1. Camera Calibration from Planar Patterns

2. Camera Calibration M c y c x m = [Camera Projection Matrix] M A [R t]
Object Space
Image Space M y c m x c
m = [Camera Projection Matrix] M f x y
alpha* o 1
A [R t]
extrinsics
camera intrinsics

3. Camera Calibration M c y c x m = [Camera Projection Matrix] M A [R t]
Object Space
Image Space M y c m x c
m = [Camera Projection Matrix] M f x y
alpha* o 1
Camera calibration is about
finding the camera intrinsics
But, why do we need them?
A [R t]
extrinsics
camera intrinsics

4. Camera Calibration
Common approach
Non-planar pattern
Planar pattern

5. Camera Calibration from Planar Patterns
ICCV Zhang’99: “Flexible Calibration by Viewing a Plane From Unknown Orientations”
m = [Camera Projection Matrix] M
A [R t]
Minimize:
estimate: A [R t] M
observed

6. Camera Calibration from Planar Patterns
ICCV Zhang’99: “Flexible Calibration by Viewing a Plane From Unknown Orientations”
m = [Camera Projection Matrix] M
A [R t]
Two steps:
Find an initial solution
for A [R t]
Minimize the objective function
using the initial solution
Minimize:
estimate: A [R t] M
observed

7. Camera Calibration from Planar Patterns
Finding an initial solution
First step
Estimate the image homography matrix H for each image
[u, v, 1]T
Minimize:
Initial solution for minimization: L
x is the eigenvector of LTL
with smallest eigenvalue

8. Camera Calibration from Planar Patterns
Finding an initial solution
First step
Estimate the image homography matrix H for each image
Second step
Solve for b in the linear system:
V b = 0
V =
B = A –T A -1
b is the eigenvector of VTV
with smallest eigenvalue

9. Camera Calibration from Planar Patterns
Finding an initial solution
First step
Estimate the image homography matrix H for each image
Second step
Solve for b in the linear system:
b yields the intrinsic parameter matrix A.
Rotation matrix [r1 r2 r3] and translation t is computed from:
V b = 0

10. Camera Calibration from Planar Patterns
Finding an initial solution
First step
Estimate the image homography matrix H for each image
Second step
Solve for b in the linear system:
b yields the intrinsic parameter matrix A.
Rotation matrix [r1 r2 r3] and translation t:
But the computed rotation matrix does not satisfy the properties of rotation matrix: RTR=RRT=I.
One can it enforce by: min||Rnew - R||,
[U D V] = SVD(R),
Rnew = UVT
V b = 0

11. Camera Calibration from Planar Patterns
m = [Camera Projection Matrix] M
A [R t]
Two steps:
Find an initial solution
for A [R t]
Minimize the objective function
using the initial solution
Minimize:
use “lsqnonlin” in Matlab
estimate: A [R t] M
observed