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[Next](DefaultICPConfig.md) | +| [Tutorials Home](index.md) | [Previous](OutlierFiltersFamilies.md) | [Next](DefaultICPConfig.md) | | :--- | :---: | ---: | # An Introduction to ICP Registration diff --git a/doc/OutlierFiltersFamilies.md b/doc/OutlierFiltersFamilies.md new file mode 100644 index 00000000..18445d7f --- /dev/null +++ b/doc/OutlierFiltersFamilies.md @@ -0,0 +1,377 @@ +| [Tutorials Home](index.md) | [Previous](ApplyingDataFilters.md) | [Next](ICPIntro.md) | +| :--- | :---: | ---: | + +# Outlier Filters Families + +This section adresses the various filters and tries to organise them in families. The families are based on the influence functions of the cost function associated to an outlier filter. The purpose of outlier filtering, applied to point cloud registration, is to reduce the effect of outliers, improve the error minimisation and improve the crispness of the registered point clouds. + +Appart for the Other family, the presentation of the families and functions is the following. The functions were plotted with an error variable (noted e) ranging from [-4,4] with a step of 0.01. The plot of a family is made with a value of k = 1. The first plot of a function is made with values of k = 0.5, k = 2 and k = 4. Then a second plot is proposed for values of k ranging from [0,2] with a step of 0.1. + +This catalog was made from thanks to Philippe Babin's Mémoire on the analysis of error functions for the Iterative closest point algorithm: P.Babin, "Analysis of Error Functions for the Iterative Closest Point Algorithm", pp.70-72. + +Libpointmatcher provides an implementation of the following outlier filters: + +* Cauchy +* Welsch +* Switchable-Constaint +* Geman-McClure +* Tukey +* Huber +* L1 +* Student + +## Filter Index +### [Pseudo-Sinus Family](#pseudosinfamilyhead) +1. [Cauchy](#cauchyhead) +2. [Geman-McClure (GM)](#gemanmcclurehead) +3. [Welsch](#welschhead) +4. [Tukey](#tukeyhead) +5. [Andrew](#andrewhead) +6. [Kernel-Correlation (KC)](#kernelcorrelationhead) +7. [Fast Global Registration (FGR)](#fastglobalregistrationhead) + +### [Linear Family](#linearfamilyhead) +8. [L2](#l2head) +9. [Switchable-Constraint (SC)](#switchableconstrainthead) +10. [Maximum-Distance (MaxDist)](#maximumdistancehead) +11. [Generalized ICP (GICP)](#gicphead) + +### [Step Family](#stepfamilyhead) +12. [L1](#l1head) +13. [Huber](#huberhead) +14. [Logistic](#logistic) +15. [LM-ICP](#lmicphead) +16. [Fair](#fairhead) + +### [Percentile Family](#percentilefamilyhead) +17. [Trimmed](#trimmedhead) +18. [Median](#medianhead) +19. [FICP](#ficphead) +20. [AICP](#aicphead) + +### [Other Family](#otherfamilyhead) +21. [Student](#studenthead) +22. [Sparse-ICP](#sparsehead) +23. [EM-ICP](#emicphead) +24. [RANSAC](#ransachead) +25. [Relative Motion Threshold (RMT)](#rmthead) +26. [BiDistance](#bidistancehead) + +## Descriptions + +Outlier filters come in different forms. However the most common is a cost function noted p(e,k) associated with a weight function noted w(e,k). The value e is the error associated with a pair of matches and k is a tuning parameter. An influence function is the first derivative of a cost function p'(e,k). For most cases the influence function can be approximated by p'(e,k) = ew(e,k). + +### Effect of the tuning parameter k + +The weighted linear regression was used in order to illustrate the effect of k on outlier filtering. For this example, the chosen outlier filter is Cauchy and the problem is to have better results than an unweighted linear regression. The tuning of the k parameter will determine the efficiency of the outlier filtering. In the following example, a value of k which is nearly equal to the maximum error on the first iteration results nearly as poor as un unweighted linear regression. + +![cauchy example](images/cauchy_weighted_linear_regression.png) + +The following plot represents how the outliers and inliers are affected by the cost function and weight function. The poor results of k = 4 can also be seen in this plot. Indeed, the weight of some outliers is the same as the weight of some inliers for lower values of k. This means that for values of k too big (relative to the point cloud) some outliers will be considered as inliers. The opposite effect can happen where the value of k is so low that some inliers are downweighted like outliers. This means that some prior knowledge of the point cloud is necessary for a fine tuning of the k parameter. + +![cauchy with error points](images/cauchy_with_points.png) + +## Families Catalogue + +### Pseudo-Sinus Family + +This family of outlier filters is characterized by an influence function that looks like one period of the sinusoid function. The value of k affect the width of the period. Past the sinusoidal behavior, the influence function becomes null. Usually, inliers are affected by the sinusoidal part while the outliers are affected by the null part of the function. + +The sinusoidal aspect of this family is well presented by Andrew and Tukey which provides a sharp transition between the effect on inliers and outliers. Welsch, Cauchy and Geman-McClure provide an influence function with a smooth transition between the effect on inliers and outliers. Kernel-Correlation and Fast Global Registration are identical to other functions of the family. Therefore they overlap with those functions, Welsch and Geman-McClure respectively, and can't be seen on the plot. + +![Pseudo-Sin Family plot](images/pseudo_sin_family.png) + +#### Cauchy + +Equations: +![eq cauchy](images/eq_cauchy.JPG) + +Plots: +![cauchy threek](images/cauchy_threek.png) +![cauchy multik](images/cauchy_multik.png) + +Original reference: +M. Bosse and R. Zlot, "Continuous 3D scan-matching with a spinning 2D laser", 2009 IEEE International Conference on Robotics and Automation,pp.4312-4319,2009. + +#### Geman-McClure + +Equations: +![eq geman](images/eq_geman.JPG) + +Plots: +![geman threek](images/geman_threek.png) +![geman multik](images/geman_multik.png) + +Original reference: +S. Geman, D.E. McClure, and D. Geman, "A nonlinear filter for film restoration and other problems in image processing", CVGIP: Graphical Models and Image Processing, vol. 54, no.4, pp.281-289, 1992. + +#### Welsch + +Equations: +![eq welsch](images/eq_welsch.JPG) + +Plots: +![welsch threek](images/welsch_threek.png) +![welsch multik](images/welsch_multik.png) + +Original reference: +R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. + +#### Tukey + +Equations: +![eq tukey](images/eq_tukey.JPG) + +Plots: +![tukey threek](images/tukey_threek.png) +![tukey multik](images/tukey_multik.png) + +Original reference: +R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. + +#### Andrew + +Equations: +![eq andrew](images/eq_andrew.JPG) + +Plots: +![andrew threek](images/andrew_threek.png) +![andrew multik](images/andrew_multik.png) + +Original reference: +R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. + +#### Kernel-Correlation (KC) + +Kernel Correlation is a registration algorithm known to use the Welsch outlier filter. + +Equations: +![eq kc](images/eq_kc.JPG) + +Original reference: +Y.Tsin and T.Kanade, "A Correlation-Based Approach to Robust Point Set Registration", in 8th European Conference on Computer Vision, vol.0 , 2004, pp.558-569 + +#### Fast Global Registration (FGR) + +This registration algorithm is known to use the Geman-McClure outlier filter. + +Equations: +![eq fgr](images/eq_fgr.JPG) + +Original reference: +Q.Y. Zhou, J. Park, and V.Koltun, "Fast global registration", Lecture notes in Computer Science, vol.9906 LNCS, pp. 766-782,2016 + +### Linear Family + +This family of outlier filters is characterized by an influence function that looks like a linear function. Appart from L2, the value of k affects the width of the linear part. Past the linear behavior, the influence function becomes null. Usually, inliers are affected by the linear part of the function while the outliers are affected by the null part of the function. The Switchable-Constraint outlier filter provides a smooth transition between the inlier and outlier part while Maximum Distance provides a sharp transition between those two behaviors. L2 represents the ideal linear influence function. On the plot Maximum Distance and GICP are overlapping. + +![linear Family plot](images/linear_family.png) + +#### L2 + +L2 is a function which is a part of the Lp space family of functions. The L2 outlier filter can't be tuned with a k parameter. + +Equations: +![eq l2](images/eq_l2.JPG) + +Plot: +![L2 nok](images/L2_nok.png) + +#### Switchable-Constraint + +Equations: +![eq sc](images/eq_sc.JPG) + +Plots: +![switchableconstraint threek](images/switchableconstraint_threek.png) +![switchableconstraint multik](images/switchableconstraint_multik.png) + +Original reference: +N. Süderhauf and P. Protzel,"Switchable constraint for robust pose graph slam", in 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2012, pp. 1879-1884. & P Argawal, G.D. Tipaldi, L. Spinello, C.Stachniss, and W.Burgard, "Robust map optimization using dynamic covariance scaling", IEEE international Conference on Robotics and Automation, pp. 62-69, 2013. + +#### Maximum Distance (MaxDist) + +The Maximum Distance outlier filter is used to compute the unscaled distance (e': euclidean error). + +Equations: +![eq maxdist](images/eq_maxdist.JPG) + +Plots: +![maximumdistance threek](images/maximumdistance_threek.png) +![maximumdistance multik](images/maximumdistance_multik.png) + +#### Generalized ICP (GICP) + +This registration algorithm is known to use the Maximum Distance outlier filter on the scaled error (e). + +Equations: +![eq gicp](images/eq_gicp.JPG) + +Original reference: +A.Segal, D.Haehnel, and S.Thrun, "Generalized-ICP", in Robotics: Science and Systems, vol.5, 2009, pp.168-176 + +### Step Family + +This family of outlier filters is characterized by an influence function that looks like a linear function. Apart from L1, the value of k affects the width of the linear part linking the step. Past the linear behavior, unlike the linear family the influence function becomes equal to -k or k. Usually, inliers are affected by the linear part while the outliers are affected by the -k/k part of the function. The Logistic outlier filter provides a smooth transition between the inliers and outliers parts while Huber provides a sharp transition between the two behaviors. L1 represents the ideal step function therefore it does not have a linear part. The LMICP outlier filter overlaps with the plot of Huber. + +![step family plot](images/step_family.png) + +#### L1 + +L1 is a function which is a part of the Lp space family of functions. The L1 outlier filter can't be tuned with a k parameter. + +Equations: +![eq l1](images/eq_l1.JPG) + +Plot: +![l1 nok](images/l1_nok.png) + +#### Huber + +Equations: +![eq huber](images/eq_huber.JPG) + +Plots: +![huber threek](images/huber_threek.png) +![huber multik](images/huber_multik.png) + +Original reference: +P.J. Huber, "Robust Estimation of a Location Parameter", The Annals of Mathematical Statistics, vol.35, no. 1, pp. 73-101, 1964. + +#### Logistic + +Equations: +![eq logistic](images/eq_logistic.JPG) + +Plots: +![logistic threek](images/logistic_threek.png) +![logistic multik](images/logistic_multik.png) + +Original reference: +R.E. Welsch, "Robust Regression Using Iteratively reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no.9, pp.813-827, 1977. + +#### LM-ICP + +This registration algorithm is known to use the Huber outlier filter. + +Equation: +![eq huber](images/eq_huber.JPG) + +Original reference: +A.W. Fitzgibbon, "Robust registration of 2D and 3D point sets", Image and Vision Computing, vol.21, no.13-14, pp.1145-1153, 2003. + +#### Fair + +Equations: +![eq fair](images/eq_fair.JPG) + +Plots: +![fair threek](images/fair_threek.png) +![fair multik](images/fair_multik.png) + +Original reference: +R.E. Welsch, "Robust Regression Using Iteratively reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no.9, pp.813-827, 1977. + +### Percentile Family + +This family of outlier filters uses the percentile in order to affect a binary weight. The values past the f percentile are considered as outliers. Trimmed, AICP, and FICP are the same outlier filters and have a tunable value of f. Median is the same as the other percentile filters, however it's value is non-tunable and equal to 50%. The value of f must be equal to the overlap ratio parameter. + +Equation: +![eq percentile](images/eq_percentile.JPG) + +This filtering method doesn't rely on a function. In order to illustrate it's effect, a Gaussian density function was applied to the range of errors [-4,4] with a step of 0.1. With this simplified problem the overlap ratio parameter can't be computed therefore we affect arbitrary values to f. Respectively for Trimmed, AICP and FICP the values of f are 30%, 60% and 90%. + +![percentile family plot](images/percentile_family.png) + +Original Reference: +Trimmed: +D.Chetverikov, D.Svirko, D.Stepanov and P.Krsek, "The Trimmed Iterative Closest Pointalgorithm", Object recognition supported by user interaction for service robots, vol.3, no. c, pp.545-548, 2002. +Median: +A.Censi, "An ICP variant using point-to-line metric", IEEE International Conference on Robotics and Automation (ICRA), pp. 19-25, 2008. +FICP (i.e. Var. Trim.): +J.M.Phillips, R.Liu, and C.Tomasi, "Outlier robust ICP for minimizing fractional RMSD", in 3DIM 2007 - Proceedings 6th International Conference on 3-D Digital Imaging and Modeling, 2007, pp.427-434. +AICP: +S.Nobili, R.Scona, M.Caravagna, and M.Fallon, "Overlap-based ICP tuning for robust localization of a humanoid robot", in Proceedings - IEEE International Conference on Robotics and Automation, 2017, pp.4721-4728. + +### Other Family + +This family of function regroups outlier filters that can't be categorized. This is due to their behavior charging drastically (Student, Sparse ICP), dependent of a value and k (RMT, BiDistance) or a different algorithm that doesn't rely on functions (RANSAC). Due to the diversity of those functions a plot for the family can't be provided. + +The outlier filters RANSAC, RMT and BiDistance requires a problem in order to be computed. Therefore, the algorithm was applied to a linear regression problem on a cloud of points presenting 17% of outliers. + +#### Student + +Equations: +![eq student](images/eq_student.JPG) + +Plots: +![student fourk](images/student_fourk.png) + +Original reference: +G.Agamennoni, S.Fontana, R.Y. Siegwart, and D.G. Sorrenti, "Point Clouds Registration with Probabilistic Data Association", IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.4092-4098, 2016. + +#### Sparse ICP + +Sparse ICP is an outlier function based on the Lp norm therefore it uses a tuning parameter p. In the plots the parameter is noted k so that it can't be mistaken for the result of the cost function p(e,k). This function is only applicable for values of k in the range [0,1]. It is relevant to point out that Sparse ICP for k = 1 is equal to L1. + +Equations: +![eq sparse](images/eq_sparse.JPG) + +Plots: +![sparseICP fourk](images/sparseicp_fourk.png) + +Original reference: +S.Bouaziz, A.Tagliasacchi, and M.Pauly, "Sparse iterative closest point", Eurographics Symposium on Geometry Processing, vol.32, no.5, pp. 113-123, 2013. + +#### EM-ICP + +The EM-ICP algorithm presents an effect on inliers similar to the step function. This is due it being based on L1. The effect on outliers is the same as the functions in the linear or sin families. Since the outlier filtering method is a trimmed version on L1 and it shares behavior with the step and linear families, it is placed in the other family. + +Equations: +![eq emicp](images/eq_emicp.JPG) + +Plots: +![emicp nok](images/emicp_nok.png) + +Original reference: +S.Granger, and P.Xavier, "Multi-scale EM-ICP: A Fast and Robust Approach for Surface Registration", Proceedings of the European Conference on Computer Vision, pp. 418-432. + +#### RANSAC + +The RANSAC outlier filtering algorithm or RANdom SAmple Consensus relies on a voting system that uses inliers. The first step of the algorithm is to randomly sample the cloud of points. Then a model is applied depending on the problem. This model is applied to the sampled points. After the model is computed the algorithm counts the number of values contained within a [-k,k] range from the model. These values are considered as inliers. The model with the highest number of inliers is considered as the result of the algorithm. The RANSAC algorithm uses three parameters: +* (s): the number of sampled points +* (c): the number of computations +* (k): the range containing the inliers +Since the algorithm is based on random sampling, the results can be quite independent of k as shown with the plots. Indeed, for the same amount of points and the same number of computations the result for k = 0.5 ended up being poorer than the example using k = 1.5. This can be explained by the fact that (s) is the predominant tuning parameter for this algorithm. + +Plots: +![ransac threek](images/ransac_threek.png) + +Original reference: +M.A.Fischler and R.C. Bolles, "Random sample consensus: a paradigm for the model fitting with applications to image analysis and automated cartography", Communications of the ACM, vol.24, no.6, pp.381-395, 1981. + +Relative motion threshold (RMT): + +The relative motion threshold (RMT) uses a threshold based on translation between two transforms. In our simplified case this motion threshold was the mean of the absolute values of the translation on the y axis between two iterations. + +Equations: +![eq rmt](images/eq_rmt.JPG) + +Plots: +![rmt nok](images/rmt_nok.png) + +Original reference: +F.Pomerleau, F.Colas, F.Ferland, and F.Michaud,"Relative Motion Threshold for Rejection in ICP Registration", Field and Service Robotics, vol.62 p.229-238, Jul.2015. + +Bidisance: + +The BiDistance outlier filtering method affects an uniform weight based on the tuning parameter k and a parameter computed at each iteration noted B. This computed parameter is the bidirectional correspondence between the two point sets. Since this outlier filter is a weight function with two variables it can be plotted as a 3D graph. For the example the plot was made by values of k and B in the range [-3,3] with a step of 0.1. The weight function results can get really high due to the exponential in the equation, therefore the axis of the weight is limited to [0,4]. + +Equation: +![eq bidistance](images/eq_bidistance.JPG) + +Plots: +![bidistance multikB](images/bidistance_multikB.jpg) + +Original reference: +J.Zhu, D.Wang, X.Bai, H.Lu, C.Jin, and Z.Li, "Regression of point clouds based on the ratio of bidirectional distances", Proceedings of the 4th international Conference on 3D vision, pp.102-107, 2016. \ No newline at end of file diff --git a/doc/images/L2_nok.png b/doc/images/L2_nok.png new file mode 100644 index 0000000000000000000000000000000000000000..54dfc1695d8d32de876a916897cfbd4bc3093c10 GIT binary patch literal 24614 zcmZ^L1z1&GwC$m!JEcRBE~Q&iDM1vZLqJj*6hu-FEg~QYA|WACBGO$-gS0eCBOodH z=Jvk0@7??O{_p!0&pCVVwdR_0j4{VN;kUHahzaNkP$(4f4RvKb6bcImg~A}my8yr8 zsUw_%|6zMvzoCx@KmK_4(eU?+o+^f(w;$Si`dGT#pzK^8I@|DjSi9TUxOhBx=(&m2 zAcsP+qHZWF>iedyPx%;9E!E?0ZR)-Ed6lD1PKHfLkcXG@j$Fg)27761=MCX@y7DxJ!`Q5wT z?K7S>{^VcKe=o9^pus_Y zp-dlxyhJp-tnBL9+3}`PUt3292Ol3%Vq#)RX=&3=pL}d;Dq|c41-b6+QZ82ZOqv>x zB@%WH4!4~}t?b;~meyA6?>SnO(lqcXFJs0q&}j8NVivjDk%jRxJB~}2l(VlXD-Y?{ z>cI`%^f}zJ^!4+LWC}r}>B9{O;Klj9Y0}SCB55o;VwkMd*w`~w=Mym5+4WK!M~f^E zx8@A%{Uw7#LfT@Kl$E1fGXv4+SM0K~Ogi$X*W`|tIW*tO+*_A`D_}dx!Q*IgCfwA$ zcQ4;J_05||Hl=44`|IPrzY+|THC2_Bd)Uaa70INr$;tCc?Ck8|?nkA2zJARcS@`tn z(-$Ge|jcq+nYLWyXH66nZPUT z^Lx^^?k;=W>_oYv*V~4t=W;T#vT^C@GlRZE5$+xy!VmgcU%!4myHPRIo|_&JaU-5H zoT*@dxo0LZDmvP2p*>p6)b0s8JA28K<9*JC-?L32uAZJBeI}|rmipo25>+X24(w9w zb{2c+q&y#FEt*!iFkK*F9LwB0{kxRz>h2y4_tyJX_D80TsLJhWJI8N*ea~}quki5j zq?}CT-_D(HA(e;E`|+?L8lI-~{kHyv$Y9QiPd0=nr>B-qPDIku(nKtBjB2sW&I2N8 zX=&RBGf%}OOnDNlrfVOKH9VE??Ci9#u^G@lmC?8IqUE5 zm-=CM@7_fd6B8rnv^`1RE!OKwO4PA^5E(yz-j_LEFS}_PTU;#I-P2R*I4ba7prSNK zJ8k0QsN2&VsrtitLb&1D5s%-$fA7gvc&&|e2sqBShQqx!=(!J96v{iCo%}t7=+?b^ z_nBGUV{V(e*y=|+oR=u@KNJ*D_i^t$5gUDZSwqNcRiBQIj?1fRXQ{7aDg9~8hYvh( zu{tZ=er$~-fwsvikL@O07FKa_+6tGcX2;^XkyW1!xGNPKo6EIzb;e$@8{cA>q-R%F z@Nsc*-wPhUPcmyh+FNrwKYQ|C@Xad@h}@|fpXbW#E=4W0~BhhFTLaB@6MagM8Es=JKiaVxSkxkeZ4P&Qqa~W z$=2hB-8NpULXJ!SJ4GwiLb!-&RtzO`B}CsyvH1*-LHx%0I^0uunzRqklkpAm$Y50- zZtkWImZwqFymu|u$4Z4nM1qTo1XQEx!;0&V2o!o~J$_GCl~hzbud9`f~e)1{@N zviSKmPDot*`TO^m;YhM2bqPQEbAPhhi!e(y8i~95lRsTXYjuzJcw}T`P^iVlMLLoD z>U9Z6Eq3J0CeKpkpUUy^@%5KG8nm{yaxL=n@HBt6|8DW{At@mtq0GUg2bZ8=$JSg+ zTv{5jwY4=J0|SS&bd7ddG~9Uej~@*1fq2BkZSOUbsJ^E{WbTccw~UU`+S=OsZnx1o zMq8hi!0uZ%24cLH^%Lvv?nWNEvQnaL_w{SK(9qDK$6EgWGDioSq#hm~hkyUh!SRYd zT909tCAf0sO4~_HOiW2hiGshsf0mg}=RxJ>@=*Tm(ty*){(pZZ35kgnd3a^i^BR$w zhrQ_fW%%hnzKN-+rHxG(t(h+dgf{BzoNp9%d!Wk0&d}I+p{!5tHlG@2{2XM_=LH44 znwpyZxjHQO@83tE1kCHBN=k$bjf|RGTCf_6H&3k$=^(Rwp7yKg$Pe+)O%($z&sLqkI+=42S6lp?nZ$w1}c_f%YR zGX6VTb6Q&3+})7b^>wEx25}NGF|p-|a^CvWePfiuQg52D%cKM(-HYcZJEQ?8+iJB3 z(|+n28X;9xRr8yYjw5_y6B9g%;t&{o#$Sj~2X-H_AI{XjxcOfF=1oi#qog}_2rlWf z=g-v&myp$a_%Qb6ON#E-!dTwk-mb2$jn7#E^0=6pohd#)JrD5p^=<6#CMvGmr@wyv z`fBQF@k2kS2gXll&rXkFKb~oFhNP#nz~^8LIeYJK1uQqoEx&nKA5>V#zrMa+;=N(i zR!dDy?R&gYG4Ni4_H}Zy}`SleHsqdv57n3zKzzwD7#`sgfRB7OuVUk!_EMnc)0Mxy@tq@bU`VcE7%` z^mo0ikBo&SHaq*$r7KrB*w`>4X$3E!{C9f9A^R%_1js?@*oOSB#2#k>HDY

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introduction to ICP](ICPIntro.md) - [The ICP chain configuration and its variants](DefaultICPConfig.md) - [Configuring libpointmatcher using YAML](Configuration.md) From 7b8f0be846b1b7de99865b32431144ea826eda4d Mon Sep 17 00:00:00 2001 From: nimisbert Date: Sun, 2 Aug 2020 16:27:17 +0200 Subject: [PATCH 2/4] outlier filter catalog corrections --- doc/OutlierFiltersFamilies.md | 436 +++++++++++------- doc/images/bidistance_multikB.jpg | Bin 454060 -> 34286 bytes .../cauchy_weighted_linear_regression.png | Bin 27024 -> 31240 bytes doc/images/cauchy_with_points.png | Bin 62258 -> 64436 bytes doc/index.md | 2 +- 5 files changed, 275 insertions(+), 163 deletions(-) diff --git a/doc/OutlierFiltersFamilies.md b/doc/OutlierFiltersFamilies.md index 18445d7f..35679f34 100644 --- a/doc/OutlierFiltersFamilies.md +++ b/doc/OutlierFiltersFamilies.md @@ -1,13 +1,17 @@ | [Tutorials Home](index.md) | [Previous](ApplyingDataFilters.md) | [Next](ICPIntro.md) | | :--- | :---: | ---: | -# Outlier Filters Families +# Families of Outlier Filters -This section adresses the various filters and tries to organise them in families. The families are based on the influence functions of the cost function associated to an outlier filter. The purpose of outlier filtering, applied to point cloud registration, is to reduce the effect of outliers, improve the error minimisation and improve the crispness of the registered point clouds. +This section adresses the various filters and tries to organise them in families. +The families are based on the influence functions of the cost function associated to an outlier filter. +The purpose of outlier filtering, applied to point cloud registration, is to reduce the effect of outliers, improve the error minimisation and improve the crispness of the registered point clouds. -Appart for the Other family, the presentation of the families and functions is the following. The functions were plotted with an error variable (noted e) ranging from [-4,4] with a step of 0.01. The plot of a family is made with a value of k = 1. The first plot of a function is made with values of k = 0.5, k = 2 and k = 4. Then a second plot is proposed for values of k ranging from [0,2] with a step of 0.1. +Appart from the section [Other family](#otherfamilyhead), the presentation of the families and functions is the following. +Functions were plotted with an error variable (noted e) ranging from [-4,4] with a step of 0.01. +The plot of a family is made with a value of k = 1. The first plot of a function is made with values of k = 0.5, k = 2 and k = 4. Then a second plot is proposed for values of k ranging from [0,2] with a step of 0.1. -This catalog was made from thanks to Philippe Babin's Mémoire on the analysis of error functions for the Iterative closest point algorithm: P.Babin, "Analysis of Error Functions for the Iterative Closest Point Algorithm", pp.70-72. +This catalog was made from thanks to Philippe Babin's master thesis on the analysis of error functions for the Iterative closest point algorithm: P.Babin, "Analysis of Error Functions for the Iterative Closest Point Algorithm", pp.70-72. Libpointmatcher provides an implementation of the following outlier filters: @@ -19,6 +23,29 @@ Libpointmatcher provides an implementation of the following outlier filters: * Huber * L1 * Student +* VarTrimmed +* Trimmed +* Median +* Surface Normal + +Filters not implemented yet are: + +* Andrew +* Kernel-Correlation +* Fast Global Registration +* L2 +* Maximum-Distance +* Generalized ICP +* Logistic +* LM-ICP +* Fair +* FICP +* AICP +* Sparse-ICP +* EM-ICP +* RANSAC +* Relative Motion Threshold +* BiDistance ## Filter Index ### [Pseudo-Sinus Family](#pseudosinfamilyhead) @@ -27,351 +54,436 @@ Libpointmatcher provides an implementation of the following outlier filters: 3. [Welsch](#welschhead) 4. [Tukey](#tukeyhead) 5. [Andrew](#andrewhead) -6. [Kernel-Correlation (KC)](#kernelcorrelationhead) -7. [Fast Global Registration (FGR)](#fastglobalregistrationhead) ### [Linear Family](#linearfamilyhead) -8. [L2](#l2head) -9. [Switchable-Constraint (SC)](#switchableconstrainthead) -10. [Maximum-Distance (MaxDist)](#maximumdistancehead) -11. [Generalized ICP (GICP)](#gicphead) +6. [L2](#l2head) +7. [Switchable-Constraint (SC)](#switchableconstrainthead) +8. [Maximum-Distance (MaxDist)](#maximumdistancehead) ### [Step Family](#stepfamilyhead) -12. [L1](#l1head) -13. [Huber](#huberhead) -14. [Logistic](#logistic) -15. [LM-ICP](#lmicphead) -16. [Fair](#fairhead) +9. [L1](#l1head) +10. [Huber](#huberhead) +11. [Logistic](#logistic) +12. [Fair](#fairhead) ### [Percentile Family](#percentilefamilyhead) -17. [Trimmed](#trimmedhead) -18. [Median](#medianhead) -19. [FICP](#ficphead) -20. [AICP](#aicphead) +13. [Trimmed](#trimmedhead) +14. [Median](#medianhead) +15. [VarTrimmed (Fractional ICP - FICP)](#ficphead) +16. [Articulated ICP](#aicphead) ### [Other Family](#otherfamilyhead) -21. [Student](#studenthead) -22. [Sparse-ICP](#sparsehead) -23. [EM-ICP](#emicphead) -24. [RANSAC](#ransachead) -25. [Relative Motion Threshold (RMT)](#rmthead) -26. [BiDistance](#bidistancehead) +17. [Student](#studenthead) +18. [Sparse-ICP](#sparsehead) +19. [EM-ICP](#emicphead) +20. [RANSAC](#ransachead) +21. [Relative Motion Threshold (RMT)](#rmthead) +22. [BiDistance](#bidistancehead) ## Descriptions -Outlier filters come in different forms. However the most common is a cost function noted p(e,k) associated with a weight function noted w(e,k). The value e is the error associated with a pair of matches and k is a tuning parameter. An influence function is the first derivative of a cost function p'(e,k). For most cases the influence function can be approximated by p'(e,k) = ew(e,k). +The ICP algorithm uses least square minimization to solve the outlier filtering problem. +For this type of problem M-estimator are common outlier filters. + M-estimators are usually expressed by three functions: the cost function p(e), the influence function p'(e) and w(e). + Although some outlier filters don't rely on M-estimators to perform the outlier filtering. + Some outlier filters use random sampling ([RANSAC](#ransachead)) or other parameters based on motions between two iterations of the algorithms ([RMT](#rmthead)). -### Effect of the tuning parameter k +**Effect of the tuning parameter k** -The weighted linear regression was used in order to illustrate the effect of k on outlier filtering. For this example, the chosen outlier filter is Cauchy and the problem is to have better results than an unweighted linear regression. The tuning of the k parameter will determine the efficiency of the outlier filtering. In the following example, a value of k which is nearly equal to the maximum error on the first iteration results nearly as poor as un unweighted linear regression. +The weighted linear regression was used in order to illustrate the effect of k on outlier filtering. +For this example, the chosen outlier filter is [Cauchy](#cauchyhead) and the problem is to have better results than an unweighted linear regression. +The tuning of the k parameter will determine the efficiency of the outlier filtering. +In the following example, a value of k which is nearly equal to the maximum error on the first iteration gives results as poor as un unweighted linear regression. ![cauchy example](images/cauchy_weighted_linear_regression.png) -The following plot represents how the outliers and inliers are affected by the cost function and weight function. The poor results of k = 4 can also be seen in this plot. Indeed, the weight of some outliers is the same as the weight of some inliers for lower values of k. This means that for values of k too big (relative to the point cloud) some outliers will be considered as inliers. The opposite effect can happen where the value of k is so low that some inliers are downweighted like outliers. This means that some prior knowledge of the point cloud is necessary for a fine tuning of the k parameter. +The following plot represents how the outliers and inliers are affected by the cost function and weight function. +The poor results of k = 4 can also be seen in this plot. +Indeed, the weight of some outliers is the same as the weight of some inliers for lower values of k. +For values of k too big (relative to the point cloud, illustrated by k = 4 in the exemple) some outliers will be considered as inliers. +The opposite effect can happen where the value of k is so low (illustrated by k = 0.01) that some inliers are downweighted like outliers. +Indeed, low values of k will not give better results since numerous inliers will be heavily downweighted. +In the example k = 0.01 downweight most of the inliers to 0. +This means that some prior knowledge of the point cloud is necessary for a fine tuning of the k parameter. ![cauchy with error points](images/cauchy_with_points.png) -## Families Catalogue +Methods can be used to prevent the tuning of k. +If the inliers are affected by a Gaussian perturbation with a 0 mean then S-estimators can be used to fine tune k used by the M-estimator. +Libpointmatcher provides one S-estimator called Median absolute deviation (MAD). S-estimators are beyond the scope of this catalog. + +## Families Catalog ### Pseudo-Sinus Family -This family of outlier filters is characterized by an influence function that looks like one period of the sinusoid function. The value of k affect the width of the period. Past the sinusoidal behavior, the influence function becomes null. Usually, inliers are affected by the sinusoidal part while the outliers are affected by the null part of the function. +This family of outlier filters is characterized by an influence function that looks like one period of the sinusoid function. +The value of k affects the width of the period. +Past the sinusoidal behavior, the influence function becomes null. +Usually, inliers are affected by the sinusoidal part, while the outliers are affected by the null part of the function. -The sinusoidal aspect of this family is well presented by Andrew and Tukey which provides a sharp transition between the effect on inliers and outliers. Welsch, Cauchy and Geman-McClure provide an influence function with a smooth transition between the effect on inliers and outliers. Kernel-Correlation and Fast Global Registration are identical to other functions of the family. Therefore they overlap with those functions, Welsch and Geman-McClure respectively, and can't be seen on the plot. +The sinusoidal aspect of this family is well presented by Andrew and Tukey which provides a sharp transition between the effect on inliers and outliers. +Welsch, Cauchy and Geman-McClure provide an influence function with a smooth transition between the effect on inliers and outliers. +Kernel-Correlation (KC) and Fast Global Registration (FGR) are identical to other functions of the family. +Therefore, Welsch and Geman-McClure are hidden under KC and FGR and can't be seen on the plot because they overlap with these functions. ![Pseudo-Sin Family plot](images/pseudo_sin_family.png) #### Cauchy -Equations: +**Equations:** + ![eq cauchy](images/eq_cauchy.JPG) -Plots: +**Plots:** + ![cauchy threek](images/cauchy_threek.png) ![cauchy multik](images/cauchy_multik.png) -Original reference: -M. Bosse and R. Zlot, "Continuous 3D scan-matching with a spinning 2D laser", 2009 IEEE International Conference on Robotics and Automation,pp.4312-4319,2009. +**Reference of its use in ICP:** + +* M. Bosse and R. Zlot, "Continuous 3D scan-matching with a spinning 2D laser", 2009 IEEE International Conference on Robotics and Automation, pp.4312-4319, 2009. #### Geman-McClure -Equations: +**Equations:** + ![eq geman](images/eq_geman.JPG) -Plots: +**Plots:** + ![geman threek](images/geman_threek.png) ![geman multik](images/geman_multik.png) -Original reference: -S. Geman, D.E. McClure, and D. Geman, "A nonlinear filter for film restoration and other problems in image processing", CVGIP: Graphical Models and Image Processing, vol. 54, no.4, pp.281-289, 1992. +**Reference of its use in ICP:** + +* S. Geman, D.E. McClure, and D. Geman, "A nonlinear filter for film restoration and other problems in image processing", CVGIP: Graphical Models and Image Processing, vol. 54, no.4, pp.281-289, 1992. + +This outlier filter is also used in an ICP variant called Fast Global Registration (FGR): +* Q.Y. Zhou, J. Park, and V.Koltun, "Fast global registration", Lecture notes in Computer Science, vol.9906 LNCS, pp. 766-782, 2016 #### Welsch -Equations: +**Equations:** + ![eq welsch](images/eq_welsch.JPG) -Plots: +**Plots:** + ![welsch threek](images/welsch_threek.png) ![welsch multik](images/welsch_multik.png) -Original reference: -R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. +**Reference of its use in ICP:** + +* R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. + +This outlier filter is also used in the ICP variant Kernel-Correlation: +* Y.Tsin and T.Kanade, "A Correlation-Based Approach to Robust Point Set Registration", in 8th European Conference on Computer Vision, vol.0 , 2004, pp.558-569. #### Tukey -Equations: +**Equations:** + ![eq tukey](images/eq_tukey.JPG) -Plots: +**Plots:** + ![tukey threek](images/tukey_threek.png) ![tukey multik](images/tukey_multik.png) -Original reference: -R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. +**Reference of its use in ICP:** + +* R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. #### Andrew -Equations: +**Equations:** + ![eq andrew](images/eq_andrew.JPG) -Plots: +**Plots:** + ![andrew threek](images/andrew_threek.png) ![andrew multik](images/andrew_multik.png) -Original reference: -R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. +**Reference of its use in ICP:** -#### Kernel-Correlation (KC) - -Kernel Correlation is a registration algorithm known to use the Welsch outlier filter. - -Equations: -![eq kc](images/eq_kc.JPG) - -Original reference: -Y.Tsin and T.Kanade, "A Correlation-Based Approach to Robust Point Set Registration", in 8th European Conference on Computer Vision, vol.0 , 2004, pp.558-569 - -#### Fast Global Registration (FGR) - -This registration algorithm is known to use the Geman-McClure outlier filter. - -Equations: -![eq fgr](images/eq_fgr.JPG) - -Original reference: -Q.Y. Zhou, J. Park, and V.Koltun, "Fast global registration", Lecture notes in Computer Science, vol.9906 LNCS, pp. 766-782,2016 +* R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. ### Linear Family -This family of outlier filters is characterized by an influence function that looks like a linear function. Appart from L2, the value of k affects the width of the linear part. Past the linear behavior, the influence function becomes null. Usually, inliers are affected by the linear part of the function while the outliers are affected by the null part of the function. The Switchable-Constraint outlier filter provides a smooth transition between the inlier and outlier part while Maximum Distance provides a sharp transition between those two behaviors. L2 represents the ideal linear influence function. On the plot Maximum Distance and GICP are overlapping. +This family of outlier filters is characterized by an influence function that looks like a linear function. +Appart from L2, the value of k affects the width of the linear part. +Past the linear behavior, the influence function becomes null. +Usually, inliers are affected by the linear part of the function while the outliers are affected by the null part of the function. +The Switchable-Constraint outlier filter provides a smooth transition between the inlier and outlier part, while Maximum Distance provides a sharp transition between those two behaviors. +L2 represents the ideal linear influence function. On the plot Maximum Distance and GICP are overlapping. ![linear Family plot](images/linear_family.png) #### L2 -L2 is a function which is a part of the Lp space family of functions. The L2 outlier filter can't be tuned with a k parameter. +L2 is a function which is a part of the Lp space family of functions. This outlier filter is parameter less, so no parameter needs to be tuned. + +**Equations:** -Equations: ![eq l2](images/eq_l2.JPG) -Plot: +**Plot:** + ![L2 nok](images/L2_nok.png) #### Switchable-Constraint -Equations: +**Equations:** + ![eq sc](images/eq_sc.JPG) -Plots: +**Plots:** + ![switchableconstraint threek](images/switchableconstraint_threek.png) ![switchableconstraint multik](images/switchableconstraint_multik.png) -Original reference: -N. Süderhauf and P. Protzel,"Switchable constraint for robust pose graph slam", in 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2012, pp. 1879-1884. & P Argawal, G.D. Tipaldi, L. Spinello, C.Stachniss, and W.Burgard, "Robust map optimization using dynamic covariance scaling", IEEE international Conference on Robotics and Automation, pp. 62-69, 2013. +**Reference of its use in ICP:** + +* N. Süderhauf and P. Protzel,"Switchable constraint for robust pose graph slam", in 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2012, pp. 1879-1884. +* P Argawal, G.D. Tipaldi, L. Spinello, C.Stachniss, and W.Burgard, "Robust map optimization using dynamic covariance scaling", IEEE international Conference on Robotics and Automation, pp. 62-69, 2013. #### Maximum Distance (MaxDist) -The Maximum Distance outlier filter is used to compute the unscaled distance (e': euclidean error). +**Equations:** -Equations: ![eq maxdist](images/eq_maxdist.JPG) -Plots: +**Plots:** + ![maximumdistance threek](images/maximumdistance_threek.png) ![maximumdistance multik](images/maximumdistance_multik.png) -#### Generalized ICP (GICP) - -This registration algorithm is known to use the Maximum Distance outlier filter on the scaled error (e). - -Equations: -![eq gicp](images/eq_gicp.JPG) - -Original reference: -A.Segal, D.Haehnel, and S.Thrun, "Generalized-ICP", in Robotics: Science and Systems, vol.5, 2009, pp.168-176 +Maximum Distance is used in a variant of the ICP called Generalized-ICP or GICP: +* A.Segal, D.Haehnel, and S.Thrun, "Generalized-ICP", in Robotics: Science and Systems, vol.5, 2009, pp.168-176 ### Step Family -This family of outlier filters is characterized by an influence function that looks like a linear function. Apart from L1, the value of k affects the width of the linear part linking the step. Past the linear behavior, unlike the linear family the influence function becomes equal to -k or k. Usually, inliers are affected by the linear part while the outliers are affected by the -k/k part of the function. The Logistic outlier filter provides a smooth transition between the inliers and outliers parts while Huber provides a sharp transition between the two behaviors. L1 represents the ideal step function therefore it does not have a linear part. The LMICP outlier filter overlaps with the plot of Huber. +This family of outlier filters is characterized by an influence function that looks like a linear function. +Apart from L1, the value of k affects the width of the linear part linking the step. +Past the linear behavior, unlike the linear family the influence function becomes equal to -k or k. +Usually, inliers are affected by the linear part while the outliers are affected by the -k/k part of the function. +The Logistic outlier filter provides a smooth transition between the inliers and outliers parts while Huber provides a sharp transition between the two behaviors. +L1 represents the ideal step function therefore it does not have a linear part. +The LMICP outlier filter overlaps with the plot of Huber. ![step family plot](images/step_family.png) #### L1 -L1 is a function which is a part of the Lp space family of functions. The L1 outlier filter can't be tuned with a k parameter. +L1 is a function which is a part of the Lp space family of functions. +This outlier filter is parameter-less, so no parameter needs to be tuned. + +**Equations:** -Equations: ![eq l1](images/eq_l1.JPG) -Plot: +**Plot:** + ![l1 nok](images/l1_nok.png) #### Huber -Equations: +**Equations:** + ![eq huber](images/eq_huber.JPG) -Plots: +**Plots:** + ![huber threek](images/huber_threek.png) ![huber multik](images/huber_multik.png) -Original reference: -P.J. Huber, "Robust Estimation of a Location Parameter", The Annals of Mathematical Statistics, vol.35, no. 1, pp. 73-101, 1964. +**Reference of its use in ICP:** + +* P.J. Huber, "Robust Estimation of a Location Parameter", The Annals of Mathematical Statistics, vol.35, no. 1, pp. 73-101, 1964. + +A variant of the ICP called LM-ICP uses Huber as an outlier filter: +* A.W. Fitzgibbon, "Robust registration of 2D and 3D point sets", Image and Vision Computing, vol.21, no.13-14, pp.1145-1153, 2003. #### Logistic -Equations: +**Equations:** + ![eq logistic](images/eq_logistic.JPG) -Plots: +**Plots:** + ![logistic threek](images/logistic_threek.png) ![logistic multik](images/logistic_multik.png) -Original reference: -R.E. Welsch, "Robust Regression Using Iteratively reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no.9, pp.813-827, 1977. +**Reference of its use in ICP:** -#### LM-ICP - -This registration algorithm is known to use the Huber outlier filter. - -Equation: -![eq huber](images/eq_huber.JPG) - -Original reference: -A.W. Fitzgibbon, "Robust registration of 2D and 3D point sets", Image and Vision Computing, vol.21, no.13-14, pp.1145-1153, 2003. +* R.E. Welsch, "Robust Regression Using Iteratively reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no.9, pp.813-827, 1977. #### Fair -Equations: +**Equations:** + ![eq fair](images/eq_fair.JPG) -Plots: +**Plots:** + ![fair threek](images/fair_threek.png) ![fair multik](images/fair_multik.png) -Original reference: -R.E. Welsch, "Robust Regression Using Iteratively reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no.9, pp.813-827, 1977. +**Reference of its use in ICP:** + +* R.E. Welsch, "Robust Regression Using Iteratively reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no.9, pp.813-827, 1977. ### Percentile Family -This family of outlier filters uses the percentile in order to affect a binary weight. The values past the f percentile are considered as outliers. Trimmed, AICP, and FICP are the same outlier filters and have a tunable value of f. Median is the same as the other percentile filters, however it's value is non-tunable and equal to 50%. The value of f must be equal to the overlap ratio parameter. +This family of outlier filters uses the percentile in order to affect a binary weight. +The values past the f percentile are considered as outliers. +Trimmed, Articulated ICP (AICP), and Fractional ICP (FICP) are the same outlier filters and have a tunable value of f. +Median is the same as the other percentile filters, however it's value is non-tunable and equal to 50%. +The value of f must be equal to the overlap ratio parameter. + +**Equation:** -Equation: ![eq percentile](images/eq_percentile.JPG) -This filtering method doesn't rely on a function. In order to illustrate it's effect, a Gaussian density function was applied to the range of errors [-4,4] with a step of 0.1. With this simplified problem the overlap ratio parameter can't be computed therefore we affect arbitrary values to f. Respectively for Trimmed, AICP and FICP the values of f are 30%, 60% and 90%. +This filtering method doesn't rely on a function. +In order to illustrate it's effect, a Gaussian density function was applied to the range of errors [-4,4] with a step of 0.1. +With this simplified problem the overlap ratio parameter can't be computed therefore we affect arbitrary values to f. +Respectively for Trimmed, AICP and FICP the values of f are 30%, 60% and 90%. + +AICP is an ICP variants. The outlier filtering method is the same as Trimmed but an external module is used to tune the value of f. ![percentile family plot](images/percentile_family.png) -Original Reference: -Trimmed: -D.Chetverikov, D.Svirko, D.Stepanov and P.Krsek, "The Trimmed Iterative Closest Pointalgorithm", Object recognition supported by user interaction for service robots, vol.3, no. c, pp.545-548, 2002. -Median: -A.Censi, "An ICP variant using point-to-line metric", IEEE International Conference on Robotics and Automation (ICRA), pp. 19-25, 2008. -FICP (i.e. Var. Trim.): -J.M.Phillips, R.Liu, and C.Tomasi, "Outlier robust ICP for minimizing fractional RMSD", in 3DIM 2007 - Proceedings 6th International Conference on 3-D Digital Imaging and Modeling, 2007, pp.427-434. -AICP: -S.Nobili, R.Scona, M.Caravagna, and M.Fallon, "Overlap-based ICP tuning for robust localization of a humanoid robot", in Proceedings - IEEE International Conference on Robotics and Automation, 2017, pp.4721-4728. +**Reference of its use in ICP:** + +* Trimmed: D.Chetverikov, D.Svirko, D.Stepanov and P.Krsek, "The Trimmed Iterative Closest Pointalgorithm", Object recognition supported by user interaction for service robots, vol.3, no. c, pp.545-548, 2002. +* Median: A.Censi, "An ICP variant using point-to-line metric", IEEE International Conference on Robotics and Automation (ICRA), pp. 19-25, 2008. +* FICP (i.e. Var. Trim.): J.M.Phillips, R.Liu, and C.Tomasi, "Outlier robust ICP for minimizing fractional RMSD", in 3DIM 2007 - Proceedings 6th International Conference on 3-D Digital Imaging and Modeling, 2007, pp.427-434. +* AICP: S.Nobili, R.Scona, M.Caravagna, and M.Fallon, "Overlap-based ICP tuning for robust localization of a humanoid robot", in Proceedings - IEEE International Conference on Robotics and Automation, 2017, pp.4721-4728. ### Other Family -This family of function regroups outlier filters that can't be categorized. This is due to their behavior charging drastically (Student, Sparse ICP), dependent of a value and k (RMT, BiDistance) or a different algorithm that doesn't rely on functions (RANSAC). Due to the diversity of those functions a plot for the family can't be provided. +This family of function regroups outlier filters that can't be categorized. +This is due to their behavior charging drastically (Student, Sparse ICP), dependent of a value and k (RMT, BiDistance) or a different algorithm that doesn't rely on functions (RANSAC). +Due to the diversity of those functions a plot for the family can't be provided. -The outlier filters RANSAC, RMT and BiDistance requires a problem in order to be computed. Therefore, the algorithm was applied to a linear regression problem on a cloud of points presenting 17% of outliers. +The outlier filters RANSAC, RMT and BiDistance requires a problem in order to be computed. +Therefore, the algorithm was applied to a linear regression problem on a cloud of points presenting 17% of outliers. #### Student -Equations: +**Equations:** + ![eq student](images/eq_student.JPG) -Plots: +**Plots:** + ![student fourk](images/student_fourk.png) -Original reference: -G.Agamennoni, S.Fontana, R.Y. Siegwart, and D.G. Sorrenti, "Point Clouds Registration with Probabilistic Data Association", IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.4092-4098, 2016. +**Reference of its use in ICP:** + +* G.Agamennoni, S.Fontana, R.Y. Siegwart, and D.G. Sorrenti, "Point Clouds Registration with Probabilistic Data Association", IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.4092-4098, 2016. #### Sparse ICP -Sparse ICP is an outlier function based on the Lp norm therefore it uses a tuning parameter p. In the plots the parameter is noted k so that it can't be mistaken for the result of the cost function p(e,k). This function is only applicable for values of k in the range [0,1]. It is relevant to point out that Sparse ICP for k = 1 is equal to L1. +Sparse ICP is an outlier function based on the Lp norm therefore it uses a tuning parameter p. +In the plots the parameter is noted k so that it can't be mistaken for the result of the cost function p(e,k). +This function is only applicable for values of k in the range [0,1]. +It is relevant to point out that Sparse ICP for k = 1 is equal to L1. + +**Equations:** -Equations: ![eq sparse](images/eq_sparse.JPG) -Plots: +**Plots:** + ![sparseICP fourk](images/sparseicp_fourk.png) -Original reference: -S.Bouaziz, A.Tagliasacchi, and M.Pauly, "Sparse iterative closest point", Eurographics Symposium on Geometry Processing, vol.32, no.5, pp. 113-123, 2013. +**Reference of its use in ICP:** + +* S.Bouaziz, A.Tagliasacchi, and M.Pauly, "Sparse iterative closest point", Eurographics Symposium on Geometry Processing, vol.32, no.5, pp. 113-123, 2013. #### EM-ICP -The EM-ICP algorithm presents an effect on inliers similar to the step function. This is due it being based on L1. The effect on outliers is the same as the functions in the linear or sin families. Since the outlier filtering method is a trimmed version on L1 and it shares behavior with the step and linear families, it is placed in the other family. +The EM-ICP algorithm presents an effect on inliers similar to the step function. +This is due it being based on L1. +The effect on outliers is the same as the functions in the linear or sin families. +Since the outlier filtering method is a trimmed version on L1 and it shares behavior with the step and linear families, it is placed in the other family. + +**Equations:** -Equations: ![eq emicp](images/eq_emicp.JPG) -Plots: +**Plots:** + ![emicp nok](images/emicp_nok.png) -Original reference: -S.Granger, and P.Xavier, "Multi-scale EM-ICP: A Fast and Robust Approach for Surface Registration", Proceedings of the European Conference on Computer Vision, pp. 418-432. +**Reference of its use in ICP:** + +* S.Granger, and P.Xavier, "Multi-scale EM-ICP: A Fast and Robust Approach for Surface Registration", Proceedings of the European Conference on Computer Vision, pp. 418-432. -#### RANSAC +#### RANSAC -The RANSAC outlier filtering algorithm or RANdom SAmple Consensus relies on a voting system that uses inliers. The first step of the algorithm is to randomly sample the cloud of points. Then a model is applied depending on the problem. This model is applied to the sampled points. After the model is computed the algorithm counts the number of values contained within a [-k,k] range from the model. These values are considered as inliers. The model with the highest number of inliers is considered as the result of the algorithm. The RANSAC algorithm uses three parameters: +The RANSAC outlier filtering algorithm or RANdom SAmple Consensus relies on a voting system that uses inliers. +The first step of the algorithm is to randomly sample the cloud of points. +Then, a model is applied depending on the problem. +This model is applied to the sampled points. After the model is computed, the algorithm counts the number of values contained within a [-k,k] range from the model. +These values are considered as inliers. +The model with the highest number of inliers is considered as the result of the algorithm. The RANSAC algorithm uses three parameters: * (s): the number of sampled points * (c): the number of computations * (k): the range containing the inliers -Since the algorithm is based on random sampling, the results can be quite independent of k as shown with the plots. Indeed, for the same amount of points and the same number of computations the result for k = 0.5 ended up being poorer than the example using k = 1.5. This can be explained by the fact that (s) is the predominant tuning parameter for this algorithm. +Since the algorithm is based on random sampling, the results can be quite independent of k as shown with the plots. +Indeed, for the same amount of points and the same number of computations the result for k = 0.5 ended up being poorer than the example using k = 1.5. +This can be explained by the fact that (s) is the predominant tuning parameter for this algorithm. + +**Plots:** -Plots: ![ransac threek](images/ransac_threek.png) -Original reference: -M.A.Fischler and R.C. Bolles, "Random sample consensus: a paradigm for the model fitting with applications to image analysis and automated cartography", Communications of the ACM, vol.24, no.6, pp.381-395, 1981. +**Reference of its use in ICP:** + +* M.A.Fischler and R.C. Bolles, "Random sample consensus: a paradigm for the model fitting with applications to image analysis and automated cartography", Communications of the ACM, vol.24, no.6, pp.381-395, 1981. -Relative motion threshold (RMT): +### Relative motion threshold (RMT) -The relative motion threshold (RMT) uses a threshold based on translation between two transforms. In our simplified case this motion threshold was the mean of the absolute values of the translation on the y axis between two iterations. +The relative motion threshold (RMT) uses a threshold based on translation between two transforms. +In our simplified case this motion threshold was the mean of the absolute values of the translation on the y axis between two iterations. +This simplified example shows that if dt is correctly computed, even the most outrageous initial values can give a good regression result. +In the following plots we gave an example with intial values equal to an unweighted linear regression. +Then in the second plot the initial values are set to start the regression in the outlier point cloud. +In both cases the results are well within the inliers point cloud and give equal results up to the first decimal place. +Therefore the results of RMT are more dependent of the way dt is computed than the initial values given to the outlier filter. + +**Equations:** -Equations: ![eq rmt](images/eq_rmt.JPG) -Plots: +**Plots:** + ![rmt nok](images/rmt_nok.png) -Original reference: -F.Pomerleau, F.Colas, F.Ferland, and F.Michaud,"Relative Motion Threshold for Rejection in ICP Registration", Field and Service Robotics, vol.62 p.229-238, Jul.2015. +**Reference of its use in ICP:** -Bidisance: +* F.Pomerleau, F.Colas, F.Ferland, and F.Michaud,"Relative Motion Threshold for Rejection in ICP Registration", Field and Service Robotics, vol.62 p.229-238, Jul.2015. -The BiDistance outlier filtering method affects an uniform weight based on the tuning parameter k and a parameter computed at each iteration noted B. This computed parameter is the bidirectional correspondence between the two point sets. Since this outlier filter is a weight function with two variables it can be plotted as a 3D graph. For the example the plot was made by values of k and B in the range [-3,3] with a step of 0.1. The weight function results can get really high due to the exponential in the equation, therefore the axis of the weight is limited to [0,4]. +### BiDisance + +The BiDistance outlier filtering method affects an uniform weight based on the tuning parameter k and a parameter computed at each iteration noted B. +This computed parameter is the bidirectional correspondence between the two point sets. Since this outlier filter is a weight function with two variables, it can be plotted as a 3D graph. +For the example the plot was made by values of k and B in the range [-2,2] with a step of 0.02. +The weight function results can get really high due to the exponential in the equation, therefore the axis of the weight is limited to [0,4]. + +**Equation:** -Equation: ![eq bidistance](images/eq_bidistance.JPG) -Plots: +**Plots:** + ![bidistance multikB](images/bidistance_multikB.jpg) -Original reference: -J.Zhu, D.Wang, X.Bai, H.Lu, C.Jin, and Z.Li, "Regression of point clouds based on the ratio of bidirectional distances", Proceedings of the 4th international Conference on 3D vision, pp.102-107, 2016. \ No newline at end of file +**Reference of its use in ICP:** + +* J.Zhu, D.Wang, X.Bai, H.Lu, C.Jin, and Z.Li, "Regression of point clouds based on the ratio of bidirectional distances", Proceedings of the 4th international Conference on 3D vision, pp.102-107, 2016. \ No newline at end of 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zglhsLkeIl*C27e2l`DD&zl&1!_R)VM%2#-ikp#j^D&))0ZZ;^KJ7GyG1HCv}BF-E` z6$Y$tX*sz`o($`Mv|L~u&(YC4f##_9jl)kSZE(UfL-pp3KOhYwtgfwn`1A=6#L$oP z6zs<&B=mth?8{TmtilriIF4-rO6+~}=r3~zZ!0CVN|Z$olAc2472=t+B>)@=9QsZ| zVu_7FZ0^P3g8DvdNk3ps0R%2D%*(-{Y{GTZ6o5$sgFMc_iH2aTHD51xPE33RO*sd|n?NXKE@;Xk?OKI>hl1D( zgp2}dq17Ak0IKqL@W2dn`gZ}#N$Y4I1eN{;$qn13(bUw`;|;FXz>p;XtFwT;tJO8z zhsPY~5CI9nK#*C?Gvtt!p8f{p77xQx0ZT?nDJldMn)zU#E8&?UjB8;N>!`M;`hNqt>S)JN%)eT`oAII zFK|l7A+a*rjc&!`t$u#RoukPY2vf440fE8D+y(4WJ^CB}0Ct06a^i{qn%j;91S(wY zO@#wxM1c0nbd{Rv0=^iawE%M!1x5i>qu)UwTA+;oJWss7Z$E%SNC0_nHa4~*oqCo1 zvMzb4|1{wn5-8{o;LB0BGd|*pk3s(1!9g$Q=B{`CHx&Kn56;oaX#X2`Mw2OX|2GU! naM1MU|Ns8`-;wLTABSNh+zzj?SfLWafgfpc1+fwlgTVg>-ly*7 diff --git a/doc/index.md b/doc/index.md index 8b21254e..07094256 100644 --- a/doc/index.md +++ b/doc/index.md @@ -22,7 +22,7 @@ Beginner - [What can I do with libpointmatcher?](ApplicationsAndPub.md) - [What the different data filters do?](DataFilters.md) - [Example: Applying a chain of data filters](ApplyingDataFilters.md) -- [What are the different outlier filters families?](OutlierFiltersFamilies.md) +- [What are the different outlier filters?](OutlierFiltersFamilies.md) - [Example: An introduction to ICP](ICPIntro.md) - [The ICP chain configuration and its variants](DefaultICPConfig.md) - [Configuring libpointmatcher using YAML](Configuration.md) From 5d13b4a6567927775bb97c2a6124b9fa47f45dd6 Mon Sep 17 00:00:00 2001 From: nimisbert Date: Sun, 2 Aug 2020 16:46:32 +0200 Subject: [PATCH 3/4] test Andrew and L1 title correction --- doc/OutlierFiltersFamilies.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/doc/OutlierFiltersFamilies.md b/doc/OutlierFiltersFamilies.md index 35679f34..0dd0a42e 100644 --- a/doc/OutlierFiltersFamilies.md +++ b/doc/OutlierFiltersFamilies.md @@ -194,7 +194,7 @@ This outlier filter is also used in the ICP variant Kernel-Correlation: * R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. -#### Andrew +#### Andrew **Equations:** @@ -274,7 +274,7 @@ The LMICP outlier filter overlaps with the plot of Huber. ![step family plot](images/step_family.png) -#### L1 +#### L1 L1 is a function which is a part of the Lp space family of functions. This outlier filter is parameter-less, so no parameter needs to be tuned. From 21694ccad962156c162d88b75a38bb96884d6111 Mon Sep 17 00:00:00 2001 From: nimisbert Date: Sun, 2 Aug 2020 17:31:16 +0200 Subject: [PATCH 4/4] small corrections noticed during verif --- doc/OutlierFiltersFamilies.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/doc/OutlierFiltersFamilies.md b/doc/OutlierFiltersFamilies.md index 0dd0a42e..f17a3336 100644 --- a/doc/OutlierFiltersFamilies.md +++ b/doc/OutlierFiltersFamilies.md @@ -93,7 +93,7 @@ For this type of problem M-estimator are common outlier filters. The weighted linear regression was used in order to illustrate the effect of k on outlier filtering. For this example, the chosen outlier filter is [Cauchy](#cauchyhead) and the problem is to have better results than an unweighted linear regression. The tuning of the k parameter will determine the efficiency of the outlier filtering. -In the following example, a value of k which is nearly equal to the maximum error on the first iteration gives results as poor as un unweighted linear regression. +In the following example, a value of k which is nearly equal to the maximum error gives results as poor as un unweighted linear regression. ![cauchy example](images/cauchy_weighted_linear_regression.png) @@ -158,7 +158,7 @@ Therefore, Welsch and Geman-McClure are hidden under KC and FGR and can't be see * S. Geman, D.E. McClure, and D. Geman, "A nonlinear filter for film restoration and other problems in image processing", CVGIP: Graphical Models and Image Processing, vol. 54, no.4, pp.281-289, 1992. -This outlier filter is also used in an ICP variant called Fast Global Registration (FGR): +This outlier filter is also used in an registration algorithm called Fast Global Registration (FGR): * Q.Y. Zhou, J. Park, and V.Koltun, "Fast global registration", Lecture notes in Computer Science, vol.9906 LNCS, pp. 766-782, 2016 #### Welsch @@ -176,7 +176,7 @@ This outlier filter is also used in an ICP variant called Fast Global Registrati * R.E. Welsch, "Robust Regression Using Iteratively Reweighted Least-squares", Communications in Statistics - Theory and Methods, vol. 6, no. 9, pp.813-827, 1977. -This outlier filter is also used in the ICP variant Kernel-Correlation: +This outlier filter is also used in the registration algorithm called Kernel-Correlation (KC): * Y.Tsin and T.Kanade, "A Correlation-Based Approach to Robust Point Set Registration", in 8th European Conference on Computer Vision, vol.0 , 2004, pp.558-569. #### Tukey @@ -447,7 +447,7 @@ This can be explained by the fact that (s) is the predominant tuning parameter f * M.A.Fischler and R.C. Bolles, "Random sample consensus: a paradigm for the model fitting with applications to image analysis and automated cartography", Communications of the ACM, vol.24, no.6, pp.381-395, 1981. -### Relative motion threshold (RMT) +#### Relative motion threshold (RMT) The relative motion threshold (RMT) uses a threshold based on translation between two transforms. In our simplified case this motion threshold was the mean of the absolute values of the translation on the y axis between two iterations. @@ -469,7 +469,7 @@ Therefore the results of RMT are more dependent of the way dt is computed than t * F.Pomerleau, F.Colas, F.Ferland, and F.Michaud,"Relative Motion Threshold for Rejection in ICP Registration", Field and Service Robotics, vol.62 p.229-238, Jul.2015. -### BiDisance +#### BiDisance The BiDistance outlier filtering method affects an uniform weight based on the tuning parameter k and a parameter computed at each iteration noted B. This computed parameter is the bidirectional correspondence between the two point sets. Since this outlier filter is a weight function with two variables, it can be plotted as a 3D graph.

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