Meshless Surfaces. presented by Niloy J. Mitra. An Nguyen
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1 Meshless Surfaces presented by Nloy J. Mtra An Nguyen
2 Outlne Mesh-Independent Surface Interpolaton D. Levn
3 Outlne Mesh-Independent Surface Interpolaton D. Levn Pont Set Surfaces M. Alexa, J. Behr, D. Cohen-Or, S. Fleshman, D. Levn, and C. T. Slva Progressve Pont Set Surfaces S. Fleshman, D. Cohen-Or, M. Alexa, C. T. Slva
4 Outlne Mesh-Independent Surface Interpolaton D. Levn Pont Set Surfaces M. Alexa, J. Behr, D. Cohen-Or, S. Fleshman, D. Levn, and C. T. Slva Progressve Pont Set Surfaces S. Fleshman, D. Cohen-Or, M. Alexa, C. T. Slva Meshless Parametrzaton and Surface Reconstructon M. S. Floater, M. Remers
5 Movng Least Square (MLS) Approxmaton
6 Fttng Functons Gven: {x, f } Goal: fnd p such that {x, p(x )} mn p Π d 1 m ( p( x ) f ) 2 θ ( x ) error weght
7 Motvaton Gven: PCD R={r }
8 Motvaton Gven: PCD R={r } Goal: defne a projecton operator P(P(x))=P(x)
9 Motvaton Gven: PCD R={r } Goal: defne a projecton operator P mples a unque manfold surface S S {x P(x)=x}
10 MLS Approach Step 1: defne a local reference doman (somethng lke a local tangent plane) gves a local parameterzaton
11 MLS Approach Step 1: defne a local reference doman Step 2: construct a MLS approxmaton wrt a reference doman (a polynomal fttng step)
12 Notatons so far R={r } : nput PCD S : (d-1)-dm manfold n R d
13 Notatons R={r } : nput PCD S : (d-1)-dm manfold n R d r r : Eucldean dstance between r, r
14 θ : The Weght Functon Non-negatve decayng functon Typcal example Gaussan 2 2 θ ( d) = exp( d / h )
15 Notatons so far R={r } : nput PCD S : (d-1)-dm manfold n R d r r : Eucldean dstance between r, r θ : non-negatve weght functon
16 Basc MLS Procedure For a gven pont r near R, defne a local approxmatng hyper-plane
17 Basc MLS Procedure For a gven pont r near R, defne a local approxmatng hyper-plane r
18 Basc MLS Procedure For a gven pont r near R, defne a local approxmatng hyper-plane r H r
19 Notatons so far R={r } : nput PCD S : (d-1)-dm manfold n R d r r : Eucldean dstance between r, r θ : non-negatve weght functon H r : approxmatng hyper-plane at r usng R
20 Equaton of a Lne H d d = { x < a, x > D = 0, x R }, a R, a = 1 a D x
21 Basc MLS Procedure : Step 1 For a gven pont r near R, defne a local hyper-plane H r Plane H r defned by a least square formulaton r H r
22 Basc MLS Procedure : Step 1 For a gven pont r near R, defne H r mn a, D ( < 2 a, r > D) θ ( r r ) In case of multple local mnma, the plane closest to r s chosen
23 Basc MLS Procedure : Step 1 For a gven pont r near R, defne H r mn a, D ( < 2 a, r > D) θ ( r r ) r r H r
24 Basc MLS Procedure : Step 2 H r : local approxmatng plane Fnd a polynomal approx. of degree m 2 mn ( p( x ) f d ) θ ( 1 p Π m r r r ) a f r H r x q
25 Basc MLS Procedure : Step 2 2 mn ( p( x ) f d ) θ ( 1 p Π m r r ) ~ Pm ( r) r a H r f x q r p(0)
26 Basc MLS : Curve Smoothng Step 1 Step 2
27 But t s not a projecton ~ P m ~ ( P m ( r)) ~ P m ( r) Remember, θ ( r r ) r a f r H r x q
28 Problems wth ~ P m ( r) ~ P m ~ ( P m ( r)) ~ P m ( r) Basc MLS n d R Doesn t project ponts to a (d-1)-dm manfold Doesn t defne a surface
29 A Smple Fx Replace ( r r ) by θ θ ( r q ) r a f r H r x q
30 MLS Procedure : Step 1 For a gven pont r near R, defne H r mn a, D ( < 2 a, r > D) θ ( r q ) r a f r H r x q
31 MLS Projecton : Step 2 Gven a local parameterzaton, H r 2 mn ( p( x ) f d ) θ ( 1 p Π m r q ) r P m (r) a H r f x q r p(0)
32 Why does t work? θ ( r q ) r on H r depends on the projecton of r a f r H r x q
33 Why does t work? θ ( r q ) depends on the projecton of r on H r mn a, D 2 ( < a, r > D) θ ( r q ) r a f r H r x q
34 MLS defnes (d-1)-dm manfold P m (P m (x))=p m (x) Implct surface defnton S { x Pm ( x) = x} Conjecture: S nfntely smooth f θ C
35 MLS defnes Curve/Surface
36 Computng H r and p(.) Computng hyper-plane H r Non-lnear optmzaton problem Computed teratvely Computng θ (.) : tme consumng step O(N) for each teraton step Approxmate by dong a herarchcal clusterng Fttng a polynomal p(.), gven H r Solve a lnear system Sze depends on the order of approxmaton (m)
37 Weght Functon θ θ ( d) = exp( d 2 / h 2 ) small h large h
38 Samplng Condton?
39 Smplfcaton More smplfcaton algorthms/results on Nov. 19
40 Up-samplng ,000
41 Renderng Q-Splat Rusnkewcz et al., Sggraph 2000 herarchcal pont renderng based on Boundng Sphere Herarchy
42 Renderng Results
43 Progressve Pont Surface Extenson of progressve mesh Start wth base doman P 0 Add ponts mplctly to get P 1 Predct postons and transmt error ( ) Repeat
44 Progressve Pont Surface Gven pont set R={r } Start wth base pont set P 0 Get ths by clusterng method (Nov. 19)
45 Progressve Pont Surface Gven pont set R={r } Start wth base pont set P 0 Get ths by clusterng method (Nov. 19) repeat Refne P to get P +1
46 Progressve Pont Surface Gven pont set R={r } Start wth base pont set P 0 Get ths by clusterng method (Nov. 19) repeat Refne P to get P +1 Dsplace new ponts s.t. d( SP, S ) (, ) 1 R d SP SR < +
47 Refnng P p p
48 Computng a p p a r p Transmt P o, { 1 }, { 2 }, { 3 }, Do nverse process to reconstruct
49 Results
50 Conclusons MLS: Projecton-based surface defnton Surface s smooth and a manfold Surface may be bounded Representaton error depends on pont densty Adjustable feature sze h allows to smooth out nose
51 Adaptve MLS
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