11 July 2002 Reverse Engineering 1 Dr. Gábor Renner Geometric Modelling Laboratory, Computer and Automation Research Institute

11 July 2002Reverse Engineering2 n n set data point  CAD model measured data  boundary representation (incomplete, noisy, outliers) (accurate and consistent) n n intelligent 3D Scanner interpret the structure of data points in order to create an appropriate computer representation allowing redesign of objects n n applications – –no original drawing or documentation – –reengineering for constructing improved products – –reconstruct wooden or clay models – –incorporate, matching human surfaces, etc.

11 July 2002Reverse Engineering3 Classifying objects n n conventional engineering objects – –many faces; mostly simple geometry – –f(x,y,z)=0, implicit surfaces: plane, cylinder, cone, sphere, torus – – sharp (or blended) edges n n free - form shapes – –small number of faces; complex geometry – –r = r(u,v), piecewise parametric surfaces, – –smooth internal subdividing curves n n artistic objects n n natural surfaces

11 July 2002Reverse Engineering4 Conventional engineering parts

11 July 2002Reverse Engineering5 Free-form objects

11 July 2002Reverse Engineering6 Artistic objects

11 July 2002Reverse Engineering7 Natural object

11 July 2002Reverse Engineering8 Natural objects

11 July 2002Reverse Engineering9 Natural objects

11 July 2002Reverse Engineering10 Basic Phases of RE Basic Phases of RE n 1. data acquisition n 2. pre-processing triangulation,triangulation, decimation decimation merging multiple viewsmerging multiple views n 3. segmentation n 4. surface fitting n 5. CAD model creation

11 July 2002Reverse Engineering11 Triangulation

11 July 2002Reverse Engineering12 Triangulation

11 July 2002Reverse Engineering13 Decimation

11 July 2002Reverse Engineering14 Merging point clouds (registration) - 1 Merging point clouds (registration) - 1

11 July 2002Reverse Engineering15 Merging point clouds (registration) - 2 Merging point clouds (registration) - 2 FIAT

11 July 2002Reverse Engineering16 n SEGMENTATION: separate subsets of data points; each point region corresponds to the pre-image of a particular face of the object n “chicken and egg” problem given the geometry, selecting point sets is easy given the pointsets, fitting geometry is easy n to resolve this we need: interactive help iterative procedures restricted object classes n segmentation and surface fitting are strongly coupled: hypothesis  tests Segmentation and surface fitting

11 July 2002Reverse Engineering17 Reconstructing conventional engineering objects - 1 n n basic assumptions relatively large primary surfaces – –planes, cylinders, cones, spheres, tori linear extrusions and surfaces of revolution relatively small blends n n “accurate” reconstruction ”without” user assistance

11 July 2002Reverse Engineering18 Object and decimated mesh Object and decimated mesh

11 July 2002Reverse Engineering19 Reconstructing conventional engineering objects - 2 n n the basic structure can be determined n n direct segmentation decompose the point cloud into regions n n a sequential approach using filters find “stable” regions discard “unstable” triangular strips, by detecting sharp edges and smooth edges simple regions composite, smooth regions

11 July 2002Reverse Engineering20 Reconstructing conventional engineering objects - 3 n n sharp edges (and edges with small blends) computed by surface-surface intersection n n smooth edges assure accuracy and tangential continuity surface/surface intersections would fail in the almost tangential situations explicitly created by constrained fitting of multiple geometric entities

11 July 2002Reverse Engineering21 Direct segmentation - 1 n n basic principle 1. based on a given environment compute an indicator for each point 2. based on the current filter exclude unstable portions and split the region into smaller ones 3. if simple region: done 4. if linear extrusion or surface of revolution: create a 2D profile 5. if smooth, composite region: compute the next indicator and go to 1

11 July 2002Reverse Engineering22 Direct segmentation - 2 n n planarity filter: detect sharp edges and small blends n n dimensionality filter: separate planes cylinders or cones, linear extrusions, composite conical-cylindrical regions spheres or tori, surfaces of revolution, composite toroidal-spherical regions n n direction filter: separate cylinders, linear extrusions, composite conical regions n n apex filter: separate cones n n axis filter: separate spheres, tori, surfaces of revolution

11 July 2002Reverse Engineering23 Planarity filtering Angular deviation Numerical curvatures Remove data points around sharp edges

11 July 2002Reverse Engineering24 Dimensionality filtering using the Gaussian sphere

11 July 2002Reverse Engineering25 Dimensionality filtering - An example.

11 July 2002Reverse Engineering26 Dimensionality filtering based on the number of points in two concentric spheres separate data points by their dimensionality D0: planes D1: cylinders-cones-transl. surfs D2: tori-spheres-rot. surfs

11 July 2002Reverse Engineering27 Planarity and dimensionality filtering Planarity and dimensionality filtering

11 July 2002Reverse Engineering28 Planarity and dimensionality filtering

11 July 2002Reverse Engineering29 Detect translational and rotational symmetries n n translational direction normal vectors n i of a translational surface are perpendicular to a common direction minimise   n i,d  2 n n rotational axis normal lines of a rotational surface (l i, p i ) intersect a common axis  i - angle between the normal line l i and the plane containing the axis and the point p i various measures, in general: a non-linear system

11 July 2002Reverse Engineering30 Computing best fit rotational axis

11 July 2002Reverse Engineering31 Conical - cylindrical region direction estimation detects cylinders and composite linear extrusions, rest: composite conical region

11 July 2002Reverse Engineering32 Conical composite region Conical composite region fit a least squares point to the tangent planes to compute the apex

11 July 2002Reverse Engineering33 Toroidal - spherical region Toroidal - spherical region estimate a local axis of revolution if largest eigenvalue (almost) zero -> sphere otherwise torus or surface of revolution

11 July 2002Reverse Engineering34 Apex and axis filtering

11 July 2002Reverse Engineering35 Surface fitting n n given a point set and a hypothesis - find the best least squares surface n n simple analytic surfaces - f(s,p) = 0 s: parameter vector, p: 3D point n n minimise Euclidean distances - true geometric fitting n n algebraic fitting - minimise  f(s,p i ) 2 n n approximate geometric fit - f / | f ’| n n ‘faithful’ geometric distances (Pratt 1987, Lukács et al., 1998): unit derivative on the surface n n sequential least squares based on normal vector estimations series of linear steps reasonably accurate, computationally efficient

11 July 2002Reverse Engineering36 Constrained fitting n n needed for various engineering purposes n n fitting smooth profile curves for linear extrusions and surfaces of revolution n n refitting elements of smooth composite regions for B-rep model building good initial surface parameters from segmentation set of constraints edge curves - explicitly computed n n beautify the model resolve topological inconsistencies rounded values, perpendicular faces, concentric axis

11 July 2002Reverse Engineering37 Constrained fitting Constrained fitting

11 July 2002Reverse Engineering38 Constrained profiles Translational profile Rotational profile

11 July 2002Reverse Engineering39 Constrained fitting problem n primary surfaces: s  S parameter set: a parameter set: a n point sets: p  P s n individual weight:  s n k constraint equations: {c i } find a, which minimizes f while c=0 find a, which minimizes f while c=0 c(a) = 0 Constraints: tangency, perpendicularity, concentricity, symmetry, etc..

11 July 2002Reverse Engineering40 Constrained fitting techniques n standard solution: Lagrangian multipliers, n+k equations, multidimensional Newton-Raphson n problem: constraints contradict or not independent n preferred solution: sequential constraint satisfaction constraints sorted by priority c(a) = 0 and f(a) = min. is solved simultaneously by iteration c(a) = 0 and f(a) = min. is solved simultaneously by iteration

11 July 2002Reverse Engineering41 Constrained fitting - 2 n linear approximation for c, quadratic for f n in matrix form n where

11 July 2002Reverse Engineering42 Efficient representation n signed distance function n the function to be minimized n middle term needs to be computed only once

11 July 2002Reverse Engineering43 Fitting a circle - an example n center o, radius r, point p n Euclidean distance function: |p - o| - r n n faithful approximation: n n terms are now separated n n alternative parameters with a constraint:

11 July 2002Reverse Engineering44 Equations for constrained fitting of circles n circles (lines) - in Pratt’s form (1987) n tangency constraints

11 July 2002Reverse Engineering45 Using auxiliary objects 1a1b 2a2b

11 July 2002Reverse Engineering46 Simple part reconstruction Simple part reconstruction

11 July 2002Reverse Engineering47 Final CAD (B-rep) model with blends   without blends

11 July 2002Reverse Engineering48 Functional decomposition: primary surfaces + features P0P0 P3P3 P4P4 P5P5 P6P6 P7P7 P8P8 S1S1 P 10 P1P1 P2P2 S2S2 ST(S 1,S 2 ) Ignore area Reconstruction of free-form shapes

11 July 2002Reverse Engineering49 Reconstruction of free-form shapes Surface structure

11 July 2002Reverse Engineering50 Reconstruction of free-form shapes Curvature plot

11 July 2002Reverse Engineering51 Advanced Surface Fitting (BMW model)

Free form surface fitting Functional: Functional: Initialization: Initialization: data point parameters knot-distribution smoothing weight ( ) F minimization F minimization Parametercorrection Parametercorrection Knot-insertion Knot-insertion Smoothing weight opt. Smoothing weight opt.

11 July 2002Reverse Engineering53 Reconstructing free-form features (vrrb blend) Global surface fitting Functional decomposition - stable regions - constrained fitting

11 July 2002Reverse Engineering54 Reconstructing free-form features (free-form step) Global surface fittingFunctional decomposition

11 July 2002Reverse Engineering55 Conclusion n n RE: a complex process, approaches differ by model type, quality of measured data sets and ‘a priori’ assumptions n n free-form objects: functional decomposition - some user assistance needed n n conventional engineering objects: direct segmentation - basically automatic smooth edges accurate surfaces - linear extrusions - surfaces of revolution constrained fitting