Depth estimation of the absorbing structure in a slab turbid medium using point spread function

g the cap-
tured image at one of the wavelengths with the PSF
calculated by using the absorption and scattering co-
efficients from anotherwavelength. We can obtain the
correct depth that gives the minimum in terms Root-
Mean-Square (RMS) difference between the two con-
voluted images from Eq. (7) and Eq. (8). Figure 4
shows an illustration of the process using Eq. (7) and
Eq. (8).
Using a single-wavelength observed image
The depth estimation technique mentioned above us-
ing two-wavelength perspective images based on the
wavelength dependence of the optical characteristics
in biological tissue. However, due to the movement
of the animal or human, the limitation of the imag-
ing system, and the light illuminated condition, the
proposed technique as mention above may have the
limitation. To overcome these problems, we also
propose another technique to estimate the depth in-
formation of the light-absorbing structure from the
single-wavelength observed image. It will enlarge our
estimation depth technique andmake it bemore prac-
tical.
As mentioned in section Using two-wavelength per-
spective images, with a PSF calculated from Eq. (3)
at a specific depth d, we cannot restore the whole ob-
served image of the complex light-absorbing struc-
ture. We propose a novel depth estimation technique
from a single-wavelength observed image by follow-
ing steps below:
Step 1: de-convolute the observed transillumination
image using the PSF calculated from Eq. (3) with a
specific depth di, (di varies from 0.1 mm to dmax):
habsi = y
 xi; (9)
where
 denotes the deconvolution operation.
Step 2: calculate the yi by convoluting habsi obtained
fromEq. (9)with the PSF calculated fromEq. (3)with
a specific depth di, (di varies from 0.1 mm to dmax):
yi = habsi xi; (10)
where denotes the convolution operation.
Step 3: compare in terms the correlation coefficient
between the new convoluted image yi and the ob-
served image y.
Figure 5 shows the illustration the step 1 of the depth
estimation technique using a single-wavelength tran-
sillumination image. The deconvoluted image ob-
tained from Eq. (9), as shown in Figure 5, will be con-
voluted with the PSF calculated from Eq. (3) with a
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Figure 4: Illustration of the depth estimation technique using two-wavelength perspective images.
Figure 5: Illustration the step 1 of the depth estimation technique using single-wavelength transillumination
image.
specific depth di, (di varies from 0.1 mm to dmax). The
process of step 2 was illustrated as shown in Figure 6.
The estimated depth dest is the specific depth di where
the correlation coefficient got maximum value.
The technique mentioned in section Using two-
wavelength perspective images can combine with
the technique mentioned in section Using a single-
wavelength observed image to verify and improve the
estimated result.
VALIDATION IN SIMULATION
To verify the proposed techniques, the validation of
the proposed techniques was conducted in simula-
tion.
Light-absorbing object
Figure 7 shows the light-absorbing squared metal
plate placed in the transparent medium. A black-
painted squared plate (10.00  10.00 mm) was simu-
lated as an absorbing-structure and to be placed at the
depth dt = 5.00mm from the observation surface. The
observation surface of the diffusedmedium is consid-
ered as a squared surface (100.00 100.00 mm).
The simulation conditions were made as close as pos-
sible to the experimental conditions as shown in Fig-
ure 2. The simulated images were obtained by con-
volving the original light-absorbing structure image
as in Figure 7 with the corresponding PSF calculated
from Eq. (3). The depth of the absorbing-structure
varies from 1.00 to 10.00 mm with the interval 1.00
mm.
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Figure 6: Illustration the step 2 of the depth estimation technique using single-wavelength transillumination
image.
Figure 7: Image of the absorber obtained with
transparent medium
Using two-wavelength perspective images
We simulated two models: the same scattering coeffi-
cients and the same absorption coefficients.
Model A: same scattering coefficient: m 0s(l1) =
m 0s(l2) = 1:00 =mm;ma(l1) = 0:01 =mm; and
ma(l2) = 0:10 =mm:
Figure 8 shows theRMSdifference formodelA at each
depth.
Figure 8: Estimation depth of the absorbing-
structure in model A. The RMS difference for model
with the same scattering condition. The true depth
is 5.00 mm.
Model B: same absorption coefficient: ma(l1) =
ma(l2) = 0:005 =mm;m
0
s(l1) = 0:85 =mm;m
0
s(l2) =
1:15 =mm:
Figure 9 shows the RMSdifference formodel B at each
depth.
As shown in Figure 8 and Figure 9, the difference in
terms of RMS became the minimum at the given cor-
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Figure 9: Estimation depth of the absorbing-
structure in model B. The RMS difference for model
with the same absorption condition. The true depth
is 5.00 mm.
rect depth.
Using a single-wavelength observed image
The model was made with the optical properties
as following conditions: the scattering coefficient
m 0s = 1:00 =mm and the absorption coefficient ma =
0:01 =mm. The correct depth was given with dt = 5.00
mm.
Figure 10 shows the correlation coefficient calculated
at each depth. As shown in Figure 10, at the true
depth, the correlation coefficient became the maxi-
mum.
Figure 10: Estimation depth of the absorbing-
structure using a single-wavelength observed im-
age. The correct depth is 5.00 mm.
The results suggest that the depth estimation
technique mentioned above in section Using two-
wavelength perspective images and in section Using
a single-wavelength observed image is valid. The
results validated the feasibility of the proposed
technique.
VALIDATION IN EXPERIMENTWITH
TISSUE-EQUIVALENT PHANTOM
This section presents the validation of the proposed
techniques in the experiment. Figure 11 shows the
experimental setup for the experiment with a black
squared plate in the tissue-equivalent phantom.
For easy changing the position of the absorbing-
structure in the medium, the container, as shown in
Figure 11, was filled by the Intralipid suspension (Fre-
senius Kabi AG), distilled water, and black ink (INK-
30-B; Pilot Corp.).
The black squaredmetal plate (10.00 mm 10.00 mm
 1.00mm) was used as the light-absorbing structure.
The depth was changed from 4.00 mm to 6.00 mm.
The wavelength of the NIR light source was selected
at 800 nm. For making the different conditions, the
Intralipid suspension or the black ink was added for
changing the optical properties of the medium.
Using two-wavelength perspective image
The experiments were conducted as same conditions
with these models mentioned in section Using two-
wavelength perspective images. Model C: optical
properties of scattering medium 1 and 2 were made
as:
+ medium 1: m 0s(1) = 1:00 =mm and ma(1) =
0:01 =mm
+ medium 2: m 0s(2) = 1:00 =mm and ma(2) =
0:10 =mm:
TheRMS difference was calculated at each depth. The
result was shown in Figure 12 andFigure 13. Figure 12
shows the calculated result in terms of RMS with the
correct depth is 4.00 mm.
Figure 12: Estimation depth of the structure in
model C. The correct depth is 4.00 mm.
Figure 13 shows the calculated result in terms of RMS
with the correct depth is 6.00 mm.
As shown in Figure 12 and Figure 13, the difference
in terms of RMS became the minimum at the given
correct depth.
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Figure 11: Experimental setup for the experiment with a black squaredmetal plate in the tissue-equivalent phan-
tom.
Figure 13: Estimation depth of the structure in
model C. The correct depth is 6.00 mm.
Model D: optical properties of scattering medium 1
and 2 were made as:
+ medium 1: m 0s(1) = 0:80 =mm and ma(1) =
0:05 =mm
+ medium 2: m 0s(2) = 1:00 =mm and ma(2) =
0:01 =mm:
Figure 14 and Figure 15 are respectively show the re-
sult with the given correct depth is 4.00mm, 6.00mm.
Figure 15 shows the result with the true depth is 6.00
mm.
As shown in Figure 14 and Figure 15, the difference in
terms of RMS became the minimum at the given cor-
rect depth. The estimation depth is matched exactly
with the given depth.
Figure 14: Estimation depth of the structure in
model D. The correct depth is 4.00 mm.
Figure 15: Estimation depth of the structure in
model D. The correct depth is 6.00 mm.
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Using a single-wavelength observed image
The experiment was conducted as same conditions
with the model mentioned in section Using a single-
wavelength observed image. The optical proper-
ties of scattering medium were made as: m 0s(1) =
1:00 =mm and ma(1) = 0:01 =mm. The absorbing-
structure was placed at the depth dt = 5.00 mm from
the observation surface.
Figure 16: Estimation depth of the absorbing-
structure using a single-wavelength observed im-
age. The correct depth is 7.00 mm.
Figure 16 shows the correlation coefficient calculated
at each depth. As shown in Figure 16, the correlation
coefficient became the maximum at the given correct
depth.
The proposed technique was validated and examined
in experiment with a tissue-equivalent phantom. The
results shown that the proposed techniques in section
Using two-wavelength perspective image and section
Using a single-wavelength observed image can be ap-
plied in practice with a good condition of an imaging
system.
RESULTS ANDDISCUSSION
As shown in these figures in section Validation in
simulation and section Validation in experiment with
tissue-equivalent phantom, the estimated result is
exactly matched with the given value. The results
confirmed that the proposed depth-estimation tech-
niques can be applied in practice with a good condi-
tion of an imaging system. We could able to estimate
the depth information of the structure even with only
single observed image. We also can combine both
techniquesmentioned in section Estimation depth in-
formation of the structure techniques to obtain better
estimated depth result in practice.
Figure 17 shows the results of image restoration of the
observed image while the absorbing-structure placed
at a true depth of 6.00 mm using these results. For
comparison, the restored images after de-convoluted
with PSF at d = 4.00 mm and 8.00 mm were also
shown. By obtaining the correct estimated depth, it
can be seen that appropriate image restoration has
been performed. Through such analysis, the useful-
ness of the proposed scattering-suppression and esti-
mation depth techniques was confirmed.
CONCLUSION
With the view toward the realization of the transillu-
mination imagingmodality for authentication and di-
agnostic application, we proposed the scattering sup-
pression technique and the novel depth estimation
techniques to restore the clearer image and the “true
shape” of the absorbing-structure.
By observing images with two-wavelength selected at
which the scattering property of the medium is differ-
ent. The transillumination image at one of the wave-
lengths is convolved with the calculated PSF with the
optical properties of another wavelength while chang-
ing the depth. We can obtain the correct depth that
gives the minimum difference between the two con-
voluted images.
By observing the image with a single-wavelength, we
also can estimate the depth of the absorbing-structure.
By combining these techniques, we can obtain an es-
timated depth value in agree with the given depth.
The proposed techniques were examined and vali-
dated in the simulation and also in the experiments
with the tissue-equivalent phantom.
In future works, the proposed techniques will be ex-
amined with the image obtained with biological tis-
sue.
The results in this paper suggested that even with a
limited angle of view, we can obtain the transillumi-
nation image sufficiently clear and information of the
light-absorbing structure. These techniques are useful
for developing the noninvasive imaging of the light-
absorbing structure. It is useful for developing the di-
agnosis modality and the application in medical field.
ACKNOWLEDGMENTS
This presented research was supported by Ho Chi
Minh City University of Technology, VNU-HCM un-
der Grant T-KHUD-2018-83.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
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Figure17: Effectiveness of scattering suppressionusing correctly estimateddepth. The yellowdashed-line square
represents the true size of the absorber.
AUTHORS’ CONTRIBUTIONS
Conceptualization: Trung Nghia Tran, Takeshi Na-
mita, and Koichi Shimizu.
Methodology: Trung Nghia Tran, Kohei Yamamoto,
and Takeshi Namita.
Software: Trung Nghia Tran, Kohei Yamamoto, Ngoc
An Dang Nguyen, and Minh Quang Nguyen.
Validation: Trung Nghia Tran, Kohei Yamamoto,
Ngoc An Dang Nguyen, and To Ni Phan Van.
Data curation: Ngoc An Dang Nguyen, and To Ni
Phan Van
Writing—original draft preparation: Ngoc An Dang
Nguyen, and To Ni Phan Van.
Writing—review and editing: Anh Tu Tran, and
Trung Nghia Tran.
Supervision: Trung Nghia Tran, Takeshi Namita, and
Koichi Shimizu.
All authors have read and agreed to the published ver-
sion of the manuscript.
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