Geographical point cloud modelling with the 3D medial axis transform

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Delft University of Technology

Geographical point cloud modelling with the 3D medial axis transform

Peters, Ravi

DOI

10.4233/uuid:f3a5f5af-ea54-40ba-8702-e193a087f243

Publication date

2018

Document Version

Final published version

Citation (APA)

Peters, R. (2018). Geographical point cloud modelling with the 3D medial axis transform.

https://doi.org/10.4233/uuid:f3a5f5af-ea54-40ba-8702-e193a087f243

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Geographical point cloud modelling

with the 3D medial axis transform

Ravi Peters

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O ⊂ Rn

S = δO

O

O Rn

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exterior

interior

exterior

interior

O S O O B O O B _O B O B _O Rn \ O B Rn\ O B Rn \ O S S C2 S O M(O)

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O M(O) O C R O M(O) = ⟨C, R⟩ C O R C S

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a c_{∈ C} r_{∈ R} a =_{⟨c, r⟩}

R3

R2

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junction_curve medial sheets a =_{⟨c, r⟩} c r a c r B(c, r) a S r c p q c p s˜p q s⃗q ˜ sp ˜ sq b˜ θ s˜p ˜ sq

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p

f (p)

✏f (p)

S

M[S]

f (p) p M[S] ϵ ϵ f (p) p O M(O) p S S ϵ P O ϵ _O p∈ P ϵf (p) ϵ ϵ→ 0

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ϵ ϵ = 0.4 ϵ ϵ R2 R3 S

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S

O M(O)

d_{− 1} O

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S O S _M(O) O M(O) S O M(O)

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O M(O) S M(O) O S O O

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λ = 11 λ = 15

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λ = 0

λ = 6

λ = 10

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O

θ θ

10003

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s B B

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λ λ

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p1 p2 m λ ✓ p1 p2 m θ p1 p2 m p1 p2 m λ θ θ s > 1 1/s θ s

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M(O) O

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1. Surface

point cloud approximation2. Normal 3. MAT approximationfor both normal orientations 4.Unstructured MAT ⃗ n p c p c L ⃗n q p q L p q

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T p ⃗ n θ0 θ1 c r i_{← 0} r← rinit c_← p, ⃗n, r qnext← T, c rnext← p, ⃗n, qnext cnext← p, ⃗n, rnext rnext> r− ϵconv i = 0 θ0>∠pcnextqnext i > 0 θ1>∠pcnextqnext c_{← c}next r← rnext p q p, ⃗n q p, ⃗n, q p L q q qnext c p, ⃗n, qnext cnext qnext

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Po ∈ Po T _← P , ⃗n∈ Po T, p, ⃗n T, p,−⃗n c q p, ⃗n p,−⃗n p 2n

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p, ⃗n p, ⃗n, q T, q P q O(log n) P O(n log n) p, ⃗n, q O(1) p, ⃗n, r O(1) O(n log n) 2n O(log n) O(log n)

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O(n log n) O(n log n) O(n2₎

p N p

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central MAT sheet protruding sheets bumps in surface p q qnoise pnoise p

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p p pnoise qnoise qnoise p q p p q p p B p

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1

2

3

4

1

2

3

4

5

5th 4th 4th 5th q q θ θ0 θ1

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Separation angle threshold

last stable ball

noisy balls

n

p

c

i+1

q

i

c

i

q

i+1

✓

i+1

✓

i p θi+1

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Pn P Pn M(Pn) θ < 30◦ MF(Pn) MD(Pn) θ0 = 30◦ θ1 = 60◦ M(P ) A B A B A_{→ B} A_{→ B} B_{→ A} A B M(P )

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M(P )→ M(Pn) M(P )→ MF(Pn) M(P )→ MD(Pn)

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M(P )→ M(Pn) M(Pn) M(P ) M(P ) M(P ) MD(Pn) MD(Pn) M(P ) MD(Pn) M(P ) MF(Pn) M(P ) MF(P ) MD(Pn) θ1 M(S) Pn P

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10× 20 × 15

Pn M(S)

MD(Pn)

θ0= 30◦

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5 10 15 20 25 30 35 40 45 θ1 (degrees) 0.05 0.10 0.15 0.20 0.25 0.30 0.35 st and ar d d ev ia ti on in d is ta n ce to re fe re n ce (m ) M() MD() θ1 θ1 = 32◦ M(Pn) MD(Pn) M(S] MD(Pn) θ1 MD(Pn) 0.33 0.22 θ0 = 20◦ MD(P ) P

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θ1

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Without denoising F ront -vi ew T op-vi ew With denoising θ0= 20◦ θ1= 32◦

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θ < 20◦ λ < 0.05 θ0= 20◦ θ1= 32◦ λ < 0.05 rinit rinit rinit rinit

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interior

exterior

1. Unstructured

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watercourse

buildings

hill

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o = r1+ r2 d

d

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S a, b i← 0 S_{̸= ∅} S _{← 0} s← S C_← s ← i C_{̸= ∅} c_← C n c, n n C ← i S_{← S − n} i_{← i + 1} o > toverlap toverlap > 1 toverlap

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locally similar bisector direction divergent bisector direction in case of extrema in radius the sheet is into 'sub-sheets' a, b a b tbisec

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tbisec

180◦

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S tmincount tstraight tbisec tθ S a, b ← ∠ a b < tbisec S tmincount M ← a S a > tstraight a, b _{← |} a₋ b _{| < t}θ M

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adjacency graph flip graph Common feature points interior MAT exterior MAT surface

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S A A_← a S a _̸= n a < n l_{← ⟨a, n⟩} l_{← ⟨n, a⟩} l[0] = 0 l A[l]_{← 1} A[l] O(n) n

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S A A_← a, b S a _̸= b a < b l_{← ⟨a, b⟩} l_{← ⟨b, a⟩} l[0] = 0 l A[l]_{← 1} A[l]

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z Bz

Bz > 0

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surface

points

no

rmals

first

atom

second

atom

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i+ _i− r c rinit rinit rinit rinit rinit

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r

_init

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nth

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k k k ϵ _S ϵ ϵ ϵ P ∈ P B( , ϵf ( )) P P

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c d nmin nmax ϵ

favg c

ntarget=_ϵf_avgd

nmin< ntarget< nmax

c x = ntarget n n c ϵ ϵ ϵf2(p) ϵf (p) ϵ ϵ ϵ ϵ ϵ

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ϵ = 0.2 211373 ϵ = 0.2 167848

ϵ = 0.4 88835 ϵ = 0.4 86973

ϵ = 0.6 53359 ϵ = 0.6 54385

10% 74746 0.8m

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r r

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r

r r

ϵf ( ) ϵ

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p f (p) ✏f (p)

S

M[S]

p ϵ≈ 0.5 ϵ ϵf ( ) ϵ = 0.4

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ϵ = 0.4

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p0 ⃗vx ⃗vy

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p0 ~vy ~vx s|~vx| s|~vy| pm p0, ⃗vx, ⃗vy, s pm ps ⃗v_{← p}0− pm l⃗vx← ⃗v·⃗vx |⃗vx|⃗vx l⃗vy ← ⃗ v·⃗vy |⃗vy|⃗vy ps.x← (l⃗vx+|⃗vx|) s |⃗vx| ps.y← (l⃗vy +|⃗vy|) s |⃗vy| ⃗n_{← (⃗v}y× ⃗vx) ps.z← ⃗v_|n|·⃗n

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c r s D D (c, r) cs← c x _−rs +rs y −rs +rs h_←!x2_{+ y}2 h rs d′_{← c}s.z− (rs − h) d_{← D[c}s.x + x, cs.y + y] d′ d D[cs.x + x, cs.y + y]← d′ qm D qm qs← qm d_{← D[q}s.x, qs.y] qs.z d qm qm

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O((rs)2_{N )} _N

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15.0 10.0 5.0 0.0 5.0 depth

a)

b)

c)

a) b) c) d) e)

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Tree

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0 100 200 m

Identified water courses Missed water courses

0 100 200 m

Reference water courses Erroneously identified water courses

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1.Groundpoints. 2.3D skeletonof the ground points.

3. Segmentation of 3D

skeleton in sheets. 4.sheets and select lowerTriangulateexterior envelope to find the centrelines.

exterior sheet

centreline

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Exterior skeleton Interior skeleton Ground Surface Skeleton points Ground points xy

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Peat

Clay

0 m 200 m

0 m 200 m 0 m 200 m

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0 50 100 m

0 100 200 m

Reference water courses Erroneously identified water courses

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buildings

terrain buildings

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θ r

r θ

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−2 −2 −2 −2 −2 −2 −2

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