3D Shape Understanding and Reconstruction based on Line ...sodwana.uni-ak.ac.at/geom/mitarbeiter/odehnal/talk/hk_05.pdf · 3D Shape Understanding and Reconstruction based on Line

3D Shape Understanding andReconstruction based on LineElement Geometry

B. Odehnal

Vienna University of Technology

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Contents.

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Contents.

• Aims & Results

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Contents.

• Aims & Results• Equiform Kinematics

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Contents.

• Aims & Results• Equiform Kinematics• Line Elements

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Contents.

• Aims & Results• Equiform Kinematics• Line Elements• Linear Complexes of Line Elements

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Contents.

• Aims & Results• Equiform Kinematics• Line Elements• Linear Complexes of Line Elements• Classification of Surfaces

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Contents.

• Aims & Results• Equiform Kinematics• Line Elements• Linear Complexes of Line Elements• Classification of Surfaces• Recognition

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Contents.

• Aims & Results• Equiform Kinematics• Line Elements• Linear Complexes of Line Elements• Classification of Surfaces• Recognition• Segmentation

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Contents.

• Aims & Results• Equiform Kinematics• Line Elements• Linear Complexes of Line Elements• Classification of Surfaces• Recognition• Segmentation• Reconstruction

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Aims & Results.

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Aims & Results.

• new theory dealing with line elements

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Aims & Results.

• new theory dealing with line elements• much wider class of surfaces can be detected:

Euclidean kinematic surfaces (helical (rotational)surf., cylinders),equiform kinematic surfaces (spiral surf., general,cones, etc.)

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Aims & Results.

• new theory dealing with line elements• much wider class of surfaces can be detected:

Euclidean kinematic surfaces (helical (rotational)surf., cylinders),equiform kinematic surfaces (spiral surf., general,cones, etc.)

• applicable to recognition/reconstruction ofsurfaces, segmentation

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Related Work.

[1] B. ODEHNAL , H. POTTMANN , J. WALLNER : Equiform kinematics

and the geometry of line elements. Submitted to: Contributions to

Algebra and Geometry.

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Related Work.




[2] M. H OFER, B. ODEHNAL , H. POTTMANN , T. STEINER, J.

WALLNER : 3D shape understanding and reconstruction based on line

element geometry. Proc. ICCV’05, to appear.

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Related Work.




[2] M. H OFER, B. ODEHNAL , H. POTTMANN , T. STEINER, J.

WALLNER : 3D shape understanding and reconstruction based on line

element geometry. Proc. ICCV’05, to appear.

[3] H. POTTMANN , M. HOFER, B. ODEHNAL , J. WALLNER : Line

geometry for 3D shape understanding and reconstruction. Proc.

ECCV’04.

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Equiform Kinematics.

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• equiform motion

x 7→ y = αAx+ a α > 0, A ∈ SO3, a ∈ R3

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• equiform motion

x 7→ y = αAx+ a α > 0, A ∈ SO3, a ∈ R3

• one-parameter equiform motion

αt : I ⊂ R → R, At : I → SO3, at : I → R3

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• equiform motion

x 7→ y = αAx+ a α > 0, A ∈ SO3, a ∈ R3


αt : I ⊂ R → R, At : I → SO3, at : I → R3

• field of velocity vectors at timet0

v(y) = y = αAx+ αAx+ a =

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• equiform motion

x 7→ y = αAx+ a α > 0, A ∈ SO3, a ∈ R3


αt : I ⊂ R → R, At : I → SO3, at : I → R3



αα−1y + AATy − αα−1a− AATa+ a =

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• equiform motion

x 7→ y = αAx+ a α > 0, A ∈ SO3, a ∈ R3


αt : I ⊂ R → R, At : I → SO3, at : I → R3



αα−1y + AATy − αα−1a− AATa+ a =

c× y + c+ γy . . . linear iny

AATy = c× y, γ = αα−1, c = a− γa− c× a

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Line Elements.

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Line Elements.

N

po ν

n

y

n line element(N, y) =lineN + pointy ∈ N

(n, n) Plücker coordina-tes ofN , n := y × n,‖n‖ = 1, 〈n, n〉 = 0

ν = 〈n, y〉signed distance ofp, y

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Line Elements.

N

po ν

n

y




• (n, n, ν) ∈ R7 coordinates of(N, y)

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Line Elements.

N

po ν

n

y




• (n, n, ν) ∈ R7 coordinates of(N, y)

• depend ony, homogeneous=⇒ point model inP6: quadratic cone (vertex, one generator

missing, see [1]). – p.10/35

Linear Complexes of Line Elements.

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Definition:The setC of line elements(N, y) = (n, n, ν) with

〈n, c〉+ 〈n, c〉 + νγ = 0

is called a linear complex of line elements.(c, c, γ) ∈ R

7 is the coordinate vector ofC.

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〈n, c〉+ 〈n, c〉 + νγ = 0



Lemma:For givenC = (c, c, γ) (γ 6= 0) and each lineL = (l, l), there is a uniquely defined pointx, s.t.(L, x) = (l, l, λ) ∈ C.

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〈n, c〉+ 〈n, c〉 + νγ = 0



Lemma:For givenC = (c, c, γ) (γ 6= 0) and each lineL = (l, l), there is a uniquely defined pointx, s.t.(L, x) = (l, l, λ) ∈ C.

Proof: λ = −(〈c, l〉 + 〈c, l〉)/γ. �

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Linear Complexes & Equiform Kinematics.

n

v(y)

Theorem:At any regular instant ofa smooth 1-param. equi-form motion the pathnormal elements form alinear complex of lineelements.

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Linear Complexes & Equiform Kinematics.

n

v(y)

Theorem:At any regular instant ofa smooth 1-param. equi-form motion the pathnormal elements form alinear complex of lineelements.

Proof:N(y) . . . (n, n = y × n) path normal aty,N(y)⊥v(y) ⇐⇒ 0 = 〈n, c× y + c+ γy〉 =〈n, c〉 + 〈n, c〉+ γν = 0. �

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Properties of linear Complexes of Line Elements.

Given a linear complex of line elements(c, c, γ):• line elements on lines of a bundle:

points form a sphere

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points form a sphere• line elements on lines of a field:

path normals of a planar equiform motion

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points form a sphere• line elements on lines of a field:

path normals of a planar equiform motion

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Types of Complexes & corresponding uniform equif. motions.

up to equiform equivalence• C = (0, 0, 1; 0, 0, p; 0) p ∈ R

helical motion (p 6= 0), rotation (p = 0)

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helical motion (p 6= 0), rotation (p = 0)• C = (0, 0, 0; 0, 0, 1; 0)

translation

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translation• C = (0, 0, 1; 0, 0, 0; p) p ∈ R

spiral motion (p 6= 0), rotation (p = 0)

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translation• C = (0, 0, 1; 0, 0, 0; p) p ∈ R

spiral motion (p 6= 0), rotation (p = 0)• C = (0, 0, 0; 0, 0, 0; 1)

similarity transformation

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Spiral Motion .

A

s

s′o

π

A

o

c

s

s′

π

∆

• o = fixed point,A = axis,π = fixed plane

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Spiral Motion .

A

s

s′o

π

A

o

c

s

s′

π

∆

• o = fixed point,A = axis,π = fixed plane• s = orbit of a point/∈ π, s′ = orbit of a point∈ π

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Spiral Motion .

A

s

s′o

π

A

o

c

s

s′

π

∆


• Γ = s ∨ o = invariant cone of revolution

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Spiral Motion .

A

s

s′o

π

A

o

c

s

s′

π

∆


• Γ = s ∨ o = invariant cone of revolution• ∆ = invariant spiral cylinder

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Recognition of Surfaces.

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Recognition of Surfaces

Theorem:The normal elements ofa regular C1 surfaceare contained in a line-ar complex of line ele-ments if and only if it ispart of an equiform ki-nematic surface.

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Recognition / exact case.

• given:(ni, ni, νi) . . . normal elements of anequiform kinematic surfaceS

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• compute a basis in the spaceV of linear equations〈c, x〉+ 〈c, x〉+ ξγ = 0

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• C1, . . . , Ck = Basis ofV

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• C1, . . . , Ck = Basis ofV• independent basis vectors = independent unif.

equif. motionsS → S

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Invariant Surfaces.

k = 4: plane,k = 3: sphere

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Invariant Surfaces.

k = 4: plane,k = 3: sphere

k = 2: cylinder/cone of rev., spiral cylinder

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Invariant Surfaces.

k = 1 : general cone/cylinder

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Invariant Surfaces.

k = 1 : general cone/cylinder

k = 1 :

rotational/helical surface, spiral surface. – p.21/35

Invariant Surfaces.

c = 0, γ 6= 0 c = 0, γ = 0

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Invariant Surfaces.

c = 0, γ 6= 0 c = 0, γ = 0

c 6= 0, 〈c, c〉 = γ = 0 c 6= 0, γ = 0, 〈c, c〉 6= 0 c 6= 0, γ 6= 0. – p.22/35

Recognition / non-exact case.

• given:(ni, ni, νi) normal elements of a surfaceSwanted: best approx. equiform kinematic surface

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• scaling data such thatmax ‖xi‖ ≈ 1

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• minimizing the sum of squared momentsN∑

i=1

µ2

i :=N∑

i=1

(〈c, ni〉+ 〈c, ni〉+ νiγ)2

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• minimizing the sum of squared momentsN∑

i=1

µ2

i :=N∑

i=1

(〈c, ni〉+ 〈c, ni〉+ νiγ)2

• number/type of eigenvectors corresponding tosmall eigenvalues of

n∑

i=1

(ni, ni, νi)T (ni, ni, νi)

determine type of best approx. kinematic surface. – p.23/35


• using RANSAC

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• using RANSAC• downweighting outliers with

wi =1

1 + Fµ2ki

, F > 0

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Recognition / non-exact case

color of region = color of axis

centers and axis within the transparent fat line element

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Segmentation.

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Segmentation.

• removing sharp edges

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Segmentation.


• looking for plane/sphere/cylinder (cone) of rev.

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Segmentation.


• looking for plane/sphere/cylinder (cone) of rev.• looking for multiply invariant surfaces

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Segmentation.


• looking for plane/sphere/cylinder (cone) of rev.• looking for multiply invariant surfaces• looking for simply invariant surfaces

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Extracting geometric Data.

• axisA and centero of spiral motion:

v(z) = 0 = c× z + c+ γz

z =1

γ(c2 + γ2)(γc× c− γ2c− (c · c)c)

A = (c,1

γ2 + c2(c2c− (c · c)c+ γc× c))

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v(z) = 0 = c× z + c+ γz

z =1

γ(c2 + γ2)(γc× c− γ2c− (c · c)c)

A = (c,1

γ2 + c2(c2c− (c · c)c+ γc× c))

• equation of sphere/plane/cylinder (cone) of rev.already found

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v(z) = 0 = c× z + c+ γz

z =1

γ(c2 + γ2)(γc× c− γ2c− (c · c)c)

A = (c,1

γ2 + c2(c2c− (c · c)c+ γc× c))

• equation of sphere/plane/cylinder (cone) of rev.already found

• axes of cones

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Segmentation / Examples

detecting rotational surface, morphologhicaloperations

detecting a general cylinder, planar parts, and small

features . – p.29/35

Reconstruction.

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Reconstruction (spiral surfaces).

• (c, c, γ) . . . best approx. line element complex

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• (c, c, γ) . . . best approx. line element complex• computing axisA, centero, spiral parameter

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• (c, c, γ) . . . best approx. line element complex• computing axisA, centero, spiral parameter• applying unif. equif. motion to data points:

‘spiral projection’ into a plane containing the axis

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‘spiral projection’ into a plane containing the axis• fitting a generator curve

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‘spiral projection’ into a plane containing the axis• fitting a generator curve• applying unif. equif. motion

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Reconstruction.

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Reconstruction.

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Reconstruction

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Overview.• Contents• Aims• Equiform Kinematics• Line Elements• Linear Complexes of Line Elements• Recognition of Surfaces• Segmentation• Reconstruction

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3D Shape Understanding and Reconstruction based on Line ...sodwana.uni-ak.ac.at/geom/mitarbeiter/odehnal/talk/hk_05.pdf · 3D Shape Understanding and Reconstruction based on Line