CS4495/6495 Introduction to Computer Vision 3C-L2 Intrinsic camera - - PowerPoint PPT Presentation

3C-L2 Intrinsic camera calibration

CS4495/6495 Introduction to Computer Vision

Geometric Camera calibration

Composed of 2 transformations:

• From some (arbitrary) world coordinate system to

the camera’s 3D coordinate system. Extrinisic parameters (or camera pose)

Camera Pose

World Coordinates Camera Coordinates

c wT

𝑨 𝑧 𝑦 𝑧 𝑦 𝑨

From World to Camera

Homogeneous coordinates

| | 1

C C C W W W

p R t p                                             

From world to camera is the extrinsic parameter matrix (4x4)

(sometimes 3x4 if using for next step in projection – not worrying about inversion)

Geometric Camera calibration

Composed of 2 transformations:

• From some (arbitrary) world coordinate system to

the camera’s 3D coordinate system. Extrinisic parameters (or camera pose)

• From the 3D coordinates in the camera frame to the

2D image plane via projection. Intrinisic parameters

Ideal intrinsic parameters

x u f z y v f z   Ideal Perspective projection:

Real intrinsic parameters (1)

x u z y v z    

But “pixels” are in some arbitrary spatial units

Real intrinsic parameters (2)

x u z y v z     Maybe pixels are not square

Real intrinsic parameters (3)

x u z y v u v z      

We don’t know the origin of our camera pixel coordinates

Really ugly intrinsic parameters (4)

May be skew between camera pixel axes

v



u v u

sin( ) cos( ) cot( ) v v u u v u v           

Really ugly intrinsic parameters (4)

cot( ) sin( ) y x u u z v v z z y           May be skew between camera pixel axes

v



u v u v u v u u v v ) cot( ) cos( ) sin(           

Intrinsic parameters, non-homogeneous coords

cot( ) sin( ) x y u u z z y v v z          

Notice division by z

p' K

C p



Intrinsic parameters, homogeneous coords

cot( ) * * sin( ) 1 1 x u z u y z v v z z                                         

In camera- based 3D coords In homogeneous pixels Intrinsic matrix

Kinder, gentler intrinsics

• Can use simpler notation for intrinsics –

remove last column which is zero:

1

x y

f c K f c s a           

f –focal length s – skew a – aspect ratio cx,cy - offset (5 DOF)

Kinder, gentler intrinsics

• If square pixels, no skew, and optical center is

in the center (assume origin in the middle):

1 f K f           

In this case

• nly one DOF,

focal length f

Kinder, gentler intrinsics

• Can use simpler notation for intrinsics –

remove last column which is zero:

1

x y

f c K f c s a           

f –focal length s – skew a – aspect ratio cx,cy - offset (5 DOF)

Quiz

The intrinsics have the following: a focal length, a pixel x size, a pixel y size, two offsets and a skew. That’s 6. But we’ve said there are only 5 DOFS. What happened:

a)

Because f always multiplies the pixel sizes, those 3 numbers are really only 2 DOFs.

b)

In modern cameras, the skew is always zero so we don’t count it.

c)

In CCDs or CMOS cameras, the aspect is carefully controlled to be 1.0, so it is no longer modeled.

Combining extrinsic and intrinsic calibration parameters

' K

C

p p 

Intrinsic Extrinsic

| | 1

C C C W W W

p R t p                                              World 3D coordinates Camera 3D coordinates Pixels

Combining extrinsic and intrinsic calibration parameters

 

'

C C W W W

K R p p t  '

W

M p p 

K 3x3 3x4

Other ways to write the same equation

1 2 3

. . . . . . . * 1 1 . . *

W x T T T W y W z

p u s u p v s v p s m m m                                             

'

W p

M p 

pixel coordinates world coordinates

1 3 2 3

m P u m P m P v m P      

projectively similar

Conversion back from homogeneous coordinates

Finally: Camera parameters

• A camera (and its matrix) M (or Π) is described by

several parameters

• Translation T of the optical center from the origin of world

coordinates

• Rotation R of the camera system
• focal length and aspect (f, a) [or pixel size (sx, sy)] , principle

point (x’c, y’c), and skew (s)

• blue parameters are called “extrinsics,” red are “intrinsics”

* * * * * * * * * * * * 1 X sx Y sy Z s                                  

x M X

Finally: Camera parameters

• Projection equation – the cumulative effect of all parameters:

3 1 3 1 3 3 3 3 1 3 1 3

' 1 ' 1 1 1 1 1

c x x x x c x x

f s x af y                                     

R I T M

projection intrinsics rotation translation

Finally: Camera parameters

• Projection equation – the cumulative effect of all parameters:

DoFs: 5+0+3+3 = 11

(3x4)