Method for implementing depth deconvolution algorithm for enhanced thermal tomography 3D imaging
A computer-implemented method, apparatus, and computer program product implement enhanced thermal tomography three-dimensional (3D) thermal effusivity imaging. Experimental thermal imaging data is acquired. A response function is derived and a convolution formulation is constructed from the experimental thermal imaging data. A deconvolution solution procedure is implemented that includes constructing a matrix solution equation with a damping parameter, and solving the matrix solution equation with a selected number of iterations to construct a plurality of effusivity images. Using the novel depth deconvolution algorithm with experimental data acquired from a one-sided pulsed thermal-imaging system provides greater sensitivity for internal sample features substantially eliminating degradation in depth resolution.
1. A computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging comprising:
acquiring experimental thermal imaging data;
using a data acquisition and control computer, deriving a response function and constructing a convolution formulation using the experimental thermal imaging data;
performing a deconvolution solution process including:
calculating an apparent effusivity function, and
solving a matrix solution equation with a damping parameter with a selected number of iterations to construct a plurality of effusivity images.
2. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 1 wherein acquiring experimental thermal imaging data includes providing an input of heat with a flash lamp onto one surface of a sample.
3. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 2 includes measuring a transient temperature decay on said one surface of the sample.
4. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 1 wherein acquiring experimental thermal imaging data includes utilizing an infrared camera, acquiring a series of thermal images responsive to a pulse of thermal energy heating a first surface of a sample.
5. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 1 wherein deriving a response function and constructing a convolution formulation using the experimental thermal imaging data includes converting measured time variation of surface temperature into a depth distribution of thermal effusivity within a sample.
6. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 1 wherein deriving a response function and constructing a convolution formulation using the experimental thermal imaging data includes identifying thermal effusivity distribution as a function of depth is represented by E(z) and
E
(
z
)
=
{
ρ
ck
,
0
<
z
<
L
0
,
z
>
L
where z represents depth and L represents thickness of the sample, ρ is density, c is specific heat, k is thermal conductivity, z is coordinate in the depth direction, and z=0 is the sample surface receiving heating.
7. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 1 wherein deriving a response function and constructing a convolution formulation using the experimental thermal imaging data includes calculating a function R(z/l) represented by
R
(
z
/
l
)
=
1
z
l
[
1
+
2
∑
n
=
1
∞
exp
(
-
n
2
π
2
z
2
l
2
)
]
where z is coordinate in the depth direction, and z=0 is the sample surface receiving heating, and the function R(z/l) is related to diffusion at an interface at l.
8. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 7 includes constructing a matrix [Dδ i,j +R i,j ] where D is a positive constant defining a damping parameter,
where R i,j =R(z i /l j ),
δ i,j is represented by:
δ
i
,
j
=
{
1
,
if
i
=
j
0
,
if
i
≠
j
δ i,j is the Kronecker delta.
9. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 1 wherein performing a deconvolution solution process includes calculating an apparent effusivity function represented by:
e
a
(
t
)
=
Q
T
(
t
)
π
t
where Q is the pulsed energy absorbed on surface z=0 at t=0, T(t) is the surface temperature that is measured by an infrared detector during a thermal imaging test.
10. The computer-implemented method for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 1 wherein performing a deconvolution solution process includes solving said matrix solution equation represented by
e
a
,
i
=
Dp
i
n
=
∑
j
=
1
M
p
j
n
+
1
[
D
δ
i
,
j
+
R
i
,
j
]
,
i
=
1
,
2
,
3
,
…
M
,
n
=
0
,
1
,
2
,
3
,
…
where thermal effusivity E i depth distribution is identified by
E
i
=
∑
j
=
i
M
p
j
n
+
1
for i, j=1, 2, 3, . . . M uniformly discretized number (1−M) of uniform grids of set size for an entire depth of a sample,
the superscript n is the iteration number,
p j is unknown parameter to be solved, and in the first step of the iteration, p j 0 =0;
D is a positive constant defining a damping parameter,
δ i,j is represented by:
δ
i
,
j
=
{
1
,
if
i
=
j
0
,
if
i
≠
j
δ i,j is the Kronecker delta.
11. An apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging comprising:
a flash lamp applying a pulse of thermal energy heating a first surface of a sample;
an infrared camera acquiring experimental thermal imaging data responsive to said pulse of thermal energy for heating the first surface of the sample;
a data acquisition and control computer, said data acquisition and control computer deriving a response function and constructing a convolution formulation using the experimental thermal imaging data;
said data acquisition and control computer performing a deconvolution solution process including:
calculating an apparent effusivity function, and
solving a matrix solution equation with a damping parameter with a selected number of iterations to construct a plurality of effusivity images.
12. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 wherein said infrared camera acquires a series of thermal images responsive to said pulse of thermal energy heating a first surface of a sample.
13. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 wherein said data acquisition and control computer includes a control program embodied in a machine readable medium and storing a depth deconvolution algorithm; said control program used for controlling said data acquisition and control computer for constructing effusivity images.
14. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 wherein said data acquisition and control computer deriving a response function and constructing a convolution formulation using the experimental thermal imaging data includes said data acquisition and control computer converting measured time variation of surface temperature into a depth distribution of thermal effusivity within a sample.
15. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 wherein said data acquisition and control computer deriving a response function and constructing a convolution formulation using the experimental thermal imaging data includes said data acquisition and control computer identifying thermal effusivity distribution as a function of depth is represented by
E(z) and
E
(
z
)
=
{
ρ
ck
,
0
<
z
<
L
0
,
z
>
L
where z represents depth and L represents thickness of the sample, ρ is density, c is specific heat, k is thermal conductivity, z is coordinate in the depth direction, and z=0 is the sample surface receiving heating.
16. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 wherein said data acquisition and control computer deriving a response function and constructing a convolution formulation using the experimental thermal imaging data includes said data acquisition and control computer calculating a function R(z/l) represented by
R
(
z
/
l
)
=
1
z
l
[
1
+
2
∑
n
=
1
∞
exp
(
-
n
2
π
2
z
2
l
2
)
]
where z is coordinate in the depth direction, and z=0 is the sample surface receiving heating, and the function R(z/l) is related to diffusion at an interface at l.
17. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 16 includes said data acquisition and control computer constructing a matrix [Dδ i,j +R i,j ] where D is a positive constant defining a damping parameter,
where R i,j =R(z i /l j ),
δ i,j is represented by:
δ
i
,
j
=
{
1
,
if
i
=
j
0
,
if
i
≠
j
δ i,j is the Kronecker delta.
18. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 wherein said data acquisition and control computer performing a deconvolution solution process includes said data acquisition and control computer calculating an apparent effusivity function represented by:
e
a
(
t
)
=
Q
T
(
t
)
π
t
where Q is the pulsed energy absorbed on surface z=0 at t=0, T(t) is the surface temperature that is measured by an infrared detector during a thermal imaging test.
19. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 wherein said data acquisition and control computer performing a deconvolution solution process includes said data acquisition and control computer solving said matrix solution equation represented by
e
a
,
i
=
Dp
i
n
=
∑
j
=
1
M
p
j
n
+
1
[
D
δ
i
,
j
+
R
i
,
j
]
,
i
=
1
,
2
,
3
,
…
M
,
n
=
0
,
1
,
2
,
3
,
…
where thermal effusivity E i depth distribution is identified by
E
i
=
∑
j
=
i
M
p
j
n
+
1
for i, j=1, 2, 3, . . . M uniformly discretized number (1−M) of uniform grids of set size for an entire depth of a sample,
the superscript n is the iteration number,
p j is unknown parameter to be solved, and in the first step of the iteration, p j 0 =0;
D is a positive constant defining a damping parameter,
δ i,j is represented by:
δ
i
,
j
=
{
1
,
if
i
=
j
0
,
if
i
≠
j
δ i,j is the Kronecker delta.
20. The apparatus for implementing thermal tomography three-dimensional (3D) thermal effusivity imaging as recited in claim 11 includes said data acquisition and control computer constructing plane and cross-section effusivity images.