SLIDE 23 ✷✳✹ ■♠♠❡rs❡❞ ❇♦✉♥❞❛r② ▼❡t❤♦❞ ✶✼ ❚❤❡ ♠❛tr✐① M✱ ❝❛❧❧❡❞ ♠♦♠❡♥t ♠❛tr✐①✱ ❢♦r t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❜❧❡♠s r❡❛❞s ✿ M = m0,0,0 m1,0,0 m0,1,0 m0,0,1 m1,1,0 m0,1,1 m1,0,1 m2,0,0 m0,2,0 m0,0,2 m1,0,0 m2,0,0 m1,1,0 m1,0,1 m2,1,0 m1,1,1 m2,0,1 m3,0,0 m1,2,0 m1,0,2 m0,1,0 m1,1,0 m0,2,0 m0,1,1 m1,2,0 m0,2,1 m1,1,1 m2,1,0 m0,3,0 m0,1,2 m0,0,1 m1,0,1 m0,1,1 m0,0,2 m1,1,1 m0,1,2 m1,0,2 m2,0,1 m0,2,1 m0,0,3 m1,1,0 m2,1,0 m1,2,0 m1,1,1 m2,2,0 m1,2,1 m2,1,1 m3,1,0 m1,3,0 m1,1,2 m0,1,1 m1,1,1 m0,2,1 m0,1,2 m1,2,0 m0,2,2 m1,1,2 m2,1,1 m0,3,1 m0,1,3 m1,0,1 m2,0,1 m1,1,1 m1,0,2 m2,1,1 m1,1,2 m2,0,2 m3,0,1 m1,2,1 m1,0,3 m2,0,0 m3,0,0 m2,1,0 m2,0,1 m3,1,0 m2,1,1 m3,0,1 m4,0,0 m2,2,0 m2,0,2 m0,2,0 m1,2,0 m0,3,0 m0,2,1 m1,3,0 m0,3,1 m1,2,1 m2,2,0 m0,4,0 m0,2,2 m0,0,2 m1,0,2 m0,1,2 m0,0,3 m1,1,2 m0,1,3 m1,0,3 m2,0,2 m0,2,2 m0,0,4 . ✇✐t❤ mi,j,k =
- ΩI (x − s)i(y − t)j(z − v)k ˜
wd(x − s, t − y, t, z − v) dsdtdv✳
✷✳✹✳✸ ◆✉♠❡r✐❝❛❧ ■♠♣❧❡♠❡♥t❛t✐♦♥
❋✐❣✉r❡ ✷✳✹✿ ❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ s✉♣♣♦rt ❝❛❣❡ ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥✳ ❚❤❡ ❞❛s❤❡❞ ❧✐♥❡ ✐s t❤❡ ❡♠❜❡❞❞❡❞ ❝✉r✈❡ ❛♥❞ t❤❡ s♦❧✐❞ ❧✐♥❡ ✐s t❤❡ r❡❝t❛♥❣✉❧❛r s✉♣♣♦rt ❝❛❣❡ ΩI✳ • ▲❛❣r❛♥❣✐❛♥ ♠❛r❦❡rs ♦♥ t❤❡ ❡♠❜❡❞❞❡❞ ❝✉r✈❡✱ ✷ ❣r✐❞ ♣♦✐♥ts✱ ❣r✐❞ ♣♦✐♥ts ✇✐t❤✐♥ t❤❡ s✉♣♣♦rt✳ ❚❤❡ ✜st st❡♣ ✐s t♦ ❞✐s❝r❡t✐③❡ t❤❡ ❡♠❜❡❞❞❡❞ ❣❡♦♠❡tr② ✐♥t♦ ❛ ♥✉♠❜❡r ♦❢ ♥♦❞❡s Xi, i = 1, ..., Ne✳ ❆r♦✉♥❞ ❡❛❝❤ ♥♦❞❡ XI ❛ ❝✉❜✐❝✴r❡❝t❛♥❣✉❧❛r ❝❛❣❡ ΩI ✐s ❞❡✜♥❡❞ t❤❛t ❝♦♥t❛✐♥s ❛t ❧❡❛st t❤r❡❡ ♥♦❞❡s ♦❢ ✉♥❞❡r❧②✐♥❣ ♠❡s❤ ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥ ✭9 ✐♥ 2D ❛♥❞ 27 ✐♥ 3D✮ ❛♥❞ t❤❛t ✇✐❧❧ ❜❡ t❤❡ s✉♣♣♦rt ❢♦r t❤❡ ✇✐♥❞♦✇ ❢✉♥❝t✐♦♥ ✭✷✳✹✺✮✳ ❚❤❡ ❝❛❣❡ ❛♥❞ t❤❡ ✉♥❞❡r❧②✐♥❣ ♠❡s❤ ❛r❡ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✹✳ ❚❤❡ ❡❞❣❡s ♦❢ t❤❡ ❝❛❣❡ ♠❡❛s✉r❡ 3δ✱ 3η✱ 3σ ✐♥ x, y, z ❞✐r❡❝t✐♦♥ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡r❡ δ, η, σ ❛r❡ t❤❡ ❞✐❧❛t❛t✐♦♥s ❢❛❝t♦rs✳ ❚♦ ❞❡t❡r♠✐♥❛t❡ ♦❢ s✉♣♣♦rt ❝❡♥tr❡❞ ✐♥ XI✱ t❤❡ ❝❧♦s❡st ♥♦❞❡ t♦ XI✱ xˆ
i,ˆ j,ˆ k✱ ✐s ❢♦✉♥❞
❛♥❞ t❤❡ ❞✐st❛♥❝❡ ∆x = |XI − xˆ
i,ˆ j,ˆ k| ✐s ❝❛❧❝✉❧❛t❡❞❀ t❤❡♥ ❛ s❡t ♦❢ ♥♦❞❡s ♥❡✐❣❤❜♦✉r✐♥❣
xˆ
i,ˆ j,ˆ k✱ NI = xˆ i+l,ˆ j+m,ˆ k+n, l, m, n = −1, 0, 1 ✐s ❝♦♥s✐❞❡r❡❞ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t✐❡s
