Geometric Camera Models EECS 274 Computer Vision Geometric Camera Models

Geometric Camera Models Elements of Euclidean geometry Intrinsic camera parameters Extrinsic camera parameters General Form of the Perspective projection equation Reading: Chapter 2 of FP, Chapter 2 of S

Quantitative Measurements and Calibration Euclidean Geometry

Euclidean Coordinate Systems

Planes homogenous coordinate

OBP = OBOA + OAP , BP = BOA+ AP Coordinate Changes: Pure Translations OBP = OBOA + OAP , BP = BOA+ AP

Coordinate Changes: Pure Rotations 1st column: iA in the basis of (iB, jB, kB) 3rd row: kB in the basis of (iA, jA, kA)

Coordinate Changes: Rotations about the z Axis

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

A rotation matrix is characterized by the following properties: 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 R3 whose determinant is 1 Preserve length of vectors Preserve angles between two vectors Preserve orientation of space

Coordinate Changes: Pure Rotations

Coordinate Changes: Rigid Transformations

Block Matrix Multiplication What is AB ? Homogeneous Representation of Rigid Transformations

Rigid Transformations as Mappings

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 Ratios of distance along a line |p2-p1|/|p3-p2| is preserved

Shear matrix Horizontal shear Vertical shear

2D planar transformations

2D planar transformations

2D planar transformations

3D 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

The Intrinsic Parameters of a Camera Units: k,l : pixel/m f : m a,b : pixel Physical Image Coordinates (f ≠1) Normalized Image Coordinates

The Intrinsic Parameters of a Camera Calibration Matrix 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 Focal length may vary for zoom lenses Optical axis may not be perpendicular to image plane Change focus affects the magnification factor From now on, assume camera is focused at infinity

Extrinsic Parameters

Explicit Form of the 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 x 4 matrix with 11 parameters 5 intrinsic parameters: α, β, u0, v0, θ 6 extrinsic parameters: 3 angles defining R and 3 for t

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

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 projection intrinsics rotation translation identity matrix Definitions are not completely standardized especially intrinsics—varies from one book to another