Università La Sapienza Rome, Italy Scan matching in the Hough domain Andrea Censi, Luca Iocchi, Giorgio Grisetti dis.uniroma1.it lastname SIED Lab

A. Censi, L. Iocchi, G. Grisetti2 of 16 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(x t |x t-1, u t-1, y t, y t- 1 ) or others... Map portionSensor scan

A. Censi, L. Iocchi, G. Grisetti3 of 16 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]

A. Censi, L. Iocchi, G. Grisetti4 of 16 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) andrea: decoupling andrea: decoupling

A. Censi, L. Iocchi, G. Grisetti5 of 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. Andrea Censi: si può fare in modo formale Andrea Censi: si può fare in modo formale (x,y) cartesian plane Hough Domain ( ,  ) HT     r r

A. Censi, L. Iocchi, G. Grisetti6 of The Hough Transform (HT) A point in the cartesian plane  a sinusoid in the Hough domain Sinusoids of collinear points intersects. Andrea Censi: si può fare in modo formale Andrea Censi: si può fare in modo formale Cartesian plane (x,y). Hough Domain ( ,  )  

A. Censi, L. Iocchi, G. Grisetti7 of 16 HT   Feature detection with the HT andrea: in you algorithm andrea: in you algorithm

A. Censi, L. Iocchi, G. Grisetti8 of 16 Expressiveness of the HT HT -1 HT “features distributions”  

A. Censi, L. Iocchi, G. Grisetti9 of 16 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. HS g [i] g

A. Censi, L. Iocchi, G. Grisetti10 of 16 Hough Spectrum properties it is invariant to input translation it shifts on input rotation Andrea Censi: anche alla scala Andrea Censi: anche alla scala (same spectrum) T   T

A. Censi, L. Iocchi, G. Grisetti11 of 16 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  HS g [i]HS g [i’] The peaks of the cross correlation are estimates for .  +180 ° cross correlation

A. Censi, L. Iocchi, G. Grisetti12 of 16 Handling ambiguities Multi-modal global search can detect ambiguities result of correlation Input data Hough spectrum multiple hypotheses for 

A. Censi, L. Iocchi, G. Grisetti13 of 16 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: Andrea Censi: anche alla scala Andrea Censi: anche alla scala Input data Hough spectrum histogram of surface angles result of correlation

A. Censi, L. Iocchi, G. Grisetti14 of 16 HSM: estimating T   HT |T|   HT T translation T

A. Censi, L. Iocchi, G. Grisetti15 of 16 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 linear constraints d' d

A. Censi, L. Iocchi, G. Grisetti16 of 16 Example with real data Map portionLaser scan First solution (exact) Second solution

A. Censi, L. Iocchi, G. Grisetti17 of 16 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).

A. Censi, L. Iocchi, G. Grisetti18 of 16 Thanks for your attention Slides and an extended version of the paper available at censi Andrea Censi, Luca Iocchi, Giorgio Grisetti dis.uniroma1.it lastname