=Paper=
{{Paper
|id=Vol-2485/paper11
|storemode=property
|title=Photorealistic Visualization of Fluorescence Materials with Dual Surface Scattering
|pdfUrl=https://ceur-ws.org/Vol-2485/paper11.pdf
|volume=Vol-2485
|authors=Dmitry Zhdanov,Igor Potemin,Andrey Zhdanov,Vadim Sokolov,Sergey Ershov,Evgeniy Denisov
}}
==Photorealistic Visualization of Fluorescence Materials with Dual Surface Scattering==
https://ceur-ws.org/Vol-2485/paper11.pdf
Photorealistic Visualization of Fluorescence Materials with Dual Surface
Scattering
D.D. Zhdanov1, I.S. Potemin1, A.D. Zhdanov1, V.G. Sokolov2, S.V. Ershov2, E.U. Denisov2
ddzhdanov@mail.ru|ipotemin@yandex.ru|adzhdanov@itmo.ru|sokolovv@gmail.com|sergey_65@mail.ru
|eed@spp.keldysh.ru
1
ITMO University, Saint Petersburg, Russia;
2
Keldysh Institute of Applied Mathematics RAS, Moscow, Russia.
We describe a simple method to extract fluorescent characteristics of a surface by combining measurements by a “usual”
gonioreflectormeter GSCM-4 and fluorimeter FP-8600. The fluorescent BDF consists of three components: glossy near-specular peak
which is not fluorescent and white, highly diffuse “passive” part which is also not fluorescent but colored, and fluorescent part. The latter
obviates Kasha’s-Vavilov’s rule (factorization) with good accuracy. The BDFs obtained were used in rendering and shown good visual
match with the natural photographs.
Keywords: fluorescence, fluorescent emission, fluorescence efficiency, Bi-directional Scattering Function (BSDF)
1. Introduction 𝐿(𝒖, 𝜆) = ∫ 𝑓(𝒖, 𝒗; 𝜆, 𝜆′)𝐼(𝜆′)𝑑𝜆′ (1)
Usually light scattered by a surface or a turbid medium where I is also spectral density of irradiance, see [5], [6], [7], [8].
illuminated by a monochrome light has the same wavelength, as Usually besides fluorescence there is also a “passive”
the incident one. This is however not always; the effect when scattering when the light is re-emitted at the same wavelength, so
scattered light has another wavelength is named fluorescence. BDF is
Fluorescence occurs at the molecular level. Roughly 11, an 𝑓(𝒖, 𝒗; 𝜆, 𝜆′ ) = 𝑓 (𝑓) (𝒖, 𝒗; 𝜆, 𝜆′ ) + 𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝛿(𝜆 − 𝜆′ ) (2)
incident photon while interacting with a molecule, kicks it into an where 𝑓 (𝑓) is the pure fluorescent part (continuous in both
excited state and this photon “disappears” instead of being
wavelengths) and 𝑓 (𝑝) is the passive part.
elastically scattered or gone into heat (absorption). There can be
From the quantum nature of the fluorescent effect it follows
several excited energy levels, but all of them are “reachable” for
that frequently at the molecular level the spectrum of emission is
those incident wavelengths that interact with the molecule
independent from the incident wavelength, which is termed
inelastically.
Kasha–Vavilov rule [9], [10]. If so, this will also hold for a bulk
Then the molecule returns to the ground state emitting photon
material, and then the fluorescent component of BDF factors as
whose energy is thus also fixed: it is the difference between the
done in [6], [7]:
ground and the excited energy level. Ideally it means a discrete
spectrum of emission, but in reality because of the thermal motion 𝑓 (𝑓) (𝒖, 𝒗; 𝜆𝑜𝑢𝑡 , 𝜆𝑖𝑛 ) = 𝐸(𝒖, 𝒗; 𝜆𝑜𝑢𝑡 )𝐴(𝒗; 𝜆𝑖𝑛 ) (3)
and other factors, the peaks blur and emission has a continuous Here E is termed emission and A is termed excitation, or,
spectrum. Most frequently, in fluorescence a short-wave light (UV sometimes, quantum yield (for the latter we must also use the
or at least blue) is converted into a visible range; so we can see scale by the ratio of frequencies of the incident and emitted
them under an UV lamp in spite a human eye can not sense UV. photons). Emission spectrum is normalized so that
The bulk material that contain fluorescent molecules can be
∫ 𝐸(𝒖, 𝒗; 𝜆𝑜𝑢𝑡 )𝑑𝜆𝑜𝑢𝑡 = 1 (4)
homogeneous (when it consists of that molecules entirely) or not,
when there is a “passive” material and fluorescent molecules The above factorization is not the general rule and it violates in
dispersed in it. In this latter case it can be a solution or particles of some cases [9], [11].
completely made of fluorescent material.
Fluorescent emission from molecular solution is rather 2. Acquisition of fluorescent BDF
isotropic. But in case of particles the situation is different. We can
Like a usual BDF, it can be either measured or calculated. The
consider the interior of that particle as uncorrelated random source
latter requires that we know all detailed physical properties of all
(of isotropic light). Its local amplitude is proportional to the local
substances (passive and fluorescent) involved, geometry and
intensity of the incident light, diffracted in the particle. This
distribution of size, position and shape of particles and so on.
already creates some anisotropy, and diffraction of that wave field
Then we simulate light interaction with that material assuming
inside particle adds more. As a result, fluorescent emission from a
parallel monochrome illumination. Usually one must account for
particle can be anisotropic. Nevertheless its angular distribution is
diffraction (see above), and this requires wave optics. Although
quite smooth, without sharp peaks [2], [3].
this way is possible, but it is rare that all the data are known at the
The radiance of a “usual” surface under monochrome parallel
necessary detail.
illumination is calculated from its BDF [4] f as
Or one can measure this BDF, but since it depends on two
𝐿(𝒖, 𝜆) = 𝑓(𝒖, 𝒗; 𝜆)𝐼(𝜆)
wavelengths, of illumination and of observation, it can not be
where I is the spectral density of irradiance, u is direction of
measured on such devices like GSCM-4 used to measure “usual”
observation and v is direction of illumination and 𝜆 is wavelength.
BDFs. In the latter case it is enough to use one monochromator, in
A fluorescent surface can be described by an extension of BDF,
either illumination or observation channel. For a fluorescent BDFs
which now depends on two wavelengths, of illumination and of
we need two, in both channels, see Figure 1.
observation. Now the spectral density of radiance at wavelength 𝜆
In principle it is possible to take a device like GSCM-4 and
is
place additional monochromator in the illumination channel. This
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
device would measure dependence on illumination wavelength, wavelength of illumination, which due to final bandpass of the
observation wavelength, illumination direction and observation filter spreads over some interval near 𝜆𝑖𝑛 .
direction. The authors of [5] just followed that way and assembled
a reduced version of that device which operates in the plane of
light monochromator 1
incidence.
sensor with lenses etc
We did not have a possibility for an optical device
manufacturing and decided to use a ready fluorimeter available on
the market instead. Regrettably most of them do not measure monochromator 2 lamp
angular dependence. with lenses etc
We have access to FP-8600 manufactured by JASCO. Besides sample
it also measures only one combination (𝒖, 𝒗), there is yet another
problem with this device. It does not output the ready-to-use
values of BDF at least for a single illumination/observation
Figure 1. Scheme of the device FP-8600
condition. Its output is in such units that one needs do some
calibration of the device and postprocessing of data to get the
necessary values. Specification of the device does not say it clearly what it
This approach is more accurate and detailed than the one used outputs, but because of presence of the word “intensity” in the
in [6], [7] whose authors used an RGB measuring camera and a output file one can assume it is the spectral density of the power
set of varying polychrome illuminations. The use of a polychrome flow of reflected light, up to a constant scale. That is, the records
illumination instead of a monochrome one only required a more in the output file are
complex processing procedure because mathematically an 𝑅(𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 ) = 𝐶(𝜆𝑜𝑢𝑡 )𝐿𝑜𝑢𝑡 (𝒖, 𝜆𝑜𝑢𝑡 |𝜆𝑖𝑛 )
acquisition of a linear operator requires measuring of its action on where 𝜆𝑜𝑢𝑡 is wavelength set in the 2nd monochromator (for
a sufficient number of different input vectors. Using vectors with observation) and c is that scale factor, which in principle can
only one not zero component (monochrome illumination) is more depend on wavelength. Cross-sections for just three incident
straightforward and no more. But the use of an RGB camera wavelength are shown in Figure 2.
makes it impossible to measure the spectrum of emission.
3. Samples
For this experiment we used two fluorescent samples of thin
paper-like opaque sheets. Although GSCM-4 can not measure
fluorescent BDFs, it still can measure the angular dependence,
though the result is some mixture in wavelengths and also it has
wrong scale (its total reflection may exceed 100%). Although
these measurements were helpful. They shown the angular
distribution of scattered light consists of two parts. One is a sharp
near-specular peak, which comes from reflection of the rather
glossy front surface. It is not fluorescent because fluorescent
emission has a rather smooth angular distribution. The second
component is, on the contrary, close to Lambert.
The gloss peak is nearly not affected by the smooth
fluorescent emission, so its measurement by GSCM-4 is reliable.
We used this part, zeroing the off-specular area.
As to the off-specular part, we assume it is Lambert. As to the Figure 2. Raw output spectrogram of fluorescent “sample1” for
wavelength dependence, it was calculated from measurements by just three wavelengths of illumination: 400, 450 and 500 nm.
FP-8600. Below we shall explain how we did that.
Combining, we have
4. What happens in FP-8600 𝑅(𝜆𝑜𝑢𝑡 ) = 𝐶(𝜆𝑜𝑢𝑡 ) ∫ 𝑓(𝒖, 𝒗; 𝜆𝑜𝑢𝑡 , 𝜆′ )𝐹1 (𝜆′, 𝜆𝑖𝑛 )𝐼(𝜆𝑖𝑛 )𝑑𝜆′
In FP-8600, the sample is illuminated by nearly parallel light Applying decomposition (2),
at 𝜎 = 30° . This light passes the first monochromator which
leaves only a narrow spectral interval. The detector collects light 𝑅(𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 ) = 𝐶(𝜆𝑜𝑢𝑡 )𝐼(𝜆𝑖𝑛 ) ∫ 𝑓 (𝑓) (𝒖, 𝒗; 𝜆𝑜𝑢𝑡 , 𝜆′ )𝐹1 (𝜆′ , 𝜆𝑖𝑛 )𝑑𝜆′
in a narrow cone about observation direction at 𝛾 = 60° and this + 𝐶(𝜆𝑜𝑢𝑡 )𝑓 (𝑝) (𝒖, 𝒗; 𝜆𝑖𝑛 )𝐹1 (𝜆𝑜𝑢𝑡 , 𝜆𝑖𝑛 )𝐼(𝜆𝑖𝑛 )
light passes the second monochromator, see Figure 1. For a good monochromator, 𝐹1 ≠ 0 only in a narrow interval,
Since observation is in the off-specular area and BRDF is
while 𝑓 (𝑓) depends on wavelengths smoothly. So
rather smooth, we can forget angular distribution of illumination
and angular distribution the sensor sensitivity and assume 𝑅(𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 ) = 𝐶(𝜆𝑜𝑢𝑡 )𝐼(𝜆𝑖𝑛 )𝑓 (𝑓) (𝒖, 𝒗; 𝜆𝑜𝑢𝑡 , 𝜆𝑖𝑛 )𝐹̅1 (𝜆𝑖𝑛 )
(5)
illumination is parallel and observation too. + 𝐶(𝜆𝑜𝑢𝑡 )𝑓 (𝑝) (𝒖, 𝒗; 𝜆𝑖𝑛 )𝐹1 (𝜆𝑜𝑢𝑡 , 𝜆𝑖𝑛 )𝐼(𝜆𝑖𝑛 )
Spectral density of radiance of light reflected by the sample in where
direction u at wavelength is therefore
𝐹̅1 (𝜆𝑖𝑛 ) ≡ ∫ 𝐹1 (𝜆′ , 𝜆𝑖𝑛 )𝑑𝜆′
𝐿𝑜𝑢𝑡 (𝒖, 𝜆; 𝜆𝑖𝑛 ) = ∫ 𝑓(𝒖, 𝒗; 𝜆, 𝜆′ )𝐹1 (𝜆′, 𝜆𝑖𝑛 )𝐼(𝜆𝑖𝑛 )𝑑𝜆′
Since 𝐶, 𝐹1 and I are unknown, we need some “calibration” to get
Here 𝜆𝑖𝑛 is wavelength set in the 1st monochromator (for the BDF 𝑓 (𝑝) , 𝑓 (𝑓) from the measurement results. To this end, we
illumination), 𝐹1 is transmission of that monochromator, 𝜆′ is used measurement of a diffuse etalon w/o fluorescence, but with
known BDF1. Its cross-sections for just three incident wavelength 𝜆𝑖𝑛 +𝜖
are shown in Figure 3. So, for the passive etalon the above ∫ 𝑅𝑒 (𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 )𝑑𝜆𝑜𝑢𝑡 = 𝐶(𝜆𝑜𝑢𝑡 )𝑓𝑒 (𝒖, 𝒗; 𝜆𝑖𝑛 )𝐹̅1 (𝜆𝑖𝑛 )𝐼(𝜆𝑖𝑛 )
𝜆𝑖𝑛 −𝜖
equation yields
from what it follows that
𝑅𝑒 (𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 ) = 𝐶(𝜆𝑜𝑢𝑡 )𝑓𝑒 (𝒖, 𝒗; 𝜆𝑖𝑛 )𝐹1 (𝜆𝑜𝑢𝑡 , 𝜆𝑖𝑛 )𝐼 (6) 𝑓𝑒 (𝒖, 𝒗; 𝜆𝑖𝑛 )
𝑓 (𝑝) (𝒖, 𝒗; 𝜆𝑖𝑛 ) ≈ (7)
𝑅̅𝑒 (𝜆𝑖𝑛 )
5. Processing of data while away from diagonal (5) yields
To begin with, one can see that (5) consists of two 𝑅(𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 )
𝑓 (𝑓) (𝒖, 𝒗; 𝜆𝑜𝑢𝑡 , 𝜆𝑖𝑛 ) = , |𝜆𝑜𝑢𝑡 − 𝜆𝑖𝑛 | > 𝜖 (8)
components. The first which comes from passive scattering, is 𝑅̅𝑒 (𝜆𝑖𝑛 )
nearly singular, i.e. it is sharp peak near the diagonal 𝜆𝑜𝑢𝑡 = 𝜆𝑖𝑛 .
The near diagonal values of 𝑓 (𝑓) (𝒖, 𝒗; 𝜆𝑜𝑢𝑡 , 𝜆𝑖𝑛 ) are unknown,
The second is smooth, see Figure 4, and near diagonal the values
but this function is smooth and we can interpolate them.
of the first component are much higher. Similarly, (6) is also a
Spectrograms of the passive part obtained this way are shown
sharp diagonal peak.
in Figure 5.
Figure 3. Raw output spectrogram of etalon sample for just three
wavelengths of illumination: 400, 450 and 500 nm. Figure 5. Spectrograms of the passive part 𝑓 (𝑝) (𝒖, 𝒗; 𝜆𝑖𝑛 )
6. Factorization of the fluorescent BDF
Applying factorization (3) to (8) we have
𝑅(𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 )
𝐴(𝒗; 𝜆𝑖𝑛 ) = ∫ 𝑑𝜆𝑜𝑢𝑡
| 𝜆𝑜𝑢𝑡 −𝜆𝑖𝑛 |>𝜖 𝑅̅𝑒 (𝜆𝑖𝑛 )
𝑅(𝜆𝑜𝑢𝑡 ;𝜆𝑖𝑛 ) (9)
𝑅̅𝑒 (𝜆𝑖𝑛 )
𝐸(𝒖, 𝒗; 𝜆𝑜𝑢𝑡 ) = ∫ 𝑑 𝜆 , |𝜆 − 𝜆𝑖𝑛 | > 𝜖
𝐴(𝒗; 𝜆𝑖𝑛 ) 𝑖𝑛 𝑜𝑢𝑡
Notice that usually emission is zero in the wavelength range
where A > 0, so E is all the same 0 in the near-diagonal area.
Results of this procedure are shown in Figures 6 and 7.
As said in the end of Section 1, this factorization may violate,
so it was not evident whether it can or can not be applied to other
samples. It happened it can, i.e. (3) is satisfied with good
accuracy. The shape of spectral emission graphs for both samples
is the same, so the difference is the visible color (see Figure 8) is
Figure 4. Raw output spectrogram of fluorescent “sample1” after due to, first, excitation, and, second, passive part of BDF.
zeroing the near-diagonal components, shown for just three
wavelengths of illumination: 400, 450 and 500 nm. 7. Putting all together
So, if we integrate over a narrow spectral interval around 𝜆𝑖𝑛 We therefore have all components of BDF. First we clear the
assuming the sensor sensitivity etc. smoothly depend on off-specular value (outside of the cone 10° about the specular
wavelength so 𝐶(𝜆) does not vary much over [𝜆𝑖𝑛 − 𝜖, 𝜆𝑖𝑛 + 𝜖], direction) in the GSCM-4 results. This gives us the gloss peak.
then Second, we take the passive part of the smooth BDF
𝜆𝑖𝑛 +𝜖 component from (7). This gives us BDF for single combination
∫ 𝑅(𝜆𝑜𝑢𝑡 ; 𝜆𝑖𝑛 )𝑑𝜆𝑜𝑢𝑡 ≈ 𝐶(𝜆𝑜𝑢𝑡 )𝑓 (𝑝) (𝒖, 𝒗; 𝜆𝑖𝑛 )𝐹̅1 (𝜆𝑖𝑛 )𝐼(𝜆𝑖𝑛 ) (𝒖, 𝒗) but since (we assumed that) Lambert angular dependence,
𝜆𝑖𝑛 −𝜖 it applies to all of them.
1
Obtained by measurement in GSCM-4
divergence which can be in results of tone mapping, gamma
correction and other post-processing procedures is excluded from
comparison.
Figure 6. Spectrograms of excitation 𝐴(𝒗; 𝜆).
Figure 8. Top to bottom: natural photo, rendering with the
BDF taken “as is” from the GSCM-4 measurements, rendering
with the BDF obtained by our method.
9. Conclusion
As a result of the current research, we found out that the
Figure 7. Spectrograms of emission 𝐸(𝒖, 𝒗; 𝜆). method of fluorescence support can be successfully used in 3D
simulation software. It gives noticeable improvements in color
Third, we take the fluorescent part of the smooth BDF reproduction of simulated objects having fluorescent properties.
component from (8). In fact we even used factorization described The main advantage of the method is its simplicity. Simple
in Section 6, because this decreases various random errors mathematical description based on diffuse reflection allows to use
it in any ray tracing techniques from forward Monte-Carlo ray
because the components E and A are averages over one
tracing up to bidirectional ray tracing technique with combination
wavelength. This gives us BDF for single combination (𝒖, 𝒗) but
of forward and backward ray tracing, using photon maps etc.
since (we assumed that) Lambert angular dependence, it applies to
More significant advantage is simplicity of measuring technique
all of them.
which can be applied for fluorescent materials. It is combination
8. Rendering of measurements of BSDF and usual spectrograms which can be
executed with well-known measuring devices available in market.
Once we know all components of BDF we can use it in
rendering. They were compared with the natural photograph made 10. Appendix. Fluorescent BDF in MCRT
by placing the two samples in the colour evaluation device “Judge
In MCRT, after a ray hits a surface, we first choose at random
II”. Rendering was done for the scene which is the model of that
its new direction (after scattering), and then, knowing the
setup.
direction, the color of the new ray is calculated deterministically
The results are shown in Figure 8. One can see serious visible
because it is a unique function of direction and illumination color.
improvements in color reproduction after the use of the BDF
In many variants of MCRT, rays have constant (unit) energy
obtained with our method.
and absorption is simulated by killing rays at random with
The natural photo of the samples in the Judge II has been
“Russian Roulette”. Since rays scattered by BDF have all unit
obtained by Spectroradiometer Konica Minolta CA-2000. To
energy, their angular density equals (up to a constant scale) the
avoid possible influence of camera software, specific tone
angular density of scattered energy.
mapping, spectral sensitivity of CCD and so on indirect approach
So for a non-fluorescent BDF the probability of ray killing is
was chosen for images preparation. An output in XYZ
chromaticity coordinates measured by Konica Minolta was 𝑃 = 1 − ∫ 𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝐸𝑖𝑛 (𝜆)𝑑𝜆𝑑2 𝒗
converted to format supported by our optical simulation software
the angular density is
with next conversion to RGB images. The same approach was
applied to the results of rendering. So the same technique was ∫ 𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝐸𝑖𝑛 (𝜆)𝑑𝜆
𝑝(𝒗) =
used for transformation of XYZ data to images and any ∫ 𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝐸𝑖𝑛 (𝜆)𝑑𝜆𝑑2 𝒗
and the spectrogram of the scattered ray is a diagonal matrix, until it hit a fluorescent surface
𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝐸𝑖𝑛 (𝜆) a product |𝐶𝐸 〉〈𝐶𝐴 | after that.
𝐸𝑜𝑢𝑡 (𝜆) = In the former case, we need only 𝑁𝜆 elements, in the latter one
∫ 𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝐸𝑖𝑛 (𝜆)𝑑𝜆
where 𝐸𝑖𝑛 (𝜆) and 𝐸𝑜𝑢𝑡 (𝜆) are spectra of the incident and scattered 2𝑁𝜆 elements which is still admissible.
rays.
For fluorescent BDF, the angular density and the probability 11. Acknowledgments
of ray killing are given by the same expressions if substitute The research was partially supported by RFBR grants No. 19-
instead of 𝑓 (𝑝) (𝒖, 𝒗; 𝜆) we substitute the integral over wavelength 01-00435 and 17-01-00363.
of emission:
∫ 𝑓 (𝑓) (𝒖, 𝒗; 𝜆′′, 𝜆)𝑑𝜆′′ 12. References
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∫ 𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝐸𝑖𝑛 (𝜆)𝑑𝜆 of an Opaque Surface. // Applied Optics, vol. 4(7), 1965, p.
𝑝(𝒖) =
∫ 𝑓 (𝑝) (𝒖, 𝒗; 𝜆)𝐸𝑖𝑛 (𝜆)𝑑𝜆𝑑2 𝒖 767-775.
(notice that in BMCRT 𝒖 is the incident and 𝒗 is the scattered ray [5] Matthias B. Hullin, Johannes Hanika, Boris Ajdin, Hans-
directions). Peter Seidel, Jan Kautz, and Hendrik P. A. Lensch. 2010.
But what to do with a fluorescent BDF? Its transformation of Acquisition and analysis of bispectral bidirectional
spectrum is matrix multiplication. So, like in BRT in crystals [12], reflectance and reradiation distribution functions. In ACM
camera ray “color” becomes a matrix. For a passive BDF it is SIGGRAPH 2010 papers (SIGGRAPH '10), Hugues Hoppe
diagonal. When camera ray is scattered by a surface, this matrix (Ed.). ACM, New York, NY, USA, Article 97, 7 pages. DOI:
transforms as https://doi.org/10.1145/1833349.1778834
𝐶̂ ↦ 𝐶̂ 𝑓̂(𝒖, 𝒗) [6] Ying Fu, Antony Lam, Yasuyuki Kobashi, Imari Sato,
where 𝑓̂ is BDF “reradiance” matrix. Notice BDF is multiplied by Takahiro Okabe, and Yoichi Sato. 2014. Reflectance and
the ray “color” from the left. Multiplication by the illumination Fluorescent Spectra Recovery Based on Fluorescent
spectrum is from the right (1), so it first interacts with this surface Chromaticity Invariance under Varying Illumination. In
BDF and after that the color transformation matrix from that Proceedings of the 2014 IEEE Conference on Computer
surface to camera is applied. Vision and Pattern Recognition (CVPR’14). IEEE Computer
In BMCRT, transformation of ray “color” must take into Society, Washington, DC, USA, 2171-2178. DOI:
account the number of rays, i.e. their angular density, and https://doi.org/10.1109/CVPR.2014.278
BMCRT ray color transforms as [7] Zhang, Cherry & Sato, Imari. (2011). Separating reflective
1 and fluorescent components of an image. Proceedings of the
𝐶̂ ↦ 𝐶̂ 𝑓̂(𝒖, 𝒗) IEEE Computer Society Conference on Computer Vision and
𝑝(𝒗|𝒖)
where 𝑝(𝒗|𝒖) is the angular density of scattered ray direction 𝒗 Pattern Recognition. p. 185–192
when before scattering the ray has direction 𝒖. https://doi.org/10.1109/CVPR.2011.5995704
In the not fluorescent case, the density is constructed like this: [8] Wilkie, Alexander & Weidlich, Andrea & Larboulette,
it is proportional to energy (sum over spectrum) brought to the Caroline & Purgathofer, Werner. (2006). A reflectance
camera pixel from given scattering direction, if the scattered ray model for diffuse fluorescent surfaces. P. 321–331.
collects “white” (with constant spectrum) illumination. https://doi.org/10.1145/1174429.1174484
In the polarized case, this can be done as well and gives [9] Kasha’s rule. Wikipedia:
https://en.wikipedia.org/wiki/Kasha%27s_rule.
𝑝(𝒗|𝒖) = 𝑐𝑜𝑛𝑠𝑡 × ∑ 𝐶𝜆,𝜆′′ 𝑓𝜆′′,𝜆′ (𝒖, 𝒗) [10] IUPAC. Compendium of Chemical Terminology, 2nd ed.
𝜆′′ (the "Gold Book"). Compiled by A. D. McNaught and A.
where the scale factor is chosen so that ∫ 𝑝(𝒗|𝒖)𝑑2 𝒗 = 1. Wilkinson. Blackwell Scientific Publications, Oxford (1997).
In presence of fluorescence, RGB simulation is impossible Online version (2019-) created by S. J. Chalk. ISBN 0-
and we must operate spectral domain. Even if just 40 wavelength 9678550-9-8 https://doi.org/10.1351/goldbook
are used to have wavelength interval 10 nm (and frequently it is [11] del Valle, Juan Carlos and Catalán, Javier. Kasha’s rule: a
insufficient!), camera ray “color” is a 40x40 matrix, i.e. it contains reappraisal. // Phys. Chem. Chem. Phys., vol. 21, no 19,
1600 elements. With such data size, photon maps for camera ray 2019, p. 10061–10069.
become hardly feasible. http://dx.doi.org/10.1039/C9CP00739C.
And here factorization of BDF is a salvation. Indeed, now the [12] Dmitry Zhdanov, Sergey Ershov, Leo Shapiro, Vadim
reradiance matrix is a product of two spectrograms, emission E Sokolov, Alexey Voloboy, Vladimir Galaktionov, Igor
and excitation A, which in Dirac’s notations is written as |𝐸〉〈𝐴|. Potemin. Realistic rendering of scenes with anisotropic
Since for not fluorescent surfaces the reradiation matrix is media // Optical Engineering, 58(8), 082413 (2019)
diagonal, the camera ray color is either https://doi.org/10.1117/1.OE.58.8.082413