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

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 , Feb

ICP Besl and McKay’s ICP Algorithm 1 st : match points between images using the criterion of closest point 2 nd : determine the optimal registration for that match by first estimating the rotation, then the translation 3 rd : rotate the 1 st image by the estimation 4 th : Repeat the 1 st, 2 nd, and 3 rd 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)~N 2

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 , Nov

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. Fourier Transform 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

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°.

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

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

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

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

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] [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 , Ottawa, ON, May [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 , 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.

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

Some Results Execution times of the frequency domain registration algorithm presented in this paper, compared to that of ICP Avg Nb of Points Avg Time for ICP (sec) Avg Time for FFT (sec) Data Set Data Set Factor

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