1.4 Example: Raster Scan at Three Frequencies
15
Since we are able to use exact line searches in this algorithm, most of the methods
which we use will always yield a direction in which Φ is decreasing. In fact, since
we normalize the direction vector, the constant in the cubic equation, a 0 , is equal to
the directional derivative of Φ at the point (J k , ρ k ) in the direction
v
u
. Therefore,
a 0 should always be negative. We have found, however, that even in the methods
which theoretically always yield descent directions, we sometimes will obtain a
positive a 0 . Usually this phenomenon occurs when we are ‘close’ to the actual
solution. We suggest that if a 0 ≥ 0, then the algorithm should be restarted by going
to the initialization step, and setting (J 0 , ρ 0 ) = (J k , ρ k ). This is equivalent to using
the steepest descent direction (i.e., letting β k+1 = 0) at Step k + 1.
Now define the new iteration to be
⎛
⎜
⎜
⎜
⎝
J
(x)
k+1
J
(y)
k+1
J
(z)
k+1
ρ k+1
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
J
(x)
k
J
(y)
k
J
(z)
k
ρ k
⎞
⎟
⎟
⎟
⎠
+ α k
⎛
⎜
⎜
⎝
v (x)
v (y)
v (z)
u
⎞
⎟
⎟
⎠ ,
check the residuals for convergence, and either quit or continue with the iteration.
This requires a stopping rule, which could be a variation of what we do now in
VIC-3D ® , or perhaps stopping when the relative error,
Φ(J, ρ)
(J, ρ)
, is less than some
prescribed value.
1.4 Example: Raster Scan at Three Frequencies 2
Consider Fig. 1.1, which shows a 4 × 4 × 4 mm 3 through-wall anomaly and a
T/R-scan system in which the transmitter occupies two positions, (−14, −14)mm
and (14, −14) mm, and the receiver undergoes a two-dimensional raster scan of 11
points, with equal intervals of 0.75 mm in each direction. The anomaly grid consists
of 16 × 16 × 4 = 1024 cells, with the four z-layers numbered 0,1,2,3. The flaw that
is to be reconstructed is shown in Fig. 1.2.
The model coil parameters are given in Table 1.1. It is clear that the transmit
coil is much larger than the grid cell-size, and is quite remote from the anomalous
region. The receive coil scans over the region with dimensions that are comparable
to the cell-size, though this is not a requirement.
The transmitter is excited at three frequencies of 10 2 , 10 4 and 10 5 Hz, which,
with the two-point transmitter scan, produces six ‘experiments,’ in the language of
set-theoretic estimation. Thus, the total impedance data set that is submitted to the
bilinear conjugate-gradient inversion algorithm comprises 6×11×11 = 726 values.
2 See [102–104] for additional examples.
15
Since we are able to use exact line searches in this algorithm, most of the methods
which we use will always yield a direction in which Φ is decreasing. In fact, since
we normalize the direction vector, the constant in the cubic equation, a 0 , is equal to
the directional derivative of Φ at the point (J k , ρ k ) in the direction
v
u
. Therefore,
a 0 should always be negative. We have found, however, that even in the methods
which theoretically always yield descent directions, we sometimes will obtain a
positive a 0 . Usually this phenomenon occurs when we are ‘close’ to the actual
solution. We suggest that if a 0 ≥ 0, then the algorithm should be restarted by going
to the initialization step, and setting (J 0 , ρ 0 ) = (J k , ρ k ). This is equivalent to using
the steepest descent direction (i.e., letting β k+1 = 0) at Step k + 1.
Now define the new iteration to be
⎛
⎜
⎜
⎜
⎝
J
(x)
k+1
J
(y)
k+1
J
(z)
k+1
ρ k+1
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
J
(x)
k
J
(y)
k
J
(z)
k
ρ k
⎞
⎟
⎟
⎟
⎠
+ α k
⎛
⎜
⎜
⎝
v (x)
v (y)
v (z)
u
⎞
⎟
⎟
⎠ ,
check the residuals for convergence, and either quit or continue with the iteration.
This requires a stopping rule, which could be a variation of what we do now in
VIC-3D ® , or perhaps stopping when the relative error,
Φ(J, ρ)
(J, ρ)
, is less than some
prescribed value.
1.4 Example: Raster Scan at Three Frequencies 2
Consider Fig. 1.1, which shows a 4 × 4 × 4 mm 3 through-wall anomaly and a
T/R-scan system in which the transmitter occupies two positions, (−14, −14)mm
and (14, −14) mm, and the receiver undergoes a two-dimensional raster scan of 11
points, with equal intervals of 0.75 mm in each direction. The anomaly grid consists
of 16 × 16 × 4 = 1024 cells, with the four z-layers numbered 0,1,2,3. The flaw that
is to be reconstructed is shown in Fig. 1.2.
The model coil parameters are given in Table 1.1. It is clear that the transmit
coil is much larger than the grid cell-size, and is quite remote from the anomalous
region. The receive coil scans over the region with dimensions that are comparable
to the cell-size, though this is not a requirement.
The transmitter is excited at three frequencies of 10 2 , 10 4 and 10 5 Hz, which,
with the two-point transmitter scan, produces six ‘experiments,’ in the language of
set-theoretic estimation. Thus, the total impedance data set that is submitted to the
bilinear conjugate-gradient inversion algorithm comprises 6×11×11 = 726 values.
2 See [102–104] for additional examples.