Camera Parameters, Calibration and Radiometry. Readings Forsyth & Ponce- Chap 1 & 2 Chap & 3.2 Chap 4

Transcription

1 Camera Parameters, Calibration and Radiometry Readings Forsyth & Ponce- Chap 1 & 2 Chap & 3.2 Chap 4

2 Camera parameters U V W " # $ $ $ % & ' ' ' = Transformation representing intrinsic parameters " # $ $ $ % & ' ' ' " # $ $ $ % & ' ' ' Transformation representing extrinsic parameters " # $ $ $ % & ' ' ' X Y Z T " # $ $ $ $ % & ' ' ' ' From last time.

3 Homogeneous Coordinates (Again)

4 Extrinsic Parameters: Characterizing Camera position in world coordinates Chasles's theorem: Any motion of a solid body can be composed of a translation and a rotation.

5 3D Rotation Matrices

6 3D Rotation Matrices cont d Euler s Theorem: An arbitrary rotation can be described by 3 rotation parameters For example: R = More Generally: Most General:

7 Rodrigue s Formula Take any rotation axis a and angle θ: What is the matrix? with R = e Aθ

8 Rotations can be represented as points in space (with care) Turn vector length into angle, direction into axis: Useful for generating random rotations, understanding angular errors, characterizing angular position, etc. Problem: not unique Not commutative

9 Other Properties of rotations NOT Commutative R1*R2 R2*R1

10 Rotations To form a rotation matrix, you can plug in the columns of new coordinate points For Example: The unit x-vector goes to x :

11 Other Properties of Rotations Inverse: R -1 = R T rows, columns are othonormal r T i r j = 0 if i j, else r T i r i = 1 Determinant: det( R ) = 1 The effect of a coordinate rotation on a function: x = R x F( x ) = F( R -1 x )

12 Extrinsic Parameters p = R p + t R = rotation matrix t = translation vector In Homogeneous coordinates, p = R p + t =>

13 Rotation 90 o Scaling *2 Translation 1 y x p = House Points 2D Example

14 Intrinsic Parameters Differential Scaling Camera Origin Offset

15 The Whole (Linear) Transformation " U $ V $ # W " X% % " Transformation %" % " Transformation % $ ' ' ' = $ ' $ ' $ ' representing representing $ Y ' $ $ ' $ ' $ ' Z' & # intrinsic parameters& # & # extrinsic parameters& $ ' # T & Final image coordinates u =U/W v =U/W

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24 Non-linear distortions (not handled by our treatment)

25 Camera Calibration You need to know something beyond what you get out of the camera Known World points (object) Traditional camera calibration Linear and non-linear parameter estimation Known camera motion Camera attached to robot Execute several different motions Multiple Cameras

26 Example: Known world points vs. known camera motion

27 Classical Calibration

28 Classical Camera Calibration Take a known set of points. (Typically 3 orthogonal planes) Treat a point in the object as the World origin Points x1, x2, x3,. Project to y1,y2,y3,..

29 Calibration Patterns

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34 Matlab code is simple Form the matrix P by appending each 3D point and the corresponding pixel locations in the arrangement given shown in the previous slide: P = [ Px1 Py1 Pz u1*px1 -u1*py1 -u1*pz1 -u1; Px1 Py1 Pz1 1 -v1*px1 -v1*py1 -v1*pz1 -v1; : : Pxn Pyn Pzn un*pxn -un*pyn -un*pzn -un; Pxn Pyn Pzn 1 -vn*pxn -vn*pyn -vn*pzn -vn]; Compute eigenanalysis on P *P [V,D]=eig(P *P); eigenvalues = diag(d); j = find(eigenvalues ==min(eigenvalues)); m=v(:,j);

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36 Real Calibration Real calibration procedures look quite different Weight points by correspondence quality Nonlinear optimization Linearizations Non-linear distortions Etc.

37 Camera Motion

38 Calibration Example (Zhang, Microsoft) 8 Point matches manually picked Motion algorithm used to calibrate camera

39 Applications of Calibrated Cameras Image based rendering: Light field -- Hanrahan (Stanford)

40 Virtualized Reality

41 Projector-based VR UNC Chapel Hill

42 Shader Lamps

43 Is classical camera calibration necessary to to computer vision? NO. For example, Shape recovery without calibration (Fitzgibbons, Zisserman) Fully automatic procedure: Video in, VRML out. Allows for irregular motion: angle between views can vary, and it doesn't have to be known. Recovery of the angle is automatic, and accuracy is about 40 millidegrees standard deviation. No calibration targets: features on the objects themselves are used to determine where the camera is, relative to the turntable. Right shows a shape model automatically extracted from a dinosaur image sequence without any additional information. Camera parameters are determined as well.

44 NEXT: Measuring light in images Geometric: (what we ve been briefly covering) how positions in the image relate to 3-d positions in the world. Photometric/Radiometric: how the intensities in the image relate surface and lighting properties in the world.