Spectral Absorption from Two-view Hyperspectral Images
Kenta Kageyama, Ryo Kawaharaaand Takahiro Okabeb

Department of Artificial Intelligence, Kyushu Institute of Technology,
680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan

Keywords:

Spectral Imaging, Passive Measurement, Spectral Absorption Coefficient, Matrix Factorization.

Abstract:

When light passes through a liquid, its energy is attenuated due to absorption. The attenuation depends bothon the spectral absorption coefficient of a liquid and on the optical path length of light, and is described by theLambert-Beer law. The spectral absorption coefficients of liquids are often unknown in real-world applications
and to be measured/estimated in advance, because they depend not only on liquid media themselves but also
on dissolved materials. In this paper, we propose a method for estimating the spectral absorption coefficientof a liquid only from two-view hyperspectral images of an under-liquid scene taken from the outside of theliquid in a passive and non-contact manner. Specifically, we show that the estimation results in Non-negative
Matrix Factorization (NMF) because both the objective variables and the explanatory variables are all nonnegative, and then study the ambiguity in matrix factorization. We conducted a number of experiments using
real hyperspectral images, and confirmed that our method works well and is useful for reconstructing shape ofan under-liquid scene.

1

INTRODUCTION

When light passes through a liquid, a part of the lightis often absorbed and scattered by the liquid, and thenits energy is attenuated in general. For transparent
liquids with negligible scattering, it is known that theattenuation of light energy due to absorption depends
both on the spectral absorption coefficient of a liquid
and on the optical path length of light in the liquid,
and is described by the Lambert-Beer law (Reinhard
et al., 2008).
The absorption due to liquid is an important clue
to solving computer vision problems; shape recovery of underwater objects (Asano et al., 2016; Murai
et al., 2019; Takatani et al., 2021; Kuo et al., 2021)
and liquid detection on unknown surfaces (Wanget al., 2021; Wang and Okabe, 2021) are achieved under the assumption that the spectral absorption coefficients of liquids are known. Unfortunately, however,the spectral absorption coefficients of liquids are often
unknown in real-world applications and to be measured/estimated in advance, because they depend not
only on liquid media themselves but also on dissolved
materials.
Conventionally, the spectral absorption coeffiabhttps://orcid.org/0000-0002-9819-3634
https://orcid.org/0000-0002-2183-7112

cients of liquids are measured via absorption spectroscopy (Jones and Kao, 1969; Kao and Davies,
1968) in an active and contact manner. Specifically,
when the Spectral Power Distributions (SPDs) of thelight both before and after transmitting a liquid of interest are known, its spectral absorption coefficient isderived from the logarithm of the ratio of those SPDson the basis of the Lambert-Beer law. In the above
computer vision problems with hyperspectral imagesof a scene, however, the SPD of the light before transmitting a liquid, i.e. the spectral radiance on an object
surface under the liquid is unknown.
Accordingly, we propose a method for estimating
the spectral absorption coefficients of liquids in a passive and non-contact manner. Our proposed method
estimates the spectral absorption coefficient of a liquid only from two-view hyperspectral images of anunder-liquid scene taken from the outside of the liquid. We make use of the fact that the absorption coefficient depends on the wavelengths but the optical
path length depends on the scene points. Specifically,
our method estimates the spectral absorption coefficient on the basis of Non-negative Matrix Factorization (NMF) (Berry et al., 2007), because both theobjective variables described by the observed spectral radiance values and the explanatory variables described by the spectral absorption coefficients, optical
path lengths, and Fresnel terms are all non-negative.
715

Kageyama, K., Kawahara, R. and Okabe, T.
Spectral Absorption from Two-view Hyperspectral Images.
DOI: 10.5220/0010917600003124
In Proceedings of the 17th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2022) - Volume 4: VISAPP, pages
715-721
ISBN: 978-989-758-555-5; ISSN: 2184-4321
Copyright c 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

VISAPP 2022 - 17th International Conference on Computer Vision Theory and Applications

Moreover, we study the ambiguity of our proposed
method based on matrix factorization. We show that
our method using only two-view hyperspectral images can estimate the spectral absorption coefficient
up to a scale and an offset. We also show that theestimated spectral absorption coefficient is useful forshape recovery even though it has the ambiguity; weshow that the shape of an object surface under liquid
is recovered up to a scale.
The main contributions of this paper are threefold.
First, we propose a novel method for estimating thespectral absorption coefficient of a liquid in a passive and non-contact manner. The proposed method
achieves the estimation of a spectral absorption coefficient only from two-view hyperspectral images without requiring the SPD of the light before transmitting
a liquid. Second, we show that our method can estimate the spectral absorption coefficient up to a scale
and an offset. In addition, we show that the spectral
absorption coefficient with the ambiguity is useful forunder-liquid shape recovery. Third, we conducted anumber of experiments using real hyperspectral images, and confirmed that our method works well andis useful for reconstructing shape of an under-liquid
scene.

2

RELATED WORK

2.1 Absorption Measurement
Absorption spectroscopy (Jones and Kao, 1969; Kaoand Davies, 1968) is a classical method for measuring the spectral absorption coefficient of a liquidof interest in an active and contact manner. When
the SPDs of the light both before and after transmitting the liquid are known, its spectral absorption
coefficient is derived from the logarithm of the ratio of those SPDs on the basis of the Lambert-Beerlaw (Reinhard et al., 2008). In the computer vision
problems with hyperspectral images of a scene suchas shape recovery (Asano et al., 2016; Murai et al.,2019; Takatani et al., 2021; Kuo et al., 2021) and liquid detection (Wang et al., 2021; Wang and Okabe,2021), however, the SPD of the light before transmitting a liquid, i.e. the spectral radiance on an object
surface under the liquid is unknown. Therefore, we
achieve the estimation of the spectral absorption coefficient of a liquid only from two-view hyperspectral
images without requiring the SPD of the light before
transmitting the liquid.
In the community of computer vision, the threeband (RGB) attenuation coefficient, i.e. the summation of the absorption and scattering coefficients716

of a liquid is often measured or estimated. It is
known that the attenuation coefficient can be estimated from an image of a known calibration target at
known distances (Tsiotsios et al., 2014; Murez et al.,2015; Akkaynak and Treibitz, 2019), but such estimation requires external hardware and distance measurement. The attenuation/absorption coefficient can be
estimated from multiple images of the same object located at different distances; the distances are assumed
to be known (Yamashita et al., 2007), or measured bya sonar (Kaeli et al., 2011), or recovered via structure from motion (Jordt-Sedlazeck and Koch, 2013;
Bryson et al., 2016). In contrast, our proposed method
estimates the spectral absorption coefficient only fromtwo-view hyperspectral images with neither a known
calibration target nor know distances nor geometriccalibration.

2.2 Computer Vision Applications
The absorption due to liquid is an important clue toshape recovery. Asano et al. (Asano et al., 2016)
make use of the fact that water absorbs Near InfraRed(NIR) light (Curcio and Petty, 1951), and show thatthe shape (depth) of an under-water scene can berecovered from two single-view images at different
wavelengths in NIR range. Takatani et al. (Takatani
et al., 2021) extends the above method by using an
event-based camera with temporally modulated illumination, and then achieve robust shape reconstruction in water. Murai et al. (Murai et al., 2019) reconstruct both the surface normals and depth of a dynamic object in water by using multi-directional NIR
lighting. Furthermore, Kuo et al. (Kuo et al., 2021)
achieve shape reconstruction of a dynamic and nonrigid object in water.
The absorption due to liquid is useful also for liquid detection. Wang et al. (Wang et al., 2021) makeuse of the fact that the absorption due to water decreases the apparent spectral reflectance on an object
surface. They achieve per-pixel water detection on
surfaces with unknown reflectance by using the lowdimensional linear model of spectral reflectance fromvisible to NIR wavelengths. Further, Wang and Okabe (Wang and Okabe, 2021) extends their method towater and oil detection on unknown surfaces by simultaneously estimating the types of liquids and optical path lengths.
The above applications assume that the spectral
absorption coefficients of liquids of interest (waterand oils) are known. Unfortunately, however, thespectral absorption coefficients of liquids are often
unknown in real-world applications and to be measured/estimated in advance. The spectral absorption

Spectral Absorption from Two-view Hyperspectral Images

coefficient of the liquid, and the optical path length
in the liquid.
Taking the logarithm of the ratio between i1 (xx, λ)and i2 (xx, λ), we obtainln

f1 (xx)r1 (xx, λ)e−α(λ)l1 (xx)i1 (xx, λ)= lni2 (xx, λ)f2 (xx)r2 (xx, λ)e−α(λ)l2 (xx)= α(λ){l2 (xx) − l1 (xx)} + ln

Figure 1: The illustration of our setup for spectral absorption recovery; our proposed method uses two-view hyperspectral images of an under-liquid scene taken from the outside of the liquid.

coefficients estimated by using our proposed method
in a passive and non-contact manner are effective forthose computer vision applications.

3

PROPOSED METHOD

We propose a method for estimating the spectral absorption coefficient of a liquid of interest in a passive
and non-contact manner. As shown in Figure 1, ourproposed method uses two-view hyperspectral imagesof an under-liquid scene taken from the outside of theliquid. We assume that the liquid is spatially uniform
and transparent with negligible scattering and that theattenuation of light energy due to absorption obeys theLambert-Beer law.
When we observe a point x on an under-liquid object surface from the v-th (v = 1, 2) viewpoint, thespectral radiance iv (xx, λ) of the point x at the wavelength λ seen through the liquid is given byiv (xx, λ) = fv (xx)rv (xx, λ)e−α(λ)lv (xx)(1)

according to the Lambert-Beer law (Reinhard et al.,2008). Here, fv (xx), rv (xx, λ), α(λ), and lv (xx) are theFresnel term 1 , the spectral radiance of the point before transmitting the liquid, the spectral absorption
1 We assume that the Fresnel term is independent of the
wavelength of light. Actually, the Fresnel term depends on
the wavelength via the refractive index, but the refractive indexes of liquids such as water are almost constant for visible
wavelengths.

f1 (xx). (2)f2 (xx)Here, we assume that the under-liquid object surface
obeys the Lambert model, and therefore the spectral radiance is independent of viewpoints: r1 (xx, λ) =r2 (xx, λ).
The above equation holds for the corresponding
point x p (p = 1, 2, 3, ..., P) between the two-view
hyperspectral images with the wavelength λw (w =1, 2, 3, ...,W ). Then, we can rewrite eq.(2) by using
matrices as
S = AB.
(3)
Here, the W × P observation matrix S, the W × 2 matrix A depending on the wavelengths, and the 2 × P
matrix B depending on the scene points are given bys11 s12 · · · s1P
 s21 s22 · · · s2P 
(4)
S=
.. 
..
..
,
 ..
.
.
.
.
sW 1 sW 2 · · · sW P
α1 1
 α2 1 
A=
(5)
.. 
 ..
,
.
.
αW 1
 ′

l1 l2′ · · · lP′B=,(6)f1′ f2′ · · · fP′
where swp = ln{i1 (xx p , λw )/i2 (xx p , λw )}, αw = α(λw ),l ′p = l2 (xx p ) − l1 (xx p ), and f p′ = ln{ f1 (xx p )/ f2 (xx p )}.
When we take the first/second images fromdeep/shallow angles as shown in Figure 1, we can assume that i1 (xx p , λw ) ≥ i2 (xx p , λw ), l2 (xx p ) ≥ l1 (xx p ), andf1 (xx p ) ≥ f2 (xx2 ). Then, all the elements in the matrices S , A , and B in eq.(3) are non-negative. Therefore,
the estimation of the spectral absorption coefficientresults in NMF: factorizing the non-negative observation matrix S into the product of the non-negative
matrices A and B . Hence, our method can estimate
not only the spectral absorption coefficient α(λw ) but
also the difference of the optical path lengths l2 (xx p ) −l1 (xx p ) and the ratio of the Fresnel term f1 (xx p )/ f2 (xx p )as byproducts.
In our current implementation, we factorize theobservation matrix via alternative least squares (Berry717

VISAPP 2022 - 17th International Conference on Computer Vision Theory and Applications

et al., 2007). Specifically, we minimize the sum ofsquares(7)∑(swp − αwl ′p − f p′ )2w,pwith respect to the non-negative unknown variable
αw , l ′p , and f p′ . We give random initial values for{l ′p , f p′ }, and then iteratively fix one set of variables({l ′p , f p′ } or αw ) and update the other set of variables
via least squares and vice versa.

4

AMBIGUITY ANALYSIS

4.1 Ambiguity in Matrix Factorization
In general, matrix factorization has ambiguity. Since
our proposed method results in the factorization of the
W × P observation matrix S into the W × 2 matrix A
and the 2 × P matrix B , the ambiguity is represented
by using a 2 × 2 arbitrary invertible matrix C as
AC )(CC −1 B ).
S = A B = ACC −1 B = (A(8)

A, B } and the matrices
In other words, the matrices {AAC ), (CC −1 B )} yield the same matrix S .
{(A
Then, from eq.(5), the relationship between the
spectral absorption coefficient estimated by our proposed method α̂w = α̂(λw ) and its ground truth αw =α(λw ) is given by




α1 1
α̂1 1


 α2 1 c11 c12
 α̂2 1
 =  .
 .
.
.
 .
 .
..  c21 c22
.. 
.
.
α̂W

1

αW 1

c11 α1 + c21
 c11 α2 + c21
= 
..

.

c11 αW + c21

c12 α1 + c22
c12 α2 + c22  .(9)..
.

c12 αW + c22

Here, ci j is the element of the matrix C at the i-th row
and j-th column.
We can derive that c12 = 0 and c22 = 1 from theabove equation, because the spectral absorption coefficients are not constant with respect to wavelengths
in general. Therefore, the ambiguity of our proposed
method is represented by the 2 × 2 invertible matrix C
defined by


c11 0
.
(10)
C=
c21 1
Thus, our method can estimate the spectral absorption coefficient up to an unknown scale c11 and an
unknown offset c21 :α̂(λ) = c11 α(λ) + c21 .
718

(11)

4.2 Ambiguity in Shape Recovery
Asano et al. (Asano et al., 2016) show that the shape(depth) of an under-water scene can be recoveredfrom two single-view images at different wavelengths
λ1 and λ2 . Specifically, the depth l(xx ) of a surface
point x is described as1i(xx , λ1 )l(xx) =ln,
(12)2{α(λ2 ) − α(λ1 )} i(xx , λ2 )where i(xx, λ1 ) and i(xx, λ2 ) are the spectral radiancevalues seen through water with the two wavelengths.
Therefore, the depth of an under-liquid (under-waterin this case) scene can be recovered if the spectral absorption coefficient of the liquid is known.
We show that the spectral absorption coefficientestimated by our proposed method can be used for reconstructing the shape of an under-liquid scene. Replacing the ground truth spectral absorption coefficient in eq.(12) with the estimated one in eq.(11), theˆ x) is given by
estimated depth l(x1i(xx, λ1 )ˆ x) =l(xln
2{α̂(λ2 ) − α̂(λ1 )} i(xx, λ2 )1l(xx).
(13)=
c11
This is because the offset c21 in the denominator iscanceled out due to subtraction. Hence, the shape ofan under-liquid scene can be recovered up to an unknown scale.

5

EXPERIMENTS

5.1 Setup
To confirm the effectiveness of our proposed method,
we conducted a number of experiments using real
hyperspectral images. We tested three scenes in an
acrylic tank: (A) a textured board in dilute methylene blue, (B) gravel in dilute methylene blue, and (C)gravel in dilute soy sauce. We captured those scenes
from the outside of the tank as shown in Figure 2. We
used a halogen lamp and a hyperspectral camera fromEBA Japan that can capture the range of near UV (380nm) to near IR (1,000 nm) with the interval of 5 nm,
i.e. 125 spectral channels in total.

5.2 Spectral Absorption CoefficientFirst, we studied the performance of our proposed
method by comparing the estimated spectral absorption coefficients with their ground truths. We measured the ground truth of the spectral absorption coefficient of a liquid from the hyperspectral images of a

Hyperspectral camera
Light source

Absorption CoefficientSpectral Absorption from Two-view Hyperspectral Images

1
0.8
0.6
0.4
0.2
0
500

600
700
800
Wavelength
ground truth
our method with ambiguity
our method w/o ambiguity

Liquid900

Under-liquid scene

Figure 2: Our setup for spectral absorption recovery; we
captured an under-liquid scene in an acrylic tank from theoutside by using a hyperspectral camera.

Absorption Coefficient

(a)
1
0.8
0.6
0.4
0.2
0
500

600

700

800

900

Wavelength
ground truth
our method with ambiguity
our method w/o ambiguity

Figure 3: The two-view images of (B) the gravel in dilute
methylene blue at 530 nm and the corresponding points between them.

target object seen through the liquid with known andvariable depths.
As described in Section 3, our proposed method
requires the correspondence between two hyperspectral images taken from different viewpoints. In ourcurrent implementation, we used MSER (Maximally
Stable Extremal Regions) features (Matas et al., 2004)
in MATLAB in order to achieve the correspondence
between those images. Figure 3 shows the two-view
images of (B) the gravel in dilute methylene blue at
530 nm and the corresponding points between them.
In general, the minimization of the sum of squares in
eq.(7) depends on the random initial values. In ourcurrent implementation, we tested 100 sets of random
initial values, and found out the best solution with theminimum sum of squares.
Figure 4 shows the spectral absorption coefficients; (a), (b), and (c) show the results of the threescenes (A), (B), and (C). The black solid line, thered dashed line, and the blue dotted line stand for the
ground truth 2 , the estimated one with the ambiguity,
2 We normalize the ground truth of a spectral absorption
coefficient so that its maximum value with respect to wavelength is equal to 1.

Absorption Coefficient

(b)
1
0.8
0.6
0.4
0.2
0
500

600

700

800

900

Wavelength
ground truth
our method with ambiguity
our method w/o ambiguity(c)
Figure 4: The spectral absorption coefficients; (a), (b), and(c) show the results of the three scenes (A), (B), and (C).
The black solid line, the red dashed line, and the blue dottedline stand for the ground truth, the estimated one with andwithout the ambiguity.

and the estimated one without the ambiguity, i.e. withthe optimal scale and offset respectively. Since our
proposed method can estimate the spectral absorption
coefficient up to a scale and an offset, we supposedthat the ground truth is known and computed the optimal ones via least squares for comparison. The rangeof the wavelength within which the spectral absorption coefficient is estimated is different; from 405 nm
to 940 nm for the dilute methylene blue and from 450
nm to 940 nm for the dilute soy sauce. This is because

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VISAPP 2022 - 17th International Conference on Computer Vision Theory and Applications

5.3 Application to Shape Recovery

(a)

(b)
Figure 5: The pseudo color images (left) and the reconstructed depth maps (right) within the yellow boxes; (a) and(b) show the results of the two scenes (A) and (B).

we could not estimate the spectral absorption coefficient at some wavelengths where the observed spectral radiance values are too small due to weak lightsource and/or strong absorption.
Comparing the ground truths and the estimatedabsorption coefficients with the ambiguity in Figure 4,
we can see that the estimated ones capture the properties of the liquids; the dilute methylene blue absorbs
red and green wavelengths and looks bluish 3 , and the
dilute soy sauce absorbs blue and green wavelengths
and looks reddish. Since both the liquids are diluted
by water, we can see the absorption due to water in
NIR wavelengths. Then, we can see that the estimated absorption coefficients with the optimal scaleand offset are almost the same as the ground truths.
The RMS (Root-Mean-Square) errors of the spectral
absorption coefficients are 0.108, 0.108, and 0.070
for (a) the first dilute methylene blue, (b) the second
dilute methylene blue, and (c) the dilute soy sauce
respectively. Those results show that our method
can accurately estimate the spectral absorption coefficients up to a scale and an offset.

3 The spectral absorption coefficients of the methylene
blue in (a) and (b) are different because their concentrations
are different.

720

Second, we studied the effectiveness of the estimatedspectral absorption coefficient with the ambiguity forunder-liquid shape recovery. According to Asano etal. (Asano et al., 2016), we placed the hyperspectral camera and the light source at almost the same
location. In addition, their method assumes that, at
two wavelengths λ1 and λ2 in eq.(13), the spectral reflectances are almost the same, but the spectral absorption coefficients are significantly different. It is
known that the spectral reflectances of most materials are almost constant for NIR wavelengths (Choeet al., 2016). Therefore, we chose the two wavelengths λ1 = 825 nm and λ2 = 900 nm where thespectral absorption coefficients are significantly different as shown in Figure 4.
Figure 5 shows the pseudo color images (left) andthe reconstructed depth maps (right) within the yellow boxes; (a) and (b) show the results of the twoscenes (A) and (B). Here, the color bar shows the relationship between the relative depth and the color fromshallow (yellow) to deep (blue). The white pixels in
those depth maps stand for the pixels where the reflectance is low and then the radiance values are too
small to estimate the depth. We can see that the reconstructed depth maps qualitatively show the effectiveness of the estimated spectral absorption coefficientwith the ambiguity; the depth of the textured board
increases linearly from the top to the bottom, and thedepth of gravel gradually increases from the top left
to the bottom right in the images.

6

CONCLUSION AND FUTURE
WORKIn this paper, we proposed a novel method for estimating the spectral absorption coefficient of a liquid only from two-view hyperspectral images of anunder-liquid scene taken from the outside of the liquid in a passive and non-contact manner. Specifically,
we showed that the estimation results in NMF, andthen studied the ambiguity in matrix factorization. We
conducted a number of experiments using real hyperspectral images, and confirmed that our method
works well and is useful for reconstructing shape ofan under-liquid scene.
Our future work includes the extension to scattering medium: the estimation of absorption and scattering coefficients. The integration of spectral imaging with polarimetric imaging (Schechner and Karpel,
2004) and the use of the prior knowledge with respect to attenuation/absorption coefficients (Akkay-

Spectral Absorption from Two-view Hyperspectral Images

nak et al., 2017) are other directions of our future study. In addition, the integration of camerabased spectral imaging with illumination-based spectral imaging (Kitahara et al., 2015; Kobayashi andOkabe, 2016; Wang and Okabe, 2017; Torii et al.,2019; Koyamatsu et al., 2019) is an interesting direction to be addressed.

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI Grant
Number JP20H00612.

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