Moving Least Squares Outline The Approximation Power of Moving - - PDF document

Moving Least Squares

David Levin

presented by Niloy J. Mitra

Moving Least Squares CS 468

Outline

• The Approximation Power of Moving Least-

Squares

• D. Levin
• Mesh-Independent Surface Interpolation
• D. Levin
• Defining point-set surfaces
• N. Amenta and Y. Kil

Moving Least Squares CS 468

Problem

• Collection of point
• Source of data : laser scanner
• Points are unorganized
• Usually no information about normal
• But not always the case (next paper)

Moving Least Squares CS 468

Applications

• Implicit surface definition
• Projection operator
• Noise removal / Thinning
• Upsampling
• Ray tracing

Moving Least Squares CS 468

Interpolation vs Smoothing

Moving Least Squares CS 468

One Approach (Mesh based)

• Smooth interpolation by joining local patches

each being an approximation in local reference domain.

• Piecewise polynomial patches.
• In most cases, result depends on the mesh

defining the patches.

Moving Least Squares CS 468

Example

350 pieces/patches

Moving Least Squares CS 468

Alternative Approach (Meshless)

• Implicit definition of surface.
• S = f({pi})

Moving Least Squares CS 468

Roadmap

Given

R = {xi}

Goal

• Define a projection operator P
• Unique manifold

S x P x P x

d

∈ → ℜ ∈ ) ( :

} ) ( | { x x P x S = ≡

Moving Least Squares CS 468

MLS Approach

• Step 1
• Define a local/reference domain

(like a tangent plane)

• Local parameterization

Moving Least Squares CS 468

MLS Approach

• Step 1
• Define a local/reference domain
• Step 2
• MLS approximation wrt reference domain

(polynomial fitting)

Moving Least Squares CS 468

Fitting Functions

Given (functional setting)

{xi, fi}

Goal

Find p in Πm such that {xi, fi} satisfies

∑

−

∏ ∈ i i i i

x f x p ||) (|| ) ) ( ( min

2 p

1

• d

m

θ

error weight fi pi xi

Moving Least Squares CS 468

θ : The Weight Function

• Non-negative decaying function
• Typical example
• Gaussian kernel θ(d) = exp(-d2/h2)

Moving Least Squares CS 468

Basic MLS

• For a given point r near R, define a local

approximating hyper-planer Hr

r Hr

Moving Least Squares CS 468

Equation of a line

1 || || , }, , , | {

,

= ℜ ∈ ℜ ∈ = − > < = a a x D x a x H

d d D a

Moving Least Squares CS 468

Basic MLS

• For a given point r near R, define a local

approximating hyper-planer Hr

r Hr

Moving Least Squares CS 468

Basic MLS

• For a given point r near R, define Hr
• In case of multiple local minima, the plane

closest to r is chosen.

||) (|| ) , ( min

2 ,

r r D r a

i i i D a

− − > <

∑

θ

r ri Non-linear optimization

Moving Least Squares CS 468

Basic MLS

• For a given point r near R, define Hr

Find a polynomial approx. of degree m

∑

− −

∏ ∈ i i i i

r r f x p ||) (|| ) ) ( ( min

2 p

1

• d

m

θ

Moving Least Squares CS 468

MLS

Step 2 Step 1

Moving Least Squares CS 468

Projection?

) ( ~ )) ( ~ ( ~ r P r P P

m m m

≠

||) (|| r r

i −

θ

||) (|| ) , ( min

2 ,

r r D r a

i i i D a

− − > <

∑

θ

ri r fi p(q)

) ( ~ r P

m

q xi

Moving Least Squares CS 468

Basic MLS

) ( ~ )) ( ~ ( ~ r P r P P

m m m

≠

• Doesn’t project points to a (d-1)-dim manifold.
• Doesn’t define a surface.

Moving Least Squares CS 468

Simple fix

||) (|| r r

i −

θ ||) (|| q r

i −

θ

fi xi r p(q)

) ( ~ r P

m

q

Moving Least Squares CS 468

Non-linear Optimization

||) (|| ) , ( min ) , (

2 ,

q r D r a a q I

i i i D a

− − > < =

∑

θ ) ( || ) ( q a q r − ) ( )) ( , ( ) (

) (

= ∂ = q J q a q I q J

q a

c

• n

s t r a i n t s

Moving Least Squares CS 468

Basic MLS

• For a given point r near R, define Hr
• In case of multiple local minima, the plane

closest to r is chosen.

||) (|| ) , ( min

2 ,

q r D r a

i i i D a

− − > <

∑

θ

fi xi r p(q)

) ( ~ r P

m

q

Moving Least Squares CS 468

MLS

Given

R = {xi}

MLS

• Define a projection operator P(P(x))=P(x)
• Unique manifold S ≡ {x|P(x)=x}
• Conjecture S is C∞

Moving Least Squares CS 468

MLS surface

Moving Least Squares CS 468

Computing Hr and p

• Computing hyper-plane Hr
• Non-linear optimization problem
• Computed iteratively
• Computing θ(): time consuming step
• O(N) for each iteration step
• Approximate by doing a hierarchical clustering
• Fitting a polynomial p(.), given Hr
• Solve a linear system
• Size depends on the order of approximation (m)

Moving Least Squares CS 468

Applications : Denoising

Moving Least Squares CS 468

Applications : Upsampling / Hole Filing

Moving Least Squares CS 468

Applications : Ray Tracing

Moving Least Squares CS 468

Sampling Condition?

Moving Least Squares CS 468

Conclusions

• Surface is smooth and a manifold
• Adjustable feature size h allows to smooth
• ut noise
• The surface changes with addition of points.