三维数据点的曲率计算

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4 [8]. X [9] Hohe VoxelMa CTMRI

5 VoxelMa Voxel Ma VoxelMa [7] Maya Clc 7 CT MRI 3D. MRI 3D CT

6 3

7 .... m {(, )} =, φ = spa ϕ, ϕ,..., ϕ } x y * * P ( x) = a ϕ ( x) ( < m) = { m * [ P ( x ) f ( x )] = m w( x )[ P( x ) f ( x ] m = w( x ) ) = = δ () f f : R R ) y = f x m ( 4

8 w(x) [a,b] ( x y ) w x ), ( P * ( x) m I( a,... a ) = w( x )[ a jϕ j ( x ) f ( x )] = j= * * ( a,..., a ) I a m m = w( x )[ a ϕ ( x ) f ( x )] ϕ ( x ) = =,,..., 3 = j= j j m ( ϕ, ϕ ) = w( x ) ϕ ( x ) ϕ ( x ) j = m ( f, ϕ ) = w( x ) f ( x ) ϕ ( x ) = j j= ( ϕ, ϕ ) a = ( f, ϕ ) j j =,,... 4 G = ( ϕ, ϕ ) ( ϕ, ϕ )... ( ϕ, ϕ ) ( ϕ, ϕ ) ( ϕ, ϕ )... ( ϕ, ϕ ) ( ϕ, ϕ ) ( ϕ, ϕ )... ( ϕ, ϕ ) } = { ϕ G 4 a = j =,,... * j a j * * P ( x) = a ϕ ( x) = 5

9 .. l m m {(( xj ) =, y j )} j= { w } = l φ = spa ϕ, ϕ,... ϕ } ( < m +) { l * * P ( x, x,... x ) = a ϕ ( x, x I( a =,..., x ) m *, a,..., a ) = w [ P ( x, x,... xl ) y ] = = m w [ a ϕ ( x = = *, x l,..., x l ) y ] { 4 a} = m ( ϕ, ϕ j ) = wϕ ( x, x,..., xl ) ϕ j ( x, x,..., xl ), = m ( f, ϕ j ) = w yϕ j ( x, x,..., xl ), =, j =,,..., j =,,..., 4 { l a} = * * P ( x, x,..., x ) = a ϕ ( x, x,..., xl ) =..3 6

10 {( y, z )} x, = φ = spa{, x, y, x, y } ( > 3) z = f + ( x, y) = a + ax + a y + a3x a4 y 4 { a } =.3.3. r. {, P,... P } P. P m P P P... P ds( Pj P ) ds( Pj P )... ds( Pj ) j j j m P m P 7

11 P m P P t, P N 8

12 P L { } = L P } = P L 3 { 4 e >t e > L N, L = L 3 a P L { P } = L { P } = e false true t false L > N true L = L b. 9

13 { O; xyz} E x = x( u, v) y = y( u, v) z = z( u, v) u, v ( u, v) R D 3 D E ( u, v) ( x( u, v), y( u, v), z( u, v)) 3 D E S ( u,v) S S r = r( u, v) = ( x( u, v), y( u, v), z( u, v)) P S, ) v = v ( u v u S P

14 r = r( u, v ) P u u u = u P v v r = r( u, v) S u v S P r u u= u v= v r v u= u v= v r u = u u v= v r v u= u v= v r u= u r u= u P S u v= v v v= v S P( u, v) r u r v 3. S C u = u( t) v = v( t) a t b t = u = u t ), v = v( ) t P u, ) t ( t C C P r = r( u( t), v( t)) ( v

15 dr dt r du r dv r du r dv = 3 ( + ) t= t = ( ) ( ) t= t + t= t u u u u t t u dt v dt u = dt v = = dt v= v v= v P r r P ω = αr u βr α, β u= u + v= v v u= u v= v P ω S P u = u v = v S + α( t t ) + β ( t t ) r = r( u + α ( t t ), v + β ( t t )) P dr dt r r = α + β = ω u u= u v u= u t= t v= v v= v P S P T T S u v P P P r u= u u v= v r T v u= u v= v P 3 P S E a, b a b a,b a,b a, b=a b T a b P T P E P T P

16 r u r v r r r ± u r v u v r = r( u, v) r r r 3. u r v u v 3. S = r r u u v r r v 3..4 r(u P ( u, v) P '( u + u, v + v) S r( u, v) + u,v + v) PP' = r = r( u + u, v + v) r( u, v) = r u + r v +... u u, v r dr = r du u + r dv v v 3

17 ' P P u, v PP' E 3 ds S ds ds = dr = dr dr = ( ru du + rv dv) ( ru du + rv dv) = ru ru ( du) + ru rv dudv + rv rv ( dv ) E = r u r u F r u rv = G = r v rv E, F, G ds = Edu + Fdudv + Gdv E, F, G Ι P P' ( u + u, v + v) P T δ 3.3 P ( r 3.3 PP' = r = r( u + u, v + v) r( u, v) = r u + r v + u v uu ( u) + r uv u v + r vv ( v) ) +... u, v δ = PP ' = ( ruu ( u) + ruv u v + rvv ( v) ) +... (4) 4

18 L = r, M = r N = r uu uv δ = L( u) + M u v + N v +... δ > ' P u, v P T δ < ' P P vv δ C = Ldu + + L M N Mdudv Ndv 3..5 Ldu + Mdudv + Ndv = C 5 Gdv Ι = Edu + Fdudv + 4 C 5 P 5 ( E L) du + ( F M ) dudv + ( du dv = µ 6 dv ( G N) dv = 6 E L) µ + ( F M ) + ( G N) = 7 µ 5

19 µ 7 µ d ( E L) µ + ( F M ) + ( Eµ + Fµ + G) = 8 dµ d = 8 dµ 7 ( E L) µ + F M = 9 µ [( E L) µ + F M ] + [( F M ) + G N] = µ 9 ( F M ) µ + G N = 9 µ E L F M F M G N = () ( EG F ) ( EN FM + GL) + ( LN M ) = () Edu + Fdv Ldu + Mdv Fdu + Gdv Mdu + Ndv = ( EM FL) du w = µ r u + ηr v + ( EN GL) dudv + ( FN GM ) dv = φ = µ φ η ( LF EM ) φ + ( LG EN) φ + ( MG FN) = 6

20 3 K = H = ( + ), K H = = LN EG M F EN FM + GL ( EG F ) 3. z = f ( x, y) p ( x, y) = ( x, y, z) = ( x, y, f ( x, y)) f ( x, y) u = x f ( x, y) v = y p x = (,, u) p y = (,, v) p xx u u v v = (,, ) p xy = (,, ) = (,, ) p yy = (,, ) x y x y 7

21 = ( u, v, ) + u + v E = + u F = u v G = + v u L = x M + u + v u v = y = x, + u + v + u + v N = + v y + u v z = f = ( x, y) a ax a y a3x a4 y p ( x, y) = ( x, y, f ( x, y)) p x =,, a + a ), p y =,, a + a ) ( 3x ( 4 y p xx =,,a ), p = (,,), p yy =,,a ) ( 3 xy ( 4 = ( a + ( a + a x, a 3 + a 3 x) + a + ( a 4 y, ) + a 4 y) E, F, G, L, M, N 8

22 FastSca Vrual C++6. OpeGLMatlab FastSca , 4.6, λ,λ g m e t r 9

23 r t e

24 z = a + a x + a y + a x + a 3 4 y a a a a3 a r t e

25 z = a + a x + a y + a x + a 3 4 y a a a a3 a r = 4. t = 3. e =.456 = 7 =.349 =.7945 g = m =

26 λ = (.4568,.99837,.36937), λ = (.54,.445,.8575) z = x y.99x.5393y + r = 4. t =. e =.437 = 4 = = g = m = λ = (.5888, ,.96), λ = ( ,.53,.86596) z = x.587y.6768x.69896y + r = 4. t =. e =.8776 = =.657 =.665 g = m =

27 λ = ( ,.996,.6), λ = (.46399,.55989,.9979) z = x.87y.86788x.7985y r =. t =. e =.9557 = 39 =.3999 = g = m =.6489 λ = ( ,.6765,.79553), λ = (.435,.9984,.4667) z = x.97943y.57x.53455y r =. t =. e = = 35 = =.3988 g = m = λ = (.9966,.8575,.858), λ = (.74,.9993,.4446) z = x.44y.3736x.3499y 4

28 4.6 r = 3., t = 3. 5

29 4.7 r = 3., t =. 6

30 7

31 [] Hogb ZhaStudes O 3-D Feature Measuremet Ad Object Recogto Based O Surface Characterzato. PhD Thess,Kyushu Uversty,99 []. 979 [3]. 989 [4]. 985 [5]. 996 [6]. 989 [7] Dael Sord ad Samuel K. Moore,The Vrtual Surgeo.IEEE Spectrum, pp.6-3, [8] Steve Seger,Vsualzg Ad Segmetg Large Volumetrc Data Sets. IEEE Computer Graphcs ad Applcatos,pp.3-37,999 [9],

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