msu cse 240 fall 2003 stockman cv: 3d to 2d mathematics perspective transformation; camera...

MSU CSE 240 Fall 2003 Stockman

CV: 3D to 2D mathematics

Perspective transformation; camera calibration;stereo computation;

and more


Roadmap of topics Review perspective transformation Camera calibration Stereo methods Structured light methods Depth-from-focus Shape-from-shading


Review coordinate systems

World or global W

Object or model M

Camera or sensor C

Camera or sensor D


Convenient notation for points and transformations

This point P has 3 real world coordinates in coordinate system W

This point P has 2 real coordinates in the image

This transformation maps each point in the real world W to a point in the image I


Current goal

Develop the theory in terms of modules (components) so that

concepts are understood and can be put into practical application


Perspective transformation

X and Y are scaled by the ratio of focal length to depth Z

Camera origin is center of projection, not lens


In next homework & project

fit camera model to image with jig jig has known precise 3D

coordinates examine accuracy of camera model use camera model to do graphics use two camera models to compute

depth from stereo


Notes on perspective trans. 3D world scaled according to ratio of

depth to focal length scaling formulas are in terms of real

numbers with the same units e.g. mm in the 3D world and mm in the image plane real image coordinates must be further

scaled to pixel row and column entire 3D ray images to the same 2D

point


Goal: General perspective trans to be developed (accept for now)

Camera matrix C transforms 3D real world point into image row and column using 11 parameters


The 11 parameters Cij model internal camera parameters: focal length f ratio of pixel height and width any shear due to sensor chip alignment external orientation parameters: rotation of camera frame relative to world frame translation of camera frame relative to world

The 11 parameters of this model are NOT independent.

Radial distortion is not linear and is not modeled.


Camera matrix via least squares

Minimize the residuals in the image plane. Get 2 equations for each pair ((r, c), (x, y, z))


2 equations for each pair

Here, (u, v) is the point in the image where 3D point (x,y,z) is projected. The 11 unknowns d jk form the camera matrix.

Known image points

Known 3D points

Camera parameters


2n linear equations from n pairs ((u,v) (x,y,z))

Standard linear algebra problem; easily solved in Matlab or by using a linear algebra package.

Often, package replaces b’s with the residuals.


Use a jig for calibration Jig has known set

of points Measure points in

world system W or use the jig to define W

Take image with camera and determine 2D points

Get pairings ((r, c) (x, y, z))


Example calibration data# # IMAGE: g1view1.ras# # INPUT DATA | OUTPUT DATA# |Point Image 2-D (U,V) 3-D Coordinates (X,Y,Z) | 2-D Fit Data Residuals X Y | A 95.00 336.00 0.00 0.00 0.00 | 94.53 337.89 | 0.47 -1.89 B 0.00 6.00 0.00 | | C 11.00 6.00 0.00 | | D 592.00 368.00 11.00 0.00 0.00 | 592.21 368.36 | -0.21 -0.36 N 501.00 363.00 9.00 0.00 0.00 | 501.16 362.78 | -0.16 0.22 O 467.00 279.00 8.25 0.00 -1.81 | 468.35 281.09 | -1.35 -2.09 P 224.00 266.00 2.75 0.00 -1.81 | 224.06 266.43 | -0.06 -0.43 # CALIBRATION MATRIX 44.84 29.80 -5.504 94.53 2.518 42.24 40.79 337.9 -0.0006832 0.06489 -0.01027 1.000


3D points on jig

Dimensions in inches


Jig set in workspace

Mapping is established between 3D points (x, y, z) and image points (u, v)


Other jigs used at MSU frame with wires and beads placed in

car instead of the driver seat (to do stereo measurements of car driver)

frame with wires and beads as big as a harp to calibrate space for people walking (up to 6 cameras, persons wear tight body suit with reflecting disks, cameras compute 3D motion trajectory)


Least squares set up

2n x 11

11 x 1 2n

x 1

=

A X = B


Least squares abstraction


Justify the form of camera matrix Another sequence of slides Rotation, scaling, shear in 3D real

world as a 3x3 (or 4x4) matrix Projection to real 2D image as 4x4

matrix Scaling real image coordinates to [r,

c] coordinates as 4x4 matrix Combine them all into one 4x4

matrix


Other mathematical models

Two camera stereo


Baseline stereo: carefully aligned cameras


Computing (x, y, z) in 3D from corresponding 2D image points


2 calibrated cameras view the same 3D point at (r1,c1)(r2,c2)


Compute closest approach of the two rays: use center point V

Shortest line segment between rays


Connector is perpendicular to both imaging rays


Solve for the endpoints of the connector

Scaler mult. Fix book


Correspondence problem: more difficult aspect


Correspondence problem is difficult

Can use interest points and cross correlation

Can limit search to epipolar line Can use symbolic matching (Ch

11) to determine corresponding points (called structural stereopsis)

apparently humans don’t need it


Epipolar constraint

With aligned cameras, search for corresponding point is 1D along corresponding row of other camera.


Epipolar constraint for non baseline stereo computation

If cameras are not aligned, a 1D search can still be determined for the corresponding point. P1, C1, C2 determine a plane that cuts image I2 in a line: P2 will be on that line.

Need to know relative orientation of cameras C1 and C2


Measuring driver body position

4 cameras were used to measure driver position and posture while driving: 2mm accuracy achieved

msu cse 240 fall 2003 stockman cv: 3d to 2d mathematics perspective transformation; camera...

Documents

msu cse

d point slide

stockman camera matrix

d world

d points camera parameters

sensor d slide

z slide

camera models