# estimateCameraParameters

Calibrate a single or stereo camera

## Description

example

[cameraParams,imagesUsed,estimationErrors] = estimateCameraParameters(imagePoints,worldPoints) returns cameraParams, a cameraParameters object containing estimates for the intrinsic and extrinsic parameters and the distortion coefficients of a single camera. The function also returns the images you used to estimate the camera parameters and the standard estimation errors for the single camera calibration. The estimateCameraParameters function estimates extrinsics and intrinsics parameters.

example

[stereoParams,pairsUsed,estimationErrors] = estimateCameraParameters(imagePoints,worldPoints) returns stereoParams, a stereoParameters object containing the parameters of the stereo camera. The function also returns the images you used to estimate the stereo parameters and the standard estimation errors for the stereo camera calibration.

example

cameraParams = estimateCameraParameters(___,Name,Value) configures the cameraParams object properties specified by one or more Name,Value arguments, using any of the preceding syntaxes. Unspecified properties have their default values.

## Examples

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Create a set of calibration images.

images = imageSet(fullfile(toolboxdir('vision'),'visiondata',...
'calibration','mono'));
imageFileNames = images.ImageLocation;

Detect the calibration pattern.

[imagePoints, boardSize] = detectCheckerboardPoints(imageFileNames);

Generate the world coordinates of the corners of the squares.

squareSizeInMM = 29;
worldPoints = generateCheckerboardPoints(boardSize,squareSizeInMM);

Calibrate the camera.

imageSize = [size(I, 1),size(I, 2)];
params = estimateCameraParameters(imagePoints,worldPoints, ...
'ImageSize',imageSize);

Visualize the calibration accuracy.

showReprojectionErrors(params);

Visualize camera extrinsics.

figure;
showExtrinsics(params);

drawnow;

Plot detected and reprojected points.

figure;
imshow(imageFileNames{1});
hold on;
plot(imagePoints(:,1,1), imagePoints(:,2,1),'go');
plot(params.ReprojectedPoints(:,1,1),params.ReprojectedPoints(:,2,1),'r+');
legend('Detected Points','ReprojectedPoints');
hold off;

Specify calibration images.

leftImages = imageDatastore(fullfile(toolboxdir("vision"),"visiondata", ...
"calibration","stereo","left"));
rightImages = imageDatastore(fullfile(toolboxdir("vision"),"visiondata", ...
"calibration","stereo","right"));

Detect the checkerboards.

[imagePoints,boardSize] = ...
detectCheckerboardPoints(leftImages.Files,rightImages.Files);

Specify the world coordinates of the checkerboard keypoints. Square size is in millimeters.

squareSize = 108;
worldPoints = generateCheckerboardPoints(boardSize,squareSize);

Calibrate the stereo camera system. Both cameras have the same resolution.

imageSize = [size(I,1) size(I,2)];
params = estimateCameraParameters(imagePoints,worldPoints, ...
"ImageSize",imageSize);

Visualize the calibration accuracy.

showReprojectionErrors(params)

Visualize camera extrinsics.

figure
showExtrinsics(params)

Create a set of calibration images.

imds = imageDatastore(fullfile(toolboxdir('vision'),'visiondata',...
'calibration','circleGrid','mono'));
calibrationImages = cat(4,calibrationImages{:});

Define the circle grid pattern dimensions.

patternDims = [8 11];

Detect the pattern in the calibration images.

imagePoints = detectCircleGridPoints(calibrationImages, patternDims,...
'PatternType','symmetric');

Specify the world coordinates for the circle grid keypoints. Center distance is in millimeters.

centerDistance = 18;
worldPoints = generateCircleGridPoints(patternDims,centerDistance,...
'PatternType','symmetric');

Calibrate the camera using the calibration images.

imageSize = size(calibrationImages,1:2);
params = estimateCameraParameters(imagePoints,worldPoints,...
'ImageSize',imageSize);

Plot the detected pattern grid and the reprojected points.

figure
imshow(calibrationImages(:,:,:,1))
hold on
plot(imagePoints(:,1,1), imagePoints(:,2,1),'gx','MarkerSize',8)
plot(params.ReprojectedPoints(:,1,1),params.ReprojectedPoints(:,2,1),'r+','MarkerSize',8)
legend('Detected Points','ReprojectedPoints')
hold off

## Input Arguments

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Key points of calibration pattern, specified as an array of [x,y] intrinsic image coordinates.

CalibrationInput Array of [x,y] Key Points
Single Camera

M-by-2-by-numImages array of [x,y] points.

• The number of images, numImages, must be greater than or equal to 2.

• The number of keypoint coordinates in each pattern, M, must be greater than 3.

Partially detected patterns are only supported for single camera calibration. To include partially detected patterns in the estimate, use [NaN,NaN] as x-y coordinates for missing keypoints.

Stereo Camera

M-by-2-by-numPairs-by-2 array of [x,y] points.

• numPairs is the number of stereo image pairs containing the calibration pattern.

• The number of keypoint coordinates in each pattern, M, must be greater than 3.

• imagePoints(:,:,:,1) are the points from camera 1.

• imagePoints(:,:,:,2) are the points from camera 2.

Data Types: single | double

Key points of calibration pattern in world coordinates, specified as an M-by-2 array of M number of [x,y] world coordinates. The pattern must be planar; therefore, z-coordinates are zero.

Data Types: single | double

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: "WorldUnits","mm" sets the world point units to millimeters.

World points units, specified as a character vector or string scalar.

Estimate skew, specified as a logical scalar. When you set this property to true, the function estimates the image axes skew. When set to false, the image axes are exactly perpendicular and the function sets the skew to zero.

Number of radial distortion coefficients to estimate, specified as the value 2 or 3.

Radial distortion is the displacement of image points along radial lines extending from the principal point.

• As image points move away from the principal point (positive radial displacement), image magnification decreases and a pincushion-shaped distortion occurs on the image.

• As image points move toward the principal point (negative radial displacement), image magnification increases and a barrel-shaped distortion occurs on the image.

The radial distortion coefficients model this type of distortion. The distorted points are denoted as (xdistorted, ydistorted):

xdistorted = x(1 + k1*r2 + k2*r4 + k3*r6)

ydistorted= y(1 + k1*r2 + k2*r4 + k3*r6)

• x, y — Undistorted pixel locations. x and y are in normalized image coordinates. Normalized image coordinates are calculated from pixel coordinates by translating to the optical center and dividing by the focal length in pixels. Thus, x and y are dimensionless.

• k1, k2, and k3 — Radial distortion coefficients of the lens.

• r2 = x2 + y2

Typically, two coefficients are sufficient for calibration. For severe distortion, such as in wide-angle lenses, you can select three coefficients to include k3.

Tangential distortion flag, specified as a logical scalar. When you set this property to true, the function estimates the tangential distortion. When you set it to false, the tangential distortion is negligible.

Tangential distortion occurs when the lens and the image plane are not parallel. The tangential distortion coefficients model this type of distortion.

The distorted points are denoted as (xdistorted, ydistorted):

xdistorted = x + [2 * p1 * x * y + p2 * (r2 + 2 * x2)]

ydistorted = y + [p1 * (r2 + 2 *y2) + 2 * p2 * x * y]

• x, y — Undistorted pixel locations. x and y are in normalized image coordinates. Normalized image coordinates are calculated from pixel coordinates by translating to the optical center and dividing by the focal length in pixels. Thus, x and y are dimensionless.

• p1 and p2 — Tangential distortion coefficients of the lens.

• r2 = x2 + y2

Initial guess for camera intrinsics, specified as a 3-by-3 matrix. The matrix has this format:

$\left[\begin{array}{ccc}{f}_{x}& s& {c}_{x}\\ 0& {f}_{y}& {c}_{y}\\ 0& 0& 1\end{array}\right]$

The coordinates [cx cy] represent the optical center (the principal point), in pixels. When the x- and y-axes are exactly perpendicular, the skew parameter s equals 0.

fx = F*sx

fy = F*sy

• F is the focal length in world units, typically expressed in millimeters.

• sx and sy are the number of pixels per world unit in the x- and y-direction respectively.

• fx and fy are expressed in pixels.

If you do not provide an initial value, the function computes the initial intrinsic matrix using linear least squares.

Initial guess for radial distortion coefficients, specified as a 2- or 3-element vector. If you do not provide an initial value, the function uses 0 as the initial value for all the coefficients.

Image size produced by camera, specified as a 1-by-2 [mrows, ncols] vector.

## Output Arguments

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Camera parameters, returned as a cameraParameters object.

Images you use to estimate camera parameters, returned as a P-by-1 logical array. P corresponds to the number of images. The array indicates which images you used to estimate the camera parameters. A logical true value in the array indicates which images you used to estimate the camera parameters.

The function computes a homography between the world points and the points detected in each image. If the homography computation fails for an image, the function issues a warning. The points for that image are not used for estimating the camera parameters. The function also sets the corresponding element of imagesUsed to false.

Standard errors of estimated parameters, returned as a cameraCalibrationErrors object or a stereoCalibrationErrors object.

Camera parameters for stereo system, returned as a stereoParameters object. The object contains the intrinsic, extrinsic, and lens distortion parameters of the stereo camera system.

Image pairs used to estimate camera parameters, returned as a P-by-1 logical array. P corresponds to the number of image pairs. A logical true value in the array indicates which image pairs you used to estimate the camera parameters.

## Algorithms

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### Calibration Algorithm

You can use the Camera Calibrator app with cameras up to a field of view (FOV) of 95 degrees. The calibration algorithm assumes a pinhole camera model.

This equation provides the transformation that relates a world coordinate [X Y Z] and the corresponding image point [x y]:

$w\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=K\left[\begin{array}{cc}R& t\end{array}\right]\left[\begin{array}{c}X\\ Y\\ Z\\ 1\end{array}\right]$

• w: arbitrary scale factor

• K: camera intrinsic matrix

• R: matrix representing the 3-D rotation of the camera

• t: translation of the camera relative to the world coordinate system

Camera calibration estimates the values of the intrinsic parameters, the extrinsic parameters, and the distortion coefficients. There are two steps involved in camera calibration:

1. Solve for the intrinsics and extrinsics in closed form, assuming that lens distortion is zero. [1]

2. Estimate all parameters simultaneously including the distortion coefficients using nonlinear least-squares minimization (Levenberg–Marquardt algorithm). Use the closed form solution from the preceding step as the initial estimate of the intrinsics and extrinsics. Then set the initial estimate of the distortion coefficients to zero. [1][2]

## References

[1] Zhang, Z. "A Flexible New Technique for Camera Calibration." IEEE Transactions on Pattern Analysis and Machine Intelligence 22, no. 11 (November 2000): 1330–34. https://doi.org/10.1109/34.888718.

[2] Heikkila, J., and O. Silven. “A Four-Step Camera Calibration Procedure with Implicit Image Correction.” In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1106–12. San Juan, Puerto Rico: IEEE Comput. Soc, 1997. https://doi.org/10.1109/CVPR.1997.609468.

[3] Bouguet, J.Y. “Camera Calibration Toolbox for Matlab”, Computational Vision at the California Institute of Technology.

[4] Bradski, G., and A. Kaehler. Learning OpenCV : Computer Vision with the OpenCV Library. Sebastopol, CA: O'Reilly, 2008.

## Version History

Introduced in R2014b

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