11 july 2002 reverse engineering 1 dr. gábor renner geometric modelling laboratory, computer and...

11Reverse EngineeringReverse Engineering11 July 200211 July 2002

Reverse EngineeringReverse Engineering

Dr. GDr. Gábor Rennerábor Renner

Geometric Modelling Laboratory,Geometric Modelling Laboratory,

Computer and Automation Research Institute Computer and Automation Research Institute


Reverse Engineering

set data point CAD model measured data boundary representation (incomplete, noisy, outliers) (accurate and consistent)

intelligent 3D Scanner • interpret the structure of data points in order to create

an appropriate computer representation allowing redesign of objects

applications– no original drawing or documentation– reengineering for constructing improved products– reconstruct wooden or clay models– incorporate, matching human surfaces, etc.


Classifying objects

conventional engineering objects– many faces; mostly simple geometry– f(x,y,z)=0, implicit surfaces:

• plane, cylinder, cone, sphere, torus

– sharp (or blended) edges free - form shapes

– small number of faces; complex geometry– r = r(u,v), piecewise parametric surfaces, – smooth internal subdividing curves

artistic objects natural surfaces


Conventional engineering partsConventional engineering parts


Free-form objects


Artistic objectsArtistic objects


Natural objectNatural object


Natural objects


Basic Phases of REBasic Phases of RE

1. data acquisition1. data acquisition 2. pre-processing2. pre-processing

• triangulation,triangulation,• decimationdecimation• merging multiple viewsmerging multiple views

3. segmentation3. segmentation 4. surface fitting4. surface fitting 5. CAD model creation5. CAD model creation

11 July 200211 July 2002 Reverse EngineeringReverse Engineering 1111

TriangulationTriangulation


TriangulationTriangulation


Decimation Decimation


Merging point clouds Merging point clouds (registration) - 1(registration) - 1


Merging point clouds Merging point clouds (registration) - 2(registration) - 2

FIAT


SEGMENTATION: separate subsets of data points; each point region corresponds to the pre-image of a particular face of the object

“chicken and egg” problem• given the geometry, selecting point sets is easy• given the pointsets, fitting geometry is easy

to resolve this we need:• interactive help• iterative procedures• restricted object classes

segmentation and surface fitting are strongly coupled: hypothesis tests

Segmentation and surface fitting


Reconstructing conventional engineering objects - 1

basic assumptions• relatively large primary surfaces

– planes, cylinders, cones, spheres, tori

• linear extrusions and surfaces of revolution• relatively small blends

“accurate” reconstruction ”without” user assistance


Object and decimated meshObject and decimated mesh



the basic structure can be determined direct segmentation

• decompose the point cloud into regions

a sequential approach using filters• find “stable” regions• discard “unstable” triangular strips, by detecting sharp

edges and smooth edges• simple regions• composite, smooth regions



sharp edges (and edges with small blends)• computed by surface-surface intersection

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


Direct segmentation - 1

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


Direct segmentation - 2

planarity filter: detect sharp edges and small blends dimensionality filter: separate

• planes• cylinders or cones, linear extrusions, composite

conical-cylindrical regions• spheres or tori, surfaces of revolution, composite

toroidal-spherical regions direction filter: separate

• cylinders, linear extrusions, composite conical regions apex filter: separate cones axis filter: separate

• spheres, tori, surfaces of revolution


Planarity filtering

Angular deviationNumerical curvatures

Remove data points aroundsharp edges


Dimensionality filtering using Dimensionality filtering using the Gaussian spherethe Gaussian sphere


Dimensionality filtering - An example.Dimensionality filtering - An example.


Dimensionality filteringDimensionality filtering

• based on the number of points in two concentric spheres

• separate data points by their dimensionality

D0: planesD1: cylinders-cones-transl. surfsD2: tori-spheres-rot. surfs


Planarity and dimensionality Planarity and dimensionality filteringfiltering


Detect translational and rotational symmetries

translational direction

• normal vectors ni of a translational surface are perpendicular to a common direction

• minimise ni,d2

rotational axis

• normal lines of a rotational surface (li, pi) intersect a common axis

i - angle between the normal line li and the plane containing the axis and the point pi

• various measures, in general: a non-linear system


Computing best fit rotational axis


Conical - cylindrical regionConical - cylindrical region

direction estimation detects cylinders and composite linearextrusions, rest: composite conical region


Conical composite regionConical composite region

fit a least squares pointto the tangent planes tocompute the apex


Toroidal - spherical regionToroidal - spherical region

estimate a localaxis of revolution

if largest eigenvalue(almost) zero ->sphere

otherwise torus orsurface of revolution


Apex and axis filteringApex and axis filtering


Surface fitting

given a point set and a hypothesis - find the best least squares surface

simple analytic surfaces - f(s,p) = 0• s: parameter vector, p: 3D point

minimise Euclidean distances - true geometric fitting algebraic fitting - minimise f(s,pi )2

approximate geometric fit - f / | f ’| ‘faithful’ geometric distances (Pratt 1987, Lukács et al.,

1998): unit derivative on the surface sequential least squares

• based on normal vector estimations

• series of linear steps

• reasonably accurate, computationally efficient


Constrained fitting Constrained fitting

needed for various engineering purposes fitting smooth profile curves for linear extrusions

and surfaces of revolution refitting elements of smooth composite regions

for B-rep model building• good initial surface parameters from segmentation• set of constraints• edge curves - explicitly computed

beautify the model• resolve topological inconsistencies• rounded values, perpendicular faces, concentric axis


Constrained fittingConstrained fitting


Constrained profilesConstrained profiles

Translational profileRotational profile


Constrained fitting problemConstrained fitting problem

primary surfaces:primary surfaces: s s S S parameter set:parameter set: aa point sets:point sets: p p P Pss

individual weight:individual weight: ss

k k constraint equations:constraint equations: {c {cii}} findfind aa, , which minimizeswhich minimizes f f

whilewhile c=0 c=0

c(a) = 0

Constraints: tangency, perpendicularity, concentricity, symmetry, etc..


Constrained fitting techniques Constrained fitting techniques

standard solution: Lagrangian multipliers, standard solution: Lagrangian multipliers, n+kn+k equations, multidimensional Newton-Raphson equations, multidimensional Newton-Raphson

problem: constraints contradict or not problem: constraints contradict or not independentindependent

preferred solution: sequential constraint preferred solution: sequential constraint satisfaction constraints sorted by satisfaction constraints sorted by prioritypriority

c(a) = 0c(a) = 0 and and f(a) = min.f(a) = min. is solved simultaneously is solved simultaneously by iterationby iteration


Constrained fitting - 2 Constrained fitting - 2

linear approximation forlinear approximation for c, c, quadratic forquadratic for f f

in matrix formin matrix form

wherewhere


Efficient representation Efficient representation

signed distance functionsigned distance function

the function to be minimizedthe function to be minimized

middle term needs to be computed only oncemiddle term needs to be computed only once


Fitting a circle - an exampleFitting a circle - an example

centercenter o, o, radiusradius r, r, pointpoint p p Euclidean distance function:Euclidean distance function: |p - o| - r |p - o| - r faithful approximation:

terms are now separated

alternative parameters with a constraint:


Equations for constrained fitting Equations for constrained fitting of circlesof circles

circles (lines) - in Pratt’s form (1987) circles (lines) - in Pratt’s form (1987)

tangency constraintstangency constraints


Using auxiliary objectsUsing auxiliary objects

1a 1b

2a 2b


Simple part reconstructionSimple part reconstruction


Final CAD (B-rep) modelFinal CAD (B-rep) model

with blends

without blends


Functional decomposition:

primary surfaces + features

P0

P3

P4

P5

P6

P7

P8S1

P10

P1

P2

S2

ST(S1,S2)

Ignore area

Reconstruction of free-form shapes



Surfacestructure



Curvatureplot


Advanced Surface Fitting (BMW model)

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Free form surface fittingFree form surface fitting

Functional:Functional:

F F F

F u v

F v u

lsq sm

lsq i ii

n

i

sm uu uv vvv

v

u

u

( ) ( ( , ) )

( )min

max

min

max

S S P

S S S S

1

2

2 2 2 d d

Initialization:Initialization: • data point parametersdata point parameters• knot-distributionknot-distribution• smoothing weight (smoothing weight ())

F minimizationF minimization ParametercorrectionParametercorrection Knot-insertionKnot-insertion Smoothing weight opt.Smoothing weight opt.


Reconstructing free-form features (vrrb blend)

Global surface fitting

Functional decomposition- stable regions- constrained fitting


Reconstructing free-form features(free-form step)

Global surface fitting Functional decomposition


Conclusion

RE: a complex process, approaches differ by model type, quality of measured data sets and ‘a priori’ assumptions

free-form objects: functional decomposition - some user assistance needed

conventional engineering objects: direct segmentation - basically automatic• smooth edges

• accurate surfaces - linear extrusions - surfaces of revolution

• constrained fitting

11 july 2002 reverse engineering 1 dr. gábor renner geometric modelling laboratory, computer and...

Documents

reverse engineeringdr

point sets

point region

sharp edges

basic structure

data acquisition2

structure of data points

surface fitting5