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