A Frequency-Domain Approach to Registration Estimation in 3-D Space

1. A Frequency-Domain Approach to Registration Estimation in 3-D Space Phillip Curtis Pierre Payeur Vision, Imaging, Video and AutonomousSystems Research Laboratory University of Ottawa, Canada

2. Coverage • Prior Art • Introduction to Frequency-Domain Registration • Our Contributions to Frequency-Domain Registration • Selected Experimental Results • Conclusions • Future Work

3. What is Registration? • A registration procedure determines an estimate of the affine transform of data acquired between different points of view

4. What is Needed? • A registration technique for autonomous applications must be: • Quick, with a low computational burden • Flexible (precision adjusted to task) • Accurate • Scalable

5. 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.

6. 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

7. 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

8. 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.

9. 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

10. 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

11. Frequency Domain Registration • Fourier Transform allows for the effective segregation of the rotation parameters from the translation parameters. Fourier Transform Magnitude Phase

12. 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

13. 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°.

14. 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

15. Estimation of Translation • The location of the impulse of the phase correlation corresponds to the estimate of the translation parameters

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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] [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.

22. 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.

23. Some Results • Histogram of rotation error of simulated house frame (top) and real house frame (bottom) data sets.

24. Some Results • Selected registration point clouds of registered data sets (top simulated house frame, bottom is real house frame)

25. Some Results • Execution times of the frequency domain registration algorithm presented in this paper, compared to that of ICP

26. 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)

27. 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.

28. 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

29. Future Work • Improve solution selection mechanism • Investigate other transform domains • Test with enhanced data sets containing multiple data attributes

30. Questions