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