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

EECS 274 Computer Vision

Geometric Camera Models


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


Eecs 274 computer vision

Quantitative Measurements and Calibration

Euclidean Geometry


Eecs 274 computer vision

Euclidean Coordinate Systems


Eecs 274 computer vision

Planes

homogenous coordinate


Eecs 274 computer vision

Coordinate Changes: Pure Translations

OBP = OBOA + OAP ,BP = BOA+ AP


Eecs 274 computer vision

Coordinate Changes: Pure Rotations

1st column:

iA in the basis of (iB, jB, kB)

3rd row:

kB in the basis of (iA, jA, kA)


Eecs 274 computer vision

Coordinate Changes: Rotations about the z Axis


Rotation matrix

Rotation matrix

Elementary rotation

R=R x R y R z , described by three angles


Eecs 274 computer vision

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 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


Eecs 274 computer vision

Coordinate Changes: Pure Rotations


Eecs 274 computer vision

Coordinate Changes: Rigid Transformations


Eecs 274 computer vision

Block Matrix Multiplication

What is AB ?

Homogeneous Representation of Rigid Transformations


Eecs 274 computer vision

Rigid Transformations as Mappings


Eecs 274 computer vision

Rigid Transformations as Mappings: Rotation about the k Axis


Affine transformation

Affine transformation

  • Images are subject to geometric distortion introduced by perspective projection

  • Alter the apparent dimensions of the scene geometry


Affine transformation1

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

Shear matrix

Horizontal shear

Vertical shear


2d planar transformations

2D planar transformations


2d planar transformations1

2D planar transformations


2d planar transformations2

2D planar transformations


3d transformation

3D transformation


Eecs 274 computer vision

Idealized coordinate system


Camera parameters

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


Eecs 274 computer vision

The Intrinsic Parameters of a Camera

Units:

k,l :pixel/m

f :m

a,b

: pixel

Physical Image Coordinates (f ≠1)

Normalized Image

Coordinates


Eecs 274 computer vision

The Intrinsic Parameters of a Camera

Calibration Matrix

The Perspective

Projection Equation


In reality

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


Eecs 274 computer vision

Extrinsic Parameters


Eecs 274 computer vision

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


Eecs 274 computer vision

Explicit Form of the Projection Matrix

Note:

: i-th row of R

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


Eecs 274 computer vision

Theorem (Faugeras, 1993)


Camera parameters1

  • 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

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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