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Changjae Oh

Computer Vision
– Calibration –

Semester 1, 22/23

Objectives

• Understanding the concept of camera calibration

• Understanding the relationship between image coordinate,
camera coordinate, and world coordinate

• Understanding a linear method for camera calibration

Our goal: Recovery of 3D structure

J. Vermeer, Music Lesson, 1662

A. Criminisi, M. Kemp, and A. Zisserman,Bringing Pictorial Space to Life: computer techniques for the analysis of painti

ngs, Proc. Computers and the History of Art, 2002

http://research.microsoft.com/apps/pubs/default.aspx?id=67260

Things aren’t always as they appear…

http://en.wikipedia.org/wiki/Ames_room

Single-view ambiguity

shadow sculptures

Rashad Alakbarov

Our goal: Recovery of 3D structure

• When certain assumptions hold, we ca
n recover structure from a single view

•In general, we need multi-view geometry

Image source

• But first, we need to understand the geometry of a single camera…

https://www.3dflow.net/elementsCV/S4.xhtml

Camera calibration

• Camera calibration:

̶ figuring out transformation from world coordinate system to image coordinate system

• Normalized (camera) coordinate system: camera center is at the origin,
the principal axis is the z-axis;
x and y axes of the image plane are parallel to x and y axes of the world

world coordinate system

)/,/(),,( ZYfZXfZYX 

Review: Pinhole camera model

Principal point

• Principal point (𝒑): point where principal axis intersects the image plane

• Normalized coordinate system: origin of the image is at the principal point

• Image coordinate system: origin is in the corner

Principal point offset

pZYfpZXfZYX ++

We want the principal point to
map to (px, py) instead of (0,0)

Principal point offset

principal point: ),(

Principal point offset

calibration matrix

 0|IKP =

principal point: ),(

projection matrix

Pixel coordinates

• mx pixels per meter in horizontal direction,
my pixels per meter in vertical direction

Pixel size:

pixels/m m pixels

( )C~X~RX~

Camera rotation and translation

• In general, the camera coordin
ate frame will be related to th
e world coordinate frame by a
rotation and a translation

coords. of point
in camera frame

coords. of camera center
in world frame

coords. of a point
in world frame

• Conversion from world to camera coordinate system
(in non-homogeneous coordinates):

camera coordinate
system world coordinate

Camera rotation and translation

( )C~X~RX~

3D transformation
matrix (4 x 4)

camera coordinate system world coordinate system

Camera rotation and translation

( )C~X~RX~

3D transformation
matrix (4 x 4)

camera coordinate system world coordinate system

Camera rotation and translation

3D transformation
matrix (4 x 4)

perspective project
ion matrix (3 x 4)

2D transformatio
n matrix (3 x 3)

camera coordinate system world coordinate system

Camera rotation and translation

camera coordinate system world coordinate system

Camera rotation and translation

camera coordinate system world coordinate system

Camera parameters

• Intrinsic parameters

̶ Principal point coordinates

̶ Focal length

̶ Pixel magnification factors

̶ Skew (non-rectangular pixels), Radial distortion

Camera parameters

• Intrinsic parameters

̶ Principal point coordinates

̶ Focal length

̶ Pixel magnification factors

̶ Skew (non-rectangular pixels), Radial distortion

• Extrinsic parameters

̶ Rotation and translation relative
to world coordinate system

̶ What is the projection of the
camera center?

 CRRKP ~−=

coords. of camera center
in world frame

CRRKPC The camera center is the null space

of the projection matrix!

Camera calibration

 XtRKx =

Camera calibration

• Given n points with known 3D coordinates Xi and known image projections
xi, estimate the camera parameters

Camera calibration: Linear method

• 𝑷 has 11 degrees of freedom

• One 2D/3D correspondence gives us two linearly independent equations

̶ 6 correspondences needed for a minimal solution

Camera calibration: Linear method

• 𝑷 has 11 degrees of freedom

• One 2D/3D correspondence gives us two linearly independent equations

̶ 6 correspondences needed for a minimal solution

Recall: Week1 quiz

Camera calibration: Linear method

• Directly estimate 11 unknowns in the P matrix using known 3D points
(𝑋, 𝑌, 𝑍) and measured (𝑋𝑖 , 𝑌𝑖 , 𝑍𝑖) and measured feature positions (𝑢𝑖 , 𝑣𝑖)

Camera calibration: Linear method

• Directly estimate 11 unknowns in the P matrix using known 3D points
(𝑋, 𝑌, 𝑍) and measured (𝑋𝑖 , 𝑌𝑖 , 𝑍𝑖) and measured feature positions (𝑢𝑖 , 𝑣𝑖)

Camera calibration: Linear method

• Solve for Projection Matrix 𝑷 using least-square techniques

Camera calibration: linear vs. nonlinear

• Linear calibration is easy to formulate and solve, but it doesn’t directly
tell us the camera parameters

• In practice, non-linear methods are preferred

̶ Write down objective function in terms of intrinsic and extrinsic parameters

̶ Define error as sum of squared distances between measured 2D points and estimated
projections of 3D points

̶ Minimize error using Newton’s method or other non-linear optimization

̶ Can model radial distortion and impose constraints such as known focal length and or
thogonality

 XtRKx =vs.

Application?

• Calibration is fundamental task for various computer vision tasks

https://ch.mathworks.com/help/vision/ug/single-camera-calibrator-app.html

Application?

• Calibration is fundamental task for various computer vision tasks

Xiang et al. “PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes.” RSS, 2018

Changjae Oh

Computer Vision
– Single-view Modeling –

Semester 1, 22/23

Objectives

• To understand calibration from vanishing points

• To understand measuring height without ruler

Application: Single-view modelling

A. Criminisi, I. Reid, and A. Zisserman, Sin

gle View Metrology, IJCV 2000

http://dhoiem.cs.illinois.edu/courses/vision_spring10/sources/criminisi00.pdf

Camera calibration revisited

• What if world coordinates of reference 3D points are not known?

• We can use scene features such as vanishing points

Camera calibration revisited

• What if world coordinates of reference 3D points are not known?

• We can use scene features such as vanishing points

Vertical vanishing

(at infinity)

Slide from Efros, Photo from Criminisi

Recall: Vanishing points

• All lines having the same direction share the same vanishing point

image plane

line in the scene

vanishing point v

Computing vanishing points

̶ X∞ is a point at infinity, v is its projection: v = PX∞
̶ The vanishing point depends only on line direction

̶ All lines having direction d intersect at X∞