a frequency-domain approach to registration estimation in 3-d space phillip curtis pierre payeur...
TRANSCRIPT
A Frequency-Domain Approach to Registration Estimation in 3-D Space
Phillip Curtis
Pierre Payeur
Vision, Imaging, Video and Autonomous Systems Research Laboratory
University of Ottawa, Canada
Coverage
Prior Art Introduction to Frequency-Domain
Registration Our Contributions to Frequency-Domain
Registration Selected Experimental Results Conclusions Future Work
What is Registration?
A registration procedure determines an estimate of the affine transform of data acquired between different points of view
RImage 1
RImage 2
Bounding Box 1
Bounding Box 2
QIm1 Im2QIm1 BB1
QIm2 BB2
QBB1 BB2
What is Needed?
A registration technique for autonomous applications must be: Quick, with a low computational burden Flexible (precision adjusted to task) Accurate Scalable
Registration Prior Art
Classical approaches Three Point Problem
Requires direct knowledge of point correspondence and 3-D spatial locations: P2=Q*P1, solve for Q
Iterative solutions Classic iterative closest point (ICP) algorithm by
Besl and McKay [1]. Most research in the field of range image registration
is centred on modifications on the ICP approach
[1] P.J. Besl, N.D. McKay, “A Method for Registration of 3-D Shapes”, IEEE Transactions on Pattern Analysis and Machine Intelligence , Vol. 14, pp. 239-256, Feb. 1992.
ICP
Besl and McKay’s ICP Algorithm 1st : match points between images using the
criterion of closest point 2nd : determine the optimal registration for that
match by first estimating the rotation, then the translation
3rd : rotate the 1st image by the estimation 4th : Repeat the 1st, 2nd, and 3rd steps until the
error delta between iterations is small enough
ICP
Advantages Allows for arbitrary data sampling structures Simple and precise Solves the point correspondence problem
Disadvantages Tends toward local minima, unless a precise
initial estimate is used Slow due to its matching algorithm - O(N)~N2
Frequency Domain Registration
Well known in 2-D registration Extended to 3-D by Lucchese et al. [2]
Takes advantage of the fact that the Fourier transform decouples the estimation of the rotational parameters from that of the translational parameters
Uses correlation and geometric projection techniques to extract rotational and translational parameters
[2] L. Lucchese, G. Doretto, G.M. Cortelazzo, “A Frequency Domain Technique for Range Data Registration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 24, No. 11, pp. 1468-1484, Nov. 2002.
Frequency Domain Registration
Advantages No initial estimate No matching of features required Avoid local minima solutions that are inherent
in ICP Disadvantages
Lucchese et al. require many transformations of the data (FFT and correlation histograms) to achieve results
Frequency Domain Benefits
The availability of Fast Fourier Transform (FFT) algorithms provides for a low computational burden
The frequency domain techniques scale well to an increase in dataset size due to the scalability of the FFT (O(N)~N log(N) )
Adjusting the FFT resolution adjusts the precision of the resulting estimation of registration
Frequency Domain Registration
Fourier Transform allows for the effective segregation of the rotation parameters from the translation parameters.
TnRn
21 ImImFourier Transform MTkRj
T
ekFkRF
2ImIm 12
kFkR
12 ImImF MTkRkFkRT
2F12 ImIm
Magnitude Phase
Determining The Axis of Rotation
All 3-D objects which rotate have an axis of rotation. When rotated, the only points in space which remain constant
lie on the axis of rotation Subtracting two frequency domain magnitude images provides a
zero line crossing through the frequency origin which is the axis of rotation
Axis of Rotation Axis of Rotation
Determining the Angle of Rotation
The angle of rotation can be determined via a minimum sum of the difference of squares search of possible rotation values about the axis of rotation.
Due to the Hermitian symmetry property of the Fourier transform, there are two possible rotation angles, separated by 180°.
Rotation by 45° or by -135°?
Solution Selection
A phase correlation between the first image, and the second image derotated by both possible solutions is performed.
The proper solution will yield a more impulse-like result when transformed to the space domain
Im2aIm1 Im2
b
Correlation of Im1
and Im2a
Correlation of Im1
and Im2b
Plot of Im1 Plot of Im2a Plot of Im2
b
n n n
n n
Estimation of Translation
The location of the impulse of the phase correlation corresponds to the estimate of the translation parameters
Correlation of Im1 and Im2
a
nT=Location
What Needed to be Done
Lucchese et al. provide a nice rigorous start to frequency domain registration, but to be practical for robotics applications the following must be improved A more efficient method for the estimation of the axis of
rotation A more efficient and flexible method for the estimation
of the angle of rotation
Determining the Axis of Rotation
Minimize calculation time, while maintaining accuracy comparable to Lucchese et al.
Solution was to develop the normalised percentage difference equation (below) to find the difference between F-D images
Use a moving window search technique to find the axis
2
Im
Im
Im
Im
Im
Im
Im
Im
0,
0
00
2
2
1
1
2
2
1
1
F
kF
F
kFMAX
F
kF
F
kF
kSE
Determine the Angle of Rotation Lucchese et al. use a correlation histogram technique using the
projections of rotated then re-transformed data to estimate the angle of rotation Huge computational penalty
Our method uses a coarse to fine minimum of least squares iterative approach
-π,π
π/3
7π/1813π/36
Solution Selection
Observations of Correct solution vs. complementary solution Correct solution is more
impulsive, and that impulse is higher than the average energy
Uses peak energy / average energy measure along the projections of each dimension
Translation Estimate
The solution with the highest ratio “wins” The location of the maximal peak in the winning
solution is the estimate of the translation parameters
Experimental Setup
Combines CRS 6 degree of freedom serial robotic arm with a track containing an additional degree of freedom, plus a laser range line scanner, and a standard PC [3][4]
Windows 2000 Workstation
RS-232 Link
RS-232 Link
Servo-Robot Cami-Box
CRS Robotics C500C
Servo-Robot Jupiter Laser Range Finder Mounted with 2 Sony XC-999 Cameras on a
CRS Robotics CRS-F3 Robotic arm and track
BNC to Matrox OrionVideo Card
VRex VRMUX2N
[3] P.Curtis, C.S. Yang, P. Payeur, “An Integrated Robotic Multi-Modal Range Sensing System”, Proceedings of the IEEE International Instrumentation and Measurement Technology Conference, Vol. 3, pp. 1991-1996, Ottawa, ON, 17-19 May 2005.
[4] P. Curtis, P. Payeur, “An Integrated Robotic Laser Range Sensing System for Automatic Mapping of Wide Workspaces”, Proceedings of the IEEE Canadian Conference on Electrical and Computer Engineering , Vol. 2, pp. 1135-1138, Niagara Falls, ON, 2-5 May 2004.
Test Data Used both simulated data sets and real data sets
The simulated house frame was selected to evaluate the performance of the algorithm using objects with a high degree of symmetry
The real house frame data was selected to see how the algorithm performed under “real” data vs. simulated data.
Some Results
Histogram of rotation error of simulated house frame (top) and real house frame (bottom) data sets.
Division of Results according to Rotation Error (FFT=128, Matches=1600, DataSet=maison_simul_xx)
0
10
20
30
40
50
60
70
80
90
100
0.0 to 0.5 0.5 to 1.0 1.0 to 1.5 1.5 to 2.0 2.0 to 2.5 2.5 to 3.0
Rotation Error
Sinc
Rect
Triangle
Gaussian
RaisedCos
InverseDecay
Division of Results according to Rotation Error (FFT=128, Matches=1600, DataSet=house_actual_ xx)
0
10
20
30
40
50
60
70
80
90
100
0.0 to 0.5 0.5 to 1.0 1.0 to 1.5 1.5 to 2.0 2.0 to 2.5 2.5 to 3.0
Rotation Error
Sinc
Rect
Triangle
Gaussian
RaisedCos
InverseDecay
Some Results
Selected registration point clouds of registered data sets (top simulated house frame, bottom is real house frame)
Front View Top View
Some Results
Execution times of the frequency domain registration algorithm presented in this paper, compared to that of ICP
Avg Nb of PointsAvg Time for
ICP (sec)Avg Time for
FFT (sec)
Data Set 1 7526.82 247.20 10.30
Data Set 2 3668.00 52.01 8.87
Factor 2.05 4.75 1.16
Results
The implementation as described in this paper is accurate, and flexible
Have improved computational efficiency, compared to Lucchese et al. without observable loss of accuracy
More scalable than ICP (execution time is faster and does not grow as rapidly as ICP)
Conclusion
Proposed, implemented, and tested an automatic registration estimation algorithm that: does not require human intervention does not require an initial estimate is independent of the geometry of the object is scalable with regards to data set size, and desired
precision is more efficient than that of Lucchese et al. and of
Besl and McKay.
Conclusion
The following innovations were contributed to the area of frequency domain registration research More computationally efficient difference equation for
calculating the difference between frequency domain magnitude images
Moving window to determine axis of rotation Coarse to fine approach to determine the angle of
rotation
Future Work
Improve solution selection mechanism Investigate other transform domains Test with enhanced data sets containing multiple data
attributes
Questions