EIT i
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- あゆみ あわび
- 7 years ago
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1 CT
2 EIT i
3 A 58 B 59 C 61 D 66 D1 66 D2 68 D ii
4 % [1] [2] *1 206% 153% 112 [3] 3 *1 1
5 (2 4 ) ([4] ) X CT MRI X CT MRI X CT 40 [5] CT CT( EIT) [6] EIT 2
![EIT X CT MRI EIT 11 11 EIT [7] 113 EIT EIT](/docs-images/48/22464379/images/6-0.png "EIT Leroy X CT X CT CT X ( ) X CT EIT ( 12)")
6 EIT X CT MRI EIT EIT [7] 113 EIT EIT EIT Leroy X CT X CT CT X ( ) X CT EIT ( 12) ( 3
7 と呼ばれる) を配置し 対象物内部の電場を制御することである程度積分領域を狭めるこ とができる (図 13) が 限界がある また 電極の配置 対象物内部の導電率分布によっては電流が曲がることがあるが 曲 がり具合は導電率分布に依存するため 保護電極による電場制御は難しい 電流分布の例 電極は上下中央部 図 13 電流分布の例 図 12 と同様であ に配置されている 主な積分領域を着色し るが 保護電極が主電極の左右にそれぞれ て示す 配置されている 図 12 大路らによる研究 [9] 図 14 大路らによる亀裂検出 前項とは異なり 電流が直線的に流れないことを受け入れた理論を使用している 構造 物 (鉄材など) に生じた亀裂の位置を同定することが目的である 電極に電圧を印加した とき 表面に生じる電位分布はポアソンの式 φ = 0 4 (11)
8 φ 11 Murai [11] b = T((f)) (12) b ( ) f T f = T 1 (b) (13) EIT T Murai T 1 2 Murai ( Murai[11] [13] ) Barber[10] ( [12]) 5
9 Murai 12 EIT 2 [13] EIT EIT 3 EIT 2 3 6
10 2 EIT 21 EIT EIT EIT 7
11 21 Γ φ J 21 EIT Murai ( ) ( ) ( ) ( ) ( ) 221 D(x, t) = ρ e (x, t), (21) B(x, t) = 0, (22) D(x, t) H(x, t) = i e (x, t), t (23) B(x, t) E(x, t) + = 0 t (24) E B D H ρ e i e x 3 x = (x, y, z) t = 8 x y z (25)
12 A = (A x, A y, A z ) A A A = A x x + A y y + A z z ( Az A = y A y z, A x z A z x, A y x A ) x y (26) (27) i e (x, t) + ρ e(x, t) t = 0 (28) D(x, t) = ɛe(x, t) (29) B(x, t) = µh(x, t) (210) i e (x, t) = σ(x, t)e(x, t) (211) ( khz ) t 0 (212) i e (x) = 0 (213) E(x) = φ(x) (214) {σ(x) φ(x)} = 0 (215) 215 φ σ σ = const 9
13 222 Ω 2 σ i (i = 1, 2) σ 1 σ (σ i φ i ) = 0 (i = 1, 2) (216) (φ 2 σ 1 φ 1 ) = φ 2 (σ 1 φ 1 ) + φ 2 (σ 1 φ 1 ) (217) 216 (φ 2 σ 1 φ 1 ) = φ 2 (σ 1 φ 1 ) (218) Ω φ 2 σ 1 φ 1 nds = σ 1 φ 1 φ 2 dv (219) Γ Γ Ω 1 2 φ 1 σ 2 φ 2 nds = σ 2 φ 1 φ 2 dv (220) Γ 219,220 φ 1 σ 2 φ 1 nds φ 1 σ 2 φ 2 nds = (σ 1 σ 2 ) φ 1 φ 2 dv (221) Γ Γ Γ {Γ A, Γ B, Γ C, Γ D } Γ 1 A-B I 1 A B 2 C-D I 2 C D I 1 = σ 1 φ 1 nds = σ 1 φ 1 nds (222) Γ A Γ B I 2 = σ 2 φ 2 nds = σ 2 φ 2 nds (223) Γ C Γ D Ω Ω Ω 10
14 Φ 2 (A) 221 Γ ) Φ 2 σ 1 φ 1 nds = I 1 Φ 2 (x)ds + Φ 2 (x)ds ( Γ A Γ B (224) = (φ 2 (A) φ 2 (B))I 1 (225) Φ 1 σ 2 φ 2 nds = (φ 1 (C) φ 1 (D))I 2 (226) Γ φ 2 (A) = 1 Φ 2 (x)ds (227) Γ A Γ A *1 σ 1 = σ φ 2 σ 1 φ 1 nds = φ 1 σ 2 φ 2 nds (228) Z Γ Γ Z = φ 2(A) φ 2 (B) I 2 = φ 1(C) φ 1 (D) I 1 (229) (229) Z σ 1 σ 2 A, B, C, D σ 1 = σ Z 2 σ 2 = ˆσ Ẑ Z Ẑ = (σ ˆσ) φ 1(σ) φ 2(ˆσ) dv (230) I 1 I 2 Ω *1 EIT
15 230 ˆσ Ẑ φ 2 /I 2 φ 1 /I 1 ˆσ Z Ẑ = (σ ˆσ) φ 1(σ) φ 2(ˆσ) dv + O(( σ) 2 ) (231) I 1 I 2 Ω σ = σ ˆσ O(( σ) 2 ) Z = Z Ẑ N *2 Z = N σ n n=1 Ω n φ 1 I 1 φ 2 I 2 dv (232) φ 1 φ 2 dv = Ω n φ 1 φ 2 (233) Ω n I 1 I 2 I 1 I 2 Ω n (233) S n (232) Z = N S n σ n (234) n=1 N (233) S n N σ n S n (234) SVD[14] σ n ˆσ n + σ n 21 EIT 229 *2 12
16 ( ) ( ) 2 4 EIT 234 n n C 2 n 2 C φ σ EIT ( ) 215 {σ(x) φ(x)} = 0 (x Ω) 215 φ 2 Ω Γ Γ D Γ N φ(x) = g D (x) (Γ D : ) (235) σ(x) φ(x) n(x) = g N (x) (Γ N : ) (236) g D,N 13
17 s I s I s = σ(x) φ(x) n(x)ds (237) Γ s w, σ, φ (w(σ φ)) = w (σ φ) + w (σ φ) (238) w Γ D Ω wσ φ nds = σ w φdv + w (σ φ)dv (239) Γ N Ω Ω 236 w wg n ds = Γ N Ω σ w φdv (240) φ = g D (Γ D ) (241) Ω Ω e Γ Ω s Γ s Γ N Γ s Ω e σ 241 g s s Γ s wds = e σ e Ω e w φdv (242) ( ) j x j i ψ i = ψ i (x) ψ i ψ i (x j ) = δ i,j δ Γ D D Γ N Ω N φ(x) φ j = φ(x j ) φ(x) = j D φ j ψ j (x) + j N φ j ψ j (x) (243) 14
18 w = ψ i, i N 242 g s ψ i ds = ψ i φ j ψ j + φ j ψ j dv s Γ s e Ω e j D j N = φ j σ e ψ i ψ j dv + φ j σ e ψ i ψ j dv j D e Ω e j N e Ω e s g s B (s) i = j D φ j e σ e A (e) i,j + j N φ j e σ e A (e) i,j (i N) (244) A (e) i,j = Ω e ψ i ψ j dv (245) B (s) i = ψ i ds (246) Γ s 244 φ j, j N ( ) ψ j x Ω e φ(x) Ω e φ(x) = i [e] φ i ψ i (x) (247) [e] Ω e [e] = {1, 2, 3, 4} Ω e ψ i (x)(i [e]) a i, b i, c i, d i ψ i (x) = a i + b i x + c i y + d i z (x Ω e ) (248) 1 x 1 y 1 z 1 a 1 a 2 a 3 a 4 1 x 2 y 2 z 2 b 1 b 2 b 3 b 4 1 x 3 y 3 z 3 c 1 c 2 c 3 c 4 1 x 4 y 4 z 4 d 1 d 2 d 3 d 4 ψ i (x j ) = δ i,j (i, j [e]) (249) = (250) 15
19 ψ i = 1 (251) i [e] ψ i = 0 (252) i [e] ψ i = (b i, c i, d i ) A (e) i,j = (b ib j + c i c j + d i d j ) Ω e (i, j [e]) (253) B (s) i = 1 3 Γ s (254) Ω e Γ s Ω e = 1 6 ((x 2 x 1 ) (x 3 x 1 )) (x 4 x 1 ) (255) Γ s = 1 2 (x 2 x 1 ) (x 3 x 1 ) (256) 254 [15] ψ n 1 1 ψn 2 2 ψn 3 Γ s 3 ds = 2 Γ n 1!n 2!n 3! s (n 1 + n 2 + n 3 + 2)! (257) n 1, n 2, n 3 [s] N = 1, 2, 3 16
20 3 3 EIT 31 Linux (Fedora13) CPU Intel Core i GB NVIDIA Tesla C1060 Tesla C C GCC445 * ( ) Z *1 17
21 31 Z = S σ σ 321 EIT ( 234) S n n n n O(n 3 ) 234 S n 233 φ 1, φ N N N O(N 3 ) 32 n = 2025, N =
![32 N n 31 1 O(n 6 ) ( O(N 6 )) 2 3 n 3 I-SVD[16] 234 (SVD) 234 O(n 2 ) 244 SVD LU SVD 31 NVIDIA Tesla C1060 (](/docs-images/48/22464379/images/22-0.png "33) NVIDIA CUDA *2 *3 CPU *2 http://developernvidiacom/category/zone/cuda-zone *3 NVIDIA Compute Capability 13")
22 32 N n 31 1 O(n 6 ) ( O(N 6 )) 2 3 n 3 I-SVD[16] 234 (SVD) 234 O(n 2 ) 244 SVD LU SVD 31 NVIDIA Tesla C1060 ( 33) NVIDIA CUDA *2 *3 CPU *2 *3 NVIDIA Compute Capability 13 19
23 33 Tesla C1060 CUDA cufft cublas( ) cusparse( cublas) curand( ) NPP( ) Thrust( ) CUDA CULA *4 CULA LAPACK SVD LU CULA 244 CULA sparse ( 32) medit *5 medit mesh 34 Listing 31 *4 *5 frey/softwarehtml 20
24 Listing 31 medit 1 MeshVersionFormatted Dimension # Set o f mesh v e r t i c e s 7 V e r t i c e s # Set o f T r i a n g l e s 19 T r i a n g l e s
25 # Set o f Tetrahedra 41 Tetrahedra End FreeFEM3D *6 GiD *7 CAD GiD medit 35 *6 *7 22
26 34 medit 35 23
27 34 ( ) φ [13] *8 ( ) *9 SVD 36 * 10 *8 *9 [25] Mersenne Twister(MT) MT *10 36 Z ( ) 24
28 36 25
29 [22] TX-151 Oil Center Research *1 *1 26
30 TX mm
31 1 24 #1000 *2 OHP Agilent 34970A ( AG-203D) *2 mm 28
32 Phantom Altenator Computer RS 232C Measurement Current change Plug in module Data Acquisition / Switch Unit Agilent 34970A 43 50Ω Ω A 34901A 20 AC/DC [19] 1% 34901A C 34901A 44 (Belling Lee,L1727A/J) 34970A RS-232C PC Agilent *3 [22] 45 * A
33 図 44 端子台の全景 図 45 実験装置の全景 30
34 [20] [20] [21] Spielrein (ln E) = 2H (51) n E H n E H 41 31
35 kHz 30mA mV : =1:3 1 Actual Simulational Normalized Voltage [AU] Electrode number
36 41 NaCl : =1:5 : =1: [23] 52, [23] 53 [23]
37 Y 1α Y 1β Y 1γ Y 2α Y 2β Y 2γ Y nα Y nβ Y nγ φ α φ β φ γ = i α i β i γ (52) 52 α, β, γ 1 φ, i Y 244 φ α = φ β = φ γ α,β,γ Y 1j α,β,γ Y 2j α,β,γ Y nj α,βγ φ j = α,β,γ i j (53) φ β,γ φ α 32 2 E = n i (σ 1 σ 2 ) 2 (54)
38
![53 2 [24] 54 24 2025 55 1 5](/docs-images/48/22464379/images/39-0.png "56 54 55 56 170 *1 2 *1 B")
39 53 2 [24] *1 2 *1 B 36
40 56 54 SVD ( ) (= ) ( ) 57 Coarse mesh 54 Fine 5 45 Fine mesh Coarse mesh RMS error at Number of singular values 57 [24] 37
41 mesh Fine mesh Coarse mesh 55 Fine mesh 58 59,510 ( ) 59,510 Fine mesh Coarse mesh 38
42 59 Coarse mesh 510 Fine mesh 57 Fine mesh 56 CT 234 Z 511 Z 2 55 E = N (σ 1 (i) σ 2 (i)) 2 (55) i ( 2 ) ( 1 3 ) 55 Coarse mesh 2 24 % 1% 1% 39
43 Z Z 10% 54 Coarse mesh
44 2 19 Without noise With 10% noise 18 RMS error Iteration
45
46 6 Z EIT
47 * ( ) RS-232C PC 220g 1kHz *1 44
48 62 8mA (012mmφ) 100mΩ * % 56 *2 45
49 mm 10mm 68 46
50 R observed = A x + R contact (61) x R contact * *3 47
51 Ω 25 40mm 15Ω 41 Ω 48
52 , CT 49
53
54 7 5 6 CT Fine mesh Fine mesh φ σ σ 51
55 71 Fine mesh Z 234 σ Z 234 σ Z 72 σ Z σ 72 52
56 72 Z σ σ [26] 57( ) 57 σ 71 Z Z 53
57 73 TARO HANAKO 74 EIT 57 Hager[27] (4s + 1)n 2 (s ) [28] (4/3n 3 ) 54
58
59 83 56
60 9 3 57
61 A F x = y (A1) y y y y x A1 F (x + x) = y + y (A2) A1 F x y (A3) A2 F 1 y = x F 1 y x (A4) F F y 0 A3 A4 x x y cond(f ) y cond(f ) F (A5) cond(f) = F F 1 (A6) A5 58
62 B A6 F F m n F B1 F = UDV T (B1) U m m V n n D m n D m = n B1 A6 B2 condf = λ 1 λ n (B2) λ 1, λ n n D D SVD F x = F mod y (B3) 59
63 F mod = V D mod U T (B4) F mod Moore-Penerose F mod = F ( ) 60
64 C C1 C4 2 ( 100) SENSE,H SOUCE,H ( 200) CH21,L CH21 H ( 100) SENSE,L SOURCE,L GND 61
65 C H 2 02H 3 03H 4 04H 5 05H 6 06H 7 07H 8 08H 9 09H 10 10H 11 NC 12 NC 13 NC 14 NC 15 11H 16 12H 17 13H 18 14H 19 15H 20 16H 21 17H 22 18H 23 19H 24 20H 62
66 C H 2 02H 3 03H 4 04H 5 05H 6 06H 7 07H 8 08H 9 09H 10 10H 11 NC 12 NC 13 NC 14 NC 15 NC 16 NC 17 NC 18 NC 19 NC 20 NC 21 NC 22 NC 23 NC 24 NC 63
67 C NC 2 NC 3 01L 4 02L 5 03L 6 04L 7 05L 8 06L 9 07L 10 08L 11 09L 12 10L 13 11L 14 12L 15 13L 16 14L 17 15L 18 16L 19 17L 20 18L 21 19L 22 20L 23 NC 24 NC 64
68 C NC 2 NC 3 NC 4 NC 5 NC 6 NC 7 NC 8 NC 9 NC 10 NC 11 NC 12 NC 13 01H 14 02H 15 03H 16 04H 17 05H 18 06H 19 07H 20 08H 21 09H 22 10H 23 11H 24 12H 65
69 D Linux(64bit) Compute Capability 13 D1 D11 NVIDIA Graphics Driver NVIDIA Web NVIDIA Web TeslaC D12 GSL GSL(GNU Scientific Library) FEM GSL 113 Fedora yum 66
70 GSL *1 D13 SuperLU SuperLU FEM *2 SuperLU SuperLU 41 D14 LAPACK BLAS ATLAS Fedora ATLAS yum D15 SFMT SFMT *3 *1 *2 xiaoye/superlu/ *3 m-mat/mt/sfmt/ 67
71 D16 CUDA CULA *4 CUDA Toolkit GPU Computing SDK NVIDIA GPU Computing SDK (nbody ) D17 CULA CULA CULA dense( CULA R11 ) *5 D2 D21 bashrc bashrc Listing D1 bashrc 1 #CULA 2 export CULA ROOT= $HOME/ cular11 3 export CULA INC PATH= $CULA ROOT/ i n c l u d e 4 export CULA BIN PATH 32= $CULA ROOT/ bin 5 export CULA BIN PATH 64= $CULA ROOT/ bin64 6 export CULA LIB PATH 32= $CULA ROOT/ l i b 7 export CULA LIB PATH 64= $CULA ROOT/ l i b 6 4 *4 *5 68
72 8 export LD LIBRARY PATH=$CULA LIB PATH 64 : $LD LIBRARY PATH 9 10 #n u r u l i b 11 export C INCLUDE PATH=$HOME/ p r o j e c t s / n u r u l i b / i n c l u d e 12 export CPLUS INCLUDE PATH=$HOME/ p r o j e c t s / n u r u l i b / i n c l u d e 13 export LD LIBRARY PATH=$HOME/ p r o j e c t s / n u r u l i b / l i b : $LD LIBRARY PATH #CUDA 16 export LD LIBRARY PATH=/usr / l o c a l /cuda/ l i b 6 4 : $LD LIBRARY PATH 17 export PATH=/usr / l o c a l /cuda/ bin :$PATH 18 export MANPATH=$MANPATH: / usr / l o c a l /cuda/man #SFMT 21 export LD LIBRARY PATH=$HOME/ i n s t a l l /SFMT/SFMT src :$LD LIBRARY PATH nurulib sslib D3 D31 Makefile make Makefile D32 inverse /inverse -r -t -1 -i10 -s256 -x60 -o60 -k30 -m u5mesh -c conductivitydat -f small 69
73 -g50 -j75 inverse -r -t -i -s -x -o -k -m -c -f -g,-j 70
74 [1] 21, pp 99, 2009 [2] 22 pp 30, 2010 [3] TMiyawaki et al, Metabolic syndrome in Japanese diagnosed with visceral fat measurement by computed tomography, Proc Japan Acad, 81, Ser B, 2005 [4] 20(1), pp 90-99, 2008 [5] CT MEDIX, 41, pp 15-20, 2004 [6] CT , pp , 2010 [7] XZhao, et al, A New Method for Noninvasive Measurement of Multilayer Tissue Conductivity and Structure Using Divided Electrodes, IEEE Trans Biomed Eng, 32, pp , 1985 [8] Leroy R Price, ELECTRICAL IMPEDANCE COMPUTED TOMOGRAPHY (ICT): A NEW CT IMAGING TECHNIQUE, IEEE Trans Nucl Sci, 26, pp , 1979 [9] CT (A ) 85 pp [10] DCBarber and BHBrown, Applied potential tomography, JPhys E:Sci Instum, 17, 1984 [11] TMurai and Y Kagawa, Electical Impedance Computed Tomography Based on a Finite Element Model, IEEE Trans Biomed Eng, 32, 1985 [12] Pai Chi Nan, et al, AN IMPLEMENTATION OF THE BACK-PROJECTION ALGORITHM ACCORDING TO SANTOSA AND VOGELIUS, ABCM Symp 71
75 Ser Bioeng, 1, 2006 [13], CT,, 2007 [14] PCHansen, The truncated SVD as a method for regularization, BIT, 27, 1987 [15] CA Brebbia, Integration of area and volume coordinates in the Finite-Element Method, AIAA J, 7, pp 1212, 1969 [16] 46 pp , 2005 [17] (B), 85, pp , 2002 [18] 4 pp [19] 34970A 2006 [20] ADSeagar, et al, Theoretical limits to sensitivity and resolution in impedance imaging, Clin Phys Physiol Meas, 8, pp 13-31, 1987 [21] 2006 [22] CT [23] 3 CT 2009 [24] CT [25] M Matsumoto and T Nishimura, Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans on Modeling and Computer Simulation, 8, pp 3-30, 1998 [26] MAICE-DP 3 pp [27] WWHager, Condition Estimates, SIAM J Sci Comput, 5, pp , 1984 [28] 7 pp ,
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1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
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A = QΛQ T A n n Λ Q A = XΛX 1 A n n Λ X GPGPU A 3 T Q T AQ = T (Q: ) T u i = λ i u i T {λ i } {u i } QR MR 3 v i = Q u i A {v i } A n = 9000 Quad Core Xeon 2 LAPACK (4/3) n 3 O(n 2 ) O(n 3 ) A {v i }
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