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Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Aaron Bobick

School of Interactive Computing

CS 4495 Computer Vision

Calibration and

Projective Geometry

(1) Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Administrivia

•Problem set 2: •What is the issue with finding the PDF???? or descr.pdf •Today: Really using homogeneous systems to represent projection. And how to do calibration. •Forsyth and Ponce, 1.2 and 1.3 Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Last time...

Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

What is an image?

Figure from US Navy Manual of Basic Optics and Optical Instruments, prepared by Bureau of Naval Personnel. Reprinted by Dover Publications, Inc., 1969. Last time: a function - a 2D pattern of intensity values

This time: a 2D projection of 3D points

Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Modeling projection

•The coordinate system •We will use the pin-hole model as an approximation •Put the optical center (Center Of Projection) at the origin •Put the image plane (Projection Plane) in front of the COP •Why? •The camera looks down the negative z axis •we need this if we want right-handed-coordinates Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Modeling projection

•Projection equations •Compute intersection with PP of ray from ( x,y,z) to COP •Derived using similar triangles •We get the projection by throwing out the last coordinate:

Distant objects

are smaller Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick Or... •Assuming a positive focal length, and keeping z the distance: xxufz yyvfz Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Homogeneous coordinates

•Is this a linear transformation? •No - division by Z is non-linear

Trick: add one more coordinate:

homogeneous image (2D) coordinates homogeneous scene (3D) coordinates

Converting from homogeneous coordinates

Homogenous coordinates invariant under scale

Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Perspective Projection

•Projection is a matrix multiply using homogeneous coordinates:

This is known as perspective projection

•The matrix is the projection matrix •The matrix is only defined up to a scale

S. Seitz

,uv 000 0 00 0 010 1 x f y f z fx fy z f x z ,f y z Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Geometric Camera calibration

Use the camera to tell you things about the world: •Relationship between coordinates in the world and coordinates in the image: geometric camera calibration, see Forsyth and Ponce,

1.2 and 1.3. Also, Szeliski section 5.2, 5.3 for references

•Made up of 2 transformations:

•From some (arbitrary) world coordinate system to the camera's 3D coordinate system. Extrinisic parameters (camera pose)

•From the 3D coordinates in the camera frame to the 2D image plane via projection. Intrinisic paramters

Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Camera Pose

In order to apply the camera model, objects in the scene must be expressed in camera coordinates. World

Coordinates Camera

Coordinates

Calibration target looks tilted from camera

viewpoint. This can be explained as a difference in coordinate systems.

This image cannot currently be displayed.

y x z z x y Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Rigid Body Transformations

•Need a way to specify the six degrees-of-freedom of a rigid body. •Why are their 6 DOF?

A rigid body is a

collection of points whose positions relative to each other can't change

Fix one point,

three DOF 3

Fix second point,

two more DOF (must maintain distance constraint) +2

Third point adds

one more DOF, for rotation around line +1 Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Notations (from F&P)

• Superscript references coordinate frame A

P is coordinates of P in frame A

B

P is coordinates of P in frame B

A P A x A y A z OP A xi A A yj A A zk A k A j A i A O A P Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Translation Only

k A j A i A k B j B i B O B O A B BA A B BA A PP O or POP P Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Translation

• Using homogeneous coordinates, translation can be expressed as a matrix multiplication. •Translation is commutative B AB A PPO 1 011 B BA A

P IO P

Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Rotation

AB AB

AAA BBB

AB xx

OP i j k y i j k y

zz B BA A P RP B A R means describing frame A in

The coordinate system of

frame B Calibration and Projective Geometry 1 CS 4495 Computer Vision - A. Bobick

Rotation

AB AB AB

B

A AB AB AB

AB AB AB

R ii ji ki ij jj kj ik jk kk BBB AA A ijk

Orthogonal matrix!

AT B AT B AT B i jquotesdbs_dbs10.pdfusesText_16