Camera Parameters INEL 6088 Computer Vision Camera Parameters - - PowerPoint PPT Presentation

▶

Apr 11, 2023 328 likes •502 views

Camera Parameters INEL 6088 Computer Vision Camera Parameters Extrinsic parameters: define the location and orientation of the camera with respect to the world reference frame Intrinsic parameters: link the pixel coordinates of an image

SLIDE 1

Camera Parameters

INEL 6088 Computer Vision

SLIDE 2

Camera Parameters

Extrinsic parameters: define the location and orientation
f the camera with respect to the world reference frame
Intrinsic parameters: link the pixel coordinates of an

image point to the corresponding coordinates in the camera reference frame

Main reference: chapter 2 of Introductory Techniques for

3D Computer Vision by Emanuele Trucco and Alessandro Verri

SLIDE 3

EXTRINSIC PARAMETERS

Identify the transformation between the unknown

camera and known world reference frames.

3-D translation vector, T, that relate the origins of

the two frames

3×3 rotation matrix, R
rthogonal matrix (RTR=RRT=I)

brings axes of the two frames into each other Pc = R (Pw − T) Pc = RPw − T

SLIDE 4

rx =   1 cψ −sψ sψ cψ   ry =   cϕ sϕ 1 −sϕ cϕ   rz =   cθ −sθ sθ cθ 1   R =   r11 r12 r13 r21 r22 r23 r31 r32 r33   = rxryrz =   cθcϕ −sθcϕ sϕ cθsψsϕ + cψsθ −sψsϕsθ + cψcθ −sψcϕ −cψsϕcθ + sψsθ cψsϕsθ + sψcθ cψcϕ  

Rotation matrix in Wikipedia

SLIDE 5

THIN LENS

Basic Properties

1. Any ray entering the lens parallel to the

axis goes through the focus on the other side;

2. any ray entering the lens from the focus

in one side emerges parallel to the axis on the other side

Thin lens equation:

1 do + 1 di = 1 f

SLIDE 6

Perspective Projection

SLIDE 7

ox, oy = Position of image centre (Principal Point)
Focal length f
sx, sy = pixel dimensions
k1 = radial distortion coefficient

INTRINSIC PARAMETERS

SLIDE 8

Perspective Camera Model

Perspective camera model Weak-perspective camera model (difference in distance to scene points is small compared to average distance) (x, y, z) (x′, y′)

x′ f = x z ⇒ x′ = f x z y′ f = y z ⇒ y′ = f y z x′ = f x ¯ z y′ = f y ¯ z

(x, y, z) Image plane coordinates (x′, y′) World coordinates

SLIDE 9

Ignoring the lenses’ geometric distortions and assuming that the sensor array is made of a rectangular grid of photosensitive elements, where

(xim, yim): coordinates of an image point, in pixels
(ox, oy): coordinates of the image center (the principal point), in pixels
(sx, sy): effective pixel size (in millimetres) in the horizontal and vertical

directions, respectively

(x',y'): coordinates if the same point (xim, yim) in the camera reference plane

Transformation between Camera and Image frame coordinates

Pc = R (Pw − T)

Transformation between World and sensor coordinates

x′ = − (xim − ox)sx y′ = − (yim − oy)sy

SLIDE 10

x′ = − (xim − ox)sx = − f R1

T(Pw − T)

= − f [r11 r21 r31] [ x y z] − tx ty tz [r13 r23 r33] [ x y z] − tx ty tz y′ = − (yim − oy)sy = − f R2

T(Pw − T)

= − f [r12 r22 r32] [ x y z] − tx ty tz [r13 r23 r33] [ x y z] − tx ty tz x′ = − (xim − ox)sx = − f R1

T(Pw − T)

Pc = R (Pw − T)

y′ = − (yim − oy)sy = − f R2

T(Pw − T)

SLIDE 11

Mint = −f/sx

−f/sy ox 1 Mext = r11 r12 r13 −R1

r21 r22 r23 −R2

r31 r32 r33 −R3

Define: Intrinsic parameter matrix: Transformation between camera and image reference frame Extrinsic parameter matrix: Transformation between world and camera reference frame x1 x2 x3 = MintMext Xw Yw Zw 1 Linear Matrix Equation of Perspective Projections: xim = x1/x3 yim = x2/x3 (xim, yim): pixel coordinates that we measure

SLIDE 12

M = −fr11 −fr12 −fr13 f R1

−fr21 −fr22 −fr23 f R2

r31 r32 r33 −R3

For simplicity, asume

(ox, oy) = (0,0)
sx = sy = 1

Projection matrix M = MintMext: M describes the full perspective camera mode Weak-perspective camera model difference in distance to scene points is small compared to average distance M = −fr11 −fr12 −fr13 f R1

−fr21 −fr22 −fr23 f R2

T(¯

P − T)

SLIDE 13

Distortion

Radial distortion of the image

Caused by imperfect lenses
Deviations are most noticeable for rays that pass through the

edge of the lens

No distortion Pin cushion Barrel

SLIDE 14

Correcting radial distortion

from Helmut Dersch

SLIDE 15

Modeling distortion

To model lens distortion

Use above projection operation instead of standard projection

matrix multiplication

Apply radial distortion Apply focal length   translate image center Project   to “normalized”   image coordinates