(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 8707159, 143667] NotebookOptionsPosition[ 8668182, 142504] NotebookOutlinePosition[ 8668691, 142521] CellTagsIndexPosition[ 8668648, 142518] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" ", FontFamily->"Arial", FontSize->36, FontColor->RGBColor[ 0.8549019607843137, 0.3137254901960784, 0.2549019607843137]], StyleBox["Plane Fitting to Point Cloud via Gr\[ODoubleDot]bner Basis", FontFamily->"Arial", FontSize->28, FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]], StyleBox["\n", FontFamily->"Arial", FontSize->28, FontColor->RGBColor[ 0.8549019607843137, 0.3137254901960784, 0.2549019607843137]] }], "Section", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.58735618365625*^9, 3.587356216125*^9}, { 3.58735711171875*^9, 3.577169708859375*^9}, {3.577169818921875*^9, 3.57716984475*^9}, {3.577169964859375*^9, 3.577170030203125*^9}, { 3.577170142046875*^9, 3.577170145453125*^9}, {3.5771703535625*^9, 3.577170355203125*^9}, {3.577178512859375*^9, 3.577178516875*^9}, { 3.578238097734375*^9, 3.578238103828125*^9}, 3.59066929628125*^9, { 3.59066945253125*^9, 3.590669457359375*^9}, 3.5910228008867373`*^9, 3.59119921134375*^9, {3.5929813221897793`*^9, 3.5929813570714407`*^9}, { 3.5942014401724405`*^9, 3.5942014479880543`*^9}, 3.5942014871597233`*^9, 3.5946468927106476`*^9, 3.595750294809805*^9, 3.5957581889444103`*^9}, TextAlignment->Center, FontFamily->"Comic Sans MS", FontSize->22, FontColor->RGBColor[ 0.8470588235294118, 0.2549019607843137, 0.18823529411764706`]], Cell["\<\ B\[EAcute]la Pal\[AAcute]ncz, \[CapitalAAcute]rp\[AAcute]d Somogyi, Tam\ \[AAcute]s Lovas and Bence Moln\[AAcute]r Department of Photogrammetry and Geoinformatics, Budapest University of Technology and Economy 1521 Budapest, Hungary \ \>", "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.5911035465*^9, 3.591103574578125*^9}, 3.59119921134375*^9, {3.5929813605814466`*^9, 3.5929813700818634`*^9}, { 3.5942012861065702`*^9, 3.594201300442995*^9}, {3.5948013078125415`*^9, 3.594801360462634*^9}, {3.5948954610627475`*^9, 3.5948954615619483`*^9}, { 3.5948955280960655`*^9, 3.594895545708496*^9}, {3.59561121003125*^9, 3.595611225953125*^9}}, TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Abstract", FontFamily->"Arial", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Subsection", CellDingbat->None, CellGroupingRules->{GroupTogetherGrouping, 10001.}, CellChangeTimes->{{3.590157012421875*^9, 3.590157021953125*^9}, 3.591103602484375*^9, 3.591198840078125*^9, 3.591199293390625*^9}, FontColor->RGBColor[ 0.8470588235294118, 0.2549019607843137, 0.18823529411764706`]], Cell["\<\ This notebook presents a new approach for surface reconstruction from point \ clouds. A fast algorithm with zero complexity to fit a plane to point clouds \ in Total Least Squares (TLS) sense has been developed using Gaussian-type \ error distribution. The application of maximum likelihood method results a \ multivariate polynomial system which can be solved via symbolic computation \ using reduced Gr\[ODoubleDot]bner basis in analytic form. In case of plane fitting when datasets are affected heavily by outliers, \ this algorithm can be embedded into the RANdom SAmple Consensus (RANSAC) \ method. In order to increase the effectiveness of this integrated algorithm \ Self-Organizing Map (SOM) can be employed as data preprocessing method to \ eliminate the main portion of the noisy data points by creating \ representative code-book vectors. In this way the total running time can be \ decreased considerably. The study compares the performance of the suggested algorithm to that of \ other methods and demonstrates its efficiency on both synthetic data adopted \ from the literature as well as on real, laser scanned measurements.\ \>", "Text", CellGroupingRules->{GroupTogetherGrouping, 10001.}, CellChangeTimes->{{3.5905664676397285`*^9, 3.5905665059221954`*^9}, { 3.590566536545049*^9, 3.590566623530802*^9}, {3.590566666680478*^9, 3.590566683154107*^9}, {3.5905667542122316`*^9, 3.590566814319137*^9}, { 3.590566925781333*^9, 3.590566947980172*^9}, {3.5905669784158254`*^9, 3.5905669790086265`*^9}, {3.59056702657311*^9, 3.5905671954122066`*^9}, 3.590567318746023*^9, {3.590567350024078*^9, 3.590567431768222*^9}, { 3.5905674766183004`*^9, 3.5905675837280884`*^9}, {3.5905676304657707`*^9, 3.590567678077054*^9}, {3.5905677274043407`*^9, 3.5905677522083845`*^9}, { 3.590567794843259*^9, 3.590567796028861*^9}, {3.590568000826021*^9, 3.5905680392644887`*^9}, {3.5905680761429534`*^9, 3.5905680824141645`*^9}, 3.5905681164378242`*^9, {3.590568241846444*^9, 3.5905682470880537`*^9}, { 3.590668747515625*^9, 3.590668749109375*^9}, {3.5906689500625*^9, 3.5906689633125*^9}, {3.590669018234375*^9, 3.59066902609375*^9}, { 3.590669104484375*^9, 3.5906691641875*^9}, {3.590669203421875*^9, 3.59066924825*^9}, {3.590669380125*^9, 3.5906693824375*^9}, { 3.590927784211629*^9, 3.590927788096036*^9}, {3.591022811213956*^9, 3.591022814349561*^9}, {3.591022898277708*^9, 3.591022906467723*^9}, { 3.591022944110589*^9, 3.5910231231053033`*^9}, {3.5910231842730107`*^9, 3.591023341599287*^9}, {3.5910233913477745`*^9, 3.591023755514814*^9}, { 3.5910238204577284`*^9, 3.5910238549649887`*^9}, {3.591023947972352*^9, 3.591024043163719*^9}, {3.5910240819609876`*^9, 3.5910241201186543`*^9}, { 3.591024550679411*^9, 3.5910245512410116`*^9}, 3.5910245899602795`*^9, { 3.591024633359556*^9, 3.5910247167261024`*^9}, 3.591198840078125*^9, 3.591199293390625*^9, {3.5929813854634905`*^9, 3.592981582445036*^9}, { 3.592981621819505*^9, 3.592981684344415*^9}, {3.592981718056074*^9, 3.592981769941766*^9}, {3.5929818110634375`*^9, 3.5929818354930806`*^9}, { 3.5929818655855336`*^9, 3.5929822691426425`*^9}, {3.5929823689048176`*^9, 3.5929823784052343`*^9}, {3.5929824760302057`*^9, 3.592982633309682*^9}, { 3.5929826658669395`*^9, 3.592982924156593*^9}, {3.5929829564330497`*^9, 3.5929830347139874`*^9}, {3.592983100514902*^9, 3.592983116614131*^9}, { 3.5930773934821377`*^9, 3.5930773939351635`*^9}, 3.5930774540256004`*^9, { 3.5942008941090813`*^9, 3.5942009543563876`*^9}, {3.594201513352169*^9, 3.594201514927772*^9}, {3.59464704782172*^9, 3.59464706479455*^9}, { 3.5946471064934235`*^9, 3.5946471734955406`*^9}, 3.5946517776564274`*^9, { 3.5946518119296875`*^9, 3.5946518286841173`*^9}, {3.5946527277292967`*^9, 3.5946527544365435`*^9}, {3.594652792906211*^9, 3.59465279817902*^9}, { 3.5948937291877055`*^9, 3.594893747829738*^9}, {3.595604215671875*^9, 3.595604299671875*^9}, {3.595741169520177*^9, 3.59574117115818*^9}, { 3.5957522824684362`*^9, 3.595752285182841*^9}, {3.5957582716713552`*^9, 3.595758278800568*^9}, {3.5957583567539053`*^9, 3.5957583632279167`*^9}, { 3.595758397376376*^9, 3.5957583982967777`*^9}, {3.595767166015625*^9, 3.595767172359375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Keywords", FontFamily->"Arial", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Subsection", CellDingbat->None, CellGroupingRules->{GroupTogetherGrouping, 10002.}, CellChangeTimes->{{3.590157012421875*^9, 3.590157021953125*^9}, { 3.5905679241518865`*^9, 3.5905679310314984`*^9}, 3.591198830515625*^9, 3.59119921134375*^9, 3.591199305734375*^9}, FontColor->RGBColor[ 0.8470588235294118, 0.2549019607843137, 0.18823529411764706`]], Cell["\<\ Surface reconstruction, plane fitting, point cloud, LiDAR, TLS, Gr\ \[ODoubleDot]bner basis, outliers, RANSAC, SOM.\ \>", "Text", CellGroupingRules->{GroupTogetherGrouping, 10002.}, CellChangeTimes->{{3.5905679356179066`*^9, 3.59056796580396*^9}, { 3.5905681412106676`*^9, 3.590568157918297*^9}, {3.591024138074286*^9, 3.591024178961958*^9}, {3.5910247399077435`*^9, 3.591024744494151*^9}, 3.591198830515625*^9, 3.59119921134375*^9, 3.591199305734375*^9, { 3.594651832974125*^9, 3.594651834300127*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["1 - Introduction", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Section", CellGroupingRules->{GroupTogetherGrouping, 10003.}, CellChangeTimes->{{3.59110430575*^9, 3.591104305765625*^9}, { 3.59110434415625*^9, 3.59110434515625*^9}, 3.59110456334375*^9, 3.591198812671875*^9, 3.591199324375*^9}, FontFamily->"Arial", FontSize->15], Cell[TextData[{ "In many fields such as robotics (", StyleBox["Poppinga at al 2008", FontSlant->"Italic"], "), computer vision (", StyleBox["Mitra and Nguyen 2003", FontSlant->"Italic"], "), digital photogrammetry (", StyleBox["Yang and F\[ODoubleDot]rtsner 2010", FontSlant->"Italic"], "), surface reconstruction (", StyleBox["Nurunnabi, Belton and West 2012", FontSlant->"Italic"], "), computational geometry (", StyleBox["Luk\[AAcute]cs, Martin and Marshall 1998", FontSlant->"Italic"], ") as well as in the increasing applications of laser scanning (", StyleBox["Stathas et al 2003", FontSlant->"Italic"], ") , it is a fundamental task for extracting plane features from 3D point \ cloud. Since the physical limitations of the sensors, the occlusions, \ multiple reflectance and noise can produce off-surface points, robust fitting \ techniques are required. Perhaps the three most popular approaches for plane \ fitting in sense of Total Least Squares (TLS) are the Singular Values \ Decomposition (SVD), the Principal Component Analysis (PCA) and the RANdom \ Sample Consensus (RANSAC) which can eliminate the so-called outliers using a \ noise threshold. For example ", StyleBox["Huang and Tseng", FontSlant->"Italic"], " (2008) employed Danish robust estimation with TLS as well as PCA. However, \ there are other type of methods, too e.g. that are based on minimum \ covariance determinant method (", StyleBox["Russeeuw and Van Driessen, 1999", FontSlant->"Italic"], "), expectation maximization method (", StyleBox["Lakaemper and Latecki 2006", FontSlant->"Italic"], "), Bayesian techniques (", StyleBox["Diebel et al, 2006", FontSlant->"Italic"], "), region growing algorithm (", StyleBox["Chen and Stamos, 2007", FontSlant->"Italic"], ") and so on. In this computational study we introduce a new efficent method \ based on computer algebra.\n\nThe notebook is arranged as follows: Section \ 2, we consider a toy problem and solve it using SVD and PCA techniques. In \ Section 3 a new algorithm is developed based on the maximum likelihood \ estimation employing TLS error definition and Gaussian error distribution. \ The related optimization problem is converted into a multivariate polynomial \ system which can be solved via reduced Gr\[ODoubleDot]bner basis in symbolic \ form. This results a non-iterative formula which has zero complexity and \ practically is as fast the PCA method even, when the solution is carried out \ - from practical point of view- with numerical Gr\[ODoubleDot]bner basis. \ In Section 4 these 3 methods - SVD, PCA and the Gr\[ODoubleDot]bner basis \ solution - are compared on the basis of a synthetic dataset adopted from the \ literature as a benchmark. In Section 5 the problem of outliers is \ illustrated. In Section 6 the Gr\[ODoubleDot]bner basis solution is \ integrated into RANSAC and applied to an artificial dataset corrupted with \ outliers. In Section 7 SOM is employed as a data preprocessing technique \ which reduces the amount of points to some tens of representative points, so \ called code-book vectors. In this way the running time of the SOM and that of \ the following RANSAC can be half of the running time of RANSAC with the \ original point of cloud. \nIn Section 8 real laser scanner measurement of \ grassy natural slope has been investigated. Computations with and without \ RANSAC as well as with SOM data preprocessing have been carried out and their \ results have been compared. These computations also verify the efficiency of \ the suggested algorithm based on Gr\[ODoubleDot]bner basis as well as that of \ the SOM preprocessing technique. In Appendix we demonstrate the known fact \ that PCA is not proper for RANSAC, (", StyleBox["Mitra and Nguyen 2003", FontSlant->"Italic"], ") therefore our suggested algorithm is reasonable.\nThe computations have \ been carried out on HP Z 420 workstation with configuration of Xeon E5-1650, \ 3.2 GHz processor and 16 GB RAM.\n " }], "Text", CellGroupingRules->{GroupTogetherGrouping, 10003.}, CellChangeTimes->{{3.591182511*^9, 3.591182543078125*^9}, { 3.591182583109375*^9, 3.5911826234375*^9}, {3.59118266834375*^9, 3.59118270409375*^9}, {3.591182739859375*^9, 3.5911827526875*^9}, { 3.591182793125*^9, 3.591183143609375*^9}, {3.5911831898125*^9, 3.591183365140625*^9}, {3.59118340078125*^9, 3.591183647*^9}, { 3.591183723984375*^9, 3.59118382278125*^9}, {3.5911839455625*^9, 3.591183947609375*^9}, {3.591184008125*^9, 3.59118404803125*^9}, { 3.59118408721875*^9, 3.591184097390625*^9}, {3.591184129234375*^9, 3.59118416521875*^9}, {3.591184248546875*^9, 3.5911842746875*^9}, { 3.591184305421875*^9, 3.591184313703125*^9}, {3.5911843585625*^9, 3.59118444921875*^9}, {3.5911845434375*^9, 3.591184577109375*^9}, { 3.5911881735*^9, 3.591188203765625*^9}, {3.59118830996875*^9, 3.591188335140625*^9}, {3.59118836959375*^9, 3.591188394375*^9}, { 3.591188439984375*^9, 3.591188610171875*^9}, {3.591188767734375*^9, 3.5911888428125*^9}, {3.5911889548125*^9, 3.591188969375*^9}, { 3.591189006625*^9, 3.591189012546875*^9}, {3.59119007296875*^9, 3.59119010059375*^9}, {3.591190194453125*^9, 3.591190231328125*^9}, { 3.591190296875*^9, 3.59119039040625*^9}, 3.591198812671875*^9, 3.591199324375*^9, 3.594647248126072*^9, {3.594647331180618*^9, 3.5946473351586246`*^9}, {3.5946474693500605`*^9, 3.594647519956549*^9}, { 3.594647568223034*^9, 3.5946475832926607`*^9}, {3.5946476271443377`*^9, 3.59464762850154*^9}, {3.594647715190892*^9, 3.594647779853006*^9}, { 3.594647874186372*^9, 3.594648037924259*^9}, {3.594648076565527*^9, 3.5946484794206343`*^9}, {3.5946485988700447`*^9, 3.5946487077426357`*^9}, {3.594648754105917*^9, 3.5946487782235594`*^9}, { 3.5946488573624983`*^9, 3.594648967904292*^9}, {3.594648998792347*^9, 3.5946490151411753`*^9}, {3.5946490468092313`*^9, 3.5946491221885633`*^9}, {3.594649429821104*^9, 3.594649535012089*^9}, { 3.5946495907821865`*^9, 3.5946495956805954`*^9}, {3.5946496265374494`*^9, 3.594649747219261*^9}, 3.594649784160126*^9, {3.594649917711961*^9, 3.5946500166941347`*^9}, {3.594650054555401*^9, 3.5946500806074467`*^9}, { 3.5946501571723814`*^9, 3.5946505763139176`*^9}, {3.5946506224119987`*^9, 3.5946506234260006`*^9}, {3.594650704202942*^9, 3.594650707806548*^9}, { 3.5946507878814893`*^9, 3.5946508088635263`*^9}, {3.5946508397671804`*^9, 3.5946510487295475`*^9}, {3.5946510881040163`*^9, 3.5946510990240355`*^9}, {3.594651193794202*^9, 3.59465130099759*^9}, { 3.594802228292158*^9, 3.5948022654826236`*^9}, {3.59480328839202*^9, 3.5948032888444214`*^9}, {3.594803691715129*^9, 3.594803694273533*^9}, { 3.5948064718428116`*^9, 3.594806474214016*^9}, {3.5948066054726467`*^9, 3.5948066206670732`*^9}, {3.5948067096340294`*^9, 3.5948067473860955`*^9}, {3.5948068316106434`*^9, 3.5948068718743143`*^9}, {3.5948935329393606`*^9, 3.5948936306891327`*^9}, {3.594893819543064*^9, 3.594893892301592*^9}, 3.59560431621875*^9, {3.595604362046875*^9, 3.59560440415625*^9}, { 3.595604449640625*^9, 3.59560449378125*^9}, {3.595604664203125*^9, 3.5956046941875*^9}, 3.59560472959375*^9, {3.595604770078125*^9, 3.595604790390625*^9}, {3.595604846109375*^9, 3.595604858875*^9}, { 3.595604923203125*^9, 3.5956050065625*^9}, {3.59560503953125*^9, 3.59560507475*^9}, 3.59560510871875*^9, {3.595605208296875*^9, 3.595605209171875*^9}, {3.595605295453125*^9, 3.595605415796875*^9}, { 3.59560559009375*^9, 3.59560563675*^9}, 3.595611275734375*^9, 3.595611631484375*^9, {3.595647343703125*^9, 3.595647480515625*^9}, 3.5957413062076173`*^9, {3.59574138166495*^9, 3.5957413827725515`*^9}, 3.595752377207402*^9, {3.5957524268778896`*^9, 3.595752476095976*^9}, { 3.5957525101040354`*^9, 3.595752510852837*^9}, {3.595752742840844*^9, 3.595752767972489*^9}, {3.595752856393444*^9, 3.5957528628986554`*^9}, { 3.59575297376805*^9, 3.595753020552532*^9}, {3.5957531514991627`*^9, 3.5957531591119757`*^9}, {3.5957532121052685`*^9, 3.595753219016081*^9}, { 3.5957541053588376`*^9, 3.5957541169652576`*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["2 - Related works and principles", FontSize->16, FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Section", CellChangeTimes->{{3.591104462484375*^9, 3.591104486203125*^9}, { 3.591184482109375*^9, 3.591184494359375*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.591199355046875*^9, 3.591199383890625*^9}, { 3.5941939985001698`*^9, 3.594194039590642*^9}, {3.5941940740511026`*^9, 3.5941940912267327`*^9}, 3.5957407776242886`*^9, 3.5957408187303605`*^9}, FontFamily->"Arial", FontSize->15, FontWeight->"Bold", FontColor->RGBColor[ 0.8784313725490196, 0.24705882352941178`, 0.15294117647058825`]], Cell[CellGroupData[{ Cell[TextData[StyleBox["2 - 1 Total Least Squares Approach", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Subsection", CellDingbat->None, CellChangeTimes->{{3.59419404398985*^9, 3.594194064472686*^9}, { 3.5941942303789773`*^9, 3.5941942322977805`*^9}, 3.595740559442305*^9, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, FontColor->RGBColor[0.9176470588235294, 0.30196078431372547`, 0.]], Cell[TextData[{ "In case of ordinary Least Squares (OLS) the residuals are the vertical \ distances between the points and the plane to be fitted. To overcome the bias \ to one direction TLS can be employed that minimizes the squared sum of the \ orthogonal distances between the points and the plane. This is the ", StyleBox["geometrical interpretation", FontSlant->"Italic"], " of the TLS. ", "However,", " there is an equivalent ", StyleBox["algebraic interpretation", FontSlant->"Italic"], ", too. In this case one should adjust all of the measured ", StyleBox["(", FontFamily->"Times New Roman"], Cell[BoxData[ SubscriptBox["x", "i"]], FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", ", FontFamily->"Times New Roman", FontSlant->"Italic"], Cell[BoxData[ SubscriptBox["y", "i"]], FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic"], Cell[BoxData[ SubscriptBox["z", "i"]], FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[")", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic"], "coordinates in order to satisfy the plane equation, namely\n" }], "Text", CellChangeTimes->{{3.591184592640625*^9, 3.5911846699375*^9}, { 3.59118471459375*^9, 3.591184838796875*^9}, {3.59118486921875*^9, 3.59118502790625*^9}, {3.591185079734375*^9, 3.59118517871875*^9}, { 3.59118525875*^9, 3.59118528296875*^9}, 3.5911853929375*^9, 3.591198885015625*^9, 3.59119921134375*^9, {3.591199355046875*^9, 3.591199383890625*^9}, 3.594893914687631*^9, 3.595611639296875*^9, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["z", "i"], "+", SubscriptBox["\[CapitalDelta]z", "i"]}], "=", " ", RowBox[{ RowBox[{ RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ SubscriptBox["x", "i"], " ", "+", SubscriptBox["\[CapitalDelta]x", "i"]}], ")"}]}], "+", RowBox[{"\[Beta]", " ", RowBox[{"(", RowBox[{ SubscriptBox["y", "i"], "+", SubscriptBox["\[CapitalDelta]y", "i"]}], ")"}]}], "+", "\[Gamma]"}], "=", "0"}]}], ",", " ", RowBox[{"i", "=", " ", "1"}], ",", "...", ",", "\[DoubleStruckCapitalN]"}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.5839901802144375`*^9, 3.583990231215355*^9}, { 3.587478003890625*^9, 3.578238157546875*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.591199355046875*^9, 3.591199383890625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ and the sum of these squared adjustment must be minimized.\ \>", "Text", CellChangeTimes->{{3.59118533059375*^9, 3.591185395984375*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.591199355046875*^9, 3.591199383890625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, { 3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell["Therefore the objective function is", "Text", CellChangeTimes->{{3.5847863580625*^9, 3.58478642459375*^9}, { 3.59118548871875*^9, 3.59118549775*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.591199355046875*^9, 3.591199383890625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", " ", "=", "1"}], "\[DoubleStruckCapitalN]"], SuperscriptBox[ SubscriptBox["\[CapitalDelta]x", "i"], "2"]}], "+", SuperscriptBox[ SubscriptBox["\[CapitalDelta]y", "i"], "2"], "+", SuperscriptBox[ SubscriptBox["\[CapitalDelta]z", "i"], "2"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.5839899661901965`*^9, 3.583990021644368*^9}, { 3.58747789146875*^9, 3.58747789784375*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.591199383890625*^9}, 3.595605702734375*^9, 3.595740614900402*^9, 3.595740649953664*^9, { 3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "and this should be minimized under the constrains of Eq. (1). Since the \ constrains are linear, eliminating ", Cell[BoxData[ SubscriptBox["\[CapitalDelta]z", "i"]], CellChangeTimes->{{3.5839899661901965`*^9, 3.583990021644368*^9}, { 3.58747789146875*^9, 3.58747789784375*^9}}, TextAlignment->Center, FontFamily->"Times New Roman"] }], "Text", CellChangeTimes->{{3.5847864445625*^9, 3.5847864513125*^9}, 3.5847864839375*^9, {3.59118552159375*^9, 3.59118561653125*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.591199383890625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, { 3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "\[DoubleStruckCapitalN]"], SuperscriptBox[ SubscriptBox["\[CapitalDelta]x", "i"], "2"]}], "+", SuperscriptBox[ SubscriptBox["\[CapitalDelta]y", "i"], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"\[Alpha]", " ", RowBox[{"(", RowBox[{ SubscriptBox["x", "i"], " ", "+", SubscriptBox["\[CapitalDelta]x", "i"]}], ")"}]}], "+", RowBox[{"\[Beta]", " ", RowBox[{"(", RowBox[{ SubscriptBox["y", "i"], "+", SubscriptBox["\[CapitalDelta]y", "i"]}], ")"}]}], "+", "\[Gamma]", "-", SubscriptBox["z", "i"]}], ")"}], "2"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.5839899661901965`*^9, 3.583990021644368*^9}, { 3.58747789146875*^9, 3.58747789784375*^9}, {3.587478459875*^9, 3.587478492140625*^9}, 3.591198885015625*^9, 3.59119921134375*^9, { 3.5911993550625*^9, 3.591199383890625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, TextAlignment->Center, FontFamily->"Times New Roman"], Cell["\<\ we get an unconstrained problem with 2 \[DoubleStruckCapitalN]+3 variables \ instead of 3 \[DoubleStruckCapitalN]+3.\ \>", "Text", CellChangeTimes->{{3.587478511359375*^9, 3.578238209515625*^9}, { 3.591185646453125*^9, 3.59118571390625*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.591199383890625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[TextData[{ "Let us consider a toy problem of four 3D points to illustrate the method, \ see ", StyleBox["Nievergelt", FontSlant->"Italic"], " (2000). The coordinates of the measured points are," }], "Text", CellChangeTimes->{{3.58747884590625*^9, 3.58747886528125*^9}, { 3.591185776546875*^9, 3.591185838203125*^9}, {3.59118611109375*^9, 3.591186112703125*^9}, {3.591186233625*^9, 3.59118623925*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.591199383890625*^9}, {3.594807657616494*^9, 3.5948076725925207`*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"XYZ", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"11", ",", "45", ",", "38"}], "}"}], ",", RowBox[{"{", RowBox[{"47", ",", "54", ",", "38"}], "}"}], ",", RowBox[{"{", RowBox[{"17", ",", "12", ",", "14"}], "}"}], ",", RowBox[{"{", RowBox[{"21", ",", "29", ",", "58"}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.58735469765625*^9, 3.587354747859375*^9}, { 3.587477764109375*^9, 3.587477765234375*^9}, 3.587477950484375*^9, 3.591198885015625*^9, 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The problem then reduces to finding a point ", Cell[BoxData[ FormBox[ SubscriptBox["c", "0"], TraditionalForm]]], "\[Element]", StyleBox["H ", FontSlant->"Italic"], "and a non-zero normal vector ", StyleBox["x", FontSlant->"Italic"], " orthogonal to ", StyleBox["H", FontSlant->"Italic"], " that minimizes the sum of the squared distances," }], "Text", CellChangeTimes->{{3.553513920984375*^9, 3.553514138875*^9}, { 3.5535142365*^9, 3.553514331234375*^9}, {3.553514423140625*^9, 3.553514479265625*^9}, 3.556343606*^9, {3.5894339290648584`*^9, 3.5894339318580184`*^9}, 3.591198885015625*^9, 3.59119921134375*^9, { 3.5911993550625*^9, 3.59119938390625*^9}, 3.595605752796875*^9, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["D", FontSlant->"Italic"], " ", RowBox[{"(", RowBox[{ StyleBox["x", FontSlant->"Italic"], ",", SubscriptBox[ StyleBox["c", FontSlant->"Italic"], "0"]}], ")"}]}], "=", " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "m"], FractionBox[ SuperscriptBox[ RowBox[{"\[LeftBracketingBar]", RowBox[{"\[LeftAngleBracket]", RowBox[{ RowBox[{ SubscriptBox[ StyleBox["c", FontSlant->"Italic"], "i"], "-", SubscriptBox[ StyleBox["c", FontSlant->"Italic"], "0"]}], ",", StyleBox["x", FontSlant->"Italic"]}], "\[RightAngleBracket]"}], "\[RightBracketingBar]"}], "2"], RowBox[{"\[LeftAngleBracket]", RowBox[{ StyleBox["x", FontSlant->"Italic"], ",", StyleBox["x", FontSlant->"Italic"]}], "\[RightAngleBracket]"}]]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.55351449971875*^9, 3.55351455346875*^9}, { 3.553514607484375*^9, 3.553514672265625*^9}, {3.55351587615625*^9, 3.5535159193125*^9}, 3.591198885015625*^9, 3.59119921134375*^9, { 3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ "Introducing a matrix ", Cell[BoxData[ FormBox[ SubscriptBox["C", SubscriptBox["c", "0"]], TraditionalForm]]], " with ", StyleBox["i-", FontSlant->"Italic"], "th rows ", Cell[BoxData[ FormBox[ SubscriptBox["c", "i"], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ SubscriptBox["c", "0"], TraditionalForm]]], "," }], "Text", CellChangeTimes->{{3.5535147834375*^9, 3.5535149659375*^9}, { 3.553516977390625*^9, 3.55351697865625*^9}, {3.5894339626787815`*^9, 3.589433969674181*^9}, 3.591198885015625*^9, 3.59119921134375*^9, { 3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", SubscriptBox["c", "0"]], "=", SuperscriptBox[ RowBox[{"[", " ", RowBox[{ RowBox[{ SubscriptBox["c", "1"], "-", SubscriptBox["c", "0"]}], ",", "...", ",", " ", RowBox[{ SubscriptBox["c", "m"], "-", SubscriptBox["c", "0"]}]}], "]"}], StyleBox["T", FontSlant->"Plain"]]}], TraditionalForm]], "DisplayFormulaNumbered", CellChangeTimes->{{3.553514977265625*^9, 3.553515034015625*^9}, { 3.55351684971875*^9, 3.553516851421875*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, TextAlignment->Center], Cell["we obtain that", "Text", CellChangeTimes->{{3.553515051671875*^9, 3.553515066484375*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, { 3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{ StyleBox["D", FontSlant->"Italic"], RowBox[{"(", RowBox[{ StyleBox["x", FontSlant->"Italic"], ",", SubscriptBox[ StyleBox["c", FontSlant->"Italic"], "0"]}], ")"}]}], FontFamily->"Times New Roman"], "=", FractionBox[ SuperscriptBox[ SubscriptBox[ RowBox[{"\[LeftDoubleBracketingBar]", StyleBox[ RowBox[{ SubscriptBox["C", SubscriptBox["c", "0"]], "x"}], FontFamily->"Times New Roman", FontSlant->"Italic"], "\[RightDoubleBracketingBar]"}], "2"], "2"], SuperscriptBox[ SubscriptBox[ RowBox[{"\[LeftDoubleBracketingBar]", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic"], "\[RightDoubleBracketingBar]"}], "2"], "2"]]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.553515070203125*^9, 3.553515070828125*^9}, { 3.553515166375*^9, 3.5535152534375*^9}, {3.55351529025*^9, 3.553515305515625*^9}, 3.55351594271875*^9, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, TextAlignment->Center], Cell[TextData[{ "Now let us consider two lemmas. \n", StyleBox["Lemma 1", FontSlant->"Italic"], " : The optimal hyperplane must pass through the centroid of the data. Namely" }], "Text", CellChangeTimes->{{3.553515500265625*^9, 3.553515604171875*^9}, { 3.553516581578125*^9, 3.553516583109375*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, {3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ SubscriptBox[ StyleBox["c", FontSlant->"Italic"], "0"], "=", RowBox[{ FractionBox["1", StyleBox["m", FontSlant->"Italic"]], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "m"], SubscriptBox[ StyleBox["c", FontSlant->"Italic"], "i"]}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.5535156329375*^9, 3.55351570415625*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, { 3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}, TextAlignment->Center, FontFamily->"Times New Roman"], Cell[TextData[{ StyleBox["Lemma 2", FontSlant->"Italic"], " : For every point ", Cell[BoxData[ FormBox[ SubscriptBox["c", "0"], TraditionalForm]]], ", let ", StyleBox["v", FontSlant->"Italic"], " be a singular vector corresponding to the smallest singular value \[Sigma] \ of matrix ", Cell[BoxData[ FormBox[ SubscriptBox["C", SubscriptBox["c", "0"]], TraditionalForm]]], ", then" }], "Text", CellChangeTimes->{{3.553515500265625*^9, 3.553515604171875*^9}, 3.553515761875*^9, {3.553515827984375*^9, 3.553515847890625*^9}, 3.553515965625*^9, {3.553516009125*^9, 3.553516139328125*^9}, 3.591198885015625*^9, 3.59119921134375*^9, {3.5911993550625*^9, 3.59119938390625*^9}, 3.595740614900402*^9, 3.595740649953664*^9, { 3.5957407119481726`*^9, 3.5957407589042554`*^9}, 3.5957408187303605`*^9}], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["D", FontSlant->"Italic"], " ", RowBox[{"(", RowBox[{ StyleBox["x", FontSlant->"Italic"], ",", SubscriptBox[ StyleBox["c", FontSlant->"Italic"], "0"]}], ")"}]}], "\[GreaterEqual]", " ", RowBox[{ 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Let us consider the values of \ the formal objective function based on the different real solutions\ \>", "Text", CellChangeTimes->{{3.58920258534375*^9, 3.589202680078125*^9}, { 3.590929636137682*^9, 3.590929693202582*^9}, 3.59119900196875*^9, 3.591199062921875*^9, 3.5911990958125*^9, 3.59119914215625*^9, 3.591199211390625*^9, 3.591199257578125*^9, 3.59119946440625*^9, 3.591199537828125*^9, 3.591199597203125*^9, {3.591199888265625*^9, 3.59119989640625*^9}, 3.5912000905625*^9, 3.591200260078125*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L\[Alpha]\[Beta]\[Gamma]", "=", RowBox[{ RowBox[{ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"L", "/.", "#"}], "&"}], ",", "selsol"}], "]"}], "/.", RowBox[{"\[Sigma]", "\[Rule]", "1"}]}], "//", "N"}]}]], "Input", CellChangeTimes->{{3.5891821625*^9, 3.58918223871875*^9}, { 3.58918349096875*^9, 3.589183579625*^9}, 3.58920256578125*^9, { 3.589202650265625*^9, 3.589202654796875*^9}, {3.589203423796875*^9, 3.589203437078125*^9}, 3.59119900196875*^9, 3.591199062921875*^9, 3.5911990958125*^9, 3.59119914215625*^9, 3.591199211390625*^9, 3.591199257578125*^9, 3.59119946440625*^9, 3.591199537828125*^9, 3.591199597203125*^9, {3.591199888265625*^9, 3.59119989640625*^9}, 3.5912000905625*^9, 3.591200260078125*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "829.2749288217645`"}], ",", RowBox[{"-", "165.67575413281892`"}], ",", RowBox[{"-", "393.0765794438715`"}]}], "}"}]], "Output", CellChangeTimes->{{3.589182187421875*^9, 3.58918224025*^9}, {3.589183476*^9, 3.58918358259375*^9}, 3.58920329728125*^9, 3.589203440359375*^9, { 3.589204845046875*^9, 3.578843137515625*^9}, 3.578845348765625*^9, 3.578888944125*^9, 3.578890112484375*^9, 3.57889130109375*^9, 3.5893783611877813`*^9, 3.589434313503847*^9, 3.589437085338387*^9, 3.5896372772800446`*^9, 3.5902404776300573`*^9, 3.5909296976953897`*^9, 3.5910114467655954`*^9, 3.5910247913878336`*^9, 3.5910250985835733`*^9, 3.59119900196875*^9, 3.591199062921875*^9, 3.5911990958125*^9, 3.59119914215625*^9, 3.591199211390625*^9, 3.591199257578125*^9, 3.59119946440625*^9, 3.591199537828125*^9, 3.591199597203125*^9, { 3.591199888265625*^9, 3.59119989640625*^9}, 3.5912000905625*^9, 3.591200260078125*^9, 3.5929838037485375`*^9, 3.5930784465623703`*^9, 3.5930791598951707`*^9, 3.5941964310904427`*^9, 3.5948099206424694`*^9, 3.59575338370557*^9, 3.5957542178210354`*^9}] }, Open ]], Cell["\<\ The real solution will be selected accordingly which maximizes the objective \ function. The value of \[Sigma] will not influence the relative comparison of \ the real solutions, therefore we consider \[Sigma] = 1. Then the optimal \ solution, which gives the highest value for the maximum likelihood function is\ \>", "Text", CellChangeTimes->{{3.589202699390625*^9, 3.5892027625*^9}, { 3.589202821953125*^9, 3.589202941046875*^9}, {3.589203521453125*^9, 3.589203576609375*^9}, {3.5909297171642237`*^9, 3.590929721516632*^9}, { 3.5909297596274986`*^9, 3.5909298512776594`*^9}, {3.590930000257921*^9, 3.590930023517562*^9}, 3.59119900196875*^9, 3.591199062921875*^9, 3.5911990958125*^9, 3.59119914215625*^9, 3.591199211390625*^9, 3.591199257578125*^9, 3.59119946440625*^9, 3.591199537828125*^9, 3.591199597203125*^9, {3.591199888265625*^9, 3.59119989640625*^9}, 3.5912000905625*^9, 3.591200260078125*^9, {3.5956069191875*^9, 3.595606955546875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"solopt", "=", RowBox[{"selsol", "[", RowBox[{"[", RowBox[{ RowBox[{ RowBox[{"Position", "[", RowBox[{"L\[Alpha]\[Beta]\[Gamma]", ",", RowBox[{"Max", "[", "L\[Alpha]\[Beta]\[Gamma]", "]"}]}], "]"}], "//", "Flatten"}], "//", "First"}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.58918367865625*^9, 3.589183732046875*^9}, { 3.589203073875*^9, 3.589203078421875*^9}, {3.58920346515625*^9, 3.589203515359375*^9}, {3.589203550828125*^9, 3.589203555859375*^9}, 3.59119900196875*^9, 3.591199062921875*^9, 3.5911990958125*^9, 3.59119914215625*^9, 3.591199211390625*^9, 3.591199257578125*^9, 3.59119946440625*^9, 3.591199537828125*^9, 3.591199597203125*^9, { 3.591199888265625*^9, 3.59119989640625*^9}, 3.5912000905625*^9, 3.591200260078125*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"\[Alpha]", "\[Rule]", RowBox[{"-", "1.9999999998332953`"}]}], ",", RowBox[{"\[Beta]", "\[Rule]", "1.9999999998858973`"}], ",", RowBox[{"\[Gamma]", "\[Rule]", "14.999999999992681`"}]}], "}"}]], "Output", CellChangeTimes->{ 3.589183733828125*^9, 3.5892035611875*^9, {3.58920484509375*^9, 3.57884313753125*^9}, 3.57884534878125*^9, 3.578888944140625*^9, 3.5788901125*^9, 3.578891301125*^9, 3.5893783612033815`*^9, 3.5894343135478497`*^9, 3.5894370853953905`*^9, 3.5896372773112445`*^9, 3.5902404776612573`*^9, 3.59093002796357*^9, 3.5910114467967954`*^9, 3.5910247913878336`*^9, 3.5910250985991735`*^9, 3.59119900196875*^9, 3.591199062921875*^9, 3.5911990958125*^9, 3.59119914215625*^9, 3.591199211390625*^9, 3.591199257578125*^9, 3.59119946440625*^9, 3.591199537828125*^9, 3.591199597203125*^9, {3.591199888265625*^9, 3.59119989640625*^9}, 3.5912000905625*^9, 3.591200260078125*^9, 3.5929838037641373`*^9, 3.5930784465993724`*^9, 3.593079159931173*^9, 3.594196431121643*^9, 3.59480992067367*^9, 3.5957533837211704`*^9, 3.5957542178522353`*^9}] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["4 - Laser Scanning Example 1", FontSize->16, FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Section", CellChangeTimes->{{3.591197694125*^9, 3.591197709796875*^9}, { 3.59119774746875*^9, 3.591197768734375*^9}, 3.591199001953125*^9, 3.591199211359375*^9, 3.5911992575625*^9, 3.591199464375*^9, { 3.59119977821875*^9, 3.5911997938125*^9}, 3.59119983965625*^9, { 3.59120047546875*^9, 3.591200477375*^9}, 3.5941944610721827`*^9, 3.5941950945892954`*^9, 3.594195737762825*^9, {3.594649817871785*^9, 3.594649818121386*^9}, {3.5946513638501005`*^9, 3.5946513694037104`*^9}, { 3.595606968171875*^9, 3.595606968828125*^9}}, FontFamily->"Arial", FontSize->15, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[TextData[StyleBox["4 - 1 Artificial Data", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Subsection", CellDingbat->None, CellChangeTimes->{{3.587355710140625*^9, 3.5873557211875*^9}, { 3.587357062015625*^9, 3.577165121765625*^9}, {3.577165225765625*^9, 3.57716525821875*^9}, {3.5771794279375*^9, 3.57717942884375*^9}, { 3.591009662793262*^9, 3.591009667941271*^9}, {3.591198177390625*^9, 3.591198179453125*^9}, 3.594194466376192*^9}, FontFamily->"Arial", FontColor->RGBColor[ 0.8470588235294118, 0.3215686274509804, 0.24705882352941178`]], Cell[TextData[{ "LiDAR (Light Detection and Ranging) point clouds contain abundant spacial \ (3D) spatial information. Dense distribution of scanned points on object \ surfaces prominently implies surface features. Particularly, plane features \ commonly appear in a typical LiDAR dataset of artificial structures. Plane \ fitting is the key process for extracting plane features from LiDAR data.\n\ Synthetic dataset are generated from a multivariate Gauss distribution. \ Regular 3D points have means of (2, 8, 6) and variances (5, 5, 0.01), see ", StyleBox["Nurunnabi, Belton and West", FontSlant->"Italic"], " (2012). The ideal plane is z = 6." }], "Text", CellChangeTimes->{{3.5910096435584283`*^9, 3.5910097143201523`*^9}, { 3.591010715514311*^9, 3.5910108003316603`*^9}, {3.5910108479741435`*^9, 3.5910109269726825`*^9}, {3.591014046650562*^9, 3.5910140481481647`*^9}, { 3.591265349328125*^9, 3.591265616890625*^9}, {3.5929833123632746`*^9, 3.592983317792084*^9}, {3.593078537328562*^9, 3.5930785516213794`*^9}, { 3.5948100221206474`*^9, 3.59481003477227*^9}, {3.5948941522604485`*^9, 3.594894159826462*^9}, {3.595606982890625*^9, 3.595607112203125*^9}}], Cell["\<\ In this case let us consider here \[DoubleStruckCapitalN] =20000 points.\ \>", "Text", CellChangeTimes->{{3.577181162609375*^9, 3.577181273109375*^9}, { 3.589434342703517*^9, 3.5894343429315305`*^9}, 3.591265643171875*^9, { 3.59560714778125*^9, 3.595607148375*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "XYZ", "]"}]], "Input", CellChangeTimes->{{3.587482994734375*^9, 3.587483000125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[DoubleStruckCapitalN]", 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7.244898}, {34., 0.1}}, ImageSize->{Automatic, 239.29498262843498`}, PlotRange->{{-520.8333333333334, 25520.833333333332`}, {-0.13747916666666668`, 6.736479166666667}}, PlotRangeClipping->True, PlotRangePadding->Automatic]], "Input", CellChangeTimes->{{3.5910129239165897`*^9, 3.591013015691551*^9}}, TextAlignment->Center], Cell["Fig. 3 Running time vs. Number of points ", "Text", CellChangeTimes->{{3.577168969125*^9, 3.577169036515625*^9}, 3.57889046609375*^9, {3.591198345609375*^9, 3.59119834815625*^9}, { 3.5956072771875*^9, 3.595607278984375*^9}}, TextAlignment->Center, FontFamily->"Arial", FontSize->10], Cell["\<\ Both PCA and numerical Gr\[ODoubleDot]bner basis solutions have zero \ complexity with very low running time. PCA seems to be faster, but this fact \ has no practical significance. \ \>", "Text", CellChangeTimes->{{3.591265990484375*^9, 3.591266105390625*^9}, { 3.591266146390625*^9, 3.591266160375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["4 - 6 Complexity of the SVD algorithms", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Subsection", CellDingbat->None, CellChangeTimes->{{3.587355710140625*^9, 3.5873557211875*^9}, { 3.587357062015625*^9, 3.577165121765625*^9}, {3.577165225765625*^9, 3.57716525821875*^9}, {3.5771794279375*^9, 3.57717942884375*^9}, { 3.591009662793262*^9, 3.591009667941271*^9}, {3.591198177390625*^9, 3.591198179453125*^9}, 3.594194466376192*^9, {3.5941957736896877`*^9, 3.5941957921601205`*^9}, {3.5941960027916903`*^9, 3.5941960246629286`*^9}}, FontFamily->"Arial", FontColor->RGBColor[ 0.8470588235294118, 0.3215686274509804, 0.24705882352941178`]], Cell["\<\ According to Figure 3 one may approach the relation between the running time \ (T) and the number of data (N) in the following form (assuming polynomial \ complexity),\ \>", "Text", CellChangeTimes->{{3.5771686938125*^9, 3.5771687343125*^9}, { 3.591266215546875*^9, 3.591266278609375*^9}, {3.59126631603125*^9, 3.591266325796875*^9}}], Cell[TextData[{ StyleBox["T", FontSize->12, FontSlant->"Italic"], "\[Tilde] ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["N", FontSize->12], "\[Eta]"], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.577168355515625*^9, 3.57716843484375*^9}, { 3.57716868015625*^9, 3.5771686853125*^9}, {3.5771688155*^9, 3.5771688155*^9}}, TextAlignment->Center, FontSize->14], Cell["\<\ Therefore the complexity can be measured with \[Eta].\ \>", "Text", CellChangeTimes->{{3.577168772421875*^9, 3.5771687973125*^9}, { 3.577180980296875*^9, 3.5771809839375*^9}, {3.595607335953125*^9, 3.595607340453125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Fit", "[", RowBox[{ RowBox[{"dataSVD", "//", "Log"}], ",", RowBox[{"{", RowBox[{"1", ",", "LogN"}], "}"}], ",", "LogN"}], "]"}]], "Input", CellChangeTimes->{{3.577168277625*^9, 3.577168291953125*^9}, { 3.577168504125*^9, 3.577168576296875*^9}, {3.578845947046875*^9, 3.578845949421875*^9}, {3.578845995578125*^9, 3.578845998875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", "9.613967469775416`"}], "+", RowBox[{"1.0864784406731116`", " ", "LogN"}]}]], "Output", CellChangeTimes->{ 3.577168578234375*^9, 3.578846002875*^9, 3.578890579109375*^9, 3.578891478375*^9, 3.5894370892746124`*^9, 3.5910131018973026`*^9, 3.591024797986645*^9, {3.591025140516447*^9, 3.591025157192876*^9}}] }, Open ]], Cell["\<\ Therefore complexity of the SVD solution based on our numerical experiment \ is \[Eta] \[Tilde]1.1 while that of the Gr\[ODoubleDot]bner Basis as well as \ the PCA solution is \[Eta] \[Tilde] 0! \ \>", "Text", CellChangeTimes->{{3.57716862421875*^9, 3.57716865503125*^9}, { 3.57716874346875*^9, 3.577168762859375*^9}, {3.577168863359375*^9, 3.577168955296875*^9}, {3.577179498953125*^9, 3.57717950940625*^9}, { 3.57717955921875*^9, 3.577179566875*^9}, {3.577180487609375*^9, 3.577180785625*^9}, 3.577180825234375*^9, {3.577181007625*^9, 3.577181040890625*^9}, {3.5788906111875*^9, 3.5788906241875*^9}, { 3.57889148409375*^9, 3.578891485109375*^9}, {3.5910131176377296`*^9, 3.5910131780254364`*^9}}], Cell[TextData[{ StyleBox["Remark", FontSlant->"Italic"], "\nConsidering a matrix ", StyleBox["m", FontSlant->"Italic"], " \[Times]", StyleBox["n", FontSlant->"Italic"], " the theoretical computation time of the SVD algorithm (for", StyleBox[" w", FontSlant->"Italic"], " and", StyleBox[" v", FontSlant->"Italic"], " only) is \[Tilde] 4 ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["mn", FontSlant->"Italic"], "2"], TraditionalForm]]], "+8", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["n", "3"]}], TraditionalForm]]], ". Since in our case ", StyleBox["n ", FontSlant->"Italic"], "= 3, consequently the theoretical time complexity is \[Eta] = 1, see ", StyleBox["campar - wwwnavab.cs.tum.edu - 3D computer Vision Script Draft \ v.000", FontSlant->"Italic"], " (downloaded at 28.11. 2013)." }], "Text", CellChangeTimes->{{3.577180793421875*^9, 3.577180944125*^9}, { 3.577181090875*^9, 3.5771811016875*^9}, {3.578890630828125*^9, 3.578890633734375*^9}, 3.591013458201928*^9, {3.5910134930679893`*^9, 3.591013508714817*^9}, {3.5910135545632973`*^9, 3.59101360718219*^9}, { 3.5910136699098997`*^9, 3.591013725492798*^9}, {3.5910137718092794`*^9, 3.5910139089335203`*^9}, {3.59120118821875*^9, 3.591201207265625*^9}, { 3.59126637946875*^9, 3.591266400359375*^9}, 3.59464921683393*^9, { 3.595607385546875*^9, 3.595607438375*^9}, {3.595607469203125*^9, 3.59560747146875*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["5 - Outliers", FontSize->16, FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Section", CellChangeTimes->{{3.58735618365625*^9, 3.587356216125*^9}, { 3.58735711171875*^9, 3.587357120609375*^9}, {3.587477457234375*^9, 3.57716517171875*^9}, {3.5892044755625*^9, 3.589204494359375*^9}, { 3.5893778071840086`*^9, 3.5893778140168204`*^9}, {3.59119826753125*^9, 3.591198268578125*^9}, {3.59126821515625*^9, 3.591268217625*^9}, 3.594195197346676*^9}, FontFamily->"Arial", FontSize->18, FontColor->RGBColor[ 0.8470588235294118, 0.3215686274509804, 0.24705882352941178`]], Cell[CellGroupData[{ Cell[TextData[StyleBox["5 - 1 Problem of Outliers", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Subsection", CellDingbat->None, CellChangeTimes->{{3.587355710140625*^9, 3.5873557211875*^9}, { 3.587357062015625*^9, 3.577165121765625*^9}, {3.577165225765625*^9, 3.57716525821875*^9}, {3.5771794279375*^9, 3.57717942884375*^9}, { 3.591009662793262*^9, 3.591009667941271*^9}, {3.591198370328125*^9, 3.591198372203125*^9}, {3.591268236390625*^9, 3.5912682443125*^9}, 3.5941952033994865`*^9}, FontFamily->"Arial", FontColor->RGBColor[ 0.8470588235294118, 0.3215686274509804, 0.24705882352941178`]], Cell[TextData[{ "A definition of the outliers provided by ", StyleBox["Zuliani ", FontSlant->"Italic"], "(", StyleBox["2012", FontSlant->"Italic"], ") is as follows: a datum is considered to be an outlier if it will not fit \ the \[OpenCurlyDoubleQuote]true\[CloseCurlyDoubleQuote] model instantiated by \ the \[OpenCurlyDoubleQuote]true\[CloseCurlyDoubleQuote] set of parameters \ within some error threshold that defines the maximum deviation attributable \ to the effects of noise. The estimation of the parameters of a model in \ presence of outliers constitutes a very tricky and challenging problem \ although it belongs to the daily business activities of modern geodesists. \ There are many different techniques to solve this problem. Here we suggest a \ new algorithm where our numerical Gr\[ODoubleDot]bner basis solution is \ integrated into the RANdom Sample Consensus (RANSAC) algorithm. ", "However,", " first let us see, how the estimators SVD, PCA and numerical \ Gr\[ODoubleDot]bner basis can handle outliers without any data preparation." }], "Text", CellChangeTimes->{{3.591267149875*^9, 3.5912673719375*^9}, { 3.591267402546875*^9, 3.59126751859375*^9}, {3.591267585328125*^9, 3.591267609390625*^9}, {3.5912676399375*^9, 3.59126769478125*^9}, { 3.59126826228125*^9, 3.591268311671875*^9}, {3.591268398796875*^9, 3.591268713421875*^9}, {3.59126875375*^9, 3.591269001875*^9}, { 3.5948108528230247`*^9, 3.5948108611378393`*^9}, {3.5948943366527724`*^9, 3.594894342892783*^9}, {3.595607480515625*^9, 3.59560754915625*^9}, { 3.59560781440625*^9, 3.595607822265625*^9}, {3.5956078593125*^9, 3.595607869109375*^9}, 3.595607931921875*^9, 3.595608012390625*^9, { 3.595608074140625*^9, 3.59560813153125*^9}, {3.595608238453125*^9, 3.595608239859375*^9}, 3.595611646078125*^9}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["5 - 2 Artificial Data", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Subsection", CellDingbat->None, CellChangeTimes->{{3.587355710140625*^9, 3.5873557211875*^9}, { 3.587357062015625*^9, 3.577165121765625*^9}, {3.577165225765625*^9, 3.57716525821875*^9}, {3.5771794279375*^9, 3.57717942884375*^9}, { 3.59119837571875*^9, 3.591198377171875*^9}, {3.591268173703125*^9, 3.59126818246875*^9}, 3.594195214054305*^9}, FontFamily->"Arial", FontColor->RGBColor[ 0.8470588235294118, 0.3215686274509804, 0.24705882352941178`]], Cell[TextData[{ "For outliers also artificial datasets are generated from a multivariate \ Gauss distribution. The outliers have means of (15, 15, 10) and variances \ (10, 2, 1), see ", StyleBox["Nurunnabi, Belton and West", FontSlant->"Italic"], " (2012). The ideal plane is ", StyleBox["z", FontSlant->"Italic"], " = 6, see Figure 4." }], "Text", CellChangeTimes->{{3.5910096435584283`*^9, 3.5910097143201523`*^9}, { 3.591010715514311*^9, 3.5910108003316603`*^9}, {3.5910108479741435`*^9, 3.5910109269726825`*^9}, {3.5910140381485467`*^9, 3.591014122326295*^9}, { 3.591266608234375*^9, 3.59126662065625*^9}, {3.59126902896875*^9, 3.591269037234375*^9}, 3.591269477203125*^9, {3.59481090732952*^9, 3.5948109149579334`*^9}, {3.594811026342129*^9, 3.5948110441261606`*^9}, { 3.595608270375*^9, 3.5956082719375*^9}}], Cell["There are additional 3000 points, which represent outliers.", "Text", CellChangeTimes->{{3.577181162609375*^9, 3.577181273109375*^9}, { 3.5894345332834177`*^9, 3.589434584076323*^9}, {3.5894361368321357`*^9, 3.589436140478344*^9}, {3.59120127196875*^9, 3.59120129365625*^9}, 3.5941967498770027`*^9, 3.5956083243125*^9, 3.5956113845*^9}], Cell[BoxData[ RowBox[{ RowBox[{"\[DoubleStruckCapitalN]", "=", "20000"}], ";", 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"The Gr\[ODoubleDot]bner basis solution gives the same result as SVD, while \ PCA provides somewhat different value for the coefficient of \[Gamma] (5.4.. \ \[NotEqual] 4.7..). This fact indicates that", StyleBox[" PCA is more sensitive to outliers than SVD or \ Gr\[ODoubleDot]bner.", FontSlant->"Italic"], " This feature of PCA causes that it can not be employed in RANSAC reliably, \ since its result is strongly effected by anomalous observations, see ", StyleBox["Appendix", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.5894351063871975`*^9, 3.5894351338357677`*^9}, { 3.5909325930728755`*^9, 3.5909327023042674`*^9}, {3.591015310580382*^9, 3.5910154036345453`*^9}, {3.5910154513862295`*^9, 3.5910154917747*^9}, { 3.591015526749962*^9, 3.5910155299635673`*^9}, {3.5910171346915855`*^9, 3.5910171864368763`*^9}, {3.5910223239783*^9, 3.591022366519574*^9}, { 3.591022473925763*^9, 3.5910226124228067`*^9}, {3.591022664511298*^9, 3.5910227172393904`*^9}, {3.5912692499375*^9, 3.591269280875*^9}, { 3.592984347893093*^9, 3.5929843780791464`*^9}, {3.59298596288593*^9, 3.592985984710368*^9}, {3.592987273132231*^9, 3.592987298092275*^9}, { 3.593079379778747*^9, 3.593079385263061*^9}, {3.5941969968254366`*^9, 3.594197025576287*^9}, 3.594894382438853*^9, {3.5956084798125*^9, 3.595608487765625*^9}, {3.5957544323682117`*^9, 3.595754435238617*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["6 - Application of the Algorithm in RANSAC", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]]], "Section", CellChangeTimes->{{3.58735618365625*^9, 3.587356216125*^9}, { 3.58735711171875*^9, 3.587357120609375*^9}, {3.587477457234375*^9, 3.57716517171875*^9}, {3.5892044755625*^9, 3.589204494359375*^9}, { 3.5910243238082123`*^9, 3.591024324104613*^9}, {3.591198475765625*^9, 3.591198477640625*^9}, 3.5941952622583895`*^9}, FontFamily->"Arial", FontSize->16, FontColor->RGBColor[ 0.8470588235294118, 0.3215686274509804, 0.24705882352941178`]], Cell[CellGroupData[{ Cell["6 - 1 RANSAC algorithm", "Subsection", CellDingbat->None, CellChangeTimes->{{3.577168255328125*^9, 3.577168274*^9}, { 3.57889185984375*^9, 3.57889187321875*^9}, {3.5911984941875*^9, 3.59119849596875*^9}, 3.594196086267437*^9}, FontFamily->"Arial", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]], Cell[TextData[{ "Let us try to apply the RANSAC method, given in ", StyleBox["Yaniv", FontSlant->"Italic"], " (2010), which is proved to be successful for detecting outliers. The basic \ RANSAC algorithm is as follows:" }], "Text", CellChangeTimes->{{3.5719177430625*^9, 3.57191776753125*^9}, { 3.58867033321875*^9, 3.5886703369375*^9}, {3.5894351882378793`*^9, 3.5894352037707677`*^9}, {3.594886419046066*^9, 3.5948864320876884`*^9}, { 3.5956085104375*^9, 3.595608543609375*^9}}], Cell[TextData[{ "1) Pick up a model type (\[ScriptCapitalM])\n2) Input data as\n - ", StyleBox["dataQ", FontFamily->"Courier New", FontWeight->"Bold"], " - data corrupted with outliers (cardinality (", StyleBox["dataQ", FontFamily->"Courier New", FontWeight->"Bold"], ") = ", StyleBox["n", FontSlant->"Italic"], ")\n - ", StyleBox["s", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox["-", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Courier New", FontWeight->"Bold"], "number of data elements required per subset\n - \[ScriptCapitalN] \ - number of subsets to draw from the data\n - \[Tau] - \ threshold which defines if data element,", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["d", "i"]}], TraditionalForm]]], "\[Element] ", StyleBox["dataQ", FontFamily->"Courier New", FontWeight->"Bold"], ", agrees with the model \[ScriptCapitalM]" }], "Text", CellChangeTimes->{{3.57191588078125*^9, 3.571915927359375*^9}, { 3.57191596015625*^9, 3.571916081171875*^9}, {3.57191612009375*^9, 3.571916198359375*^9}, {3.571916239921875*^9, 3.57191644440625*^9}, 3.578294442421875*^9, {3.59560855859375*^9, 3.595608561796875*^9}, 3.59560866165625*^9, {3.59560874740625*^9, 3.595608751234375*^9}}], Cell["Remarks", "Text", CellChangeTimes->{{3.5719175010625*^9, 3.571917522796875*^9}}, FontSlant->"Italic"], Cell[TextData[{ "In general ", StyleBox["s", FontFamily->"Times New Roman", FontSlant->"Italic"], " can be the minimal number of the data which results a determined system \ for the unknown parameters of the model.\nThe number of subsets to draw from \ the data, \[ScriptCapitalN] is chosen high enough to ensure that at least one \ of the subsets of the random examples does not include an outliers (with the \ probability ", StyleBox["p, ", FontSlant->"Italic"], "which is", StyleBox[" ", FontSlant->"Italic"], "usually set to 0.99). Let ", StyleBox["u", FontSlant->"Italic"], " represent the probability that any selected data point is an inlier and ", StyleBox["v", FontSlant->"Italic"], " = 1 - ", StyleBox["u", FontSlant->"Italic"], " the probability of observing an outlier. Then the iterations \ \[ScriptCapitalN] can be computed as " }], "Text", CellChangeTimes->{{3.57191661534375*^9, 3.571916801140625*^9}, { 3.571916873265625*^9, 3.571917104078125*^9}, {3.5719171639375*^9, 3.571917188703125*^9}, {3.57191726565625*^9, 3.571917268484375*^9}, { 3.571917345296875*^9, 3.5719173824375*^9}, {3.595608737078125*^9, 3.595608755859375*^9}, 3.59560882403125*^9, {3.595608900828125*^9, 3.59560896009375*^9}, {3.595609133546875*^9, 3.59560913925*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\[ScriptCapitalN]", FontFamily->"Times New Roman"], " ", StyleBox["=", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[ FractionBox[ RowBox[{"log", RowBox[{"(", RowBox[{"1", "-", StyleBox["p", FontSlant->"Italic"]}], ")"}]}], RowBox[{"log", RowBox[{"(", RowBox[{"1", "-", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", StyleBox["v", FontSlant->"Italic"]}], ")"}], "s"]}], ")"}]}]], FontFamily->"Times New Roman"]}]], "Text", CellChangeTimes->{{3.571917194953125*^9, 3.5719172460625*^9}, { 3.571917288515625*^9, 3.571917304421875*^9}, {3.571917386109375*^9, 3.57191739771875*^9}}, TextAlignment->Center, FontFamily->"Courier New"], Cell[TextData[{ "3) maximalConsensusSet \[LeftArrow] \[EmptySet]\n4) Iterate \ \[ScriptCapitalN] times:\n a) ConsensusSet \[LeftArrow] \ \[EmptySet]\n b) Randomly draw a subset containing ", StyleBox["s", FontSlant->"Italic"], " elements and estimate the parameters of the model \[ScriptCapitalM]\n \ c) For each data element, ", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["d", "i"]}], TraditionalForm]]], "\[Element] ", StyleBox["dataQ", FontFamily->"Courier New", FontWeight->"Bold"], ":\n if agree (", Cell[BoxData[ FormBox[ SubscriptBox["d", "i"], TraditionalForm]]], ",\[ScriptCapitalM], \[Tau]), ConsensusSet \[LeftArrow] ", Cell[BoxData[ FormBox[ SubscriptBox["d", "i"], TraditionalForm]]], "\n d) if cardinality (maximalConsensusSet) < \ cardinality(ConsensusSet),\n maximalConsensusSet \[LeftArrow] \ ConsensusSet\n \n 5) Estimate model parameters using \ maximalConsensusSet \n " }], "Text", CellChangeTimes->{{3.57191752815625*^9, 3.571917542046875*^9}, { 3.57191759590625*^9, 3.571917671890625*^9}, {3.5719177055*^9, 3.57191773546875*^9}, {3.571917774421875*^9, 3.571918147828125*^9}, { 3.578294365765625*^9, 3.578294369625*^9}, {3.5929875396898994`*^9, 3.5929875429815054`*^9}, {3.594886953066204*^9, 3.594886995155078*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["6 - 2 The necessary number of iterations", "Subsection", CellDingbat->None, CellChangeTimes->{{3.577168255328125*^9, 3.577168274*^9}, { 3.578891929078125*^9, 3.578891950921875*^9}, {3.591198499875*^9, 3.591198501359375*^9}, 3.5941961005258617`*^9}, FontFamily->"Arial", FontColor->RGBColor[ 0.803921568627451, 0.2196078431372549, 0.15294117647058825`]], Cell["\<\ In our case we have 3 parameters to determine, so we need minimum 3 \ equations, therefore\ \>", "Text", CellChangeTimes->{{3.578892889546875*^9, 3.5788929396875*^9}, 3.595609169984375*^9}], Cell[BoxData[ RowBox[{ RowBox[{"s", "=", "3"}], ";"}]], "Input", CellChangeTimes->{{3.578892942609375*^9, 3.578892945078125*^9}, { 3.5894388916387014`*^9, 3.5894390491417103`*^9}, 3.5896375981102076`*^9, 3.5896376321338673`*^9, {3.5896380859230647`*^9, 3.589638113488313*^9}, 3.5896394482110577`*^9, 3.590932795452031*^9, {3.5948115678346806`*^9, 3.59481157329469*^9}, 3.594887711726736*^9}], Cell[TextData[{ "In this case the parameters can be computed directly from a linear system. ", "However,", " in general if ", StyleBox["s", FontFamily->"Times New Roman", FontSlant->"Italic"], " > 3 we need numerical Gr\[ODoubleDot]bner solution. Let" }], "Text", CellChangeTimes->{{3.571918532359375*^9, 3.571918539640625*^9}, { 3.594887705954726*^9, 3.594887734580776*^9}, {3.59560921484375*^9, 3.595609215484375*^9}, 3.59561165128125*^9}], Cell[BoxData[ RowBox[{ RowBox[{"p", "=", "0.99"}], ";"}]], "Input", CellChangeTimes->{{3.57191854325*^9, 3.5719185475*^9}}], Cell["\<\ In addition let us assume that the probability of observing outlier is\ \>", "Text", CellChangeTimes->{{3.57191849984375*^9, 3.571918516171875*^9}, { 3.57191855503125*^9, 3.571918564953125*^9}, {3.571918605609375*^9, 3.57191862175*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["\[Nu]", FontSlant->"Italic"], StyleBox["=", FontSlant->"Italic"], RowBox[{"1", "/", "2"}]}], ";"}]], "Input", CellChangeTimes->{{3.5719186245*^9, 3.571918637375*^9}, 3.571926064296875*^9, {3.5719262734375*^9, 3.571926276109375*^9}, { 3.571926589921875*^9, 3.57192659278125*^9}, {3.57192685378125*^9, 3.571926856703125*^9}, {3.5957544912115154`*^9, 3.5957544923191175`*^9}}], Cell["Then the 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3.577454539296875*^9, 3.571920369296875*^9}}, TextAlignment->Center, FontFamily->"Times New Roman"], ",", Cell[BoxData[ SubscriptBox["\[Gamma]", "s"]], CellChangeTimes->{{3.577453817265625*^9, 3.577453893359375*^9}, { 3.577454539296875*^9, 3.571920369296875*^9}}, TextAlignment->Center, FontFamily->"Times New Roman"], ") computed from a subset of ", StyleBox["s ", FontSlant->"Italic"], "elements," }], "Text", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.571919219734375*^9, 3.57191922046875*^9}, { 3.571919388890625*^9, 3.5719194128125*^9}, {3.5719194835*^9, 3.571919723078125*^9}, {3.571920452375*^9, 3.571920586328125*^9}, { 3.57889251278125*^9, 3.5788925461875*^9}, {3.591015626746137*^9, 3.59101566821101*^9}, {3.59101570285867*^9, 3.591015703045871*^9}, 3.59101593579828*^9}], Cell[TextData[{ " here ordinary least square (", Cell[BoxData[ FormBox[ SubscriptBox["OLS", "z"], TraditionalForm]]], ") estimation could be also applied as maximum support, see ", StyleBox["Fischler and Bolles", FontSlant->"Italic"], " (1981)." }], "Text", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.59101570827188*^9, 3.591015729924718*^9}, { 3.5910158014664435`*^9, 3.5910159308842707`*^9}, {3.594198084568547*^9, 3.59419809215016*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["6 - 4 The computation model step by step", "Subsection", CellDingbat->None, CellChangeTimes->{{3.577168255328125*^9, 3.577168274*^9}, { 3.578891929078125*^9, 3.578891950921875*^9}, {3.578892430078125*^9, 3.57889243928125*^9}, {3.578892745078125*^9, 3.578892758078125*^9}, { 3.591198511109375*^9, 3.59119851325*^9}, 3.594196111227481*^9}, FontFamily->"Arial", FontColor->RGBColor[ 0.7529411764705882, 0.22745098039215686`, 0.17254901960784313`]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"inliers", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", "}"}], ",", RowBox[{"{", "\[ScriptCapitalN]", "}"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"the", " ", "consensus", " ", "sets"}], ",", " ", RowBox[{"j", 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A more recent approach to the problem \ of surface reconstruction is that of learning based methods, see e.g.", StyleBox[" DalleMole et al", FontSlant->"Italic"], " (2010). Learning algorithms are able to process very large and/or noisy \ data, such as point clouds obtained from 3D scanners and are used to \ construct surfaces. Following this approach some studies have been employed \ SOM and their variants for surface reconstruction. SOM is suitable for this \ problem because it can form topological maps representing the distribution of \ input data. In our case this mapping occurs from 3D space to 2D one , see ", StyleBox["Kohonen", FontSlant->"Italic"], " (1998). \ \n\nTherefore we try to \ represent this cloud of data points by considerably less points using \ Kohonen-map with a lattice of SOM of size 6 \[Times] 6 code-book vectors \ with symmetrical neighborhood " }], "Text", CellChangeTimes->{{3.567689903453125*^9, 3.567689969578125*^9}, { 3.567699661*^9, 3.56769990546875*^9}, {3.5677040456875*^9, 3.56770405003125*^9}, {3.5677111803125*^9, 3.5677111853125*^9}, 3.567964207484375*^9, {3.56830887209375*^9, 3.56830918096875*^9}, { 3.56830922334375*^9, 3.56830972496875*^9}, {3.568309784703125*^9, 3.56830979646875*^9}, {3.56830984421875*^9, 3.568309845328125*^9}, 3.5789174103125*^9, {3.578985701296875*^9, 3.57829008534375*^9}, { 3.578290148984375*^9, 3.578290156640625*^9}, {3.578294255703125*^9, 3.5782942695625*^9}, {3.590237988602085*^9, 3.5902379954972973`*^9}, { 3.5941983350737867`*^9, 3.594198340408996*^9}, 3.594888935439686*^9, 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The total computation time of \ SOM + RANSAC combination is less than that of the direct RANSAC with the \ original number of points.\ \>", "Text", CellChangeTimes->{{3.591016928334423*^9, 3.591017088827505*^9}, { 3.592984747908996*^9, 3.592984748595397*^9}, {3.5929863051193314`*^9, 3.592986329704974*^9}, {3.592987983261079*^9, 3.59298801823634*^9}, { 3.594199298070278*^9, 3.5941993046066895`*^9}, {3.5941993384275494`*^9, 3.594199375072014*^9}, {3.5941995812107754`*^9, 3.5941996215524464`*^9}, { 3.594889629172904*^9, 3.5948896361929164`*^9}, {3.595609845046875*^9, 3.59560984690625*^9}, {3.5956098823125*^9, 3.595609902546875*^9}, { 3.5957551367366495`*^9, 3.595755145659865*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["8 - Laser Scanning Example 2", FontSize->16, FontColor->RGBColor[ 0.7529411764705882, 0.22745098039215686`, 0.17254901960784313`]]], "Section", CellChangeTimes->{{3.591197694125*^9, 3.591197709796875*^9}, { 3.59119774746875*^9, 3.591197768734375*^9}, 3.591199001953125*^9, 3.591199211359375*^9, 3.5911992575625*^9, 3.591199464375*^9, { 3.59119977821875*^9, 3.5911997938125*^9}, 3.59119983965625*^9, { 3.59120047546875*^9, 3.591200477375*^9}, 3.5941944610721827`*^9, 3.5941950945892954`*^9, 3.594195737762825*^9, {3.594649817871785*^9, 3.594649818121386*^9}, {3.5946513638501005`*^9, 3.5946513694037104`*^9}, { 3.594651412958987*^9, 3.594651416172593*^9}, 3.5946518645797806`*^9, { 3.59560991396875*^9, 3.595609914828125*^9}}, FontFamily->"Arial", FontSize->15, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["8 -1 Real measurement", "Subsection", CellDingbat->None, CellChangeTimes->{{3.577169241546875*^9, 3.5771692838125*^9}, { 3.58920366528125*^9, 3.589203693984375*^9}, {3.589204273625*^9, 3.5788937601875*^9}, {3.59119852553125*^9, 3.591198527328125*^9}, 3.594196217869268*^9, {3.5946514580742664`*^9, 3.5946514803355055`*^9}, 3.59465194412432*^9, {3.595609923953125*^9, 3.595609924125*^9}}, FontFamily->"Arial", FontColor->RGBColor[ 0.7529411764705882, 0.22745098039215686`, 0.17254901960784313`]], Cell["\<\ Outdoor laser scanning measurements have been carried out in a hilly park of \ Budapest, see Fig.10. 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The test also aimed to investigate the tie point \ detection capabilities of the scanner\[CloseCurlyQuote]s processing software; \ different types of spheres were deployed all over the test area. 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This measurement resulted 178.8 million points that was \ acquired in 5 and half minutes. The test data set was cropped from the point \ cloud; moreover, further resampling was applied in order to reduce the data \ size. The final data set is composed of 38\[NonBreakingSpace]318 points in \ ASCII format, and only the x, y, z coordinates were kept (no intensity \ values). 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0.22745098039215686`, 0.17254901960784313`]], Cell["\<\ In this a new algorithm for plane fitting to large cloud of data set is \ presented. This method is based on the Gr\[ODoubleDot]bner bases solution of \ the polynomial form of maximum likelihood equations with error definition in \ TLS sense. This method integrated into RANSAC algorithm is more efficient - \ faster and provide better fitting - than RANSAC with PCA, see Appendix.The \ computations carried out with artificial benchmark data as well as with real \ LiDAR measurements demonstrate that employing first SOM as data \ preprocessing reducing the thousands of points of the cloud of data to a \ dozens number of code-book vectors, and then running RANSAC with Gr\ \[ODoubleDot]bner solution, could reduce the computation time of the fitting \ process roughly by half. \ \>", "Text", CellChangeTimes->{{3.591200989078125*^9, 3.591201002140625*^9}, { 3.594199951009425*^9, 3.5942000081211257`*^9}, {3.5942000422071857`*^9, 3.594200181094229*^9}, {3.59420021558589*^9, 3.5942005883641443`*^9}, 3.594200646006246*^9, {3.5942009814692354`*^9, 3.5942010322941246`*^9}, { 3.594201567250264*^9, 3.5942016870740747`*^9}, {3.5942017276341457`*^9, 3.5942018870508256`*^9}, 3.594652589887454*^9, {3.594891267674982*^9, 3.5948913904783974`*^9}, {3.594891424829658*^9, 3.5948914687437353`*^9}, { 3.594891522329829*^9, 3.594891598489163*^9}, {3.5948916535572596`*^9, 3.59489166538208*^9}, {3.5948917007005424`*^9, 3.594891757328642*^9}, { 3.59561051375*^9, 3.59561051853125*^9}, {3.5956106055625*^9, 3.595610632484375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["References", FontSize->16]], "Section", CellChangeTimes->{{3.58735618365625*^9, 3.587356216125*^9}, { 3.58735711171875*^9, 3.587357120609375*^9}, {3.587477457234375*^9, 3.57716517171875*^9}, {3.5892044755625*^9, 3.589204494359375*^9}, { 3.5902354447836175`*^9, 3.5902354481376233`*^9}, {3.590235561237822*^9, 3.590235582001458*^9}, {3.5910092452117286`*^9, 3.5910092481133337`*^9}}, FontFamily->"Arial", FontSize->18, FontColor->RGBColor[ 0.7529411764705882, 0.22745098039215686`, 0.17254901960784313`]], Cell[TextData[{ "\nChen C C and Stamos I (2007) Range image segmentation for modeling and \ object detection in urban scenes, 3DIM2007.1\n\nDalleMole V, do Rego R and \ Ara\[UAcute]jo A (2010) The Self-Organizing Approach for Surface \ Reconstruction from Unstructured Point Clouds, in Matsopoulos G.K.(Ed.) \ Self-Organizing Map, In Tech, \n\nDiebel J R, Thrun S and Brunig M (2006) A \ Bayesian method for probable surface reconstruction and decimation, ACM \ Transactions on Graphics (TOG). 1\n\nFischler M A and Bolles R C (1981) \ Random sample consensus: A paradigm for model fitting with applications to \ image analysis and automated cartography. Communications of the ACM, 24, \ pp.381-395\n\nHuang C-M and Tseng Y-H. Plane fitting methods of Lidar point \ cloud, Dept. of Geomatics, National Cheng Kung Uni. Taiwan, \ tseng@mail.ncku.edu.tw\n\nHubert M , Rousseeuw P J and Van den Branden K \ (2005) ROBPCA: A new approach to robust principal component analysis, \ TECHNOMETRICS, Vol. 47. No. 1. pp.64 - 79.\n\nKohonen T (1998) The \ self-organizing map, Neurocomputing, Vol. 21,pp.1-6.\n\nLakaemper R and \ Latecki L J (2006) Extended EM for Planar Approximation of 3D Data, IEEE \ Int. Conf. on Robotics and Automation (ICRa), Orlando, Florida, May 2006.\n\n\ Luk\[AAcute]cs G, Martin R and Marshall D (1998) Faithful Leat-Squares \ Fitting of Spheres, cylinders, Cones and Tori for Reliable Segmentation, in \ Burkhardt H and Neumann B (Eds.) Computer Vision, ECCV \[OpenCurlyQuote]98, \ Vol I, LNCS 1406, pp.671-686, Springer-Verlag Berlin Heidelberg.\n\nMitra N J \ and Nguyen (2003) SoCG\[CloseCurlyQuote]03, June 8-10, San Diego, California, \ USA, ACM 1-58113-663-3/03/0006,pp. 322-328.\n\nNievergelt Y (2000) A \ tutorial history of least square with applications to astronomy and geodesy, \ J. Comp. and Appl. Mathematics,121, pp. 37-72.\n\nNurunnabi A, Belton D and \ West G. (2012) Diagnostic -Robust Statistical Analysis for Local Surface \ Fitting in 3D Point Cloud Data, ISPRS Annals of the Photogrammetry, Remote \ Sensing and Spatial Information Sciences, Vol.I-3,2012 XXII ISPRS Congress, \ 25 Aug. 2012, Melbourne, Australia, pp.269-275\n\nPal\[AAcute]ncz B (2014) \ Fitting Data with Different Error Models, The ", StyleBox["Mathematica", FontSlant->"Italic"], " Journal (accepted for publication, to be published in April)\n\nPoppinga \ J, Vaskevicius N, Birk A and Pathak K (2006)Fast Plane Detection and \ Polygonalization in noisy 3D Range Images, Int. Conf. on Intelligent Robots \ and Systems (IROS), Nice, France, IEEE Press 2008\n\nRose C and Smith D \ (2000) Symbolic Maximum Likelihood Estimation with ", StyleBox["Mathematica,", FontSlant->"Italic"], " The Statistician 49, pp.229 - 240\n\nRusseeuw P J and Van Driessen K \ (1999) A Fast Algorithm for the Minimum Covariance Determinant Estimator, \ TECHNOMETRICS, Vol. 41. No. 3. pp. 212-223.\n\nStathas D, Arabatzi O, \ Dogouris S, Piniotis G, Tsini D and Tsinis D (2003) New Monitoring Techniques \ on the Determination of Structure Deformation, Proc. 11th FIG Symp. on \ Deformation Measurements, Santorini, Greece. \n\n Zuliani M (2012) RANSAC for \ Dummies, vision.ece.ucsb.edu/~zuliani\n\nYang M Y and F\[ODoubleDot]rtsner W \ (2010) Plane Detection in Point Cloud Data,TR-IGG-P-2010-01, Techn.Report. \ Nr.1, Dept. of Photogrammetry Inst. of Geodesy and Geo-information, Uni., \ Bonn, Germany.\n\nYaniv Z (2010) Random Sample Consensus (RANSAC) Algorithm, \ A Generic Implementation, Georgetown University Medical Center, Washington, \ DC, USA, zivy@isis.georgetown.edu" }], "Text", CellChangeTimes->{{3.5910092618413577`*^9, 3.5910092752105813`*^9}, { 3.5910093073154373`*^9, 3.5910095805655174`*^9}, {3.591010822358899*^9, 3.591010826508506*^9}, {3.594801969581304*^9, 3.594802164394446*^9}, { 3.594802422465699*^9, 3.594802612661234*^9}, {3.5948026781501484`*^9, 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