Estimating the Principal Curvatures and the Darb oux Frame from Real D Range Data Eyal Hameiri Ilan Shimshoni Abstract As pro ducts of secondorder computations esti While these features are continuous prop erties of a regu mations of principal curvatures are highly sensitive to noise lar surface range data is discrete in nature This is where Due to the availability of more accurate D range imaging the challenge of how to extract these continuous features equipment evaluation of existing algorithms for the extrac from a discrete approximation of a surface emerges Prin tion of these invariants and other useful features from dis crete D data is now relevant The work presented here cipal curvatures are secondorder derivatives of a surface makes some subtle but very imp ortant mo dications to two whichmakes them very sensitivetonoise Thus the main such algorithms originally suggested by Taubin and goal of the work presented here is to suggest algorithms for Chen and Schmitt The algorithms have b een adjusted to deal with real discrete noisy range data The results of principal curvatures and Darb oux frame estimations from this implementation were evaluated in a series of tests on real noisy data and to test their validity synthetic and real input yielding reliable estimations Our The sensitivity of the features to noise imp osed limita conclusion is that with current scanning technology and the tions on some early works restricting their use to only the algorithms presented here reliable estimates of the prin atures and the Darb oux frame can be extracted cipal curv sign of the mean and Gaussian curvatures for segmenta from real data and used in a variety of more comprehensive In the recognition system tion tasks tasks presented in only the normal to the surface which re quires only a rst derivative of the surface without the I Introduction and Previous Related work principal directions is used to recognize D ob jects In recentyears following the availability of more accurate D The imp ortance of principal curvatures as invariantfea sensing equipment as well as the use of p olyhedral meshes tures of freeform rigid surfaces has b een widely recognized to approximate freeform surfaces more and more works in In addition the lo cal co ordinate system consisting of the volve full principal curvature estimation from discrete data normal to the surface at that p ointandthetwo principal in their suggested algorithms directions the Darboux frame can also b e exploited for a Two main approaches are b eing taken by all previous vast variety of tasks The Darb oux frame is not invariant works The more p opular one until recent years is the to rigid transformations but it can b e used to establish a analytical approach In this approach for eachD p oint lo cal co ordinate frame which is attached to the ob ject at within the input data a lo cal patch of surface is tted to each of its sampled p oints the p oint and its geometric neighbors The tted patchis These two typ es of features are lo cally asso ciated with formulated as an explicit function f u v Then the rst the surface and therefore can be used in a lo cal manner and second partial derivatives of f u v are analytically ob For example the principal curvatures can b e used to nd tained and used within a general and closed analytic form corresp onding p oints b etween a freeform scene ob ject and to compute the principal curvatures and directions or the a library ob ject Then the lo cal Darb oux frame on the two mean and Gaussian curvatures One analytic metho d dif ob jects can b e used to register them and test the validity fers from another by the technique in which the surface of the matc hed p oints But the features can also b e used patch is being tted In the patch is a general bi in a global manner For example in the case of geometric quadratic patch In an analytic approachwhich uses a primitives wehave preknowledge of the dierentcharac cubic bspline tting technique presented in is tested teristics of the two features when they are collected from In the patch is also formulated as a biquadratic for the entire surface of the primitive Then by gathering mula but the basis functions consist of a dierent triplet of the feature values from all points of a range scene evi discrete orthogonal p olynomials and not of the canonical dence for the presence or absence of geometric primitives p olynomials will emerge Furthermore when a primitive is identied in The second approach extracts the features directly or the range image its accurate dimensions and orientation almost directly from the discrete data For example in can b e recovered from the values of the features In b oth a Gaussian curvatures op erator was develop ed based typ es of uses global as well as lo cal it is therefore nec on a known expression for the Gaussian curvature see essary to accurately estimate the principal curvatures and for example as the limit of the ratio of a lo cal surface area the Darb oux frame from range data and of its corresp onding area on the Gauss map From a similar limit of ratio of areas an op erator for the mean Eyal Hameiri is with the Dept of Computer Science The T echnion Israel Inst of Technology Haifa Israel Email curvatures was develop ed directly from discrete data after eylcstechnionacil dening sp ecial lo cal areas In the algorithm is based Ilan Shimshoni is with the Dept of Industrial Engineering and on an integral formula which denes at each sampled p oint Management The Technion Israel Inst of Technology Haifa Israel Email ilansietechnionacil a matrix whose eigenvectors and eigenvalues pro duce the principal curvatures and the Darb oux frame The matrix is a nonnegativescalarand n denotes the normal to the is based on discrete data except for a continuous approxi curveatP The reader can nd more details in mations of normal sections curves The algorithm in Now consider S to be a regular surface illustrated in is based on solving a leastsquares problem at eachvertex Figure P a p ointlyingonS and C S a regular curve The leastsquares problem is formulated directly from the passing through P Let N and n b e the unit length normals discrete data except for the approximation of cross sections at P ofS and C resp ectivelyandT the unit length tangent curves as circles We based our two algorithms on the last If n is the curvature of C at P and is the of C at P two nonanalytic metho ds and therefore our suggested al angle b etween N and n then the quantity T cos P gorithms b elong to the nonanalytic approach is dened as the normal curvature of C at P Note that for any normal section of S incidentonP the curvature Most of the curvature estimation algorithms suggested of this normal section at P equals to the normal curvature so far had presented their results only on nonenoisy syn at that p oint thetic data That is the case in and for example where satisfactory results have b een achieved for synthetic ob jects Few others likethework in were also tested on noisy synthetic data and on real data But as far as real data is concerned it concluded that curvature esti mation should b e dealt with caution In full and accu rate estimation of principal curvatures and Darb oux frame is presented while the input is a mesh approximating an articial spatial surface and which is arbitrarily triangu lated There is no reference to the algorithms p erformance in case of real data or of synthetic noisy data Two algorithms presented in for the estimation of principal curvatures and Darb oux frame from range im ages have b een mo died in this work We will refer to these two algorithm as to the T algorithm and C al gorithm resp ectively The T algorithm which deals cor rectly with continuous surfaces has b een mo died to deal Fig The lo cal dierential prop erties of a curve and a freeform with real discrete surfaces op en or close represented by surface on whichitlies a cloud of noisy D p oints In the C algorithm the mo d ication improved the results for b oth synthetic and real The Meusnier theorem states that all curves in S in noisy data and also reduced the complexity of the algo tersecting P and sharing at that p oint the same unit tan rithm We have tested the p erformance of the mo died gentvector also share the same normal curvature There algorithms on syn thetic noisy range data and on real data fore the quantity T has b een established as the di P rectional curvature of S at P along the tangent direction obtained from a Cyb erware D scanner Cases of noiseless T Note that the sign of the directional curvature is deter synthetic data were also tested esp ecially for comparative reasons For each surface p oint the features are estimated mined by the orientation of N In all our discussions from and a reliability factor is applied to indicate how accurate nowonN is p ointing out in case of a closed surface or has the estimations are Wherever the groundtruth is avail a nonnegative comp onent in the direction of the viewer in case of an op en one able statistics of the errors resulting from the comparison were studied The maximal and minimal values of T along all P p ossible tangent directions to S at P are referred to as the The pap er continues as follows In Section I I we explain principal curvatures and their two asso ciated tangents as terms related to curvatures In Section I I I the two original the principal directions of S at P If the t wo principal cur algorithms are presented in detail followed by descriptions vatures share the same value then