AN ABSTRACT OF THE THESIS OF
Sudarshanram Shetty for the degree of Master of Science in Computer Science
presented on December 3, 2010.
Title: Layered Rendering Model for Human Teeth
Abstract approved:
Michael J. Bailey
This study introduces a layered model for rendering human teeth. Human teeth
exhibit complex light interaction due to their layered structure. While the lighting
responses of teeth have been studied in the dental industry for the production of
realistic looking dentures, to our knowledge this is the first study of its type in
computer graphics for the production of realistic renderings. It evaluates the role
played by the two outermost layers enamel and dentin. We approximate the color of
teeth using a set of physical properties for each layer namely scattering, absorption
coefficients, anisotropy and refractive indices. The translucency of enamel results in
subsurface light activity making dentin a major contributor to the final appearance
of tooth. The reflectance profiles extracted using optical properties based on
multipole diffusion approximation are used to the determine diffuse contribution of
teeth. The diffuse along with specular contribution give rise to final tooth color. An
interactive OpenGL based user interface with the listing of the physical properties
considered over red, green and blue color channel provides one with the flexibility
visualize the effects of these properties at interactive rates. The final rendering is
carried out on a graphics processing unit using GLSL shader language.
c
Copyright by Sudarshanram Shetty
December 3, 2010
All Rights Reserved
Layered Rendering Model for Human Teeth
by
Sudarshanram Shetty
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Presented December 3, 2010
Commencement June 2011
Master of Science thesis of Sudarshanram Shetty presented on December 3, 2010.
APPROVED:
Major Professor, representing Computer Science
Director of the School of Electric Engineering and Computer Science
Dean of the Graduate School
I understand that my thesis will become part of the permanent collection of Oregon
State University libraries. My signature below authorizes release of my thesis to
any reader upon request.
Sudarshanram Shetty, Author
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my academic and research advisor,
Dr Mike Bailey, for his encouragement, motivation, exceptional guidance and
support in helping me to complete this research work. I would like to thank the
faculty and my peers here at EECS department, Oregon State University for
providing a wonderful learning and competitive environment for graduate studies.
Finally, I would also like to thank my other committee members Dr Eugene Zhang,
Dr Toshimi Minoura and Dr Adam Branscum for their time and inputs.
TABLE OF CONTENTS
Page
1 Introduction
1
1.1
Brief Anatomy of Human Teeth . . . . . . . . . . . . . . . . . . . . .
1
1.2
Enamel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
1.3
Dentin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
1.4
Simulation in Graphics . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Background
9
2.1
Shade Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Subsurface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 BSSRDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 BRDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
10
10
2.3
Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.4
Dipole Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.5
Human Skin Simulation . . . . . . . . .
2.5.1 Biospec Model . . . . . . . . . . .
2.5.2 Heterogeneous Reflectance Model
2.5.3 GPU Rendering Model . . . . . .
2.5.4 Other Rendering Models . . . . .
17
17
18
19
20
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3 Implementation
22
3.1
Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . .
3.1.2 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . . .
22
22
23
3.2
Multipole Diffusion Approximation . . . . . . . . . . . . . . . . . . .
25
3.3
Rendering Scheme . . . . . . . . .
3.3.1 GUI Sliders . . . . . . . . .
3.3.2 Diffuse Profile Extraction . .
3.3.3 Texture Packing . . . . . . .
3.3.4 Specular Light Computation
3.3.5 Diffuse Light Computation .
28
30
31
33
35
38
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TABLE OF CONTENTS (Continued)
Page
4 Results
40
5 Discussion
42
5.1
Anisotropic Nature of Enamel and Dentin . . . . . . . . . . . . . . .
42
5.2
Need for a Range of Optical Properties . . . . . . . . . . . . . . . . .
42
6 Conclusions
45
6.1
Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.2
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.3
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
Bibliography
48
LIST OF FIGURES
Figure
1.1
Page
Left: Sections of human teeth [Zijp, Rudolf, 2001], Right: Central
incisor showing opaque dentin and translucent enamel [Terry, Geller,
2002]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Enamel surrounding dentin [Zolotarev, Grisimov, 2000]. . . . . . . . .
3
1.3
Enamel transmission spectrum [Brodbelt, O’Brien, 1981]. . . . . . . .
4
1.4
Dentin tubules. (1) Intertubular dentin (2) Peritubular dentin (3) Hydroxyapatite crystals. Here dint and dext denote the internal and
external tubule diameters, respectively. The dashed lines show an 8.5
x 8.5 mm unit cell [Zolotarev, Grisimov, 2000]. . . . . . . . . . . . . .
5
Left: Cross sectional surface of dentin [Kinney, Balooch, 1995], Right:
Dentin tubules [Kienle, Forster, 2002]. . . . . . . . . . . . . . . . . .
6
Absorption coefficient (A) and scattering coefficient (S) over visible
wavelength (400 nm-700 nm) [Bosch, Zijp, 1987]. . . . . . . . . . . .
7
Left: Vita system 3D Master tooth guide [Baltzer, Kaufmann, 2004],
Right: 5-Dimensional tooth shade [Vanini, Mangani, 2001]. . . . . . .
9
Subsurface conditions [Jensen, Marschner, 2001].Left: BRDF, same
entry and exit points, Right: BSSRDF, different entry and exit points.
10
Left: Series of scatter sites along z direction of incident light, Right:
Negative image source with extrapolated boundary [Farrell, Patterson,
1992]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Left: Volumetric point sources [Jensen, Marschner, 2001], Right: Dipole
configuration for semi-infinite geometry [Donner, Jensen, 2005]. . . .
15
Materials rendered using dipole diffusion model [Jensen, Marschner,
2001]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Left: Simulating erythema [Krishnaswamy, Baronski, 2004], Center:
Parameter maps to simulate subsurface effects [Donner, Weyrich, 2008],
Right: GPU simulation [Eon, Luebke, 2007]. . . . . . . . . . . . . . .
17
Translucency of skin for different concentrations of melanin [Krishnaswamy, Baronski, 2004]. . . . . . . . . . . . . . . . . . . . . . . . .
18
1.5
1.6
2.1
2.2
2.3
2.4
2.5
2.6
2.7
LIST OF FIGURES (Continued)
Page
Figure
2.8
Use of absorption maps between layers to simulate conditions [Donner,
Weyrich, 2008]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Weighted Gaussians simulating diffuse color [Eon, Luebke, 2007]. . . .
19
2.10 Blurring effect of weighted Gaussian profiles [Eon, Luebke, 2007]. . .
19
2.11 Face scanning dome [Weyrich, Matusik, 2006]. . . . . . . . . . . . . .
20
2.12 Simulating Caucasian, Asian and African skin types by adjusting skin
parameters [Donner, Jensen, 2006]. . . . . . . . . . . . . . . . . . . .
20
3.1
Decimation-in-time radix-2 algorithm [Jones, Selesnick, 2007]. . . . .
23
3.2
Stages where Fourier analysis is utilized in rendering scheme. . . . . .
25
3.3
Series of dipole arrangement constituting for multipole configuration
[Donner, Jensen, 2005]. . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.4
Forward and backward transmittance and reflectance [Mayorga, 2008].
28
3.5
Different implementation stages in rendering scheme. . . . . . . . . .
29
3.6
GUI sliders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.7
Enamel and dentin radial reflectance and transmittance profiles. . . .
32
3.8
Inter slab forward radial reflectance profiles. . . . . . . . . . . . . . .
33
3.9
Floating point realization of textures in graphics pipeline using the red
color channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.10 Kelemen/Szirmay-Kalos computation model [Eon, Luebke, 2007] . . .
36
3.11 Incisor mesh with specular highlight rendered to an image. . . . . . .
36
3.12 Precomputed Beckmann texture [Eon, Luebke, 2007]. . . . . . . . . .
37
2.9
3.13 Left: Diffuse irradiance, Right: Attenuation texture [Eon, Luebke, 2007]. 38
3.14 Left: Hand drawn dentin distribution map, Center: Diffuse color of
dentin, Right: Diffuse and specular color combined. . . . . . . . . .
38
LIST OF FIGURES (Continued)
Page
Figure
4.1
Images show various shades exhibited by a central incisor for different optical properties.Top Row: Enamel with high scattering and low
absorption giving it whitish blue appearance, Bottom Row: Enamel
with low scattering and high absorption making color of dentin more
prominent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Two dimensional scattering pattern [Kienle, Forster, 2002].Left: Incident beam perpendicular to surface, Right: Incident beam oblique to
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.2
Radial intensity distribution [Kienle, Forster, 2002]. . . . . . . . . . .
43
5.3
Caries lesion seen as a dark spot under laser fluorescence [Mansson,
Bosch, 1987]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Lip and gum shade influencing teeth color perception [Reno, Sunberg,
2000]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
Stretch correction in horizontal and vertical texture space [Eon, Luebke, 2007]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5.1
6.1
6.2
LIST OF TABLES
Table
Page
3.1
Operation count for Direct DFT and Radix-2 FFT operation. . . . .
24
3.2
Multipole profiles with position of real and virtual lights, ratio of refractive indices considered in calculation of extrapolation distance at
top and bottom layer. . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.3
Range of optical properties for enamel. . . . . . . . . . . . . . . . . .
31
3.4
Range of optical properties for dentin. . . . . . . . . . . . . . . . . .
32
3.5
glTexSubImage2D parameters and value. . . . . . . . . . . . . . . . .
34
Chapter 1 – Introduction
1.1 Brief Anatomy of Human Teeth
Human teeth are broadly divided into two sections namely root and crown. Clinical
crown section essentially consists of visible portion of the tooth and is the section
considered in our model while root section makes up for the embedded portion of
tooth. The two outermost layers of crown (figure 1.1) are enamel and dentin which
play a major role in imparting color to the tooth. We focus on these two layers and
neglect the contribution of pulp layer and subsequent layers in the underlying root
section in order to simplify the model.
Figure 1.1: Left: Sections of human teeth [Zijp, Rudolf, 2001], Right: Central incisor
showing opaque dentin and translucent enamel [Terry, Geller, 2002].
The root is covered by calcified structure called cementum. The junction between
enamel and cementum is called cementoenamel junction (CEJ). Similarly the junction
between enamel and dentin in crown is called dentin enamel junction (DEJ). The
2
soft central core made up of connective tissue is called pulp. Teeth is subjected to
variations both and physical and chemical for example acidic content of food taken,
artificial whiteners, high concentration of fluoride in drinking water, and conditions
like caries are some of the contributing factors. The important optical properties
which govern light interaction in teeth and which are part of interactive user interface
in our simulation model are:
• Scattering coefficient (σ s )[mm−1 ] is a measure of distance traveled by light
before it is scattered thereby obstructing it from continuing in its intended path
[Jacques, Prahl, 1998]. High scattering coefficient signifies frequent obstruction
by particles along the intended path hence decreased intensity of traveling beam
or change in direction. It can be defined as product of effective cross sectional
area and volume density of obstructing particle. As a consequence we define
mean free path as a distance traveled between two successive collisions before
it is diverted. It depends on composition of the material and wavelength of
incident light.
• Absorption coefficient (σ a )[mm−1 ] signifies how far the light penetrates
before it is absorbed [Jacques, Prahl, 1998]. It signifies effective cross sectional
area per unit volume. Like scattering coefficient it also depends on composition
of the material and wavelength of incident light.
• Anisotropy factor (g)[dimensionless] is a measure of light traveling in its
intended path after scattering event. It is calculated as the cosine of the average deflection angle. We take net isotropic contribution (1-g) to generate the
required profiles.
• Refractive index (ri)[dimensionless] is the relative speed of light in the
material compared to its speed in vacuum. Both enamel and dentin exhibit different refractive indices. This ri mismatch contributes to inter layer scattering.
It is wavelength dependent but in our simulation we consider it as a constant.
The above mentioned properties are obtained for the red (630 nm-740 nm), green
(520 nm-570 nm) and blue (400 nm-490 nm) spectrum of visible wavelength range.
3
1.2 Enamel
1.2.1 Structure
Enamel is the hardest tissue grown in nature. It envelopes the crown section and
is in direct contact with oral cavity. Sound enamel is translucent. Dental enamel
is primarily made up of ordered array of inorganic HAP (hydroxyapatite) crystals
[Ca10 (PO4 )6 (OH)2 ] arranged in key hole shaped prisms surrounded by a protein, lipid
and water matrix forming the inter prismatic area (figure 1.2).
Figure 1.2: Enamel surrounding dentin [Zolotarev, Grisimov, 2000].
Density of enamel decreases as we move inwards from the surface of the tooth.
It is characterized by weak absorption over the visible wavelength. Its scattering
coefficient decreases with increasing wavelengths. High scattering at lower wavelength
is attributed to dimensions of crystals resulting in Rayleigh scattering. One of the
conditions of enamel which alters its translucency is caries which is characterized as
white lesion. High localized crystallization results in high scattering.
Analysis done in enamel are determination of reflection coefficient of bulk material
with adaptive linear regression [Groenhuis, Bosch, 1983], laser based automated goniometric setup [Fried, Glena, 1995], asymmetry factor using laser and camera setup
[Zijp, Bosch, 1995]. A list of all these properties is also mentioned in [Zijp, Rudolf,
2001].
4
1.2.2 Optical Properties
The crystalline prismatic structure and its composition give rise to translucency of
enamel. Translucency is the relative amount of light transmitted through a material.
Transmission coefficient is a measure of translucency, defined as the relative amount
of light passing through the unit thickness of the material. Detailed analysis of
translucency (figure 1.3) of enamel was carried out by [Brodbelt, O’Brien, 1981].
Figure 1.3: Enamel transmission spectrum [Brodbelt, O’Brien, 1981].
Following were the key aspects of the analysis:
• Total transmittance determined over wavelengths ranging from 400 to 700 nm
(visible spectrum).
• Total transmission is directly proportional to wavelength i.e. human tooth
enamel is more translucent at higher wavelengths.
5
• Dehydrated enamel exhibited lower translucency due to replacement of water
around the enamel prisms by air. The larger difference in relative indices between enamel crystals ( 1.7) and air (1.33) contributes to higher scattering at
the interface thereby decreasing translucency.
• Matches well with scattering analysis which shows higher scattering at lower
wavelengths.
1.3 Dentin
1.3.1 Structure
Figure 1.4: Dentin tubules. (1) Intertubular dentin (2) Peritubular dentin (3) Hydroxyapatite crystals. Here dint and dext denote the internal and external tubule
diameters, respectively. The dashed lines show an 8.5 x 8.5 mm unit cell [Zolotarev,
Grisimov, 2000].
Dentin is composed of three main components namely collagen based organic
matrix, hydroxyapatite crystals and tubules (figure 1.4). In dentine tubules (figure
1.5) are present, extending from the dentin enamel junction to the pulp.
The tube like structure of dentin tubules makes them the primary scattering
component in this layer. The orientation of the tubules with respect to incident
beam gives rise to anisotropic response of teeth [Kienle, Forster, 2002].
6
1.3.2 Optical Properties
The diameter and density of dentin tubules decreases as we moves towards dentin
enamel junction (DEJ). In dentin, tubules are major scatterers. Scattering coefficients
are independent of mineral content in dentin.
Figure 1.5: Left: Cross sectional surface of dentin [Kinney, Balooch, 1995], Right:
Dentin tubules [Kienle, Forster, 2002].
Scattering in dentin is dependent on transmission in enamel. Some of the experiments carried out to determine optical properties of dentin (figure 1.6) are magnification effect of dentin tubules [Kienle, Michels, 2006], anisotropy dependence [Kienle,
Forster, 2002], dentin polarization effects [Zolotarev, Grisimov, 2000] and optical coherence tomography imaging technique [Wang, Milner, 1999]. The composition and
structure of dentin is discussed in detail in [Zolotarev, Grisimov, 2000], [Wang, Milner,
1999] and [Fried, Glena, 1995].
1.4 Simulation in Graphics
From computer graphics standpoint human teeth exhibits various phenomena which
are challenging to represent accurately. Apart from surface and subsurface reflection
and refraction it also exhibits fluorescence [Terry, Geller, 2002], birefringence effects
[Zijp, Rudolf, 2001] and angular dependency. The dimension of enamel crystals are
comparable to lower wavelengths of visible spectrum giving rise to Rayleigh scattering
which is approximated by our lighting model. Hence we see a blue tinge in teeth where
7
dentin is absent.
The environment in which tooth is present also plays a role in appearance like
the composition of saliva, type of food intake which deposits on the surface or percolates through the surface altering the optical properties bringing about a change in
appearance [Weatherell, Robinson, 1974]. The use of Kubelka-Munk model to determine diffuse reflectance is also discussed in detail in [Ragain, Johnston, 2001]. For an
overview of research carried out in field of dentistry to determine color of teeth refer
[Joiner, 2003].
Figure 1.6: Absorption coefficient (A) and scattering coefficient (S) over visible wavelength (400 nm-700 nm) [Bosch, Zijp, 1987].
Our model makes use of thin film of saliva layer above enamel to simulate the
actual conditions in buccal cavity. Scattering, absorption coefficients over red, green
and blue wavelengths (figure 1.6), refractive indices of two layers and anisotropy
factor are the variable optical properties used in calculation of diffuse component. A
8
suitable value of fixed specular and roughness factor are used to approximate specular
component.
9
Chapter 2 – Background
2.1 Shade Guides
Figure 2.1: Left: Vita system 3D Master tooth guide [Baltzer, Kaufmann, 2004],
Right: 5-Dimensional tooth shade [Vanini, Mangani, 2001].
Shade guide (figure 2.1, left) uses three dimensional CIE (International Commission on Illumination) L*a*b color space to identify tooth color. It involves matching
the tab with the adjacent or surrounding dentition to determine the correct shade.
The detailed mechanism of usage is described in [Ahmad, 2000] and [Baltzer, Kaufmann, 2004]. This classification is based on value, chroma and hue.
Various shades are clinically assessed with tooth to determine the best match
based on the three parameters. These matches highly depend on the lighting conditions present during match and may vary over different lighting conditions. Two
shades perfectly match under all illuminants only when their corresponding spectral
reflectance curves match [Ahmad, 2000]. Some advanced shade guides use five dimensions (figure 2.1, right) which add characterization and opalescence in addition
to three existing dimensions to capture tooth shade [Vanini, Mangani, 2001].
10
2.2 Subsurface Scattering
This refers to the scattering activity taking place below the surface due to the translucent nature of the material allowing incident light to be transmitted through the material. It implies incident light travels below the surface gets scattered and may exit
at some other point on the surface or travel deeper into the material.
Figure 2.2: Subsurface conditions [Jensen, Marschner, 2001].Left: BRDF, same entry
and exit points, Right: BSSRDF, different entry and exit points.
2.2.1 BSSRDF
The distribution function (figure 2.2, right) which relates the outgoing radiance to
incoming flux where both entry and exit points may not be same is termed as BSSRDF
(Bidirectional Subsurface Reflectance Distribution Function).
2.2.2 BRDF
BRDF (Bidirectional Reflectance Distribution Function) is a special case of BSSRDF
where entry and exit points are same (figure 2.2, left).
There is an exponential fall off as light travels between two successive collisions.
The effect of reflectance at a point contributing to irradiance at some other point
brings about a blurring effect on overall appearance which is characteristic of all
11
materials exhibiting translucency. Monte Carlo based simulations, photon mapping
with participating media and diffusion approximation based algorithms are some of
the rendering techniques used to simulate subsurface effects.
2.3 Diffusion Equation
Subsurface activity contributes towards diffuse lighting. The following derivation
completely follows the derivation in [Jensen, Marschner, 2001]. The BSSRDF (S)
relates the outgoing radiance, Lo (xo , w
~ o ) at point (xo ) along direction (~
wo ), to the
incident flux Φ(xi , w
~ i ) at point (xi ) along direction (~
wi ).
dLo (xo ,~
wo )=S(xo , w
~ o , xi , w
~ i ) dΦ(xi , w
~ i)
The outgoing radiance is computed by integrating the incident radiance over incoming directions and area (A) for given surface normal (~n)
R R
Lo (xo , w
~ o )= A 2π S(xo , w
~ o , xi , w
~ i ) Li (xi , w
~ i )(~n·~
wi ) dwi dA(xi )
Phase function [Prahl, 1988] refers to the scattering profile of a particle which is
measure of bending of light which depends on dimension of the scattering particle,
angle and wavelength of incident beam. It is generally denoted as p(~
wi , w
~ o ) which
specifies amount of light scattered in direction (~
wo ) coming from (~
wi ) direction. Since
it is usually treated as a distribution function, normalization of integral phase function
for all possible directions yields
R
4π
p(~
wi , w
~ o )=1
The average phase function (g) is assumed to be dependent on the angle of incoming and outgoing direction which is dot product of the two vectors (cosine of the
angle).
R
g= 4π (~
wi ·~
wo )p(~
wi ·~
wo )d~
wi
12
A positive value of g indicates forward scattering,negative value indicates backward scattering and a constant values specifies isotropic scattering. There is an
exponential fall off of incoming radiance (Li ) traveled for distance (s) between two
successive collisions. This is referred as reduced intensity (Lri ).
~ i)
Lri (xi +s~
wi )=e-sσt Li (xi , w
where:
σ t =extinction coefficient=σ s +σ a .
The equation for radiative transfer commonly called as volume rendering equation.
This equation relates radiance along the desired output direction (~
w’ ) to the incident
radiance along direction (~
w) in terms of radiance scattered at collision points and
scattered energy not available along the output direction.
~ w)L(x, w
(∇·~
~ )=-σ t L(x, w
~ )+σ s
R
4π
p(~
w, w
~ ’ ) L(x, w
~ ’ )dw’ +Q(x, w
~)
where:
~
∇=Vector
differential operator.
R
Q(x, w
~ )=First order scattering of reduced intensity= σ s 4π p(~
w’ , w
~ ) Lri (x, w
~ ’ ) dw’.
Taking normalization of phase function and definition of irradiance into account
the above equation reduces to
~ w)L(x, w
(∇·~
~ )=σ a L(x, w
~ )+Q(x, w
~)
Integrating the above equation for all possible directions (~
w) at point (x) yields
our equation-1
~ E(x)=-σ
~
∇·
a φ(x)+Q0 (x)
where:
R
~
E(x)=Vector
irradiance= 4π L(x, w
~) w
~ dw.
R
φ(x)=Scalar irradiance= 4π L(x, w
~ ) dw.
R
th
Q0 (x)=Zero order source term= 4π Q(x, w
~ ) dw.
13
From truncated spherical harmonic expansion we deduce the following relation for
radiance
L(x, w
~ )=
1
3 ~
φ(x)+
E(x)· w
~
4π
4π
Substituting the above equation in radiative transport equation and integrating
over direction (~
w) yields equation-2
’~
~
~
∇φ(x)=-3σ
t E(x)+Q1 (x)
where:
σ s ’ =Reduced scattering coefficient=σ s (1-g).
σ t ’ =Reduced extinction coefficient=σ s ’ +σ a .
R
~ 1 (x)=First order source term= Q(x, w
Q
~) w
~ dw.
4π
~ 1 (x) reduces to
In case when there are no sources or when sources are isotropic Q
zero yielding equation-3
1 ~
~
∇φ(x)
E(x)=3σt’
~
~
E(x)=-D
∇φ(x)
where:
D=Diffusion constant
This equation specifies a gradient of flux flow from region of higher concentration
to a region of lower concentration. From equation-1, equation-2 and equation-3 we
have the diffusion equation
~ Q
~ 1 (x)
D∇2 φ(x)=σ a φ(x)-Q0 (x)+3D∇·
The diffusion equation is a measure of fluctuation of density of flux over time
which in this case is affected by the nature of the medium in which the radiance
propagates, factors like absorption and scattering coefficients which are measures of
obstruction to the gradient flow.
14
2.4 Dipole Diffusion Model
An approximate solution to the diffusion equation as mentioned in [Jensen, Marschner,
2001] is
φ(x)=
Φ e−σtr r(x)
4πD r(x)
where:
Φ=power of point light source.
p
σtr =Effective transport equation= 3σa σt’ .
r(x)=distance to location of the point source.
Figure 2.3: Left: Series of scatter sites along z direction of incident light, Right:
Negative image source with extrapolated boundary [Farrell, Patterson, 1992].
The difference in refractive indices existing at the interface of two media contributes to refractive index dependent reflectance called Fresnel reflectance (Fdr ). An
approximate solution to Fdr is provided in [Donner, Jensen, 2005].
R
Fdr = 2π F(η, ~n · w
~ ’ )(~n · w
~ ’ ) dw’
where:
η=ratio of ri of medium containing reflected ray to the other medium.
15
The diffuse BSSRDF Rd (r) is defined as radiant exitance divided by incident flux.
Rd (r)=-D
~
(~n · ∇φ)(x
s)
dΦi (xi )
Where:
r=absolute distance between entry (xi ) and exit (xs ) points.
[Farrell, Patterson, 1992] proposed placing two volumetric point sources (figure
2.3) called as dipoles in order to satisfy diffusion equation by reducing the net flux to
zero. By placing the positive dipole at one mean free path distance from the boundary
and placing the negative image source at a distance which is sum of extrapolation
distance and mean fee path the desired zero fluence is achieved at the extrapolation
(1 + Fdr )
.
distance. The extrapolation distance is 2AD where A=
(1 − Fdr )
Figure 2.4: Left: Volumetric point sources [Jensen, Marschner, 2001], Right: Dipole
configuration for semi-infinite geometry [Donner, Jensen, 2005].
The diffuse reflectance Rd (r) computed using dipole is
Rd (r)=
α’ zr (σtr dr + 1)e−σtr dr α’ zv (σtr dv + 1)e−σtr dv
4πd3r
4πd3v
Where:
zr =Depth/placement of positive light source.
zv =Depth/placement of negative light source.
p
dr =Distance of positive light source from surface= z2r + r2 .
16
dv =Distance of negative light source from surface=
σs’
’
α =Reduced albedo= ’ .
σt
p
z2v + r2 .
Figure 2.5: Materials rendered using dipole diffusion model [Jensen, Marschner, 2001].
Advantages of dipole diffusion model:
• Relatively minimal computation time required to determine diffuse reflectance.
• Accounts for important physical properties of the material which shape the light
response of the material.
Limitations of dipole diffusion model:
• This model is valid for semi-infinite homogeneous materials (figure 2.5). It does
not cover materials which are made up of more than one layer. It assumes
scattering properties to remain same throughout the material.
• It accounts only for reflectance at a point due to an illuminant, fails to consider
the transmission within the material which affects the overall radiance exitance
at a point. As it does not account for loss due to transmission this model
predicts more energy than it should.
Improvements:
• Enhancement to this model includes extending dipoles to both end of the
medium to compute reflectance for multilayered material. Multipole diffusion
model satisfies this criteria.
17
2.5 Human Skin Simulation
Figure 2.6: Left: Simulating erythema [Krishnaswamy, Baronski, 2004], Center: Parameter maps to simulate subsurface effects [Donner, Weyrich, 2008], Right: GPU
simulation [Eon, Luebke, 2007].
Human skin like teeth is composed of layers. Skin is composed of many layers
exhibiting varying degree of physical properties greatly increasing the complexity of
accurately rendering skin and conditions of the skin. The three primary layers of
skin are epidermis, dermis and hypodermis. Amount of oil, chemical composition of
melanin and hemoglobin are some of the additional factors which influence skin color.
We briefly discuss some of the important rendering models presented on realistic
skin rendering. These models have been referred based on types of scheme employed
for rendering ranging from physically correct full scale Monte Carlo simulation to
approximate GPU based solutions.
2.5.1 Biospec Model
This is a highly elaborate spectral model made up of at least five layers of skin
with detailed optical properties and their respective concentration listed. A Monte
Carlo based rendering technique was employed to simulate skin for interaction at
each layer of skin. The availability of real world skin parameters enables this model
to show different skin conditions (figure 2.7) based on changes to the composition of
skin pigments like color change during erythema and jaundice (figure 2.6, left), in
jaundice higher secretion of bilirubin causes yellow coloration of skin. This model is
18
computationally very expensive but follows the light interaction in skin very closely
making it physically correct.
Figure 2.7: Translucency of skin for different concentrations of melanin [Krishnaswamy, Baronski, 2004].
2.5.2 Heterogeneous Reflectance Model
Figure 2.8: Use of absorption maps between layers to simulate conditions [Donner,
Weyrich, 2008].
This model makes use of parameter maps between layers of the material. These
maps represent distribution and simulate thin layered absorption effects (figure 2.8).
It makes use of sum of Gaussians to approximate reflectance profiles. This model
introduces determining profiles over visible wavelength range involving calculation
at certain wavelengths and then interpolating over other wavelengths. This model
also makes use different optical properties over different regions of the skin making it
possible to generate realistic skin.
19
2.5.3 GPU Rendering Model
Figure 2.9: Weighted Gaussians simulating diffuse color [Eon, Luebke, 2007].
Figure 2.10: Blurring effect of weighted Gaussian profiles [Eon, Luebke, 2007].
This model uses profiles extracted using multipole diffusion model to approximate
weighted sum of Gaussians (figure 2.9). These Gaussians are then applied in order to
get the required blurring effect (figure 2.10). It also introduces the concept of texture
space diffusion which involves carrying out the blurring process on a 2D texture map
instead of a mesh thereby improving efficiency. It also makes use of stretch correction
algorithms in order to overcome inherent limitations of mapped textures. We closely
follow this algorithm in our rendering model. We make use of the algorithm used for
specular color computation and the technique of performing convolution on an image
rather on a mesh.
20
2.5.4 Other Rendering Models
Figure 2.11: Face scanning dome [Weyrich, Matusik, 2006].
Some of the other prominent realistic skin rendering models are spectral BSSRDF model [Donner, Jensen, 2006] and measurement based skin reflectance model
[Weyrich, Matusik, 2006].
Figure 2.12: Simulating Caucasian, Asian and African skin types by adjusting skin
parameters [Donner, Jensen, 2006].
Spectral BSSRDF model makes use of four variables which control amount of
oil, melanin and hemoglobin in the skin which enable it to simulate different skin
types. Multipole diffusion approximation is used to generate diffusion profile while
21
Torrance-Sparrow model is used for specular computation simulating interaction with
oily layer on top of skin. By utilizing the four variables this model is able to generate
Caucasian, Asian and African skin types as shown in (figure 2.12).
Measurement based skin reflectance model uses an elaborate setup to capture
skin reflectance as shown in (figure 2.11). The face dome consists of 150 LED (Light
Emitting Diode) light sources and 16 cameras surrounding the chair. HDR (High
Dynamic Range) images are captured using different exposure settings.
22
Chapter 3 – Implementation
3.1 Fourier Analysis
A given continuous signal can be represented as a sum of sine and cosine functions.
Fourier series is associated with periodic phenomenon while transforms are meant
for non-periodic signals. Transforms analyze a continuous signal into its constituent
parts and inverse transforms synthesize this signal from its constituent parts. A signal
is represented by its amplitude (Ak ), phase (φk ) and frequency (1/t) for N samples.
N
P
Ak sin(2πkt+φk )
k=0
Fourier analysis of reflectance profiles plays a key role in implementation of multipole diffuse model. Multipole diffuse model is used to estimate the subsurface activity
in multilayered materials. Effective inter layer reflectance and transmittance can be
efficiently solved by realizing the respective profiles in fourier notion. In multipole
diffusion model layer profile varies over space representing contribution of light from
a point to its surrounding area.
3.1.1 Discrete Fourier Transform (DFT)
DFT transforms N samples of discrete time signal to N discrete frequency samples
X(k)=
N-1
P
x(n)e−(
n=0
Where:
2πjnk
e−( N ) =Complex exponential basis function.
2πjnk
N
)
23
3.1.2 Fast Fourier Transform (FFT)
Fast Fourier transform (FFT) is an efficient algorithm to compute DFT. Efficiency in
computation achieved by rearranging the terms used in evaluation of discrete samples
is the primary objective of any FFT algorithm. We use decimation-in-time (DIT)
radix-2 FFT (figure 3.1) in our model to obtain discrete samples. We limit the
number of samples in our layer profiles to 256. We follow the implementation steps
as mentioned in [Jones, Selesnick, 2007].
Figure 3.1: Decimation-in-time radix-2 algorithm [Jones, Selesnick, 2007].
We rearrange the DFT equation into two parts, one is sum of even numbered
indices and the other is sum of odd numbered indices.
N
−1
2
X(k)=
P
N
−(
x(2n)e
2πj(2n)k
N
−1
2
) + P x(2n+1)e−( 2πj(2n+1)k
)
N
n=0
N
−1
2
X(k)=
P
n=0
n=0
−
x(2n)e
2πjnk
N
2
+e−j (
2πk
N
N
−1
2
) P x(2n+1)e−
2πjnk
N
2
n=0
X(k)=DFT N [x(0), x(2)....x(N-2)]+WkN DFT N [x(1), x(3)....x(N-1)]
2
2
24
Table 3.1: Operation count for Direct DFT and Radix-2 FFT operation.
Direct DFT
Per X(k)
Total samples
Multiply
N
N
Add
N-1
N
Radix-2 FFT Per X(k) odd even pair Total samples
Multiply
N+2
(N/2)
Add
N
N2
Where: −
WkN =e
2πjnk
N
2
Count
N2
N2 -N
Count
(N2 /2)+N
(N2 /2)
.
X(k) is computed as two N2 length DFT’s of odd and even index terms where
the odd indexed sum is multiplied by a per sample constant factor. It is called
decimation-in-time because samples are rearranged and reused in computation and is
called radix-2 because it is divided into groups of odd and even numbered samples.
Let G[n] denote frequency response of even discrete inputs while H[n] represents
frequency response of odd samples. Each H and G terms are reused across the two
groups with a different per sample constant factor thereby improving efficiency (table
3.1).
Convolution theorem of fourier transform states that convolution (?) in time domain is equals multiplication in frequency domain. This property facilitates in calculation of multilayered profiles like reflectance and transmittance which tend to be
convolved combinations of individual profiles. Let z be the fourier transform then
from this property we have:
z(f?g)=z(f)·z(g).
Inverse discreet fourier transform (IDFT) is calculated using the following equation:
x(n)= N1
N-1
P
k=0
2πjnk
X(k)e( N )
25
Multipole diffusion process in our rendering model evaluates radial intensity at
discrete positions. We use fast fourier transform (FFT) to transform the discrete space
profile to spatial frequency domain. By using fourier transforms we are improving the
efficiency by converting calculations to be done using convolution to multiplication
of the profiles involved (figure 3.2).
Figure 3.2: Stages where Fourier analysis is utilized in rendering scheme.
3.2 Multipole Diffusion Approximation
This technique is used to approximate diffuse color contribution in multilayered materials. It tries to approximate the subsurface contribution of underlying layers based
on their interaction with light. It is an extension to dipole diffusion model which
uses point sources (dipoles) to approximate solution to diffusion equation. Multipole
diffusion technique uses series of dipoles (figure 3.3) for multilayered materials. It is
superior compared to dipole approximation because it takes into account energy loss
due to transmission inside the material. Both reflectance and transmittance process
are considered while evaluating overall reflectance.
For a slab of thickness (d) as in (figure 3.3) the boundary condition is satisfied
when fluence at extrapolation distance (zb ) on top is zero by arranging the point
sources as discussed in dipole model. To achieve net zero fluence at the bottom
(d+zb ) we need to mirror the top dipole about (d+zb ) which in turn needs to satisfy
the dipoles already on top which in turn leads to use of infinite set of dipoles to
satisfy the boundary conditions simultaneously. We use finite set of dipoles which
gives reasonable approximation. Here zr,i and zv,i refer to positive and negative point
26
sources. Here we assume the refractive index at top and bottom of the material under
consideration are same. For cases with refractive index mismatches refer (table 3.2).
Figure 3.3: Series of dipole arrangement constituting for multipole configuration [Donner, Jensen, 2005].
The reflectance profile for (2n+1) dipoles is given by:
α’ zr,i (σtr dr,i + 1)e−σtr dr,i α’ zv,i (σtr dv,i + 1)e−σtr dv,i
4πd3r,i
4πd3v,i
i=-n
Rd (r)=
i=n
P
The transmittance profile is obtained by adjusting for the depth of the slab.
α’ (d − zr,i )(σtr dr,i + 1)e−σtr dr,i α’ (d − zv,i )(σtr dv,i + 1)e−σtr dv,i
4πd3r,i
4πd3v,i
i=-n
Td (r)=
i=n
P
Two reflectance profiles R(r)+ , R(r)− and two transmittance profiles T(r)+ , T(r)−
are calculated per layer, four scattering profiles are evaluated per layer. Here (+)
indicates forward and (-) indicates backward direction of scattering (figure 3.4).
27
Table 3.2: Multipole profiles with position of real and virtual lights, ratio of refractive
indices considered in calculation of extrapolation distance at top and bottom layer.
zr,i
zv,i
η:zb (0) η:zb (d)
R(r)+
2i(d+zb (0)+zb (d))+l
2i(d+zb (0)+zb (d))-l-2zb (0)
Lri
Tri
Bri
Lri
R(r)−
2i(d+zb (d)+zb (0))+l
2i(d+zb (d)+zb (0))-l-2zb (d)
Tri
Lri
Lri
Bri
T(r)+
d-2i(d+zb (0)+zb (d))+l
d-2i(d+zb (0)+zb (d))-l-2zb (0)
Lri
Tri
Bri
Lri
T(r)−
d-2i(d+zb (d)+zb (0))+l d-2i(d+zb (d)+zb (0))-l-2zb (d)
Tri
Lri
Lri
Bri
Where:
Tri =Refractive index of top layer.
Lri =Refractive index of layer under consideration.
Bri =Refractive index of bottom layer.
The effective forward transmittance between two layers with inter slab reflections can
be modeled as convolution of reflectance and transmittance profiles:
+
+
+
+
−
+
+
+
−
+
−
+
T+
12 =T1 ?T2 + T1 ?R2 ?R1 ?T2 + T1 ?R2 ?R1 ?R2 ?R1 ?T2 +...
As per discussion in fourier analysis convolution in space here is multiplication in
frequency domain on evaluating the fourier response of the profiles.
+
=T1+ ·T2+ + T1+ ·R2+ ·R1− ·T2+ + T1+ ·R2+ ·R1− ·R2+ ·R1− ·T2+ +...
T12
+
T12
=T1+ ·T2+ [R2+ ·R1− +R2+ ·R1− ·R2+ ·R1− +...]
We have R2+ (r)·R1− (r) less than 1 for all (r),the geometric series reduces to
28
Figure 3.4: Forward and backward transmittance and reflectance [Mayorga, 2008].
T1+ T2+
1 − (R2+ R1− )
On similar basis we have effective forward inter layer reflectance as:
+
T12
=
+
+ +
R12
=R1+ +T12
R2
3.3 Rendering Scheme
The scheme is divided into CPU (Central Processing Unit) and GPU (Graphics Processing Unit) environments. The various stages are clearly outlined in (figure 3.5).
• GUI (Graphical User Interface) sliders to input diffuse optical properties.
• Extraction of reflectance and transmittance profiles using multipole technique.
• Packing the profiles as floating point textures.
• Computation of specular color.
• Final rendering adding specular and diffuse color.
GUI sliders interface, profile extraction and floating point texture packing are
taken care in CPU environment while specular computation and final rendering is
handled in GPU. Dividing the various stages into these two environments helps us to
utilize the advantages of these environments like performance, ease of implementation
thereby improving overall efficiency.
29
Figure 3.5: Different implementation stages in rendering scheme.
30
3.3.1 GUI Sliders
Figure 3.6: GUI sliders.
We use a range for optical properties (figure 3.6) to extract reflectance profiles.
Variation in optical properties values have been noticed in experiments carried out
mainly due underlying assumptions, organic content and quality of specimen, experimental setup and lighting conditions under which they are tested.
Taking all these discrepancies use of simple GLUI (Graphics Library User Interface) sliders which are GLUT (Graphics Library Utility Toolkit) based C++ user
interface for OpenGL applications. It gives the user the ability to visualize the profiles for a particular combination of optical properties of enamel and dentin over RGB
31
Table 3.3: Range of optical properties for enamel.
Red
Optical properties
Green
Blue
min max min max min max
Absorption coefficient
0.5
1.0
0.5
1.0
0.5
1.5
Scattering coefficient
2.5
3.0
3.0
5.0
5.0
15.0
Anisotropy factor
0.5 to 0.7
Refractive index
1.62
channels.
3.3.2 Diffuse Profile Extraction
We use the range (table 3.3) to simulate the effects of enamel light interaction. The
structure of enamel permits most of the light above 480 nm to be transmitted resulting
in minimal scattering while it scatters light in blue wavelength region heavily as
evident from the high scattering range for wavelength between 400 nm to 490 nm.
The effect of translucency of enamel can be visualized by adjusting the sliders for
scattering and absorption coefficients. The angular dependency is approximated using
anisotropy factor. These properties are then fed to multipole diffusion function which
approximates profiles for reflectance and transmittance over the three color channels.
We use the range (table 3.4) to simulate the effects of dentin light interaction.
The structure of dentin scatters light almost equally for all the three channels with
slight increase as we move towards lower wavelengths (blue color channel). Dentin
32
Table 3.4: Range of optical properties for dentin.
Red
Optical properties
Green
Blue
min max min max min max
Absorption coefficient
0.6
1.0
0.6
1.0
0.6
1.5
Scattering coefficient
6.
10.0
6.0
10.0
6.0
10.0
Anisotropy factor
0.4 to 0.7
Refractive index
1.45
Figure 3.7: Enamel and dentin radial reflectance and transmittance profiles.
33
tubule density plays an important role in overall response which is taken care by
using suitable distribution maps. These properties are then fed to multipole diffusion
function which approximates profiles (figure 3.7, 3.8) for reflectance and transmittance
over the three color channels. We have assumed refractive index of human saliva to
be 1.33 since it is made up of 98% water.
Figure 3.8: Inter slab forward radial reflectance profiles.
3.3.3 Texture Packing
From [Eon, Luebke, 2007] fast approximation of diffusion profiles is implemented by
evaluating the equivalent weighted sum of Gaussian profiles for each profile, once the
coefficients are obtained the convolution operation is realized as a series of Gaussian
blurs for a suitable kernel size. To approximate a weighted (w) sum of k number of
Gaussians to a profile we need to minimize the energy difference between them.
2
k
R∞
P
r R(r) − wi G(vi , r)
0
i=0
Where:
34
Table 3.5: glTexSubImage2D parameters and value.
OpenGL Type
Parameter
Value
GLenum
target
GL TEXTURE 2D
GLint
level
0
GLint
xoffset
0
GLint
yoffset
0
GLsizei
width
16
GLsizei
height
16
GLenum
format
GL LUMINANCE
GLenum
type
GL FLOAT
GLvoid
*data
Pointer to stored profile data
2
 −r 
1
G(vi , r)=Gaussian of variance (v)=
e 2v .
2πv
1
The value of
is chosen so that sum of all gaussians have unit total diffuse
2πv
response (Rd ). The weights can be predicted by restricting their sum to diffuse profile
R(r) calculated.

k
P
i=0
R∞
wi =Rd = 0 2πrR(r)dr
The convolution of Gaussian blurs to simulate inter slab interaction results in another
Gaussian which is the sum of individual variances. So the entire inter slab components
can be simulated as linear combinations of Gaussian blurs.
G(v1 , r)?G(v2 , r)=G(v1 +v2 , r)
35
Figure 3.9: Floating point realization of textures in graphics pipeline using the red
color channel.
Levenberg-Marquardt optimization is used to evaluate the weighted sum of Gaussians. In our model we do not make use of sum of Gaussians as the above optimization
does not converge quickly hence evaluating weighted sum of Gaussians is not possible
in real time.
We directly convolve the packed profiles with the diffuse irradiance to get diffuse
color. From OpenGL implementation standpoint floating point textures (figure 3.9)
are created using the glTexSubImage2D API call (table 3.5).
3.3.4 Specular Light Computation
The specular contribution is evaluated on an upper left incisor mesh model. We follow
the Kelemen/Szirmay-Kalos (KS) specular model computation used in [Eon, Luebke,
2007]. It is an approximation of Cook Torrance model.
The input to the KS model (figure 3.10) is roughness factor (m) and a specular
brightness factor. The KS model utilizes Schlick Fresnel approximation and Beckmann distribution function in order to compute the specular contribution (figure
3.11). For detailed analysis of this model refer [Kelemen, Kalos, 2001].
The Schlick Fresnel technique is an approximation for BRDF involving Fresnel
36
Figure 3.10: Kelemen/Szirmay-Kalos computation model [Eon, Luebke, 2007]
Figure 3.11: Incisor mesh with specular highlight rendered to an image.
37
term in specular computation given by the following equation
R(θ)=Ro +(1-Ro )(1-cos(θ))
Where:
θ=Angle of incidence.
Ro =Reflection coefficient at normal incidence.
For efficiency purpose the Beckmann distribution (figure 3.12) is evaluated over
possible ranges and stored in a 2D texture for computation.
Figure 3.12: Precomputed Beckmann texture [Eon, Luebke, 2007].
The Beckmann distribution function is a physically based microfacets model. Microfacets are minute surfaces which contribute to glossiness, considered around the
point where light calculation is done and is given by the following equation
− tan2 (α)


m2
e


πm2 α cos4 (α)
Where:
α=Dot product of normal and halfway vector.
m=Root mean square slope of surface microfacets=surface roughness.
38
3.3.5 Diffuse Light Computation
In order to balance energy conservation between specular and diffuse contribution
the light available for diffuse activity is approximated using precomputed attenuation
texture. This texture stores KS model evaluated over all possible angles and roughness
factor combinations to approximate an integral contribution.
Figure 3.13: Left: Diffuse irradiance, Right: Attenuation texture [Eon, Luebke, 2007].
Figure 3.14: Left: Hand drawn dentin distribution map, Center: Diffuse color of
dentin, Right: Diffuse and specular color combined.
The energy available for diffuse lighting is the energy left over after subtracting the
attenuation factor in precomputed texture. Once evaluated we render the image to
39
a texture to facilitate image based diffuse color computation. The product of diffuse
irradiance (figure 3.13, left), intensity distribution (figure 3.14, left) is convolved
with the profiles with a suitable kernel size over all the three color channels. The
distribution of dentin decreases as we move towards the periphery from the center
also as we move towards dentin enamel junction which can be characterized as grey
scale variation to simulate depth distribution. We modulate the overall contribution
with a diffuse attenuation factor to adjust the overall color. On changing the grey scale
intensity dentin of dentin distribution different pattern of diffuse shade is obtained
provided we maintain the relation of decreasing the density as we move towards the
periphery.
40
Chapter 4 – Results
The Rayleigh effect in enamel due to minute crystals results in bluish tinge where
dentin distribution is very minimal mostly along the periphery of tooth as seen in
(figure 4.1), blue wavelength is highly scattered compared to red and green. As we
increase absorption coefficient in enamel the blue scattering decreases considerably
making diffuse color of dentin dominant. The above mentioned effects are successfully
simulated by our model at interactive rates. In younger people, teeth tend to be whiter
due to high scattering and low absorption. With age, teeth tend to be yellowish due to
wear and tear of enamel, exposing more dentin which is categorized as mineralization
and demineralization of enamel involving changes in absorption capacity of enamel.
Multipole diffusion approximation is valid for flat surfaces and gives a fair approximation for highly curved surfaces also this method does not capture anisotropy of
the material, it uses overall isotropic contribution to simulate subsurface effects. We
have also neglected the role played by the pulp layer which might affects the overall reflectance depending on amount of light transmitted through the layer. Use of
X-ray images of dentin and enamel will give us a better insight to material distribution giving better intensity approximation. From reflectance profile point of view we
consider only three dominant wavelengths we can increase accuracy by constructing
more profiles for properties with wavelengths lying within the visible spectrum and
interpolate wherever necessary.
Simulations involving determination of profiles were carried out on Intel(R) Pentium(R) D CPU 3.20 GHz, 2.0GB RAM and the images were rendered on a NVIDIA
GeForce 8600 GTS card. The application program interface OpenGL version 2.1 and
the corresponding shader language GLSL (Graphics library shader language) version
1.2 were used as part of code development to render using graphics processing unit
(GPU). We followed the GLSL coding techniques from [Bailey, Cunningham, 2009].
The (obj) file for model of central incisor was obtained from [Kato, Ohno, 2008].
41
Figure 4.1: Images show various shades exhibited by a central incisor for different
optical properties.Top Row: Enamel with high scattering and low absorption giving it whitish blue appearance, Bottom Row: Enamel with low scattering and high
absorption making color of dentin more prominent.
42
Chapter 5 – Discussion
5.1 Anisotropic Nature of Enamel and Dentin
Figure 5.1: Two dimensional scattering pattern [Kienle, Forster, 2002].Left: Incident
beam perpendicular to surface, Right: Incident beam oblique to surface.
The orientation of hydroxyapatite crystal axis in enamel makes it anisotropic while
in dentin the tubules make scattering highly directional (figure 5.1). The orientation
of tubules gives rise to waveguide like nature aided by refractive mismatches between
tubule and surrounding media (figure 5.2). Detailed analysis of this effect was done
in [Kienle, Forster, 2002] and [Zolotarev, Grisimov, 2000].
5.2 Need for a Range of Optical Properties
We use a range for optical properties to extract reflectance profiles. Due to discrepancies in experimental methods use of simple GUI sliders enables us to use values
between minimum and maximum determined experimentally.
43
Figure 5.2: Radial intensity distribution [Kienle, Forster, 2002].
Some of the factors and setups which result in variations in determining the optical
properties are listed below:
• In [Brodbelt, O’Brien, 1981] effect of water on enamel specimen was carefully
outlined. Enamel specimen when removed from water is dried and used tends
to show higher reflectance. Some of the other factors are effect of film thickness
on surface reflection, water on bulk and water content below the tooth surface.
• Integrating sphere setup used in [Brodbelt, O’Brien, 1981] fails to consider light
transmitted by specimen reentering the sample from the integrating sphere as
pointed out in [Zijp, Rudolf, 2001].
• Biological tissue samples mostly have to be placed in a liquid to prevent drying.
Samples that contain living cells have to be placed in a saline solution, which
prevents osmotic swelling of the samples [Zijp, Rudolf, 2001].
• The effect of surface roughness can be reduced by using index matching liquid
acting as a bridge between surface and surrounding medium or polishing the
surface to make it smooth. Care should be taken in case of index matching
liquid such that it does not alter the structural configuration of the specimen
under consideration, example dentin [Zijp, Rudolf, 2001].
44
• Carious free enamel specimen should be used for testing (figure 5.3). Carious
lesions appear as white spots caused by increased scattering of the incident
light.
Figure 5.3: Caries lesion seen as a dark spot under laser fluorescence [Mansson, Bosch,
1987].
• Anisotropic nature the specimen of dentin used for optical examination might
tend to exhibit different values based on the angle with which it is cut.
45
Chapter 6 – Conclusions
6.1 Advantages
• Profiles generated closely follow the properties of the underlying material obtained from research journals in dentistry thus giving a better characteristic of
light interaction.
• Profiles evaluated over three major wavelengths namely red, green and blue in
line with color channels of the display system.
• Model evaluated per layer basis taking care of secondary layer like dentin.
• Graphics based interface gives end user an option to view the effects of change
in optical properties rapidly and range provided takes care of the discrepancies
involved in measurement of optical properties.
• Fast convolution computation for variable kernel size due to GPU based rendering.
• Standard subsurface algorithm like multipole technique employed in evaluation
of diffuse color.
6.2 Limitations
• Multipole diffusion technique is estimated for flat homogeneous layer gives reasonable approximation for curved surfaces.
• We have assumed optical properties are same per layer and change across different layers. These properties tend to vary based on density and distribution
of the inherent components.
46
• We also have neglected the role played by the lip and gum color in imparting
color perception to teeth [Reno, Sunberg, 2000].
Figure 6.1: Lip and gum shade influencing teeth color perception [Reno, Sunberg,
2000].
• We have carried out convolution on the image itself which limits the effects
due to curvature of the mesh.Improvements include carrying out the diffusion
in texture space provided a suitable texture mapping is available followed by
stretch correction.
Figure 6.2: Stretch correction in horizontal and vertical texture space [Eon, Luebke,
2007].
6.3 Conclusion
Overall conclusion of this research work was to develop a model to successfully render
realistic teeth based on their optical properties which are inputs from various research
47
papers in dentistry, highlight the importance of variations across experimental techniques to determine these properties and hardware based rendering technique to view
results at interactive rates.
48
Bibliography
[1] [Ahmad, 2000], Three dimensional shade analysis, Irfan Ahmad, Proceedings aesthetic dentistry, 2000.
[2] [Bailey, Cunningham, 2009], Graphics Shaders: Theory and Practice, Mike Bailey
and Steve Cunningham, A K Peters Ltd, 2009.
[3] [Baltzer, Kaufmann, 2004], The Determination of the Tooth Colors, Andres
Baltzer and Vanik Kaufmann-Jinoian, Vol 30, Quintessenz Zahntech, 2004.
[4] [Brodbelt, O’Brien, 1981], Translucency of Human Dental Enamel, R.H.W. Brodbelt, W.J. O’Brien, P.L. Fan, J.G. Frazer-Dib and R. Yu, Journal of Dental Research, 1981.
[5] [Donner, Jensen, 2005], Light Diffusion in Multi-Layered Translucent Materials,
Craig Donner and Henrik Wann Jensen, Proceedings of ACM Siggraph, 2005.
[6] [Donner, Jensen, 2006], A Spectral BSSRDF for Shading Human Skin, Craig
Donner and Henrik Wann Jensen, Eurographics Symposium on Rendering, 2006.
[7] [Donner, Weyrich, 2008], A Layered, Heterogeneous Reflectance Model for Acquiring and Rendering Human Skin, Craig Donner, Tim Weyrich, Eugene d’Eon, Ravi
Ramamoorthi and Szymon Rusinkiewicz, Proceedings of ACM Siggraph Asia,
2008.
[8] [Eon, Luebke, 2007], Efficient Rendering of Human Skin, Eugene d’Eon, David
Luebke and Eric Enderton, Eurographics Symposium on Rendering, 2007.
[9] [Farrell, Patterson, 1992], A diffusion theory model of spatially resolved, steady
state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo, Thomas J Farrell, Michael S Patterson and Brian Wilson, Vol.19,
No.4, Medical Physics, 1992.
49
[10] [Fried, Glena, 1995], Nature of light scattering in dental enamel and dentin at
visible and near-infrared wavelengths, Daniel Fried, Richard E. Glena, John D.B.
Featherstone and Wolf Seka, Vol.34, No.7, Applied Optics, 1995.
[11] [Groenhuis, Bosch, 1983], Scattering and absorption of turbid materials determined from reflection measurements.2: Measuring method and calibration, R.A.J.
Groenhuis, J.J. Ten Bosch and H.A. Ferwerda, Vol.22, No.16, Applied Optics,
1983.
[12] [Jacques, Prahl, 1998], Biomedical Optics-ECE 532, Steven L.Jacques and Scott
A. Prahl, Oregon Graduate Institute, 1998.
[13] [Jensen, Marschner, 2001], A Practical Model for Subsurface Light Transport,
Henrik Wann Jensen, Stephen R. Marschner, Marc Levoy and Pat Hanrahan,
Proceedings of ACM Siggraph, 2001.
[14] [Joiner, 2003], Tooth colour: A review of the literature, Andrew Joiner, Unilever
Oral Care, 2003.
[15] [Kato, Ohno, 2008], Construction of three-dimensional tooth model by microcomputed tomography and application for data sharing, A. Kato and N. Ohno,
Springer Verlag, 2008.
[16] [Kelemen, Kalos, 2001], A microfacet based coupled specular-matte BRDF model
with importance sampling, Kelemen C. and Szirmay-Kalos L., Eurographics Short
Papers, pp. 25-34, 2001.
[17] [Kienle, Forster, 2002], Light propagation in dentin: influence of microstructure
on anisotropy, Alwin Kienle, Florian K Forster, Rolf Diebolder and Raimund
Hibst, Phys.Med.Biol.48, N7 N14 PII, Institute of Physics (IOP) publishing, 2002.
[18] [Kienle, Michels, 2006], Magnification a New Look at a Long-known Optical
Property of Dentin, A. Kienle, R. Michels and R. Hibst, Journal of Dental Research, 2006.
50
[19] [Kinney, Balooch, 1995], Mineral Distribution and Dimensional Changes in Human Dentin during Demineralization, J.H. Kinney, M. Balooch, D.L. Haupt,
Jr,S.J. Marshall and G.W. Marshall,Jr, Journal of Dental Research, 1995.
[20] [Krishnaswamy, Baronoski, 2004], A biophysically-based spectral model of light
interaction with human skin, A. Krishnaswamy and G.V.G Baronoski, vol 23,
Proceedings of Eurographics, 2004.
[21] [Mansson, Bosch, 1987], Optical Methods for the Detection and Quantification
of Caries, B.Angmar-Mansson and J.J. Ten Bosch, Advances in Dental Research,
1987.
[22] [Mayorga, 2008], A Combined Approach to Subsurface Scattering-Draft, Adrian
Mayorga, University of Texas at Austin, Department of Computer Sciences, 2008.
[23] [Prahl, 1988], Light Transport in Tissue, S.A. Prahl, PhD thesis, University of
Texas at Austin, 1988.
[24] [Ragain, Johnston, 2001], Accuracy of Kubelka-Munk Reflectance Theory Applied to Human Dentin and Enamel, J.C. Ragain, JR and W.M. Johnston, Journal
of Dental Research, 2001.
[25] [Reno, Sunberg, 2000], The Influence of Lip/Gum Color on Subject Perception of
Tooth Color, E.A. Reno, R.J. Sunberg, R.P. Block and R.D. Bush, International
Association of Dental Research (IADR), 2001.
[26] [Terry, Geller, 2002], Anatomical form defines color: Function,Form and Aesthetics, Douglas A Terry, Willi Geller, Oliver Tric, Mark J Anderson, Monte Tourville
and Alvin Kobashigawsa, Proceedings Aesthetic Dentistry, 2002.
[27] [Vanini,Mangani, 2001], Determination and Communication of color using the
five color dimensions of teeth, Lorenzo Vanini and Francesco M Mangani, Proceedings Aesthetic Dentistry, 2001.
51
[28] [Wang, Milner, 1999], Characterization of dentin and enamel by use of optical
coherence tomography, Xiao-Jun Wang, Thomas E. Milner, Johannes F. de Boer,
Yi Zhang, David H. Pashley and J. Stuart Nelson, Vol.38, No.10, Applied Optics,
1999.
[29] [Weatherell, Robinson, 1974], Variations in the Chemical Composition of Human
Enamel, J.A. Weatherell, C. Robinson and A.S. Hallsworth, Journal of Dental
Research, 1974.
[30] [Weyrich, Matusik, 2006], Analysis of Human Faces using a Measurement-Based
Skin Reflectance Model, Tim Weyrich, Wojciech Matusik, Hanspeter Pfister,
Bernd Bickel, Craig Donner, Chien Tu, Janet McAndless, Jinho Lee, Addy Ngan,
Henrik Wann Jensen and Markus Gross, Proceedings of ACM Siggraph, 2006.
[31] [Zijp, Bosch, 1995], HeNe-Laser Light Scattering by Human Dental Enamel, J.R.
Zijp, J.J. ten Bosch and R.A.J. Groenhuis, Journal of Dental Research, 1995.
[32] [Zijp, Rudolf, 2001], Optical properties of dental hard tissues, J.R. Zijp and
Jacob Rudolf, Dissertations, University of Groningen, 2001.
[33] [Zolotarev, Grisimov, 2000], Architectonics and Optical Properties of Dentin
and Dental Enamel, V.M. Zolotarev and V.N. Grisimov, Vol.90, No.5, Optics and
Spectroscopy, 2000.