università la sapienza rome, italy scan matching in the hough domain andrea censi, luca iocchi,...

Università La Sapienza

Rome, Italy

Scan matching in the Hough domain

Andrea Censi, Luca Iocchi, Giorgio Grisetti

lastname @ dis.uniroma1.it

www.dis.uniroma1.it/~lastname

SIED Lab www.dis.uniroma1.it/~multirob/sied/

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Scan matching• 2D scan matching (geometric interpretation): find a

rotation and a translation T who maximize overlapping of two sets of 2D data.

• 2D scan matching (probabilistic interpretation): approximate a pdf of the robot pose; ex: p(xt|xt-1, ut-1,

yt, yt-1) or others...

Map portion Sensor scan

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Previous research• Existing methods differ by:

– assumptions about environment (ex: features?)– assumptions about sensing devices (noise, FOV)– assumptions about the search domain (local vs. “global”)– representation of uncertainty (multi-hypothesis,

continuous pdf)• Methods performing a local search:

– features based [ex: Guttman ‘96, Lingemann ‘04]– ICP family [Lu-Milios ‘94, several

extensions/optimizations]– gradient-based iterative methods [ex: Hähnel ‘02, Biber

‘03]• Methods performing a global search:

– feature based: many [ex: us, 2002]– histogram of surface angles [ex: Weiß ‘94]– extensive search: 2D correlation [Konolige-Chou ‘99]

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• Our approach:

– works in unstructured environments and with noisy range finders (we don’t do feature “detection”, we work with features “distributions”)

– global search (but if a guess is available, it performs efficient local search) and multi-modality (detects ambiguities)

– completeness: if an exact match exists, it will be included in the solution set (works in practice with very different data).

• Algorithm. Given reference and sensor data:– compute the Hough Transform (HT) for both– compute the Hough Spectrum (HS) from the HT– find hypotheses for via the cross-correlation of the HS– given an estimate , estimate T via cross-correlation of

the HT

Hough Scan Matching (HSM)

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7 - The Hough Transform (HT)• The simplest HT transforms the cartesian space X-

Y into the Hough Domain (, ). The straight line

cos()x+sin()y = r

corresponds to point ( , r) in the Hough Domain.

(x,y) cartesian plane Hough Domain (, )

HT

r

r

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7 - The Hough Transform (HT)• A point in the cartesian plane a sinusoid in the

Hough domain • Sinusoids of collinear points intersects.

Cartesian plane (x,y). Hough Domain (, )

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HT

Feature detection with the HT

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Expressiveness of the HT

HT-1HT

“features distributions”

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Definition of Hough Spectrum• We compute a “spectrum” from the Hough

Transform (applying a translation-invariant functional g to the columns of the HT)

HT

HT[i]i

• The spectrum is a a function of with 180° period.

HSg[i]g

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Hough Spectrum properties• it is invariant to input translation • it shifts on input rotation

(same spectrum)

T

T

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HSM: finding the rotation • The spectrum of an input transformed by (,Tx,Ty) is

shifted by regardless of T; we can estimate by correlating the two spectra.

T

HSg[i] HSg[i’]

The peaks of the cross correlation are estimates for .

+180°

cross correlation

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Handling ambiguities• Multi-modal global search can detect

ambiguities

result ofcorrelation

Input data

Houghspectrum

multiple hypotheses for

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Comparison with circular histogram

The histogram of surface angles has similar properties, but• HS works with noisier data (does not need orientation information) • HS can handle cases when the circular histogram fails. Example:

Input data

Houghspectrum

histogramof surfaceangles

result ofcorrelation

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HSM: estimating T

HT

|T|

HT

Ttranslation T

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T

HSM: how to estimate T• Given an estimate of , we can get linear constraints for T

comparing columns of the HT (“directions of alignment”). We choose the directions with higher expected energy = peaks of the spectrum.

d~ p(T| )

d'

T

linearconstraints

d'

d

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Example with real data

Map portion Laser scan

First solution (exact) Second solution

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Summary• Operating in the Hough space allows to

decouple the search of the rotation from the translation (3D search 3 x 1D searches )

• Does not rely on the existence of features.• Multi-modal and global search (efficient local

search).• Experimental simulation results:

– Good results in curved enviroments if sensor is accurate.

– Reliability to different kinds of sensor noise (except for high discretization).

• Future (hard) work: extension to 3D for dealing with 3D noisy sensors (stereo camera).

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Thanks for your attention• Slides and an extended version of the paper

available at www.dis.uniroma1.it/~censi

Andrea Censi, Luca Iocchi, Giorgio Grisettilastname @ dis.uniroma1.it

www.dis.uniroma1.it/~lastname