new image rectification schemes for 3d vision based on sequential virtual rotation jin zhou june 16...
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New Image Rectification Schemes for 3D Vision Based on Sequential Virtual Rotation
Jin ZhouJune 16th, 2009
Dissertation Defense
Outline
Introduction Rectification based on Virtual Sequential Rotation Image Rectification for Stereoscopic Visualization Camera Calibration Stereoscopic View Synthesis from Monocular
Endoscopic Sequences Rapid Cones and Cylinders Modeling from Single
Images Robot Vision Conclusions and Future Work
The Geometry of 3D to 2D
Images are 2D projections of the 3D world
3D Vision – The Problem
?
How do we extract 3D information from 2D images?
? ? ??
3D of the objects
3D of the cameras
3D Vision – Applications
Augmented Reality Scene Modeling Virtual Touring 3D Imaging Robots
3D Vision – A Human Perspective
Size Linear Perspective Object Connections Stereo Motion Shading Texture
3D Vision – Computational Approaches
Different approaches use different cues Different problems requires different
approaches. Structure from Motion (SfM)
Rely on point correspondences Single View Based Modeling (SVBM)
Rely on knowledge of the scene Camera calibration
All approaches requires the images are calibarated first (either manual or automatic)
3D Vision – Practical Challenges
Camera information is unavailable Point correspondences is not reliable and
time-consuming Image resolution is limited Degeneracy
3D Vision – Limitations of Current Approaches
Distortion Degeneracy
Due to high degree of freedom of geometric models
Lack of geometric meaningMost approaches are purely based on
algebraic derivations or imaginary objects. Not accurate or not convenient (SVBM).
3D Vision – Our Contributions
Novel image rectification schemes are proposed based on sequential virtual rotation
Novel approaches are proposed for the following problems Image Rectification for Stereoscopic visualization Camera Calibration Stereoscopic View Synthesis from Monocular
Endoscopic sequences Rapid Cones and Cylinders Modeling Monocular Vision Guided Mobile Robot Navigation
3D Vision – Results of Our Approaches
No affine/projective distortion Can handle degeneracy Intuitive geometric meanings
Lead to insights of particular problems Accurate and fast
Publications Jin Zhou and Baoxin Li, "Image Rectification for Stereoscopic Visualization", Journal
of the Optical Society of America A (JOSA), November, 2008. Wenfeng Li, Jin Zhou, Baoxin Li, and M. Ibrahim Sezan, "Virtual View Specification
and Synthesis for Free Viewpoint Television", IEEE Transactions on Circuit and Systems for Video Technologies (TCSVT), (in press)
Jin Zhou, Baoxin Li, "Rapid Cones and Cylinders Modeling from a Single Calibrated Image Using Minimal 2D Control Points", Machine Vision and Applications (revision)
Jin Zhou and Baoxin Li, “Stereoscopic View Synthesis from Monocular Endoscopic Image Sequences", IEEE Transactions on Medical Imaging, (submitted)
Jin Zhou, Ananya Das, Feng Li, Baoxin Li, "Circular Generalized Cylinder Fitting for 3D Reconstruction in Endoscopic Imaging Based on MRF", In 9th IEEE Computer Society Workshop on Mathematical Methods in Biomedical Image Analysis (Joint with CVPR 2008).
Jin Zhou and Baoxin Li, "A Four Point Algorithm for Fast Metric Cone Reconstruction from a Calibrated Image", In 4th International Symposium on Visual Computing (ISVC), 2008.
Xiaolong Zhang, Jin Zhou and Baoxin Li, "Robust Two-view External Calibration by Combining Lines and Scale Invariant Point features", In 4th International Symposium on Visual Computing (ISVC) 2008.
Publications (Cont’d) Jin Zhou and Baoxin Li, “Exploiting Vertical Lines in Vision-Based Navigation for
Mobile Robot Platforms”, International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2007.
Xiaokun Li, Roger Xu, Jin Zhou, Baoxin Li, "Creating Stereoscopic (3D) Video from a 2D Monocular Video Stream", In 3rd International Symposium on Visual Computing (ISVC), 2007.
Wenfeng Li, Jin Zhou, Baoxin Li, M. Ibrahim Sezan, "Virtual View Specification and Synthesis in Free Viewpoint Television Application", 3D Data Processing, Visualization and Transmission (3DPVT), 2006.
Jin Zhou and Baoxin Li, "Image Rectification for Stereoscopic Visualization without 3D Glasses", ACM International Conference on Image and Video Retrieval (CIVR), 2006.
Jin Zhou and Baoxin Li, “Homography-based Ground Detection for a Mobile Robot Platform using a Single Camera”, International Conference on Robotics and Automation (ICRA), 2006.
Jin Zhou and Baoxin Li, “Robust Ground Plane Detection with Normalized Homography in Monocular Sequences from a Robot Platform”, International Conference on Image Processing (ICIP), 2006.
Jin Zhou and Baoxin Li, “Rectification with Intersecting Optical Axes for Stereoscopic Visualization”, International Conference on Pattern Recognition (ICPR), 2006.
What is Image Rectification?
Image rectification is a process to transform the original images to new images which have desired properties.
General Image Transformations
H?
Image transformation can be defined by a 3x3 matrix H, which is called Homography.
xx' H
Image Rectification based on Virtual Rotation
Homography of camera rotation/zooming
If we normalize the coordinates
Assume R = I
[ | ], ' ' '[ | ]P KR I C P K R I C
1 1x ' ' ' '[ | ] ' '( ) [ | ] ( ' ')( ) xP X K R I C X K R KR KR I C X K R KR
1( ' ')( )H K R KR
1x xK 1x' ' x'K
ˆ 'H R Camera orientation is determined at the same time
Advantages of the New Rectification Schemes Intuitive geometric meaning Robust
Rotation parameters can be computed by various basic image features, such as points, lines and circles.
Can be used for camera calibration. Can be used for 3D information extraction. Lead to non-distorted results
Reason: Rotation do not introduce affine/projective distortion
Rotations are Decomposed as Euler Angles
1 0 0
( ) 0 cos( ) sin( )
0 sin( ) cos( )xR
cos( ) 0 sin( )
( ) 0 1 0
sin( ) 0 cos( )yR
cos( ) sin( ) 0
( ) sin( ) cos( ) 0
0 0 1zR
Rotation Parameters Can Be Estimated by Basic Image Features
Each rotation has only one degree of freedom and thus only needs one constraint.
Example: transforming a point on to y axis
ˆ (0, , )TzR p d c
cos( ) sin( ) 0a b
tan( , )arc a b
/ 2
[ ] / 2
otherwize
[ tan( , )]arc a b Ambiguity!
Normalize
Image Rectification for Stereoscopic Visualization
The Principle of Stereoscopic (3D) Visualization
Motivation
Stereo content is scarce Stereo cameras/camcorders are expensive Common users seldom use stereo
cameras/camcorders We want to generate stereo content from
images/videos taken by common cameras
The Problem
Given two arbitrary images, rectify them so that the results look like a stereo pair.
Our Approach – Rectification based on Virtual Rotation
We can “rotate” camera to standard stereo setup.
1( )i i iH KR K R
Calibrated Case
( , , )TR r s t
1 2 1 2( ) / || ||r C C C C
( ) / || ||t r p r p
s r t
1( )i i iH KR K R
Known
UnKnown
Constraints of the stereo camera pair:1. The two cameras have the same intrinsic parameters (K)
and orientation (R)2. The camera’s optical axis is perpendicular to the baseline
(C1 – C2) i.e. the camera’s x axis has the same direction with the
baseline
K1 Any vector
Uncalibrated Case
For the uncalibrated case, all K, R and C are unknown. We can only start from the fundamental matrix and point correspondences.
Estimate H2 (homography for the second image)
2
0 / 2
0 with / 2
0 0 1 ( ) / 2
x x
y y
f p p w
K K f p p h
f w h
2R I
1( )i i iH KR K R ?
Determine R based on Sequential Virtual Rotation
2ˆ (1,0,0)TRe 12 2 2e K eConstraints:
First rotate around z axis so that the point is transformed to x axis (i.e. y = 0)
y zR R R
( , , ) ( ,0, )T TzR a b c d c
( ,0, ) ( ,0,0) (1,0,0)T T TyR d c e
[arctan( , )]z b a
[arctan( , )]y c d
Rotate around y axis so that the point is transformed to infinity.
Estimate H1
1 11 1 1 2 2 2 1 1 2( ) ( )H KR K R H K R K R H M
2 0( )TM I e v M 2[ ]F e M
1 2
2 2 0
2 0 2 0
12 0 2 2 2 2 0
12 2 1 2 0 1 2
( )
( )
( ) ( = )
T
T
T
T T T
H H M
H I e v M
H M e v M
H M H e v H H M
I H e v H M v v H
12 2 2 2 [ , , ]
(1,0,0) ( ,0,0)
T
T T
H e KRK e K r s t r
K k
1 2 2 2 0 2 0( )
0 1 0
0 0 1
TA
A
H I H e v H M H H M
a b c
H
Estimate H1
Determine a, b and c Property of standard stereo setup:
For two points with the same depth, their projection on different images should have the same distance (Points with the same depth should have the same disparity).
Approach Group points by similar disparities Then compute a, b by minimizing
2
1 1 2 2,
(|| x x || | x ' x ' ||)p
i j i jp i j A
H H H H
2
,
ˆ ˆ ˆ ˆ ˆ ˆ( ( ) ( ) ( ' '))p
i i i j i jp i j A
a x x b y y x x
0 0 0ˆ ˆ ˆ 'ax by c x
Results
Original Pair Hartley’s Method Our Method
Shear distortion
Results
Original Pair Hartley’s Method Our Method
Shrink horizontally
Camera Auto-Calibration from the Fundamental Matrix
Ti i iw K K
2 2 2 1[ ] [ ] Te w e Fw F Kruppa Equations
dual image of the absolute conic
Huang-Faugeras constraints
2 2
2 1
1( ) ( ) 0
2T T
T
trace EE trace EE
E K FK
Cons: Complex and hard to understand!Derivation for degenerate cases are purely algebraic.
Traditional Approaches:
Our Approach
12 2 2 1 1 1
1 1 1 2 2 2( , , , , , , )
T T T Sz y x y z
z y x z y
F K R R F R R R K
f f
We transform the original pair to a standard stereo pair through sequential virtual rotation and zooming
0 0 0
0 0 1
0 1 0
SF
F
7 DOFs
7 Parameters
Decomposition Illustration
Solving The Equation1
2 2 2 1 1 1
1 1 1 2 2 2( , , , , , , )
T T T Sz y x y z
z y x z y
F K R R F R R R K
f f
1 12 1 2 2 1 1
T T Sz z z y x yF R FR K R F R R K
2 1
2 1
0 0 0 0
0 1 0 0 1 0
0 0 0 0z
c a b a c
F c d c
d a b a d
1 21 2
2
2 1 1 2
1 1 2
sin( ), cos( ),cos( )
cos( ), sin( )cos( ) cos( ) cos( )
x xy
x xy y y
f da f f c
f d d db d
1 1 1 1( ) ( ,0, )Tz zR e d c
2
2 22 21 2
1 22 21 2
( )
,
x
adtg
bc
acd abdf f
bd acc cd abc
2 2 22 1 1 1
1 2 21 1
2 2 22 2 2 2
2 2 22 2
( )
( )
y
y
f c acctg
d bd acc
f c abctg
d cd abc
1 1 1 1( , , )e a b c1 1 1arctan( / )z b a
1 0Fe
Degeneracy Analysis
Degeneracy Analysis
Results of Monte Carlo Simulation
Stereoscopic View Synthesis From Monocular Endoscopic Videos
3D imaging helps to enable faster and safer surgical operations
Two view image rectification can not be applied to the new problem
Challenges: 1. Image quality is poor 2. Degeneracy
The Framework
We proved:1. Affine 3D reconstruction is
sufficient.2. Linear interpolation in
normalized disparity field is equal to linear interpolation in 3D space.
Strategy for Solving Degeneracy
We assume the initial two frames have same orientation (i.e. they are rectified)
The assumption makes the DOF of the fundamental matrix from 8 to 2!
No Assumption
Assume the two frame are rectified
Degeneracy!
Interpolation
a) Shows the disparities based on the SfM results
b) We do Delaunay triangulation and interpolate each triangle
c) We pick a set of grid points from b) and do bilinear interpolation
d) We fill holes using Laplacian interpolation and do smoothing.
Results of Synthetic Data
Ground truth
Stereo images
Final disparity image
Disparity image after triangulation
Results of Real Data
Rapid Cylinders and Cones Modeling from A Single Image
Overview of Our Approach
Goal: Rapid + Accurate Camera Calibration (Orientation Estimation)
Vertical lines Vanishing line of horizontal plane A Cone
Modeling from Image Cones (two points / four points) Cylinders (two points / four points)
The Coordinate Systems
(X,Y,Z,O) -- World Coordinate SystemY is perpendicular to the ground
(x,y,z,O) -- Camera Coordinate SystemCamera center is at the origin
Observation: Most objects are standing on the ground
Orientation Estimation from Vertical Lines
(0,1,0)T( , , )Ta b c R
First rotate around z axis so that the point is transformed to y axis (i.e. x = 0)
Rotate around x axis so that the point is transformed to infinity.
Orientation Estimation from a Cone
Rx
Rx (π/2)
RzRxRz
Edges are symmetric to y axisR Cross section is a circle
Illustration
Metric Rectification of the Ground Plane1 0 0
( / 2) 0 0 1
0 1 0g xH R
Original
Original Rectified
Rectified
Modeling Cones
0Μ ( (0, ,1) , , )Tc vc y r y
Standard view Cones on Ground General Cones
0Μ ( , ,M )R d
Cone Parameters:
2D Control Points for Cones Modeling:
Cones Modeling from the Standard View
Rectify the standard view to the ground plane view by Rx(π/2) The cross section is rectified to a circle The edge line is tangent to the circle The center and radius can be determined for any
point on the edge line
Modeling Cones Standing on the Ground
Ry
Four points Standard view
Ground view 3D Mesh
Modeling General Cones
R
Five points Standard View
3D Mesh
Modeling Cylinders
0L ( (0, ,1) , , )Tb tc y r y
0L ( , ,L )R d
Standard View Cylinders on Ground General Cylinders
Cylinder Parameters:
2D Control Points for Cones Modeling:
Modeling Cylinders Standing on the Ground
Ry
two points Standard view
Ground view 3D Mesh
Modeling General Cylinders
R
Four points Standard View
3D Mesh
Screenshots of Real Data Experiments
Exploiting Vertical Lines for Monocular Vision based Mobile Robot Navigation
In man-made environments, vertical lines are omni-present: buildings, boxes, bookshelves, cubicle walls, door frames
Many vision based systems assume the image plane is perpenticular to the ground plane
We proposed methods to rectify an image plane with general pose to be vertical based on vertical lines.
Rectified Images for Ground Plane Detection
ˆ ( / )TH R I Cn d
0 (0,1,0)Tn 0 0( ,0, )TC x z
0
0
cos( ) / sin( )ˆ 0 1 0
sin( ) / cos( )
x d
H
z d
The normalized homography of the ground plane has a special form in the rectified images:
4 DOFs
Results
Special Form
General Form
4 DOFs
8 DOFs
Ground Rectification for Mosaic based SLAM
After rectification, the relationship between any two views directly indicates their relative locations and orientations.
From vertical image plane, it’s easy to get the ground plane image
Results
Vision Based Control
Vanishing line
Occupied
OccupiedTurn angle
876
5…
910
11…
Nearest left Nearest right
Given an rectified imageFind the object to trackIdentify the obstacle or ground planeOutput a turn angle.Adjust the camera to make object always stays at the center of the image.
Movie 1Movie 2ImagesObstacles
Rectified Image
Other Potential Applications
Surveillance/Activity Recognition/ Path Planning
Different object’s size
Same object’s sizeGround is rectifiedRight angles are recoveredSpeed is reflected on the image
Conclusions and Future Works
Novel image rectification schemes are proposed in the context of exploring several practical 3D vision problems
For each of the problem, we designed novel algorithms and nice results are achieved. Moreover, we gain new insights to the problems.
In the future, we should combine different approaches and exploiting more visual cues for 3D vision problems.
Questions?
Thanks!