Eecs 274 computer vision
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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.

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EECS 274 Computer Vision

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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


homogenous coordinate

Coordinate Changes: Pure Translations


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


k,l :pixel/m

f :m


: pixel

Physical Image Coordinates (f ≠1)

Normalized Image


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


: i-th row of R

M is only defined up to scale in this setting!!

Theorem (Faugeras, 1993)

  • Projection equation

    • The projection matrix models the cumulative effect of all parameters

    • Useful to decompose into a series of operations

identity matrix





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”

  • Definitions are not completely standardized

    • especially intrinsics—varies from one book to another

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