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

David Levin presented by Niloy J. Mitra Moving Least Squares

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 CS 468 Moving Least Squares

Problem • Collection of point • Source of data : laser scanner • Points are unorganized • Usually no information about normal • But not always the case (next paper) CS 468 Moving Least Squares

Applications • Implicit surface definition • Projection operator • Noise removal / Thinning • Upsampling • Ray tracing CS 468 Moving Least Squares

Interpolation vs Smoothing CS 468 Moving Least Squares

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. CS 468 Moving Least Squares

CS 468 350 pieces/patches Example Moving Least Squares

Alternative Approach (Meshless) • Implicit definition of surface. • S = f({p i }) CS 468 Moving Least Squares

Roadmap Given R = {x i } Goal • Define a projection operator P ∈ ℜ → ∈ d : ( ) x P x P x S • ≡ = • Unique manifold { | ( ) } S x P x x CS 468 Moving Least Squares

MLS Approach • Step 1 • Define a local/reference domain (like a tangent plane) • Local parameterization CS 468 Moving Least Squares

MLS Approach • Step 1 • Define a local/reference domain • Step 2 • MLS approximation wrt reference domain (polynomial fitting) CS 468 Moving Least Squares

Fitting Functions Given ( functional setting ) {x i , f i } x i p i f i Goal Find p in Π m such that {x i , f i } satisfies ∑ − θ 2 min ( ( ) ) (|| ||) p x f x i i i ∈ ∏ d - 1 p m i error weight CS 468 Moving Least Squares

θ : The Weight Function • Non-negative decaying function • Typical example • Gaussian kernel θ (d) = exp(-d 2 /h 2 ) CS 468 Moving Least Squares

Basic MLS • For a given point r near R, define a local approximating hyper-planer H r H r r CS 468 Moving Least Squares

Equation of a line = < > − = ∈ ℜ ∈ ℜ = d d { | , 0 , }, , || || 1 H x a x D x a a , a D CS 468 Moving Least Squares

Basic MLS • For a given point r near R, define a local approximating hyper-planer H r H r r CS 468 Moving Least Squares

Basic MLS • For a given point r near R, define H r Non-linear optimization ∑ < > − θ − 2 min ( , ) (|| ||) a r D r r i i , a D i • In case of multiple local minima, the plane closest to r is chosen. r i r CS 468 Moving Least Squares

Basic MLS • For a given point r near R, define H r Find a polynomial approx. of degree m ∑ − θ − 2 min ( ( ) ) (|| ||) p x f r r i i i ∈ ∏ d - 1 p m i CS 468 Moving Least Squares

CS 468 Step 2 Step 1 Moving Least Squares MLS

Projection? ~ ~ ~ ≠ ( ( )) ( ) P P r P r m m m θ i − (|| ||) r r ∑ < > − θ − 2 min ( , ) (|| ||) a r D r r i i a , D i r i r f i p(q) ~ r x i ( ) P m q CS 468 Moving Least Squares

Basic MLS ~ ~ ~ ≠ ( ( )) ( ) P P r P r m m m • Doesn’t project points to a (d-1)-dim manifold. • Doesn’t define a surface. CS 468 Moving Least Squares

Simple fix θ i − θ i − (|| ||) (|| ||) r r r q r f i p(q) ~ r x i ( ) P m q CS 468 Moving Least Squares

Non-linear Optimization ∑ = < > − θ − 2 ( , ) min ( , ) (|| ||) I q a a r D r q i i , a D i r − ( ) || ( ) q a q s t n i a r = ∂ = t s ( ) ( , ( )) ( ) 0 J q I q a q J q n o ( ) a q c CS 468 Moving Least Squares

Basic MLS • For a given point r near R, define H r ∑ < > − θ − 2 min ( , ) (|| ||) a r D r q i i , a D i • In case of multiple local minima, the plane closest to r is chosen. r f i p(q) ~ r ( ) P x i m q CS 468 Moving Least Squares

MLS Given R = {x i } MLS • Define a projection operator P(P(x))=P(x) • Unique manifold S ≡ {x|P(x)=x} • Conjecture S is C ∞ CS 468 Moving Least Squares

MLS surface CS 468 Moving Least Squares

Computing H r and p • Computing hyper-plane H r • 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) CS 468 Moving Least Squares

Applications : Denoising CS 468 Moving Least Squares

Applications : Upsampling / Hole Filing CS 468 Moving Least Squares

Applications : Ray Tracing CS 468 Moving Least Squares

Sampling Condition? CS 468 Moving Least Squares

Conclusions • Surface is smooth and a manifold • Adjustable feature size h allows to smooth out noise • The surface changes with addition of points. CS 468 Moving Least Squares