EIT i

CT 10801610 23

1 1 11 1 12 6 2 EIT 7 21 7 22 7 3 17 31 17 32 17 33 20 34 24 4 26 41 26 42 27 43 28 5 31 51 31 52 33 53 36 54 37 55 38 56 39 i

57 40 58 41 6 43 61 43 62 44 63 49 7 51 71 51 72 52 73 54 74 54 8 55 81 55 82 55 83 56 9 57 A 58 B 59 C 61 D 66 D1 66 D2 68 D3 69 71 ii

1 11 111 30% [1] [2] *1 206% 153% 112 [3] 3 *1 1

(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 EIT Leroy X CT X CT CT X ( ) X CT EIT ( 12) ( 3

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)

φ 11 Murai [11] b = T((f)) (12) b ( ) f T 12 12 f = T 1 (b) (13) EIT T Murai T 1 2 Murai ( Murai[11] [13] ) Barber[10] ( [12]) 5

Murai 12 EIT 2 [13] EIT 3 3 3 EIT 3 EIT 2 3 6

2 EIT 21 EIT 22 21 EIT EIT 7

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)

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 ) 28 211 213 t 0 (212) i e (x) = 0 (213) E(x) = φ(x) (214) {σ(x) φ(x)} = 0 (215) 215 φ σ 11 215 σ = const 9

222 Ω 2 σ i (i = 1, 2) σ 1 σ 2 215 (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

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 2 221 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 229 2 1 σ 1 = σ Z 2 σ 2 = ˆσ Ẑ 221 229 Z Ẑ = (σ ˆσ) φ 1(σ) φ 2(ˆσ) dv (230) I 1 I 2 Ω *1 EIT 227 227 11

230 ˆσ Ẑ φ 2 /I 2 φ 1 /I 1 ˆσ 230 0 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

( ) ( ) 2 4 EIT 234 n n C 2 n 2 C 2 223 233 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

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

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) = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (250) 15

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

3 3 EIT 31 Linux (Fedora13) CPU Intel Core i7 930 4 6GB NVIDIA Tesla C1060 Tesla C1060 240 C GCC445 *1 32 22 31 31 ( ) Z *1 http://gccgnuorg 17

31 Z = S σ σ 321 EIT ( 234) S n n n n O(n 3 ) 234 S n 233 φ 1, φ 2 244 244 N N N O(N 3 ) 32 n = 2025, N = 583 18

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 http://developernvidiacom/category/zone/cuda-zone *3 NVIDIA Compute Capability 13 19

33 Tesla C1060 CUDA 2012 1 cufft cublas( ) cusparse( cublas) curand( ) NPP( ) Thrust( ) CUDA CULA *4 CULA LAPACK SVD LU 234 244 CULA 244 CULA sparse 33 244 ( 32) medit *5 medit mesh 34 Listing 31 *4 http://wwwculatoolscom *5 http://wwwannjussieufr/ frey/softwarehtml 20

Listing 31 medit 1 MeshVersionFormatted 1 2 3 Dimension 4 3 5 6 # Set o f mesh v e r t i c e s 7 V e r t i c e s 8 8 9 0 0 0 0 10 1 0 0 0 11 1 1 0 0 12 0 1 0 0 13 0 0 1 0 14 1 0 1 0 15 1 1 1 0 16 0 1 1 0 17 18 # Set o f T r i a n g l e s 19 T r i a n g l e s 20 18 21 6 7 1 0 22 6 5 7 0 23 7 5 1 0 24 1 5 6 0 25 1 8 3 0 26 1 4 8 0 27 8 4 3 0 28 3 4 1 0 29 3 7 1 0 30 3 2 7 0 31 7 2 1 0 21

32 1 2 3 0 33 7 2 6 0 34 6 2 1 0 35 1 8 7 0 36 7 8 3 0 37 7 5 8 0 38 8 5 1 0 39 40 # Set o f Tetrahedra 41 Tetrahedra 42 6 43 6 7 1 5 0 44 1 8 3 4 0 45 3 7 1 2 0 46 1 7 6 2 0 47 1 7 3 8 0 48 7 8 1 5 0 49 50 End FreeFEM3D *6 GiD *7 CAD GiD medit 35 *6 http://wwwfreefemorg/ff3d/ *7 http://wwwgidhomecom/ 22

34 medit 35 23

34 ( ) 3 234 233 φ [13] *8 ( ) *9 SVD 36 * 10 *8 http://wwwtimpelcombr/si/site *9 [25] Mersenne Twister(MT) MT *10 36 Z ( ) 24

36 25

4 41 41 [22] 41 41 TX-151 Oil Center Research *1 *1 http://wwwoilcentercom 26

41 41 170 303 510 182 TX-151 86 303 86 303 18 63 30 106 42 60 25mm 24 1 27

1 24 #1000 *2 OHP 42 42 43 43 Agilent 34970A ( AG-203D) *2 mm 28

0000000 Phantom Altenator 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 00 11 Computer RS 232C 0000000 1111111 0000000 1111111 0000000 1111111 1111111 Measurement Current change Plug in module Data Acquisition / Switch Unit Agilent 34970A 43 50Ω Ω 3 20 34901A 34901A 20 AC/DC [19] 1% 34901A C 34901A 44 (Belling Lee,L1727A/J) 34970A RS-232C PC Agilent *3 [22] 45 *3 34901A 20 24 29

図 44 端子台の全景 図 45 実験装置の全景 30

5 51 511 41 215 41 [20] [20] [21] Spielrein (ln E) = 2H (51) n E H n E H 41 31

512 51 32 1 3 63 3kHz 30mA 63 10 50mV 41 41 41 : =1:3 1 Actual Simulational Normalized Voltage [AU] 08 06 04 02 0 0 5 10 15 20 25 Electrode number 51 51 32

41 NaCl : =1:5 : =1:3 52 521 227 [23] 52,53 52 52 [23] 53 [23] 244 52 33

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) 2 2 53 244 244 34

53 53 215 234 35

53 2 [24] 54 24 2025 55 1 5 56 54 55 56 170 *1 2 *1 B 36

56 54 SVD ( ) (= ) ( ) 57 Coarse mesh 54 Fine 5 45 Fine mesh Coarse mesh RMS error 4 35 3 25 2 15 123 at 130 1 0 50 100 150 200 250 300 Number of singular values 57 [24] 37

mesh 58 58 Fine mesh 24 2 57 55 57 Coarse mesh 55 Fine mesh 58 59,510 ( ) 59,510 Fine mesh Coarse mesh 38

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

511 57 56 512 Z Z 10% 54 Coarse mesh 512 2 40

2 19 Without noise With 10% noise 18 RMS error 17 16 15 14 13 0 1 2 3 4 Iteration 512 58 52 513 52 100 300 50 20 60 103 60 54 3 1 41

513 513 42

6 Z 61 61 61 2 EIT 234 43

62 4 621 62 *1 62 20 63 62 ( ) 62 64 64 RS-232C PC 220g 1kHz *1 44

62 8mA (012mmφ) 100mΩ 65 66 65 66 *2 0 66 14% 56 *2 45

63 66 622 67 67 10mm 10mm 68 46

64 65 4 0 68 67 R observed = A x + R contact (61) x R contact *3 69 67 67 *3 47

66 67 61 227Ω 25 40mm 15Ω 41 Ω 48

68 63 610, 611 20 2 611 CT 49

69 610 611 50

7 5 6 CT 71 58 Fine mesh 71 130 512 Fine mesh 2 71 10 231 2 φ σ σ 51

71 Fine mesh Z 234 σ Z 234 σ Z 72 σ Z σ 72 52

72 Z σ σ [26] 57( ) 57 σ 71 Z Z 53

73 TARO HANAKO 74 EIT 57 Hager[27] (4s + 1)n 2 (s ) [28] (4/3n 3 ) 54

8 81 82 55

83 56

9 3 57

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

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 100 100 F x = F mod y (B3) 59

F mod = V D mod U T (B4) F mod Moore-Penerose F mod = F ( ) 60

C C1 C4 2 ( 100) SENSE,H SOUCE,H ( 200) CH21,L CH21 H ( 100) SENSE,L SOURCE,L GND 61

C1 100 1 01H 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

C2 200 1 01H 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

C3 100 1 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

C4 200 1 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

D Linux(64bit) Compute Capability 13 D1 D11 NVIDIA Graphics Driver NVIDIA Web NVIDIA Web TeslaC1060 2601944 D12 GSL GSL(GNU Scientific Library) FEM GSL 113 Fedora yum 66

GSL *1 D13 SuperLU SuperLU FEM *2 SuperLU SuperLU 41 D14 LAPACK BLAS ATLAS Fedora ATLAS yum D15 SFMT SFMT *3 *1 http://wwwgnuorg/software/gsl *2 http://crd-legacylblgov/ xiaoye/superlu/ *3 http://wwwmathscihiroshima-uacjp/ m-mat/mt/sfmt/ 67

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 http://developernvidiacom/category/zone/cuda-zone *5 http://wwwculatoolscom 68

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 14 15 #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=$MANPATH: / usr / l o c a l /cuda/man 19 20 #SFMT 21 export LD LIBRARY PATH=$HOME/ i n s t a l l /SFMT/SFMT src 1 3 3 :$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

-g50 -j75 inverse -r -t -i -s -x -o -k -m -c -f -g,-j 70

[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 3 27 27, pp 135-139, 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 177-184, 1985 [8] Leroy R Price, ELECTRICAL IMPEDANCE COMPUTED TOMOGRAPHY (ICT): A NEW CT IMAGING TECHNIQUE, IEEE Trans Nucl Sci, 26, pp 2736-2739, 1979 [9] CT (A ) 85 pp 1352 1986 [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

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 299-311, 2005 [17] (B), 85, pp 582-596, 2002 [18] 4 pp 34-44 [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 3 2009 [23] 3 CT 2009 [24] CT 3 28 2011 [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 47-58 1993 [27] WWHager, Condition Estimates, SIAM J Sci Comput, 5, pp 311-316, 1984 [28] 7 pp 307-319, 1997 72