The Meaning of Redundancy – 3D Topology and Geometric ...€¦ · 9 C. Clemen – The Meaning of...

Department of Engineering Surveying and Adjustment TechniquesDepartment of Engineering Surveying and Adjustment Techniques

Technische Universität Berlin

The Meaning of Redundancy – 3D Topology and Geometric Parameterization

Christian ClemenLothar Gründig

Institute for Geodesy and Geoinformation ScienceTechnische Universität Berlin, Germany

Working Week 2008, 14-19 June 2008, Stockholm, Sweden

2 C. Clemen C. Clemen –– The Meaning of Redundancy – 3D Topology and Geometric…


Motivation

What if…

…measurements have been forgotten …

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Motivation

What if…

…measurements disagree

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Motivation

What if…

…resulting walls do not aligne

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Motivation

What if typical mistakes occur during 3D as-build-documentation ?

Staff has to return to the building for re-measuring.

Mistakes remain undetected, thus the resulting building geometry is wrong.

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Motivation

sketch observe

adjustcalculate



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Motivation

sketch observe

adjustcalculate



Agenda

• Topologytopological primitiveswhy topology ?topological normalization

• GeometryRedundancy of point-based geometry Advantages of surface-based geometry

• Adjustmentfunctional modelstatistical testing

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topological primitives

A node is a 0-dimensional topological object. A node has no boundary.

An edge is a 1-dimensional topological object. An edge is bounded by two nodes

A face is a 2-dimesional topological object. A face is bounded by a finite set of edges. The boundary is closed and has no self intersection.

A topological solid is a 3-dimensional topological object. A solid is bounded by faces. The surface (boundary) is closed, orientable, connected and free of intersection.

node edge face solid



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why topology ?

Right Question: Why explicitly specified topology in three dimensional boundary representation models ?

Database tables/attributes…XML tags…Java objects…that represent topological primitives and topological relations



deterministic redundancy vs. stochastic redundancy

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why topology ?


ISO 19103: The most important productive use of topology is to accelerate computational geometry

Surveying Engineer: The most important productive use of topology is to ensure consistency during inserting, updating and deleting operations

Database designer: Consistency is ensured by minimizing redundancy (normalization)



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why topology ?


Today: Reduce deterministic redundancy !

….Upcoming slides: stepwise reduction of deterministic redundancy



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Topological Normalization

Bildquelle: www.marions-kochbuch.de/rezept/0592.htm

Spaghetti – Polygon Soup

NoderedundancyMultiple storage of points

EdgeredundacyMultiple storage of node connectivity

LoopredundancyMultiple storage of edge loops

FaceredundancyMultiple storage of facesNow:

Stepwise „Normalization“



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Noderedundancy is avoided by swapping node composition to node

aggregation

topological normalization



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Orientation of edgesby defining start-node and end node

Problem:Each faces stores its own edges




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Orientation is stored seperately in thehalf-edge entity

… thus a face is oriented since ordering of edges (and therefore

nodes) is given




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Loops allow for unique storage of half-edge collections, Thus the interior ring

and the exterior is topologically the same entity




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By letting the face refer to the solid (not the other way around) a face

(collection of loops) is only stored once.

No redundancy any more ???




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ACIS IFC

GML/ ISO 19107

…and many more !!!!!!!

The choise of topological model strongly depends on the application




Agenda




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Geometric normalization

Planarity

Co-planarity

Paralellism

Geometric Redundancy



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Planarity

Geometric Normalization



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Planarity

Geometric Normalization

Co-planarity

Paralellism



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Point based parameterization:412 nodes -> 1236 coordinate values

Surfaced based parameterization:104 d, 14 nv-values

Reduction: 91%



Agenda




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Adjustment

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Adjustment

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Functional model observation equations: Distances between topological primitives, local spherical points on topological primitives, angular constraints (parallel, perpendicular)

Leica DISTOTM A6

BlueToothTMApplication

Total station



Adjustment

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j il v d d+ = − 2 2 2, 1x y zn n n n n n= = + + =r r r

( ) ( )1

, , , , , 0T T

i jk i jk j k j k j kl v n p d p n n n n d d−

+ = + = ×

ur uuur uuur uur uur uur uurK

( ) ( ) ( ) ( )1 1

, , , , 0 ; , , , , 0T TT T

ij kl ij i j i j i j kl k l k l k ll v p p p n n n n d d p n n n n d d− −

+ = − = × = ×

uur uur uur ur uur ur uur uur uur ur uur urK

, 0i j k ln n n n× × =ur uur uur ur

0,

1 i j k l

v perpendicularn n n n

v parallel+

= × × +

ur uur uur ur

( ) ( ) ( )( ) ( ) ( )

( ) ( )

sin cos

( , ) , sin sin , 0

cos

i r i i TT

i j j i r i i T T j j

Ti r i

r v v v xf x l v p n d r v v v y n d

zr v v

θ φ

θ φ

θ

θ φ ω

θ φ ωθ

+ + + + + = − = + + + + + − = + +

uur uur uur

Functional model (observation equations, constraints): refer to the paper



Adjustment

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Post Adjustment Statistics are applied in order to

• test the general goodness of fit and detect whether mistakes occur in observations or the functional/statistical model. (-Test

• test if two normal vectors are stochastically equal. If true, the normal vectors would be unified by interchanging the references in the data base. (Student’s t-test)

• test if two orthogonal distances d of parallel planes are stochastically equal. If true the two planes will be unified (Student’s t-test).

• Test for outliers due to bad face-plane assignment. (z-Test on residuals vi)

2χ



Workflow

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Recap




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Technische Universität Berlin

… thank you.

Christian [email protected]

The Meaning of Redundancy – 3D Topology and Geometric ...€¦ · 9 C. Clemen – The Meaning of...

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