EECS 274 Computer Vision Geometric Camera Models Geometric

EECS 274 Computer Vision Geometric Camera Models

Geometric Camera Models • • Elements of Euclidean geometry Intrinsic camera parameters Extrinsic camera parameters General form of perspective projection • Reading: Chapter 1 of FP, Chapter 2 of S

Geometric camera calibration Euclidean Geometry

Euclidean coordinate system

Planes homogenous coordinate

Pure translation OB P = O B OA + O A P , B P = B OA + A P AP: point P in frame A

Pure rotation 1 st column: i. A in the basis of (i. B, j. B, k. B) 3 rd row: k. B in the basis of (i. A, j. A, k. A)

Rotation about z axis

Rotation matrix Elementary rotation R=R x R y R z , described by three angles

Properties of rotation matrix • Its inverse is equal to its transpose, R-1=RT , and • Its determinant is equal to 1. Or equivalently: • Its rows (or columns) form a right-handed orthonormal coordinate system.

Rotation group and SO(3) • Rotation group: the set of rotation matrices, with matrix product – Closure, associativity, identity, invertibility • SO(3): the rotation group in Euclidean space R 3 whose determinant is 1 – Preserve length of vectors – Preserve angles between two vectors – Preserve orientation of space

Pure rotations

Rigid transformation

Block matrix manipulation What is AB ? Homogeneous Representation of Rigid Transformations

Rigid transformations as mappings

Rotation about the k Axis

Affine transformation • Images are subject to geometric distortion introduced by perspective projection • Alter the apparent dimensions of the scene geometry

Affine transformation • In Euclidean space, preserve – Collinearity relation between points • 3 points lie on a line continue to be collinear – Ratio of distance along a line • |p 2 -p 1|/|p 3 -p 2| is preserved

Shear matrix Horizontal shear Vertical shear

2 D planar transformations See Szeliski Chapter 2

2 D planar transformations

2 D planar transformations

3 D transformation

Idealized coordinate system

Camera parameters • Intrinsic: relate camera’s coordinate system to the idealized coordinated system • Extrinsic: relate the camera’s coordinate system to a fix world coordinate system • Ignore the lens and nonlinear aberrations for the moment

Intrinsic camera parameters Units: k, l : pixel/m f : m (See EXIF tags) a, b: pixel Physical Image Coordinates (f ≠ 1) Normalized Image Coordinates Scale parameters: k, l (image sensor may not be square) Offset: u 0, v 0 Manufacturing error: θ

Intrinsic camera parameters Calibration matrix κ The perspective projection Equation

In reality • Physical size of pixel and skew are always fixed for a given camera, and in principal known during manufacturing • Some parameters often available in EXIF tag • Focal length may vary for zoom lenses when optical axis is not perpendicular to image plane • Change focus affects the magnification factor • From now on, assume camera is focused at infinity

Extrinsic camera parameters

Explicit form of projection Matrix denotes the i-th row of R, tx, ty, tz, are the coordinates of t can be written in terms of the corresponding angles R can be written as a product of three elementary rotations, and described by three angles M is 3 × 4 matrix with 11 parameters 5 intrinsic parameters: α, β, u 0, v 0, θ 6 extrinsic parameters: 3 angles defining R and 3 for t

Explicit form of projection Matrix Note: M is only defined up to scale in this setting!! : i-th row of R

Theorem (Faugeras, 1993)

Camera parameters A camera is described by several parameters • • • Translation T of the optical center from the origin of world coords Rotation R of the image plane focal length f, principle point (x’c, y’c), pixel size (sx, sy) • blue parameters are called “extrinsics, ” red are “intrinsics” Projection equation • • The projection matrix models the cumulative effect of all parameters Useful to decompose into a series of operations identity matrix intrinsics • projection rotation translation Definitions are not completely standardized – especially intrinsics—varies from one book to another

Camera calibration toolbox • Matlab toolbox by Jean-Yves Bouguet http: //www. vision. caltech. edu/bouguetj/calib_doc/ • Extract corner points from checkerboard