22
2 Voxel-Based Inversion Via Set-Theoretic Estimation
We relate the cell conductivities to the electric field moments in the following
way:
E =
J (a) (x, y, z)
σ (a) (x, y, z)
,
(2.8)
where E(x, y, z) is the total electric field, J (a) (x, y, z) is the anomalous electric
current, and σ (a) is the anomalous flaw conductivity. Upon substituting (2.8) into
(2.5) and (2.6), we get
E
x
klm =
J (a)(x) (x, y, z)
σ (a) (x, y, z)
T
x
klm (x, y, z)dxdydz
(2.9)
E
y
klm =
J (a)(y) (x, y, z)
σ (a) (x, y, z)
T
y
klm (x, y, z)dxdydz .
(2.10)
Expanding J (a) (x, y, z) in terms of the basis functions T x
KLM (x, y, z) and
T
y
KLM (x, y, z) gives
E
x
klm =
KLM J x
KLM T x
KLM (x, y, z)
σ (a) (x, y, z)
T
x
klm (x, y, z)dxdydz (2.11)
E
y
klm =
KLM J
y
KLM T
y
KLM (x, y, z)
σ (a) (x, y, z)
T
y
klm (x, y, z)dxdydz . (2.12)
We need to rewrite T x
KLM , T
y
KLM , T x
klm , and T
y
klm in terms of functions whose
support is a single flaw cell. Thus, we write
T
x
klm (x, y, z) = R
x
klm (x, y, z) + S
x
k+1lm (x, y, z)
(2.13)
T
y
klm (x, y, z) = R
y
klm (x, y, z) + S
y
kl+1m (x, y, z) ,
(2.14)
with the definitions
R
x
klm (x, y, z) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
x − kδx
δx
for kδx ≤ x ≤ (k + 1)δx,
lδy ≤ y ≤ (l + 1)δy, mδz ≤ z ≤ (m + 1)δz
0
o t h e r w i s e
(2.15)
S
x
klm (x, y, z) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
(k + 1)δx − x
δx
for kδx ≤ x ≤ (k + 1)δx,
lδy ≤ y ≤ (l + 1)δy, mδz ≤ z ≤ (m + 1)δz
0
o t h e r w i s e
(2.16)