Provably Good Implicit MLS Surfaces Nikola Milosavljevic CS 468, - PowerPoint PPT Presentation

Provably Good Implicit MLS Surfaces Nikola Milosavljevic CS 468, Fall 2005

Implicit MLS Surfaces ◮ Zero-level-set of a function I ( x ) over R 3 x 0 ◮ Fixed points of a projection x 2 operator x 1 ◮ Weighted sum of signed distances x 3 x 4 p n T � p ( x − p ) · θ p ( x ) I ( x ) = � p θ p ( x ) x ◮ Extremal surfaces np p � I ( x ) = n ( x ) · ∂ e MLS ( y , n ( x )) � � ∂ y � � x

MLS Surfaces with Guarantees ◮ Not discussed so far ◮ What is the “ground truth” surface? ◮ How do the samples arise? ◮ How good is the reconstruction? ◮ In this talk ◮ A notion of the original surface ◮ A model of sampling and noise ◮ Study the behavior of MLS surfaces ◮ The Holy Grail ◮ Geometric accuracy ◮ Correct topology ◮ Smoothness ◮ Fast (quadratic) convergence ◮ Efficient (local) computation

Problem Statement ◮ Original smooth, closed P surface Σ ◮ Given conditions on Σ ◮ Sampling density ◮ Normal estimates ◮ Noise ◮ Design an implicit function I ( x ) whose zero set recovers Σ

Outline R. Kolluri (U.C. Berkeley), “Provably Good Moving Least Squares”, SODA 2005. ◮ Globally uniform sampling + normals ◮ Correct topology ◮ Smoothness T. Dey, J. Sun (Ohio State), “An Adaptive MLS Surface for Reconstruction with Guarantees”, SGP 2005. ◮ Feature-sensitive sampling + normals ◮ Correct topology ◮ “Smoothness”

Sampling Conditions ◮ Medial axis ◮ Points with multiple closest points on Σ ◮ Local feature size, lfs( · ) ◮ Distance from x ∈ Σ to the medial axis ◮ ǫ -sampling ◮ Every x ∈ R 3 has a ǫ lfs(˜ x ) sample p at most ǫ lfs(˜ x ) x p ◮ No oversampling, ˜ | B ( x , ǫ lfs(˜ x )) | ≤ α x

Typical Proof Outline ◮ Step 1: ◮ Analyze I ( x ) ◮ Localize the zero-set ◮ Spurious zero-crossings? I ( x ) > 0 I ( x ) < 0 ◮ Implication: Reconstruction is geometrically close

Typical Proof Outline ◮ Step 2: ◮ Analyze ∇I ( x ) close to the surface ◮ Show that ∇I ( x ) � = 0 ◮ Show that I ( x ) is strictly monotonic in the direction normal to Σ x · ∇I ( x ) > 0 n ˜ n ˜ x ◮ Implications: ◮ Reconstructed surface is a manifold ◮ Normal directions define a homeomorphism

� ✁ Typical Proof Outline ◮ A common technique ◮ Bounding the influence of points farther than a suitably chosen threshold ◮ The actual radius depends on the quantity that is being evaluated ◮ Inside — reliable ◮ Outside — negligible

An MLS Surface for a Uniformly Sampled PCD

Assumptions ◮ Uniform sampling , || x − p || ≤ ǫ ◮ Assume lfs( x ) ≥ 1 everywhere ◮ Smallest features determine density ◮ No oversampling, | B ( x , ǫ ) | ≤ α ◮ Noise, || p − ˜ p || ≤ ǫ 2 ◮ Normal estimates, ∠ ( n p , n ˜ p ) ≤ ǫ

Proposed MLS Surface ◮ Weighted sum of signed distances p n T � p ( x − p ) · θ p ( x ) θ p ( x ) = 1 e −|| x − p || 2 /ǫ 2 I ( x ) = � p θ p ( x ) α p x np p

Analysis of I ( x ) ◮ Can show that all zero-crossings are within r = d ( x, Σ) + δ + ǫ δ from Σ ◮ Fix x far away (farther x than δ ) from the d ( x, Σ) boundary ◮ Influence threshold δ r = d ( x , Σ) + δ + ǫ ◮ If p is a nearby sample , n T p ( x − p ) = d ( x , Σ) · (1 + O ( ǫ )) + O ( ǫ 2 ) ◮ n T p ( x − p ) and d ( x , Σ) have the same sign, provided δ = O ( ǫ )

Analysis of I ( x ) Far away points ǫ ◮ Divide the “distant space” into spherical shells of thickness ǫ ri = r + i · ǫ x ◮ The number of samples in the i -th shell is O ( i 2 ) r = d ( x, Σ) + δ + ǫ (uniform sampling) ◮ The influence decays as O ( e − i 2 ) ◮ The overall influence r · r 2 � ǫ 2 · e − r 2 /ǫ 2 � O

Analysis of ∇I ( x ) ǫ ◮ Fix x close (within δ ) to the boundary ri = r + i · ǫ ◮ Show n ˜ x · ∇I ( x ) > 0 where ∇I ( x ) x ∈ Σ is closest to x . ˜ x ˜ x δ ( d ( x, Σ) + ǫ ) 2 + ǫ 2 � r = ◮ Influence threshold � ( d ( x , Σ) + ǫ ) 2 + ǫ 2 = O ( ǫ ) r = ◮ Far away points negligible

Analysis of ∇I ( x ) ◮ Nearby points contribution to the gradient vector ∇I ( x ) ◮ Signed distance functions x � � θ p ( x ) θ q ( x ) · n p n p ˜ x p q p ◮ Change of weights � � θ p ( x ) θ q ( x ) · O ( n T p ( x − p )) · ( p − q ) p q ◮ The normals n p are close to n ˜ x !

Uniform Case: Conclusions ◮ The zero set of I is confined to the δ = O ( ǫ ) thickening ◮ The reconstruction is geometrically accurate ◮ Whenever I ( x ) = 0, ∇I ( x ) � = 0 ◮ The reconstruction is locally flat ◮ The gradient lines provide a “morphing function” ◮ The reconstruction is topologically correct

An MLS Surface for an Adaptively Sampled PCD

Motivation ◮ Allow variations in sampling density according to local feature size ◮ Requires adaptive Gaussian kernel ◮ Uniform sampling: kernel width ǫ ◮ Does not work for adaptive sampling

✆ ✂ ☎ ✄ Adaptive Gaussian Kernel ◮ Adapt to lfs(˜ x )? „ || x − p || 2 « − O ǫ 2 lfs(˜ x )2 θ p ( x ) ∼ e ◮ Bias toward small features ◮ Adapt to lfs(˜ p )? „ « || x − p || 2 − O ǫ 2 lfs(˜ p )2 θ p ( x ) ∼ e ◮ Influence may not decrease with distance

Adaptive Gaussian Kernel ◮ Solution: Adapt to � lfs( p ) lfs( x ) √ � � 2 · || x − p || 2 θ p ( x ) = exp − ǫ 2 lfs(˜ x ) lfs(˜ p ) ◮ Note: not smooth!

Other Assumptions ◮ No oversampling, | B ( x , ǫ lfs(˜ x )) | ≤ α ◮ Noise magnitude at most ǫ 2 lfs(˜ x ) ◮ Good normal estimates, ∠ ( n p , n ˜ p ) ≤ ǫ

Analysis ◮ Extend the proofs for adaptive thickening, width r δ lfs(˜ x ) x ◮ Fix a point x and a d ( x, Σ) threshold radius r δ ◮ Bounding the influence of far away points ◮ Small distant features ◮ Reliability of nearby points ◮ Small nearby features

Influence of Far Away Points ◮ Subdivide into cubes, accumulate the counts bottom-up ◮ Size of top-level cubes O ( ǫ lfs(˜ x )) ◮ Stop subdivision when lfs( c k ) ≥ ( ǫ/ 2 k ) lfs(˜ x ) ◮ Apply “no oversampling” to the leaves ◮ The i -th shell may contain more than O ( i 2 ) samples, but still the total contribution is O ( i 2 e − i 2 ) ◮ Total contribution to I ( x ) and n ˜ x · ∇I ( x ) O (poly( r /ǫ ) · exp( − r 2 /ǫ 2 ))

Influence of Nearby Points ◮ Works only for d ( x , Σ) ≤ 0 . 1 lfs(˜ x ) from r the surface x ◮ If x is at least d ( x, Σ) ( δ = 0 . 3 ǫ ) lfs(˜ x ) away from the surface, the sign is correct δ ◮ Can claim I ( x ) � = 0 only for 0 . 3 ǫ ≤ d ( x , Σ) ≤ 0 . 1 lfs(˜ x )

Influence of Nearby Points ◮ Works only if x is not too far from the surface ◮ May have fake zero-crossings outside 0 . 1 lfs(˜ x )! ◮ Uniform sampling ◮ Adaptive sampling

Remarks ◮ Delaunay-based estimation of normals and medial axis ◮ Maxima layers for standard PMLS surfaces ◮ Comparison with other projection methods