Geometric Image Formation

Pinhole Camera Model

Theorem

Thrm For the pinhole camera model with focal length , given the world coordinate of a point we have the following projection equation:

e.g. Given a lens with a focal length of 30, and a world point at , where the camera center sits at , and the optical axis is in direction , with the and axes aligned to the world axes, this world point will project to pixel

Intrinsic Camera Parameters

Attention

For the following contents, we will use or to denote homogeneous coordinates of the corresponding point or in Euclidean space.

Perspective Camera Projection

Assuming that the origin of points in the image plane is at principal point, the projection onto the image plane can be calculated by homogeneous camera projection matrix: If the origin of the image plane is not at principal point, we then apply an offset: Now map from image plane to pixel coordinates by matrix where , .

Definition

Def Camera Calibration Matrix The camera calibration (intrinsics) matrix is defined as follows: where is the principal point, is called the aspect ratio, and , . Clearly the result of perspective camera projection then can be represented as:

Extrinsic Camera Parameters

Comment

The image is also related to some extrinsic camera parameters, such as camera rotation and translation in the world coordinate system.

Transformation to Camera Coordinate

Suppose is the principal axis, is the camera center vector in the world coordinate, is some point, is a 3D rotation matrix, then in camera coordinates, we have

Def Complete Camera Matrix The complete camera matrix, or projective camera matrix, is defined as where is the rotation matrix, is the location vector denoting the camera center. Consequently, for any point in the world coordinate, we have its coordinate in the image by Algorithm Gold Standard for Camera Calibration

1. Prepare a calibration target/object with known 3D coordinates.
2. Capture images of the calibration target/object.
3. Find corners of the target in the images, obtain 2D–3D correspondences.
   1. Option 1: Detect edges with Canny detector, fit straight lines to detected linked edges, intersect lines to obtain 2D image coordinates, match image corners & 3D target checkerboard corners.
   2. Option 2: Apply a corner detector directly.
4. Solve for the normalised camera projection matrix using the normalised DLT algorithm.
5. Refine the normalised solution using iterative minimisation of a geometric error.
6. Denormalise solution.

Radial Lens Distortion

Definition

Def Barrel Distortion Barrel Distortion is the phenomenon that image magnification decreases with distance from optical axis.

Definition

Def Pincushion Distortion Pincushion distortion is the phenomenon that image magnification increases with distance from optical axis.

Definition

Def Mustache Distortion A mixture of both above types.

Definition

Def Model for Radial Distortion Change based on distance of point on image plane from principal point: where .