3D shape acquisition and integral compact representation using optical scanning and enhanced shape parameterization

Advanced Engineering Informatics 28 (2014) 111–126

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Advanced Engineering Informatics

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3D shape acquisition and integral compact representation using opticalscanning and enhanced shape parameterization

1474-0346/$ - see front matter � 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.aei.2014.01.002

⇑ Corresponding author. Tel.: +385 98 327 642.E-mail address: [email protected] (D. Vucina).

Milan Curkovic, Damir Vucina ⇑FESB, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, R. Boskovica bb, 21000 Split, Croatia

a r t i c l e i n f o

Article history:Received 11 January 2013Received in revised form 4 January 2014Accepted 11 January 2014Available online 4 February 2014

Keywords:3D shape acquisitionB-spline surfacesEnhanced 3D parameterization

a b s t r a c t

An efficient computational methodology for shape acquisition, processing and representation isdeveloped. It includes 3D computer vision by applying triangulation and stereo-photogrammetry forhigh-accuracy 3D shape acquisition. Resulting huge 3D point clouds are successively parameterized intomathematical surfaces to provide for compact data-set representation, yet capturing local details suffi-ciently. B-spline surfaces are employed as parametric entities in fitting to point clouds resulting fromoptical 3D scanning. Beyond the linear best-fitting algorithm with control points as fitting variables, anenhanced non-linear procedure is developed. The set of best fitting variables in minimizing the approx-imation error norm between the parametric surface and the 3D cloud includes the control points coordi-nates. However, they are augmented by the set of position parameter values which identify therespectively closest matching points on the surface for the points in the cloud. The developed algorithmis demonstrated to be efficient on demanding test cases which encompass sharp edges and slopediscontinuities originating from physical damage of the 3D objects or shape complexity.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Non-contact 3D shape sensing and especially 3D optical scan-ning methods and equipment have recently started to revolution-ize some of the quality control and monitoring procedures in theindustry. High resolution and high accuracy 3D optical scanningresults in high density point clouds (typically in the range beyond108 points) that capture and contain abundant global and localinformation about the shape of some object.

Reverse engineering and design of technical objects offer manyopportunities where computer-aided design (CAD) and artificialintelligence (AI) techniques can jointly be applied very efficientlyto provide major benefits in the design or re-design process. Asthe discipline of design and related processes operate on geometricshapes, adequate 3D modeling is a major asset, which also intro-duces the need for efficient 3D geometric data acquisition, knowl-edge representation and storage. Intelligent evaluation of theacquired geometric knowledge (after processing of raw pointclouds) should provide ground for shape-related reasoning, learn-ing and eventually diagnostics, including feature recognition andclassification.

Many authors have dealt with individual aspects of geometricdesign in the generic sense of coupling shape and function.Optimization and optimum design are used along with different

mathematical formulations and numerical methods [1–3]. Com-prehensive surveys of different approaches in geometric modelingand parameterization in the context of optimization and shapemodeling are found in literature [4,5]. Different methods of model-ing shape have a strong and profound impact on the numericalanalysis and simulation procedures which operate on those shapesin order to evaluate the corresponding functionality and perfor-mance. Hence there is a strong numerical relationship betweengeometric models and physical simulation models, as full analysisand synthesis encompass both.

The procedure developed in this paper starts from 3D shapedigitization by applying optical 3D scanning [6], including stereo-photogrammetry and triangulation [7,8]. There are many differentdigitization techniques and devices that can be used [9], and sev-eral procedures that can be adopted [6,10,11]. Essentially, stereo2D images are combined for 3D reconstruction [8,12], resultingin large 3D point clouds representing the surface of the digitizedobject. Multiple scans at different locations are combined by apply-ing coordinate transformation and registration in the commoncoordinate system [13], providing a well-balanced compromise be-tween (i) accuracy in local detail capture and (ii) economy in largearea coverage. This allows different measurement volumes (lenssets) to be combined within the same digitization procedure. Fur-ther steps such as averaging, thinning and repairing of meshes canbe applied. In [14], the authors have applied an inexpensive stereo-vision system based on video frame streams for far-range scenesfor the purpose of monitoring. The final format is usually the

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common STL standard containing the mesh of polygonal faces de-fined by corresponding vertices and unit normals. For any object,geometry representation using point clouds is typically huge insize and inconvenient for processing, as numerical shape operatorsneed to be applied to the entire point clouds typically enclosingmany millions of individual points.

On the other hand, computer-aided-design techniques haveevolved based on parametric representation, where standard geo-metric primitives represented by corresponding parameters arecombined to provide overall geometries of objects. The CAD modelsare hence compact in terms of data storage and facilitate transfor-mations and modifications as the corresponding operators manipu-late the parameters only. The geometric component objects includesimple spheres and cylinders but also complex mathematicalcurves and surfaces such as B-splines and NURBS [15,16], providingfor high flexibility in modeling shape [17–19]. Recent develop-ments in CAD technology have actually re-positioned it to upgradedfunctionality and significance. CAD systems have increasinglytransformed to modeling tools that can generically encompass enti-ties and their relationships, constraints, etc. Beyond the traditionalshape primitives, tools for free-form entities and point clouds havealso been integrated in terms of support for geometric modeling.CAD tools now support animations, assembly simulations, kine-matic simulations for mechanisms and systems, functional model-ing, customization and specialization, systems engineering. Inmany aspects, coupling with external simulation applications isfeasible for additional functionality evaluations. Many CAD applica-tions integrate advanced analysis methods, for example finite ele-ment analysis for structural applications using advancedmaterials, in some cases also basic optimization functionality. Thedesign phase is augmented by production process simulation andintegration. The CAD systems are increasingly seen as product life-cycle management (PLM) systems which include standard CAD,CAM (manufacturing), CAE (engineering), surfacing, reverse engi-neering, visualizations and other complementary activities.

In order to combine the digitized as-is geometry with the para-metric CAD geometry, parameterization of the resulting point

Fig. 1. Procedure implemented for 3D point cloud acquisition, data processing and paramand knowledge discovery, and subsequent evolutionary optimization.

clouds into mathematical surfaces is necessary, which can beaccomplished by applying fitting procedures [20–23].

One of the key aspects of the procedure developed in this paperis efficient 3D parameterization based on adequate mathematicalsurfaces. Many authors have contributed in the area of best-fitting3D mathematical surfaces such as B-spline and NURBS surfaces togeometric data-sets acquired by measurement [22–27], typically inthe context of reverse engineering. In [28], the authors have alsoapplied directly chained Bezier surfaces in parameterizations forshape optimization.

The existing approaches are described further in Sections2.1–2.3. Section 2.4 outlines the proposed novel approach whichincludes the enhanced parameterization based on the modifiedfitting error measure and eigenvalue ratio distribution, wherebySection 3 illustrates the new procedure with real-world applicationcases.

The main reason for combining 3D shape acquisition and shaperepresentation in this paper is the fact that these two disciplinesare complementary and frequently coupled in terms of the ‘en-hanced reverse engineering’ procedure shown in Fig. 1 by jointlyproviding the initial ‘shape solution’.

Moreover, testing the performance of any shape representationalgorithm is less biased if carried out using complex real-world ob-jects rather than numerically generated surfaces or simple engi-neering objects. This is especially true if the real objects alsoencompass individual features such as damaged surfaces(gaps,cavities), like those acquired here by applying 3D acquisi-tion. Even the other way around, storing shapes of engineering ob-jects for later reference is far more efficient in parametric formthan as raw point clouds, whereby highly efficient shape represen-tation algorithms are needed.

2. Method and procedure

Parts of the overall procedure developed according to Fig. 1 canbe applied both in reverse engineering and also in the detection ofgeometric discrepancy or change in shape.

eterization for shape representation to be used in automatic detection, data-mining

M. Curkovic, D. Vucina / Advanced Engineering Informatics 28 (2014) 111–126 113

The overall procedure in Fig. 1 is centered at the notion ofshape. The essential problems in dealing with geometries of real-world objects is the capacity to represent shapes, transform shapesand identify individual features of the shapes [29,30]. In this paper,enhanced reverse engineering refers to standard reverse engineer-ing procedures [31] enhanced by engaging optimizers which willmodify the resulting objects towards improved performance.

As illustrated in Fig. 1, evolutionary optimization algorithmsand other metaheuristics [32,33] are very suitable procedures tobe employed in the enhanced reverse engineering workflow as pre-sented here. Not being gradient-based, they can handle discontinu-ity in excellence and constraint function values and therefore alsomajor change in shape and topology. Direct single- and multi-objective search methods [34], possibly using constraint penaliza-tion, are also adequate within this framework.

The procedure starts by digitizing the 3D surface geometry ofsome object and proceeds with the parameterization of the geom-etry and subsequent representation of shape based on the chosenset of parameters. This is where the detection of geometric changeor monitoring of difference in shape can take place, whereby thecurrent set of values of the shape parameters is compared withthe respective previous or reference values and algorithmicallyanalyzed.

If the shape is to be partially reverse-engineered aiming ataccommodating the required increased performance of the object,an evolutionary shape optimizer (EO) should be invoked [1,3]accordingly. It operates on the selected sub-set of shape variablesin order to modify the shape locally or globally. As the key se-quence within this comprehensive process, the focus of this paperis aimed at the first part of the cycle in Fig. 1, shape digitization bymeans of 3D scanning and parameterization of geometry whichmaps the point clouds into parametric surfaces. This mappingshould result in compact and efficient data-sets of the parametricentities while still providing for feature recognition and detectionof change in geometric shape.

Accordingly, the ambition of this paper is to develop an efficientnumerical procedure generically presented in Fig. 2.

The data acquisition aspect is in this case limited to generatingthe raw point clouds including some necessary processing such aspolygonization.

Fig. 2. Elements integrated in the developed shape-acquisition and processingflowchart.

The knowledge representation entity referred to in Fig. 2 is adiscipline within artificial intelligence which develops ways to for-mally represent knowledge and includes a logic system with sym-bols, operators and interpretation methodology that provides forefficient storing of knowledge, knowledge-based reasoning, anddeveloping new knowledge. In terms of this paper, knowledge isrelated to actual 3D shapes and the logic system encompasses rep-resentation based on shape parameters. The parameters can beprocessed using different mathematical operators and can bemapped to shapes using parametric curves as the interpretationmethodology. This framework leads to new knowledge (newshapes) based on associated reasoning operations (for exampleevaluation of shapes based on finite element analysis). Advancedoperators of the logic system may also include feature recognitionalgorithms operating on the parametric data-set.

Once the deviation data for a surface patch are available eitherin raw format (point cloud) or as parametric surfaces, adequate cri-teria and numerical tests can be established for automatic detec-tion and classification of geometric features.

2.1. Acquisition of the raw point cloud

This section demonstrates the acquisition of raw data for theprocedure developed in this paper, which has been successfully ap-plied with several examples in Section 3. The new contributiondeveloped in this paper is the enhanced procedure presented inSection 2.4 which can be seen as a direct extension and upgradeof the methodology in Sections 2.1–2.3.

In this paper we use the ATOS system which consists of a pro-jector and two cameras [6] and Fig. 3, and provides for efficient3D scanning with high resolution and accuracy. It is an optical sys-tem based on triangulation and stereo-photogrammetry which ap-plies structured light patterns projection and uses time-basedcoding for addressing the positions of individual points.

During scanning, light patterns in the form of stripes are pro-jected onto the surface of the object such that a large number ofpoints on the object surface can be uniquely identified andmatched. Individual stripes projected on the object surface cangenerally be identified by respective counting or by projectingtime-sequenced stripe patterns resulting in binary coding ofstripes on the time axis (Fig. 3).

The intensity of light of any point on the object surface varieswith time according to some predefined function. These patternsprovide for the phase shift needed for time-based coding of posi-tion, hence, if read in the time domain, the pixels receive a codingsequence based on their position. The optical resolution generallydepends on the width of the stripes, which is limited by the reso-lution of the projector and the cameras, and procedures of phase-shifting the stripes in the sequence of projections for betterresolutions.

The scans originating from the two cameras are combined toprovide the 3D model of the object surface. The procedure involvesthe spatial coordinate system of the object and both planar coordi-nate systems of the cameras. A mathematical model can be estab-lished based on geometric considerations which maps 3Dcoordinates of the object points into 2D points on the cameras. Gi-ven the 2D coordinates of the same point on the two cameras, themathematical model provides the three coordinates of the respec-tive spatial physical point.

The simple camera model provides the necessary expressions torelate 3D entities and their 2D projections. Three coordinate sys-tems (CS) exist (Fig. 4): the 3D world CS with origin Cw, the 3Dcamera CS with origin Cc, and the 2D image CS on the camera planeP. If the position of some point P(X,Y,Z) is given in the world coor-dinate system as Pw(X,Y,Z), its coordinates Pc(x,y,z) in the cameracoordinate system are obtained using the translation matrix

Fig. 3. (a) Optical device for 3D digitization of an object, scanner (large object setup), object, calibration plate ATOS MV 250, (b) detail of projecting of structured lightpatterns, (c) time-based coding of spatial position of points.

Fig. 4. Basic camera-world geometry, (a) camera model for point P, (b) stereo camera setup, epipolar constraint.


T = Cw–Cc and rotation matrix R between the systems (Fig. 4a),Pc = R � (Pw � T) whereby the parameters contained in T and Rare the extrinsic parameters of the camera.

The projection point p(px,py) on the camera plane P of the phys-ical point P(X,Y,Z) follows from x = f � X/Z, y = f � Y/Z, z = f, with f asthe focal length.

The intrinsic parameters of the camera with the focal point Cc

are the focal length f, the principal point C(cx,cy) as the projectionof Cc on P, and the camera pixel dimensions (hx,hy). The mappingbetween the camera coordinates Pc(x,y, f) and the 2D image coordi-nates (px,py) is x = (px � cx)hx, y = (py � cy)hy.

The final equation of the basic camera model is obtained bycombining the expressions, yielding

P ¼M � P ð1Þ

where M is the projection matrix which can be partitioned into sep-arate matrices with intrinsic and extrinsic parameters. The param-eters needed in Eq. (1) such as the focal length and other data aretaken from the specifications of the respective lens set and systemconfiguration and subsequently calibrated using certified calibra-tion objects.

In Fig. 4b, P represents the physical 3D point which by perspec-tive projection on the two camera planes PL and PD gives the 2Dprojection points PL and PD. The focal points CL and CD are con-nected by the baseline LB and project onto the other image planesinto epipolar points EL and ED forming epipolar lines LEL and LED

and the epipolar plane PE.If for some point P the left projection point PL is known, then

also the epipolar plane PE is known but also both epipolar lines,since they are the intersection of PE with PL and PD respectively.


Consequently, the projection point PD that corresponds to PL mustlie on the right epipolar line and can hence be identified as thecounterpart of PL. This provides the epipolar constraint for thetwo projection points that correspond to the same 3D point. There-fore the search for a matching image of some point on the othercamera only has to go through the corresponding epipolar lineand not the entire image plane. Moreover, if both matching projec-tion points PL and PD are known, their projection lines are alsoknown and they intersect at the corresponding 3D point P whichcan consequently be calculated (3D point reconstruction) basedon triangulation.

These constraint equations can be expressed in matrix termsand the 3D surface reconstruction can be carried out via a linearoperator L,

P ¼ LðPL;PDÞ ð2Þ

2.2. Processing the raw point clouds

Polygonization is the process where huge point clouds contain-ing points originating from multiple scans of partially overlappingregions are reduced to non-overlapping meshes of points of certaindensity, and polygonal surface patches joining them. Generally, themeshes consist of planar faces with C0 continuity, whereby meshesof triangular faces are used most frequently. The process involvesthinning of the point cloud with least-square optimization in over-lapping areas, adaptive size of the polygons (e.g. smaller size-high-er density in edge regions with high curvature, etc.) smoothing ofthe mesh, regularization, etc.

If necessary, local mesh repairs can be carried out in small voidareas without scanned points which manifest as tiny holes in theobject surface or areas hidden by reference points. These voidscan be repaired by adding a few triangles to the mesh which fillthe holes such that the surface continuity and surface gradientcontinuity with adjacent (successfully scanned) areas is preserved,also imposing curvature continuity if required.

In many cases, different parts are scanned using different mea-surement volumes to provide both for large areas coverage andhigh-accuracy insight into local details. Different scans may beassociated with different coordinate systems, which impliescoordinate transformation before joining the point clouds ormeshes. Where different scans cannot be merged via commonlyrecognizable reference points, best-fitting of clouds is necessary.Registration is used to combine multiple parts of geometry(CAD-generated meshes, scanned meshes or hybrid) defined with-in different coordinate systems.

The rotation and translation of one of the coordinate systems(six degrees of freedom-best fit variables) are determined for bestalignment based on the minimum total offset of the mesh points inthe least-square sense. Frequently the iterative-closest-pointalgorithm is applied as described in Section 2.3.

2.3. Parameterization into 3D mathematical surfaces

For many purposes it is necessary to replace the resulting pointcloud with mathematical surfaces which are typically applied incomputational geometry and CAD, thus enabling:

– compact representation of geometric knowledge,– non-demanding storage and exchange of geometric data,– easy imposition of partial changes in geometry,– parametric optimization of geometry,– application of geometric operators modifying the shape

locally or globally,– detection of change and deviation of the shape,– simulation and animation.

Parameterization by fitting B-spline curves and surfaces is ap-plied in this paper. The knowledge representation and storage ofthe shape are based on storing the matrix of control points Q inEqs. (3) and (4), which provides the indicated properties.

A B-spline curve of degree d is defined for a set of (n + 1) 2D con-trol points Q as

CðtÞ ¼Xn

i¼0

Ni;dðtÞ � Q i; t 2 ½0;1� ð3Þ

and the B-spline surface is defined for a 3D array-grid Q of controlpoints, (n0 + 1) � (n1 + 1) by

Cðu; vÞ ¼Xn0

i0¼0

Xn1

i1¼0

Ni0;d0ðuÞ � Ni1;d1ðvÞ � Q i0i1; u;v 2 ½0;1� ð4Þ

Ni;0ðtÞ ¼1; ti 6 t < tiþ1

0; otherwise

� �; 0 6 i 6 nþ d

Ni;jðtÞ ¼t � ti

tiþj � tiNi;j�1ðtÞ þ

tiþjþ1 � ttiþjþ1 � tiþ1

Niþ1;j�1ðtÞ;

1 6 j 6 d; 0 6 i 6 nþ d� j

where d0 and d1 are the respective degrees of the surface, and N arethe basis functions defined recursively using a non-decreasingsequence of scalars-knots ti, which can be chosen as uniform

ti ¼0; 0 6 i 6 d

i�dnþ1�d ; dþ 1 6 i 6 n

1; nþ 1 6 i 6 nþ dþ 1

8><>:9>=>; ð5Þ

or periodic.The resulting curve and surface, Eqs. (3) and (4), possess the

property of local control since the support (non-zero value) ofNi,j(t) is the interval [ti, ti+j+1], therefore the curve is influenced onlylocally by a few adjacent control points in Q.

A simple fitting method of a B-spline on (m + 1) data points Passumes that they are ordered with increasing sample times se-quence sk. Since the B-spline parameter t varies between 0 and 1,the sample times can be chosen to correspond to parameter valuesaccording to tk = (sk � s0)/(sm � s0).

The curve fitting problem is defined as determining the controlpoints Q such that the least square error

EðQ Þ ¼ 12

Xm

k¼0

Xn

j¼0

Nj;dðtkÞ � Q j � Pk

!2

ð6Þ

is minimized. The minimum of this quadratic error function isobtained from the necessary conditions [26] for the extreme by dif-ferentiating with respect to the control points which yields

Xm

k¼0

Xn

j¼0

akiakjQ j �Xm

k¼0

akiPk ¼ 0; i ¼ 0;n ðarc ¼ Nc;dðtrÞÞ ð7Þ

A simple fitting method of a B-spline surface on (mo + 1) �(m1 + 1) points assumes ordered and increasing sample timessequences rj0, sj1 of data points Pj0j1. Since the B-spline parametersvary between 0 and 1, the sample times may be set to correspondto the parameter values uj0 = (rj0 � r0)/(rm0 � r0), vj1 = (sj1 � s0)/(sm1 � s0).

With B-spline surface fitting, the unknown (no + 1) � (n1 + 1)control points Q are to be determined by minimizing the least-squares error function

EðQ Þ ¼ 12

Xm0

j0¼0

Xm1

j1¼0

Xn0

i0¼0

Xn1

i1¼0

Ni0;d0ðuj0Þ � Ni1;d1ðv j1Þ � Q i0i1 � Pj0j1

!2

ð8aÞ


The minimum of this quadratic error function [27] is obtainedfrom the necessary conditions for the extreme by differentiatingwith respect to the control points, oE(Q)/oQ = 0 which yields a lin-ear system of equations that provides the control points values forthe minimum least square error.

The motivation for this research has emerged based on thestate-of-the-art in this area. Free-form surface fitting is usuallybased on the ICP algorithm [13,35–38]. The Iterative Closest Point(ICP) algorithm tries to minimize the difference between two pointclouds by transforming (translation and rotation) one of the coor-dinate systems relatively to the other. This is done by iterativelyidentifying the matching points in the two clouds (nearest neigh-bors) and aligning the neighbor points based on the mean squarecriterion, which typically converges provided a good initial ‘guess’.

The first point cloud is some discretization of the initial (forexample 3D-scanned) surface, while in the second point cloud cor-responding points closest to individual points of the first pointcloud need to be identified. During the initial phase of fitting witha relatively uniform distribution of control points, the problem isnot very hard. However, as the fitting procedure has advancedand the control points aggregate in areas of significant change inshape, finding matching points on the second surface becomes verydemanding and time-consuming. The reason for this is the fact thatfor many points of the first point cloud the matching points on theparametric surface are obtained with very small increments Du,which leads to high algorithmic complexity.

Paper [13] discusses the computational complexity of the ICPalgorithm and develops the concept of subdividing the parametricsurface into triangular partitions to accelerate the search formatching points Areas with major change of shape are locationswhere smaller partitions join and where multi-dimensional bisec-tion could be applied. In [35], the authors describe the ICP algo-rithm as a method that does not converge globally, and changethe topology of the initial surface as a preparatory step for theICP algorithm. Monte-Carlo simulation has been introduced in[36] to upgrade the performance of the ICP algorithm. Several re-lated numerical interventions are also presented in [37,38].

The motivation of this paper is to improve the fitting accuracyof the existing methods such that improved shape representationis possible without increasing the number of shape parameters.This will be implemented by augmenting the variables in the fit-ting procedure by the u and v vectors and by redistributing the fit-ting points mesh to provide more impact to areas with significantchange in shape, as follows in Section 2.4.

2.4. Development of an enhanced parameterization procedure

The proposed enhanced parameterization approach is alsobased on B-spline surfaces, Eq. (4). However, the upgraded algo-rithm of fitting the parametric surface to the point cloud is devel-oped based on the conception that additional best-fitting capacityof B-spline surfaces might be accomplished by having the fittingerror between the point cloud and the fitted parametric surface de-fined and evaluated differently.

With this scheme, the matching point on the fitting surface (Fij)used in error evaluation of any particular point (Pij) in the pointcloud is not identified by simple assignment of the correspondingui and vj parameters as defined in reference to Eqs. (6) and (8a), orby linearly interpolating the ui and vj parameters within the range[0,1] based on the Pij coordinates. Instead, the optimizer will beadditionally engaged to determine the corresponding best ui andvj values in the course of minimizing E(Q). The price to be paidfor the expected improvement in representation capability is how-ever the fact that the best-fitting procedure is no longer linear. Spe-cifically, E(Q,u,v) is quadratic in Q and commonly higher-ordernon-linear in u or v, making the rE(Q,u,v) = 0 conditions for the

minimum of the respective surface fitting error generally non-linear. Moreover, the potential improvement of the fitting errorE(Q) will only partly be due to better fitting. Part of the enhance-ment will also be due to the different definition and measurementof error as the assignment of Fij to each Pij in the proposed methodis done differently than in common least-square fitting (Fig. 5).

While the error e1 in Fig. 5 is obvious and adequate in approx-imating functions to experimental data (for example response sur-faces), e2 seems more reasonable in approximating geometricshape. The error e1 is standard in least square fitting, while e2 is re-lated to the ICP algorithm and matching closest points.

In the enhanced best fitting scheme, the error will be evaluatedon the point cloud P with a different assignment of the matching Fij

to each Pij in the point cloud in the error evaluation expression inEq. (8a), Fig. 5. The redistribution variables u and v which definethe matching Fij partners to the Pij points in the cloud will consti-tute the augmented part of the best-fitting variables, in addition toQ. Hence the optimal values of Q will emerge as a consequence ofminimizing the fitting error in Eq. (8b) with respect to Q along withu and v. The latter are not needed any longer after obtaining theoptimized Q vector of control points. In the standard best fittingprocedures in Eq. (8a), the corresponding ui and vj parametersare simply allocated as defined in reference to Eqs. (6) and (8a),or by linearly interpolating the ui and vj parameters within theinterval [0,1] based on the Pij coordinates relatively to the patchfootprint range.

The developed best fitting procedure is consequently upgradedaccording to

EðQ ;u;vÞ ¼ 12

Xm0

j0

Xm1

j1

Xn0

i0

Xn1

i1

Ni0;d0ðuj0Þ � Ni1;d1ðv j1Þ � Q i0i1 � Pj0j1

!2

ð8bÞ

where the fitting error E(Q,u,v) is made a function of CP (controlpoint) coordinates Q and the locations of the Fij points (matchingpoints to Pij on the best-fitting surface) u and v, withu ¼ u0; . . . ; um0; v ¼ v0; . . . ;vm1. The best fitting problem is nownon-linear in u and v, and iterative gradient-based numerical min-imization has to be applied.

The scalar entities u and v as parameters in Eq. (4) indeed rep-resent the position coordinates of a point on the B-spline surface.However, the bold entities (vectors) u and v as related to (8a)and (8b) represent the mapped coordinates of the matching pointsfor individual points in the cloud in the least square error defini-tion in Eqs. (8a) and (8b).

The standard approach assigns the vectors u and v of the match-ing points on the surface to those in the given cloud as sums of lin-ear segments. The new definition in Eq. (8b) however lets theoptimizer decide on u and v values and pick the matching pointsitself. For any point Pij in the cloud, the scalar values ui and vj with-in [0,1], for which we assume C(ui,vj) = Pij, need to be identified,

where Cðui;v jÞ ¼Pn0

i0¼0

Pn1i1¼0Ni0;d0ðuiÞ � Ni1;d1ðv jÞ � Q i0i1 is a point

on the parametric surface. They are initially chosen as

ui ¼Pi�1

k¼0kPkþ1�PkkPm0

k¼0kPkþ1�Pkk

; i ¼ 0; . . . ;m0 and analogously for vj. Neverthe-

less, solving Eq. (8a) with such initial pre-setting of ui and vj valuesneed not necessarily provide the best solution, i.e. control points Q.

Therefore, we consider u ¼ ½uo; . . . ;um0�T ; v ¼ ½vo; . . . ;vm1�T asadditional variables of the fitting problem which leads to (8b).

In fact, Eq. (8b) can also be seen as a mechanism to relocate thecontrol points’ Q (nCP � 3) positions on the patch footprintembedded within best-fitting, where the re-positioning takes placeautomatically by virtue of minimizing E(Q, u, v) in Eq. (8b). This isdifferent to some other enhanced fitting approaches where the fit-ting error is minimized with respect to Qz (nCP � 1) only, while the

Fig. 5. (a) Conceptual illustration of the fitting error for the individual point Pij in the point cloud, e1 for common least-square fitting and e2 for the ICP-based definition oferror, (b) illustration of linear fitting based on Eq. (8a), (c) illustration of LM-based non-linear fitting based on Eq. (8b) (P are points of the given cloud and Q are the controlpoints after fitting).


CP positions Qxy (nCP � 2) are moved and adjusted towards thesharp edges contained within the point cloud. The latter is donein order to accomplish their higher population density in the vicin-ity of such edges, and an example of such a procedure is given in[28]. The latter procedure provides the benefit that the best-fittingprocedure remains a linear procedure. However, it also adds com-plexity since edge detection algorithms external to the best-fittingprocedure must be employed. This fact is very adverse for examplein topology- and shape optimization, where the changes in shapeimposed by the optimizer move and re-shape the edges of the tar-get object. Sometimes the changes even enforce the elimination ofsome edges and initiate the genesis of new edges, all of which ariseas a consequence of shape redesign.

The term genesis of new edges implies that as a numerical opti-mizer operates on the shape parameters of some object, and theshape may change even topologically such that new edges ariseat originally ‘smooth’ zones. As a radical visualization case, an ini-tially spherical object might transform to a cube or the other wayaround. Changes in shape may map to changes in overall surfacetopology.

The effect of relocating CPs in the proposed procedure based onEq. (8b) also contributes significantly to shrinking the fitting erroras demonstrated in Section 3.2.

The Levenberg–Marquardt method (LM) was applied because ofits advantages in unconstrained non-linear optimization. Essen-tially, it combines the 2nd order Newton method which convergesfast when close to the optimum point with the 1st order gradientdescent which converges in a robustly persistent manner for dis-tant trial solutions. While the standard Newton method in eachiteration (i) minimizes the function (in this case Eq. (8b), the fittingerror) using the search direction S ¼ �J�1

i � rf i, the LM method re-places the Hessian J by using eJi ¼ Ji þ ai � I. Decreasing a from largeto small values in fact changes the search strategy from steepestdescent to Newton. In this expression, I is the identity matrixand f is the error function in Eq. (8b), while ai is the LM damping

factor. The gradient rf and the Hessian J are the gradient and theHessian of this error function with respect to the fitting variables[Q,u,v] respectively.

The initial solution applied in Eq. (8b) is the solution obtainedby linearly minimizing E(Q) in Eq. (8a) with uniformly distributedu and v arrays, after which the LM non-linear gradient-based min-imization algorithm is applied with Eq. (8b). The standard best-fit-ting procedure according to Eq. (8a) is typically a linear systemwith {(n0 + 1) � (n1 + 1) � 3} variables in Q. The enhanced fittingprocedure based on Eq. (8b) is generally non-linear with {(n0

+ 1) � (n1 + 1) � 3 + (m0 + 1) + (m1 + 1)} variables as the values inu and v are assumed to apply as collective parameters to entirerows/columns in the grid on the P footprint. Generally, each Pij

might have its own associated uij and vij values, but this would leadto an extremely high number of variables and excessive numericaleffort.

Therefore, we propose a modified best-fitting method with theobjectives of (i) delivering improved best-fitting quality (reducedcumulative error), while (ii) preserving the relative spatial struc-ture of the point cloud data-set. Such multi-objective optimalparameterization was obtained by letting u and v vary within therange of the patch as resulting from Eq. (8b), but restricted to re-tain the initially given relative positions, hence the adjacency andconnectivity matrix of Fij preserve the same spatial structure. Thisis achieved as u and v (sized (m0 + 1) and (m1 + 1) respectively) ap-ply to entire grids of matching points Fij.

Another enhancement of the procedure is also developed andintroduced in the numerical procedure. The original point clouddata-set is redistributed according to the spatial eigenvalue distri-bution, providing for heaping-up of the point cloud in areas withsignificant geometric change. This provides a pondering effect forsuch regions in the overall best-fitting procedure by effectivelyassigning them heavier weight factors. The redistribution of thepoint cloud is implemented by changing the point locations withinthe patch and obtaining the relocated point coordinates from the


planar tesselated triangular mesh that constitutes the surfacepolygonial faces (STL). This is essentially a linear interpolationoperating on the original point cloud.

This enhancement of the procedure is introduced along with Eq.(8b). The original points in the point cloud resulting from 3D scan-ning are redistributed (linear mapping from Pij to P0ij) to providepondering in regions with significant geometric change, whichshould receive larger weighting factors in best-fitting. This isimplemented by locally denser aggregation of the point cloud inthose areas. The redistribution of the point cloud is implementedaccording to the spatial eigenvalue distribution. The new pointcoordinates P0ij are obtained by linear interpolation from the origi-nal planar polygonial mesh faces (Pij, STL format). This does notdeteriorate the accuracy unless sections containing edges or signif-icant curvature are skipped. The redistribution of the point cloud inP0ij also needs to preserve the topology of the point cloud such thatthe adjacency and incidence matrices of the data-set are preserved.

The term spatial eigenvalue distribution refers to the distribu-tion of the ratio of the eigenvalues across the point cloud, whereat each location only a partial point cloud in the local vicinity ofthat point is taken into account. The term pondering effect refersto the fact that by virtue of aggregating points in some regionand reducing their number in other regions the impact of the for-mer region with respect to the impact of the latter during the fit-ting procedure is effectively increased. Consequently, the formerregion is represented by a disproportionally larger number ofpoints than the latter. The redistribution of the point cloud doesnot move the points out of the surface as the point cloud is essen-tially a network of planar (triangular) faces for which a linear inter-polation within the faces provides new points which remain withinthe same faces.

From the locality property of B-splines it is evident that achange of the control point Qi has a localized impact on the respec-tive curve, only in the segment between the knots [ti, ti+d+1], Eq. (4).Similarly with B-spline surfaces, a change in Qi0,i1 only affects theinterval ½tu

i0; tui0 þ d0þ 1� � ½tv

i1; tvi1 þ d1þ 1�. This implies that the

part of error in Eq. (8b) which can be attributed to Qi0,i1 is relatedto the redistributed points Pj0,j1 in that interval, uj0 e [tu

i0; tui0þ

d0þ 1], vj1 e [tvi1; t

vi1 þ d1þ 1]. Consequently, if the objective is to

have Qi0,i1 located in an area with significant change in shapeand exhibit a strong impact on the local error, then the redistrib-uted points should be located in that area.

The essential novel elements of the proposed approach can besummarized by:

– existing papers do not apply the approach of having u and v asvariables in the fitting procedure of 3D surfaces in addition tothe control points P, which has been successfully developedhere,

– this paper develops the novel approach of redistributing the fit-ting point cloud (thinning and aggregating the mesh accordingto the features such as edges) to(i) provide more relative weight to the areas with significant

change in shape during the overall fitting procedure, as wellas

(ii) to provide a higher density of control points in such areassince u and v are used as augmented fitting variables.

3. Case studies of parameterizations of 3D-scanned surfaces

If geometrically complex shapes are to be scanned and capturedin parametric form (for example edged surfaces), a single mathe-matical surface may not be fully adequate. In such cases, the over-all domain should first be decomposed into mutuallyinterconnected parametric patches, possibly linked by correspond-

ing connectivity conditions such as slope and curvature constraintswhere applicable.

This situation can also be automatically recognized by thegeometry processing program based on the fact that without do-main partitioning the error benchmark in Eqs. (8a) and (8b) maynot be within acceptable limits. In a general case, the geometryof an object may be complicated and include arbitrary topologicalstructure and regions separated by edges. In such cases, networksof surface patches may be adequate or even necessary. Generally,the overall surface is represented by a set of surfaces, their mutualconnectivities and respective cross-continuities. The term segmen-tation implies the subdivision of the overall point set into subsetscorresponding to natural surfaces, which can be implemented asan edge-based or face-based procedure. The former proceeds byscanning the points data and seeking for edges where the normals,estimated from the points data, change intensively or abruptly. Thelatter approach tries to expand some initially chosen region interms of point data as long it keeps encompassing points that be-long to the same type of surface, whereby the intersections of suchsurfaces identify edges. The positions of sharp edges can also bedetermined in different ways within the point cloud itself. Forexample, finite differences or related approximations can be em-ployed on the point cloud after regularization and polygonizationto yield estimates of the first and second derivatives which areconclusive in terms of slopes and curvatures contained in the pointcloud, however measurement noise has to be filtered out and elim-inated first. The detection of edges may be based on some thresh-old values of these estimates, providing a tool for feature detectionin the raw data-scanned point cloud.

In this context, regularization refers to deriving a structuredmesh from the unstructured point cloud and adjusting the meshsize and density appropriately according to local geometric cir-cumstances such as curvatures.

Generally, networked partial surfaces are likely to perform bet-ter than overall-stretching surfaces. However, this dominantly ap-plies to static shapes and invariant geometries.

This paper actually uses single surfaces for the complex-shapedexamples provided in Section 3. At first glance, it may not seemreasonable to use a single surface for the cylinder head and similarcomplex objects. However, the objective has been to expose theproposed method to very demanding tests of irregular geometries,as passing such tests will prove the procedure to be very adaptiveand robust. The method can subsequently be expected to representindividual segments of partitioned surfaces in a trustworthy andefficient manner, as typically those possess much simpler shapes.

The second reason for applying integral surfaces is the ambitionto use the proposed representation within the framework of gener-ic shape optimization. In such a capacity, holes and protrusionsmay appear during some iterations and disappear later. Generally,edges may arise, be eliminated or move across the shape in optimi-zation iterations quasi-time.

The proposed method was developed as one that could poten-tially cope with such scenarios of changing shapes and variableshape topologies with success.

3.1. 3D scanning and parametric representation of mechanical objects

The particular test-objects in this paper, which will be used todemonstrate the developed procedure, are (i) several mechanicalcast alloy components, (ii) blades of a small wind turbine, as wellas (iii) numerically generated complex point clouds. The reference3D shapes of the physical objects are digitized in high-definitionbefore the start of exploitation.

The basic features of the ATOS optical system which was usedhere are as follows:

Fig. 6. (a) Object prepared for scanning, (b) partially completed 3D scanning, (c) alignment of point clouds towards registration and merging of partial point clouds.

Fig. 7. (a) Point cloud of the 3D-scanned wind turbine blade in STL format, (b) local detail revealing local irregularity in shape.


Measurement volume: 65 � 50 � 30–1000 � 800 � 800 [mm3].Assembly of individual point clouds via reference points.Distance between scanner and objects: 650–1300 [mm].3D scanning rate: approximately 1 s for 1032 � 776 Pixels.Spatial resolution: approximately 0.07–1 [mm].Compatibility and calibration for VDI2634 norm.

The choice of surface measurement technology is generallybased on many factors such as accuracy, cost, robustness, flexibilityand speed. Traditionally, surface measurement used to be imple-mented using tactile methods, which even nowadays providesomewhat higher accuracy than optical methods, typically in thelm range. However, they are very slow and touching the surfaceof an object may not be feasible or may cause problems in manycases. Non-contact technologies include the time-of-flight systemswhereby a laser beam travels to the object and back resulting insignal delay related to distance. Nevertheless, contactless opticalmethods based on laser-beam triangulation tend to be more con-venient. Photogrammetry-based systems provide wide area cover-age within a short measurement time. Structured-light systemssuch as the ATOS system project light fringes onto the surface thatappear distorted from the cameras’ viewpoints, which is used for3D surface reconstruction. They offer high accuracy and resolutionat an adequate expense and enable efficient combining of multiplescans using un-coded reference points.

Fig. 6 presents some elements of the implemented procedure.Fig. 6a shows one of the objects that were 3D scanned in orderto provide the required high-accuracy 3D data for testing purposesof the different parameterization algorithm settings. Fig. 6b showsthe partially completed 3D point cloud on one side of the object,along with the un-coded reference points needed to align individ-ual scans. Fig. 6c presents the scanned model of the object afterregistration and alignment of the point clouds in the commoncoordinate system and merging of the partial point clouds.

After the scanning procedure is completed, the polygonizationprocess results in a regularized point cloud. This is shown on theexample of a small wind turbine blade shown in Fig. 7a, presentinga large set of non-overlapping planar polygonial faces defined byindividual points and normals in the STL format. Fig. 7b showsan enlarged view of a local detail, indicating a potential shapedeviation.

The STL model in Fig. 7a is parameterized into B-spline surfacesaccording to Eqs. (4) and (8a) or alternatively Eq. (8b). An exampleof a parameterized windowed partial point cloud is shown in Fig. 8along with corresponding control points.

The procedure developed in Section 2 will now be tested on farmore demanding objects, where the parametric surfaces need torepresent point clouds which include

holes and localized voids in the scanning point cloud,sharp edges,slope discontinuities,very low local curvatures,

and similar troublesome features. Such advanced level of shapemodeling capacity and ability of capturing local details is necessaryif the parameterized surface is to be subjected to feature extractionand shape deviation detection algorithms.

3.2. Evaluation of the enhanced parameterization procedure

3.2.1. Narrow longitudinal gapIn order to explore different options in extracting essential

information relevant for the detection and classification of damagemodes on the surface of some object, a point cloud representing agap (Fig. 9) will be processed into parametric form by applying theapproach developed in Section 2.4.

Fig. 8. Parametric surface model of a windowed point cloud (small points), a partialspatial patch with B-spline surface control points (larger points on top of bars).


Fig. 9a shows the 100 � 100 point cloud considered for this testwhich represents a surface with a gap. The best-fitted B-spline sur-face of order 2 � 2 and the corresponding 15 � 15 control pointsaccording to Eq. (8a) are shown in Fig. 9b, whereby the cumulativeerror in Eq. (8a) amounts to the value of 98.2227 mm. Fig. 9c pre-sents the same problem with the only difference being the degreeof the B-spline surface, which is now 4 � 4. Unexpectedly, the lat-ter case did not contribute to reducing the cumulative fitting error,which in fact increased to the value of 116.185 mm.

Fig. 9 demonstrates that the corresponding B-spline surfacedoes not yield increased representation accuracy when the respec-tive degree is augmented, hence the parameterization based on Eq.(8a) has been linear and fast, but also limited in modeling perfor-mance and accuracy. This fact has provided motivation towardsdeveloping the enhanced best fitting scheme with the modified fit-ting error definition and different allocation of the matching Fij toeach Pij in the cloud according to Section 2.4, Eq. (8b).

The method developed in Eq. (8b) leads to a significant decreasein cumulative error in this test-case (Fig. 11), from 98.2227 mm to5.20368 mm, although only part of this enhancement may beacknowledged to the better best-fitting procedure. The remainingmerit is attributed to the different error benchmark since the

Fig. 9. (a) The 100 � 100 point cloud representing the surface gap, (b) corresponding 15 �98.223 mm), (c) 15 � 15 control points for the 4 � 4 degree B-spline surface (total error

Pij � Fij pairs are formed differently altogether, as symbolicallyshown in Fig. 5. In this case, if the optimum values of Q derivedby minimizing Eq. (8b) are introduced back into the standard errordefinition in Eq. (8a), the total error value for the 100 � 100 pointcloud increases to 1065.038 mm. This is empirical evidence for thefact that the assignment of Fij to each Pij in the point cloud exhibitsmajor impact on the generation of the best-fitting surface.

The reduction of the total error value also shows that the allo-cation of the ui and vj parameters as defined in reference to Eqs.(6) and (8a), that is by linearly interpolating the ui and vj parame-ters based on the Pij coordinates within the patch, may not be ade-quate. It is based on projection of Pij on the patch footprint or onsummation of linear segments connecting the control points. Thismay be a poor estimate of the curvilinear u and v coordinates ofthe parametric surface, especially if significant curvature is pres-ent. Releasing u and v and letting them be augmented fitting vari-ables as in Eq. (8b) also contributes to letting the CPs relocatewithin the domain and inherently aggregate in the edge regions,thereby reducing the total fitting error.

Yet another intervention was developed in the parameteriza-tion scheme in Eq. (8b), and it is the redistribution of the originalpoint cloud. Fig. 10 presents the spatial distribution of the covari-ance-matrix eigenvalues ratio on the patch, each point on thegraph taking into consideration all points encompassed by a cer-tain radius around the point’s location (r-vicinity).

This intervention introduces the redistribution of the points inthe cloud by performing linear mapping from Pij to P0ij to providelocally denser aggregation of the points in geometrically significantareas according to the spatial eigenvalue distribution, Fig. 10. Thenew points P0ij are linearly interpolated from the original tesselatedpolygonial triangular mesh stretched over Pij.

This intervention along with best-fitting based on Eq. (8b) re-sults in the surface in Fig. 11.

The resulting parameterization surface in Fig. 11 can be the ba-sis for different additional shape deviation diagnostics criteria thatcan operate as part of autonomous quality control expert systems.The re-distribution of the point cloud also contributes to filling thevoids in the cloud at locations of sudden change in shape. Theenhancement of representation accuracy by the developed param-eterization procedure is demonstrated in Fig. 12.

Fig. 13 shows the isolated orthogonal z-error part of the totaloffset distance norm from Fig. 12.

15 control points for the 2 � 2 degree B-spline surface parameterization (total error116.185 mm).

Fig. 11. Best-fitting the B-spline surface according to Eq. (8b) to point cloud P0ij redistributed according to Fig. 10 (total error 5.203 mm).

Fig. 12. Parameterization: total error distance norm for Fig. 9a by (a) applying the standard procedure Eq. (8a), and (b) by applying the developed procedure in Eq. (8b) alongwith eigenvalue-based point cloud redistribution.

Fig. 10. Spatial distribution of the ratio of eigenvalues of covariance matrix, emine1þe2þe3

, value at each point calculated from partial (local) point cloud encompassed by radius raround that point.


Such parameterization also enables the creation of 2D images ofarbitrary resolution (such as Fig. 14) where different additional im-age processing algorithms can be applied (Edge detection, Corner

detection, Gaussian convolution, Image binarization, Connected-component labeling, etc.). Consequently different features andtheir corresponding attributes can be extracted.

Fig. 13. Parameterization: orthogonal z-error for Fig. 9a by (a) applying the standard procedure Eq. (8a), (b) by applying the developed procedure in Eq. (8b) along witheigenvalue-based point cloud redistribution.

Fig. 14. Gray-scale pixelized representation of orthogonal z-coordinate in Fig. 13.


Fig. 14 also indicates that the procedure developed here couldalso be applied to problems where a distribution of some physicalquantity across a 2D domain may be given instead of a geometricz-coordinate. Such global interpolation functions could for exam-ple be used in numerically solving field problems using meshlessmethods. Such a distribution may also be represented by a para-metric surface entity as developed in Section 2.4.

3.2.2. Deep local cavityThe second case considered in testing the performance of the

parameterization procedure developed in Section 2 is the case ofa point cloud corresponding to a small-footprint deep local cavitylocated within the surface patch. It is also a very demandingtest-case for parameterization due to steep changes in coordinatevalues orthogonal to the patch with an amplitude of 12.82 mm,the shape of which closely resembles the localized mathematicalspatial step function.

The point cloud originating from high-resolution optical 3Dscanning of the local cavity is presented in Fig. 15.

Fig. 15. (a) 3D-scanning point cloud of a surface patch containing a cavity, (b)

Since it is very important to apply sparing parameterizations ascompact-size control point data-sets are numerically less expen-sive to process, the initial setting is 10 � 10 CPs with the respectivedegree of the B-spline curve of 2 � 2. Fig. 16a and b demonstratethe parameterization according to Eq. (8a) to be inadequate inmodeling the cavity since there is insufficient local shape controlin the vicinity of the edges.

However, if the fitting algorithm in Eq. (8b) with Levenberg–Marquardt (LM) non-linear minimization is applied, much betterrepresentation of the cavity points is accomplished, Fig. 17a and b.Much better local control around the steep edges could be achieved,even considering the given doubtlessly scarce CP data-set.

A similar effect of the parameterization algorithm on the totalfitting error norm can be observed in Fig. 18 vs. Fig. 19, where15 � 15 CP data-sets are applied. The former is based on linearbest-fitting according to Eq. (8a), the latter on non-linear LM min-imization of Eq. (8b). Again, the latter algorithm achieves muchbetter local control in the steep edges region with much lower er-ror function amplitudes and average values.

Figs. 9–19 demonstrate the developed fitting procedure to beefficient in handling demanding 3D scanning problems where thepoint clouds include voids, edges, discontinuities and other prob-lematic features potentially contained in raw point clouds obtainedby optical 3D scanning. It proves that even if the surface parame-terization data-set is humble in size, an efficient parameterizationprocedure as developed here can produce far better shape repre-sentation results without increasing the data-set size which stan-dard approaches do. An expert system can thus be implementedto perform 3D data acquisition, shape knowledge representation,shape-related numerical processing and compact data storageautonomously and efficiently in industrial process real-time.

3.2.3. Cylinder head of a small combustion engineTo demonstrate the performance and representation capacity

of the developed method even further, a real-world test case

gray-scale representation of the patch-orthogonal z-values of point cloud.

Fig. 16. (a) B-spline parametric surface of degree 2 � 2 with 10 � 10 control points, linear best fit to point cloud in Fig. 15a according to Eq. (8a), (b) graph presenting thez-error between the point cloud in Fig. 15a and surface in figure (a).

Fig. 17. (a) B-spline parametric surface of degree 2 � 2 with 10 � 10 control points, non-linear best fit to point cloud in Fig. 15a according to Eq. (8b), (b) graph presenting thez-error between the point cloud in Fig. 15a and surface in figure (a).

Fig. 18. (a) B-spline parametric surface of degree 2 � 2 with 15 � 15 control points, linear best fit to point cloud in Fig. 15a according to Eq. (8a), (b) graph presenting the z-error between the point cloud in Fig. 15a and surface in figure (a).

Fig. 19. (a) B-spline parametric surface of degree 2 � 2 with 15 � 15 control points, non-linear best fit to point cloud in Fig. 15a according to Eq. (8b), (b) graph presenting thez-error between the point cloud in Fig. 15a and surface in figure (a).



exhibiting a highly complex geometric shape was selected. This ob-ject is the cast cylinder head of a small gasoline engine propellingan electric generator, ‘Gude GS 950’. It is a 2 HP (1.5 kW) enginepower, 63 cm3 volume, single-cylinder, two-stroke engine drivinga 50 Hz–230 V AC generator with 650 W permanent output.

Again, the objective is to parameterize the object using a singleintegral B-spline entity. The engine head was scanned in 3D usingthe high resolution and accuracy optical scanner Atos [5], wherebya large point cloud of approximately 108 points has been generatedto serve as the raw data-set for parameterization.

Subsequently, integral parameterization of the highly complexgeometry in Fig. 20b was carried out using B-spline surfaces with30 � 20 control points. Linear fitting based on Eq. (8a) resulted inthe fitting surface shown in Fig. 21.

The non-linear parameterization algorithm developed herebased on Eq. (8b) and using Levenberg–Marquardt fitting errorminimization has resulted in the integral surface shown in Fig. 22.

As in Section 3.2.2, the aggregation of the point cloud towardsthe areas with edges or major dimensional change in some direc-tion, identified based on the eigenvalue ratio in Fig. 20c, was alsointroduced in this case.

The error distribution for the parameterizations shown in Figs.21 and 22 respectively is shown in Fig. 23.

The error improvement seems very satisfactory for this complexcase. The error reduction factor is approximately 0.5 at the errorpeak locations, and the overall error reduction distribution inFig. 23 is also significant.

Non-linear fitting based on Eq. (8b), while providing a lower fit-ting error, inherently comes at a numerical expense. As an illustra-tion, linear fitting based on Cholesky decomposition resulted in

Fig. 20. (a) Raw point cloud of the 3D-scanned engine cylinder head, (b) 3D-windoweigenvalues emin

e1þe2þe3for the points encompassed in the respective local vicinity (r = 3 mm

Fig. 21. Three views of the fitting surface and respective control points for th

0.65 s of fitting run-time (generally, complexity of the order n3,where n is the order of the system, but in this case reduced sincealgorithms for sparse matrices can be applied). Using LM-based fit-ting and Eq. (8b), the fitting problem was solved in 17.4 s on a stan-dard PC.

The numerical procedure developed in this paper is autono-mous and self-navigated in terms of its ability to adapt efficientlyto different shapes and features such as edges or peaks which maybe contained in the geometric data-set. The number of controlpoints can be automatically evaluated based on required cumula-tive fitting error across the overall shape as well as the spatial dis-tribution of the fitting error in local zones. Of course, the proposedmethod (Section 2.4) is self-navigated itself and adjusts autono-mously to demanding shapes as demonstrated in Section 3.2.Non-linear fitting involving the Q, u and v vectors as variables isperformed by the LM optimizer and hence fully autonomous. Theredistribution of the points according to the eigenvalue ratio distri-butions is also automatic, while the size of the neighborhoodswhere the ratios are evaluated can be pre-set as desired.

While the individual execution times were given above, anadditional comment should be provided here. Compared to the fit-ting procedure using the ICP algorithm, the re-distribution of thepoint cloud and introduction of the augmented set of variables intothe now non-linear fitting procedure results in a significantlyshorter computational time. The point redistribution is executedonly once and does not introduce more complexity than an averageICP iteration. Moreover, the re-distribution of the point cloudcauses the transformed solution space to have such local minimaof the fitting problem which are all close and numericallyacceptable.

ed section of the point cloud to be parameterized, (c) distribution of the ratio of).

e linear parameterization procedure based on Eq. (8a) with 30 � 20 CPs.

Fig. 22. Three views of the fitting surface and respective control points for the non-linear parameterization procedure developed based on Eq. (8b) with 30 � 20 CPs.

Fig. 23. Fitting surface error distributions for 30 � 20 CPs B-spline parameterization of the cylinder head section in Fig. 20b, (a) linear fitting procedure based on Eq. (8a),(b) non-linear fitting developed here based on Eq. (8b).


Solving individual iterations of the non-linear fitting procedureimplies solving the linear system whose matrix is symmetric andpositive definite, such that Cholesky decomposition can be applied(complexity O(n3)). It is also a banded matrix, hence banded Chole-sky factorization can be applied O((k2 + 3k) � n), where k is thewidth of the band.

4. Conclusions

An enhanced autonomous and self-navigated procedure forautomatic processing of 3D point clouds is developed for the pur-pose of faithful, yet compact, representation of complex 3D shape.The procedure includes best-fitting of B-spline surfaces as part of anumerical workflow. An upgraded B-spline fitting method is pro-posed based on an extended set of fitting variables and redistribu-tion of the point cloud data, providing for more fitting accuracy atthe numerical expense of non-linear fitting of the developedalgorithm.

Empirical evidence using demanding surfaces resulting fromhigh-end 3D optical scanning was used to demonstrate the effi-ciency of the proposed method. These point clouds represent a nar-row longitudinal gap and a deep local cavity on the surface of thetest-object, as well as a complex-shaped engine cylinder head. Allthree cases provide evidence for a significant reduction of the sur-face fitting error without increasing the parametric data-set, themerit of which can be attributed both to the developed fitting pro-cedure and proposed error metrics.

An autonomous monitoring expert system can potentially bedeveloped based on the developed procedure such that it operateson the compact control points instead on large point clouds.

Acknowledgements

This work was supported by the Ministry of Science, Educationand Sports, Croatia, Project 023-0231744-3113, Intelligent andEvolutionary Algorithms in Optimization.

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