Projective geometry is the tool that is used to describe and characterize, in mathematical terms, the process of image formation. In this article, we will introduce you to some of the fundamental projective relations that exist in multiview imagery and explain how these can be used in computer vision programming. You will learn how matching can be made more accurate through the use of projective constraints and how a mosaic from multiple images can be composited using twoview relations. Before we start the recipe, let's explore the basic concepts related to scene projection and image formation.
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Image formation
Fundamentally, the process used to produce images has not changed since the beginning of photography. The light coming from an observed scene is captured by a camera through a frontal aperture; the captured light rays hit an image plane (or an image sensor) located at the back of the camera. Additionally, a lens is used to concentrate the rays coming from the different scene elements. This process is illustrated by the following figure:
Here, do is the distance from the lens to the observed object, di is the distance from the lens to the image plane, and f is the focal length of the lens. These quantities are related by the socalled thin lens equation:
In computer vision, this camera model can be simplified in a number of ways. First, we can neglect the effect of the lens by considering that we have a camera with an infinitesimal aperture since, in theory, this does not change the image appearance. (However, by doing so, we ignore the focusing effect by creating an image with an infinite depth of field.) In this case, therefore, only the central ray is considered. Second, since most of the time we have do>>di, we can assume that the image plane is located at the focal distance. Finally, we can note from the geometry of the system that the image on the plane is inverted. We can obtain an identical but upright image by simply positioning the image plane in front of the lens. Obviously, this is not physically feasible, but from a mathematical point of view, this is completely equivalent. This simplified model is often referred to as the pinhole camera model, and it is represented as follows:
From this model, and using the law of similar triangles, we can easily derive the basic projective equation that relates a pictured object with its image:
The size (hi) of the image of an object (of height ho) is therefore inversely proportional to its distance (do) from the camera, which is naturally true. In general, this relation describes where a 3D scene point will be projected on the image plane given the geometry of the camera.
Calibrating a camera
From the introduction of this article, we learned that the essential parameters of a camera under the pinhole model are its focal length and the size of the image plane (which defines the field of view of the camera). Also, since we are dealing with digital images, the number of pixels on the image plane (its resolution) is another important characteristic of a camera. Finally, in order to be able to compute the position of an image's scene point in pixel coordinates, we need one additional piece of information. Considering the line coming from the focal point that is orthogonal to the image plane, we need to know at which pixel position this line pierces the image plane. This point is called the principal point. It might be logical to assume that this principal point is at the center of the image plane, but in practice, this point might be off by a few pixels depending on the precision at which the camera has been manufactured.
Camera calibration is the process by which the different camera parameters are obtained. One can obviously use the specifications provided by the camera manufacturer, but for some tasks, such as 3D reconstruction, these specifications are not accurate enough. Camera calibration will proceed by showing known patterns to the camera and analyzing the obtained images. An optimization process will then determine the optimal parameter values that explain the observations. This is a complex process that has been made easy by the availability of OpenCV calibration functions.
How to do it...
To calibrate a camera, the idea is to show it a set of scene points for which their 3D positions are known. Then, you need to observe where these points project on the image. With the knowledge of a sufficient number of 3D points and associated 2D image points, the exact camera parameters can be inferred from the projective equation. Obviously, for accurate results, we need to observe as many points as possible. One way to achieve this would be to take one picture of a scene with many known 3D points, but in practice, this is rarely feasible. A more convenient way is to take several images of a set of some 3D points from different viewpoints. This approach is simpler but requires you to compute the position of each camera view in addition to the computation of the internal camera parameters, which fortunately is feasible.
OpenCV proposes that you use a chessboard pattern to generate the set of 3D scene points required for calibration. This pattern creates points at the corners of each square, and since this pattern is flat, we can freely assume that the board is located at Z=0, with the X and Y axes wellaligned with the grid. In this case, the calibration process simply consists of showing the chessboard pattern to the camera from different viewpoints. Here is one example of a 6x4 calibration pattern image:
The good thing is that OpenCV has a function that automatically detects the corners of this chessboard pattern. You simply provide an image and the size of the chessboard used (the number of horizontal and vertical inner corner points). The function will return the position of these chessboard corners on the image. If the function fails to find the pattern, then it simply returns false:
// output vectors of image points std::vector<cv::Point2f> imageCorners; // number of inner corners on the chessboard cv::Size boardSize(6,4); // Get the chessboard corners bool found = cv::findChessboardCorners(image, boardSize, imageCorners);
The output parameter, imageCorners, will simply contain the pixel coordinates of the detected inner corners of the shown pattern. Note that this function accepts additional parameters if you needs to tune the algorithm, which are not discussed here. There is also a special function that draws the detected corners on the chessboard image, with lines connecting them in a sequence:
//Draw the corners cv::drawChessboardCorners(image, boardSize, imageCorners, found); // corners have been found
The following image is obtained:
The lines that connect the points show the order in which the points are listed in the vector of detected image points. To perform a calibration, we now need to specify the corresponding 3D points. You can specify these points in the units of your choice (for example, in centimeters or in inches); however, the simplest is to assume that each square represents one unit. In that case, the coordinates of the first point would be (0,0,0) (assuming that the board is located at a depth of Z=0), the coordinates of the second point would be (1,0,0), and so on, the last point being located at (5,3,0). There are a total of 24 points in this pattern, which is too small to obtain an accurate calibration. To get more points, you need to show more images of the same calibration pattern from various points of view. To do so, you can either move the pattern in front of the camera or move the camera around the board; from a mathematical point of view, this is completely equivalent. The OpenCV calibration function assumes that the reference frame is fixed on the calibration pattern and will calculate the rotation and translation of the camera with respect to the reference frame.
Let's now encapsulate the calibration process in a CameraCalibrator class. The attributes of this class are as follows:
class CameraCalibrator { // input points: // the points in world coordinates std::vector<std::vector<cv::Point3f>> objectPoints; // the point positions in pixels std::vector<std::vector<cv::Point2f>> imagePoints; // output Matrices cv::Mat cameraMatrix; cv::Mat distCoeffs; // flag to specify how calibration is done int flag;
Note that the input vectors of the scene and image points are in fact made of std::vector of point instances; each vector element is a vector of the points from one view. Here, we decided to add the calibration points by specifying a vector of the chessboard image filename as input:
// Open chessboard images and extract corner points int CameraCalibrator::addChessboardPoints( const std::vector<std::string>& filelist, cv::Size & boardSize) { // the points on the chessboard std::vector<cv::Point2f> imageCorners; std::vector<cv::Point3f> objectCorners; // 3D Scene Points: // Initialize the chessboard corners // in the chessboard reference frame // The corners are at 3D location (X,Y,Z)= (i,j,0) for (int i=0; i<boardSize.height; i++) { for (int j=0; j<boardSize.width; j++) { objectCorners.push_back(cv::Point3f(i, j, 0.0f)); } } // 2D Image points: cv::Mat image; // to contain chessboard image int successes = 0; // for all viewpoints for (int i=0; i<filelist.size(); i++) { // Open the image image = cv::imread(filelist[i],0); // Get the chessboard corners bool found = cv::findChessboardCorners( image, boardSize, imageCorners); // Get subpixel accuracy on the corners cv::cornerSubPix(image, imageCorners, cv::Size(5,5), cv::Size(1,1), cv::TermCriteria(cv::TermCriteria::MAX_ITER + cv::TermCriteria::EPS, 30, // max number of iterations 0.1)); // min accuracy //If we have a good board, add it to our data if (imageCorners.size() == boardSize.area()) { // Add image and scene points from one view addPoints(imageCorners, objectCorners); successes++; } } return successes; }
The first loop inputs the 3D coordinates of the chessboard, and the corresponding image points are the ones provided by the cv::findChessboardCorners function. This is done for all the available viewpoints. Moreover, in order to obtain a more accurate image point location, the cv::cornerSubPix function can be used, and as the name suggests, the image points will then be localized at a subpixel accuracy. The termination criterion that is specified by the cv::TermCriteria object defines the maximum number of iterations and the minimum accuracy in subpixel coordinates. The first of these two conditions that is reached will stop the corner refinement process.
When a set of chessboard corners have been successfully detected, these points are added to our vectors of the image and scene points using our addPoints method. Once a sufficient number of chessboard images have been processed (and consequently, a large number of 3D scene point / 2D image point correspondences are available), we can initiate the computation of the calibration parameters as follows:
// Calibrate the camera // returns the reprojection error double CameraCalibrator::calibrate(cv::Size &imageSize) { //Output rotations and translations std::vector<cv::Mat> rvecs, tvecs; // start calibration return calibrateCamera(objectPoints, // the 3D points imagePoints, // the image points imageSize, // image size cameraMatrix, // output camera matrix distCoeffs, // output distortion matrix rvecs, tvecs, // Rs, Ts flag); // set options }
In practice, 10 to 20 chessboard images are sufficient, but these must be taken from different viewpoints at different depths. The two important outputs of this function are the camera matrix and the distortion parameters. These will be described in the next section.
How it works...
In order to explain the result of the calibration, we need to go back to the figure in the introduction, which describes the pinhole camera model. More specifically, we want to demonstrate the relationship between a point in 3D at the position (X,Y,Z) and its image (x,y) on a camera specified in pixel coordinates. Let's redraw this figure by adding a reference frame that we position at the center of the projection as seen here:
Note that the y axis is pointing downward to get a coordinate system compatible with the usual convention that places the image origin at the upperleft corner. We learned previously that the point (X,Y,Z) will be projected onto the image plane at (fX/Z,fY/Z). Now, if we want to translate this coordinate into pixels, we need to divide the 2D image position by the pixel's width (px) and height (py), respectively. Note that by dividing the focal length given in world units (generally given in millimeters) by px, we obtain the focal length expressed in (horizontal) pixels. Let's then define this term as fx. Similarly, fy =f/py is defined as the focal length expressed in vertical pixel units. Therefore, the complete projective equation is as follows:
Recall that (u0,v0) is the principal point that is added to the result in order to move the origin to the upperleft corner of the image. These equations can be rewritten in the matrix form through the introduction of homogeneous coordinates, in which 2D points are represented by 3vectors and 3D points are represented by 4vectors (the extra coordinate is simply an arbitrary scale factor, S, that needs to be removed when a 2D coordinate needs to be extracted from a homogeneous 3vector).
Here is the rewritten projective equation:
The second matrix is a simple projection matrix. The first matrix includes all of the camera parameters, which are called the intrinsic parameters of the camera. This 3x3 matrix is one of the output matrices returned by the cv::calibrateCamera function. There is also a function called cv::calibrationMatrixValues that returns the value of the intrinsic parameters given by a calibration matrix.
More generally, when the reference frame is not at the projection center of the camera, we will need to add a rotation vector (a 3x3 matrix) and a translation vector (a 3x1 matrix). These two matrices describe the rigid transformation that must be applied to the 3D points in order to bring them back to the camera reference frame. Therefore, we can rewrite the projection equation in its most general form:
Remember that in our calibration example, the reference frame was placed on the chessboard. Therefore, there is a rigid transformation (made of a rotation component represented by the matrix entries r1 to r9 and a translation represented by t1, t2, and t3) that must be computed for each view. These are in the output parameter list of the cv::calibrateCamera function. The rotation and translation components are often called the extrinsic parameters of the calibration, and they are different for each view. The intrinsic parameters remain constant for a given camera/lens system. The intrinsic parameters of our test camera obtained from a calibration based on 20 chessboard images are fx=167, fy=178, u0=156, and v0=119. These results are obtained by cv::calibrateCamera through an optimization process aimed at finding the intrinsic and extrinsic parameters that will minimize the difference between the predicted image point position, as computed from the projection of the 3D scene points, and the actual image point position, as observed on the image. The sum of this difference for all the points specified during the calibration is called the reprojection error.
Let's now turn our attention to the distortion parameters. So far, we have mentioned that under the pinhole camera model, we can neglect the effect of the lens. However, this is only possible if the lens that is used to capture an image does not introduce important optical distortions. Unfortunately, this is not the case with lower quality lenses or with lenses that have a very short focal length. You may have already noted that the chessboard pattern shown in the image that we used for our example is clearly distorted—the edges of the rectangular board are curved in the image. Also, note that this distortion becomes more important as we move away from the center of the image. This is a typical distortion observed with a fisheye lens, and it is called radial distortion. The lenses used in common digital cameras usually do not exhibit such a high degree of distortion, but in the case of the lens used here, these distortions certainly cannot be ignored.
It is possible to compensate for these deformations by introducing an appropriate distortion model. The idea is to represent the distortions induced by a lens by a set of mathematical equations. Once established, these equations can then be reverted in order to undo the distortions visible on the image. Fortunately, the exact parameters of the transformation that will correct the distortions can be obtained together with the other camera parameters during the calibration phase. Once this is done, any image from the newly calibrated camera will be undistorted. Therefore, we have added an additional method to our calibration class:
// remove distortion in an image (after calibration) cv::Mat CameraCalibrator::remap(const cv::Mat &image) { cv::Mat undistorted; if (mustInitUndistort) { // called once per calibration cv::initUndistortRectifyMap( cameraMatrix, // computed camera matrix distCoeffs, // computed distortion matrix cv::Mat(), // optional rectification (none) cv::Mat(), // camera matrix to generate undistorted image.size(), // size of undistorted CV_32FC1, // type of output map map1, map2); // the x and y mapping functions mustInitUndistort= false; } // Apply mapping functions cv::remap(image, undistorted, map1, map2, cv::INTER_LINEAR); // interpolation type return undistorted; }
Running this code results in the following image:
As you can see, once the image is undistorted, we obtain a regular perspective image.
To correct the distortion, OpenCV uses a polynomial function that is applied to the image points in order to move them at their undistorted position. By default, five coefficients are used; a model made of eight coefficients is also available. Once these coefficients are obtained, it is possible to compute two cv::Mat mapping functions (one for the x coordinate and one for the y coordinate) that will give the new undistorted position of an image point on a distorted image. This is computed by the cv::initUndistortRectifyMap function, and the cv::remap function remaps all the points of an input image to a new image. Note that because of the nonlinear transformation, some pixels of the input image now fall outside the boundary of the output image. You can expand the size of the output image to compensate for this loss of pixels, but you will now obtain output pixels that have no values in the input image (they will then be displayed as black pixels).
There's more...
More options are available when it comes to camera calibration.
Calibration with known intrinsic parameters
When a good estimate of the camera's intrinsic parameters is known, it could be advantageous to input them in the cv::calibrateCamera function. They will then be used as initial values in the optimization process. To do so, you just need to add the CV_CALIB_USE_INTRINSIC_GUESS flag and input these values in the calibration matrix parameter. It is also possible to impose a fixed value for the principal point (CV_CALIB_FIX_PRINCIPAL_POINT), which can often be assumed to be the central pixel. You can also impose a fixed ratio for the focal lengths fx and fy (CV_CALIB_FIX_RATIO); in which case, you assume the pixels of the square shape.
Using a grid of circles for calibration
Instead of the usual chessboard pattern, OpenCV also offers the possibility to calibrate a camera by using a grid of circles. In this case, the centers of the circles are used as calibration points. The corresponding function is very similar to the function we used to locate the chessboard corners:
cv::Size boardSize(7,7); std::vector<cv::Point2f> centers; bool found = cv:: findCirclesGrid( image, boardSize, centers);
See also

The A flexible new technique for camera calibration article by Z. Zhang in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no 11, 2000, is a classic paper on the problem of camera calibration
Summary
In this article, we explored the projective relations that exist between two images of the same scene.
Resources for Article:
Further resources on this subject:
 Creating an Application from Scratch [Article]
 Wrapping OpenCV [Article]
 New functionality in OpenCV 3.0 [Article]