=Paper= {{Paper |id=Vol-2688/paper4 |storemode=property |title=Surface Discretisation Effects on 3D Printed Surface Appearance |pdfUrl=https://ceur-ws.org/Vol-2688/paper4.pdf |volume=Vol-2688 |authors=Alina Pranovich,Sasan Gooran,Jeppe Revall Frisvad,Daniel Nyström |dblpUrl=https://dblp.org/rec/conf/cvcs/PranovichGFN20 }} ==Surface Discretisation Effects on 3D Printed Surface Appearance== https://ceur-ws.org/Vol-2688/paper4.pdf

Surface Discretisation Effects on 3D Printed
Surface Appearance

Alina Pranovich1 , Sasan Gooran1 , Jeppe Revall Frisvad2 , and Daniel Nyström1
1
Linköping University, Norrköping, Sweden
alina.pranovich@liu.se, sasan.gooran@liu.se, daniel.nystrom@liu.se
2
Technical University of Denmark, Kongens Lyngby, Denmark
jerf@dtu.dk

Abstract. The spatial resolution of 3D printing is finite. The necessary
discretisation of an object before printing produces a step-like surface
structure that influences the appearance of the printed objects. To study
the effect of this discretisation on specular reflections, we print surfaces
at various oblique angles. This enables us to observe the step-like struc-
ture and its influence on reflected light. Based on the step-like surface
structure, we develop a reflectance model describing the redistribution
of the light scattered by the surface, and we study dispersion effects due
to the wavelength dependency of the refractive index of the material.
We include preliminary verification by comparing model predictions to
photographs for different angles of observation.

Keywords: BRDF · 3D printing · gloss · material appearance.

1 Introduction

The appearance of objects with known optical properties can be modelled with
different rendering techniques [2]. An aspect of particular interest is color ap-
pearance assessment under a certain illumination [10,15]. While one might expect
manufactured surfaces to be perfectly smooth, each manufacturing process im-
prints its characteristics on the object including mechanical properties [9] and
appearance [2]. An important part of object appearance is gloss (reflection high-
lights) [5]. Highlights are directly affected by the surface structure as it changes
the magnitude and directions of specular light reflections. Even with the same
raw material, the surface can be perceived as having slightly different colour
caused by different manufacturing technology and surface finishing. One exam-
ple is objects produced by 3D printing [12].
In 3D printing, the surface requires discretisation. For example, in Fused
Deposition Modelling 3D printing (FDM), the object is created by applying
melted material in a layer-by-layer flow. Any oblique surface to be printed will
Copyright c 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0). Colour and Visual Com-
puting Symposium 2020, Gjøvik, Norway, September 16-17, 2020.
2 A. Pranovich et al.

be affected by this slicing. As a result, any oblique surface will be approximated
by a staircase-like shape. In this paper, we investigate plane surfaces of FDM-
printed wedges and the effect of the surface discretisation on the appearance for
different wedge angles. For this, we consider wedges as schematically shown in
Fig. 1, a–c). We limit our study to surface reflection only. Once the scattering
of light by the surface has been modelled, it can be combined with subsurface
scattering using volumetric light transport simulation [6,7].
Reflectance models for ridges such as the symmetric V-groove cavity [17]
have been studied for a long time. Most models however study the bidirectional
reflectance distribution function (BRDF) resulting from a statistical distribu-
tion of microfacet normals [14], where the microfacets form V-grooves. Multiple
reflections were recently included in this type of model [16], also for nonsymmet-
ric V-grooves [8], but our interest is reflectance from a periodic ridged surface
rather than a distribution of microfacets. The symmetric V-groove model was
recently investigated for a periodic ridged surface [13]. Our staircase-like ridges
are, however, both nonsymmetric and periodic. In other related work, a BRDF
was developed for ridged surfaces like ours [11]. There, the authors focused on
single scattering and incorporated statistical distribution of surface normals to-
gether with the ridges to model surface imperfection. We include retroreflection
due to two scattering events, and we parameterise our model in a different way
to make the model suitable for the staircase structures observed in 3D printing.
A model for retrieval of a shading normal following the step-like structure of 3D
printed objects has also been suggested [3]. This model, however, was intended
for close-up rendering of the layers rather than calculation of a BRDF.

2 Method

Angular redistribution of incident light after single or multiple reflections can
be quantitatively estimated with geometrical ray optics, together with the Fres-
nel equations [1]. The range of validity of this approach is determined by two
main assumptions. First, the structures must be substantially larger than the
wavelength (considering visible light, 380-780 nm) to prevent visible interference
effects. Hence, at least a few micrometers. Second, the size of individual surface
elements shall not substantially exceed the resolution of the human eye, which,
for example, at a viewing distance of 50 cm is on the order of 150 μm.
For the wedges described above, together with a given printing resolution, the
unit cell considered for calculation is defined as illustrated in Fig. 1, e). A unit
cell is chosen as a concave right-angled structure with two slopes defined by the
vertical printing resolution. The length of one side corresponds to the printing
layer thickness l1 (vertical printing resolution). The second side l2 is given by l1
and the wedge angle β2 . The side corresponding to the layer thickness l1 forms
the angle β1 = 90◦ − β2 with the wedged plane (Fig. 1, a,e). We thus have

l1 sin β1
l2 = = l1 cot β2 . (1)
sin β2
Surface Discretisation Effects on 3D Printed Surface Appearance 3

a) b) c)
l2
l1
) Wedge angle

Model 3D Print Discretised surface
(side view) (red)

d) e) f)
θi-2β1

+- -+ θi+2β2
-θi

)
β1
β2
Incidence Reflection ))
)
θi θo
l

Fig. 1. a): Side view of a 3D model of a wedge with defined wedge angle. b): Side view
of resulting printed wedges with discretised surface, l1 is the layer thickness in the z
direction, l2 is a step length in the y direction. c): Examined discretised surface. The
unit cell chosen for calculation is highlighted with a dashed square. d): Sign conventions
for the angles of incidence and observation with respect to the surface normal. e): The
surface unit cell considered in calculations is illuminated by rays hitting the section of
length l between the tips of the slopes. f): Exemplary ray reflection scenarios. Each
case of double reflection is a retroreflection due to the mutually perpendicular slopes.

It is sufficient to consider a one-dimensional unit cell as the surface is constant
in the perpendicular lateral direction. The relation of the angle of incidence θi
and the angle of observation θo to 3D direction vectors describing the light-view
configuration and the surface geometry is covered by Luongo et al. [11].
Incident illumination
p is defined through a number of rays hitting the section
of the length l = l12 + l22 between the tips of the slopes. By assuming that illu-
mination consists of a number of rays homogeneously distributed with a certain
density in the plane defined by the unit cell, we can express radiance of inci-
dent light through the projection of any segment within the unit cell onto the
rayfront. The radiance illuminating each slope is then expressed through their
projections:

r1 = l1 max{sin(β2 + θi ), 0} + l2 min{cos(β2 + θi ), 0} (2)
r2 = l2 max{cos(β2 + θi ), 0} + l1 min{sin(β2 + θi ), 0} . (3)

The max and min functions help us to avoid listing three different cases. When
−β2 ≤ θi ≤ β1 , the first terms in the two expressions are nonzero, while the
second terms are zero. On the other hand, when θi < −β2 , we have r1 = 0 and
both terms active for r2 . The opposite is the case for θi > β1 . We also recall
Lambert’s cosine law: the more grazing the angle of incidence, the smaller the
4 A. Pranovich et al.

density of incident rays. The irradiance is thus proportional to the cosine of the
angle of incidence, but this term is not part of the BRDF.
Light can leave the surface after being reflected once by any of the slopes
or after a double reflection. Due to the mutually perpendicular slopes, each
case where double reflection occurs is a retroreflection, with the scattered light
direction coinciding with the direction of incidence (independently of the wedge
angle). By letting our model account for this retroreflection, masking effects do
not need to be considered explicitly, but are intrinsically accounted for. One
angle of incidence θi can be associated with several scattering angles. Depending
on the position of an incident ray along the slopes, light can be reflected with
scattering angle θo :

from left slope : θo = θi + 2β2 (4)
from right slope : θo = θi − 2β1 (5)
retroreflection : θo = −θi . (6)

The convention regarding the sign of θi and θo still follows Fig. 1, d). For a
given angle of incidence, we calculate the proportion of the rays being reflected
once or twice. Those proportions are expressed through the projections of the
corresponding sections onto the rayfront. We can use θo from Eqs. (4–5) to get
the direction of the incident light after the first reflection. Retroreflection occurs
for the part of the first reflection being incident on the other slope. Insertion of
the new directions in Eqs. (2-3) then provides the desired projections. For −β2 <
θi ≤ β1 −β2 , all rays hitting the right slope will be retroreflected. The rays hitting
the left slope will be partially directly reflected and partially retroreflected. The
direction of the light after reflection in the left slope is 180◦ − θi − 2β2 . Inserting
this direction of incidence in Eq. (2), we get the projection of the part of the left
slope from which retroreflection occurs:

r21 = l1 sin(β2 + θi ) . (7)

Similar considerations apply for β1 − β2 < θi < β1 . Here, all rays hitting the left
slope will be retroreflected. The part of the rays falling on the right slope and
being retroreflected is given by

r12 = l2 cos(β2 + θi ) . (8)

For angles of incidence satisfying θi > β1 or θi < −β2 , only direct reflection
occurs, which we calculate using Eqs. (2–3). The proportion of scattered light
in each direction can have a value between 0 and l cos θi . We thus represent the
proportions as reflectance ratios through division by l cos θi .
In addition to geometric considerations, the material refractive index influ-
ences the reflections of the individual slope surfaces. We calculate bidirectional
reflectances using the Fresnel equations with tabulated wavelength-dependent
refractive indices for the 3D printing materials obtained from spectroscopic el-
lipsometry. P- and S-polarised light is considered separately. For our final results,
we use unpolarised light incident in the plane depicted in Fig. 1. The reflectance
Surface Discretisation Effects on 3D Printed Surface Appearance 5

is then half of each type of polarisation (also in the case of retroreflection as the
plane of incidence is the same for both reflections). The amount of light entering
the material is given by one minus the Fresnel reflectance. This refracted light
can be used with a model for volumetric light transport [6,7] or subsurface scat-
tering [4] to compute the full appearance of the wedge beyond predictions by
our model. In this paper, we focus on the surface reflectance, which in terms of
appearance attributes can be interpreted as gloss and colour of gloss. In order
to model surface appearance under a certain viewing angle, light from all angles
of incidence reflected into the desired viewing direction is integrated.

3 Results and Discussion

The described approach was used to predict reflectance properties of wedges
printed with a fused deposition modelling (FDM) printer. Black polylactide
(PLA) filament was used. Slope angles of the wedges were set to the values 10◦ ,
20◦ , 30◦ , 40◦ , 45◦ , and 60◦ . Examples of printed wedges with 10◦ and 45◦ (side
view) are shown in Fig. 2. The footprint of each slope was 2 × 2 cm2 . The num-

Fig. 2. Side view of printed wedges with 10◦ (top) and 45◦ (bottom) wedge angle.

bers of layers needed for each print were obtained after slicing the models with
defined layer thickness (resolution in z direction). Larger slope angles resulted
in a larger number of layers needed. Thus the step in y direction was smaller for
the higher slope angles. Table 1 summarises used and calculated parameters.
We computed the reflectance by combining the pure geometrical effect with
the Fresnel angular reflectance for each angle of incidence and reflection. Results
at 550 nm are shown in Fig. 3. For the refractive index, a representative value
of n = 1.46 for PLA has been chosen. The wavelength-dependent values of the
6 A. Pranovich et al.

Table 1. 3D printing parameters. The layer thickness (z-step) was 150 μm.

wedge angle, ◦ 10 20 30 40 45 60
number of layers 22 47 75 109 130 225
y-step, μm 909 426 267 183 154 89

Fig. 3. Bidirectional reflectance (BRDF values) of wedges with wedge angles 10◦ (left)
and 45◦ (right) for unpolarised incident light, calculated for 550 nm.

refractive index were obtained by spectroscopic ellipsometry and are shown in
Fig. 4. As the reflectance values have been calculated for a one-dimensional
surface cross-section, i.e. one plane of incidence and reflection, they represent an
in-plane BRDF of the printed surface.
Characteristic patterns can be observed in the diagrams: The straight line
θo = −θi corresponds to retroreflection. The other two lines represent direct
reflections from the two slopes, l1 and l2 . Retroreflection is weaker because it
involves two reflections. Retroreflection is not possible for θi < −β2 or θi > β1 .
The top-left line represents direct reflection from the left slope and the bottom-
right line corresponds to direct reflection from the right slope. The intensity
of the direct reflectances becomes weaker when also retroreflection is possible.
Otherwise, the radiance distribution along the lines is a result of both the Fresnel
reflectances and the geometrically determined proportions.
Considering light incident from all directions (diffuse lighting), the reflected
contributions towards a given viewing angle can be accumulated (by integrating
over all angles of incidence). An example is shown in Fig. 5. Here, characteristic
peaks for certain angles of observation of each wedge can be observed. The peaks
correspond to the situation when the longer slope is being illuminated under
shallow angles. For example, for a 10◦ wedge with β2 = 10◦ , β1 = 80◦ , this occurs
when θi is close to −90◦ and hence, θo approaches −90◦ + 2β2 (here −70◦ ). From
Fresnel equations, larger angles of incidence (with respect to the slope’s surface
normal) lead to the higher reflectance values (for angles greater than Brewster’s
angle). However, there is no contribution if θi = ±90◦ . This causes a minimum at
Surface Discretisation Effects on 3D Printed Surface Appearance 7

Fig. 4. Real part of the refractive index of PLA for wavelengths in the visible as ob-
tained from spectroscopic ellipsometry measurements. Measured with a Dual-Rotating-
Compensator ellipsometer RC2 by J.A. Woolam.

θo = −90◦ + 2β2 . For wedge angles larger than 45◦ (β1 < 45◦ ), there is a strong
contribution of light reflected at shallow angles directly off the right slope. This
happens for θi approaching +90◦ and θo approaching 90◦ − 2β1 . Comparing
different wedge angles in Fig. 5, the longest printing step in y-direction was
made in the case of the 10◦ wedge angle. Therefore the contribution to the peak
value is the highest in this case, while the peak is least pronounced for 45◦ .
Furthermore, as expected, the BRDF for the wedge angle 45◦ is symmetric with
respect to θo = 0◦ , as the unit cell is mirror symmetric. Similarly, the BRDFs
of 30◦ and 60◦ wedges are mutually symmetric. For comparison, we calculated
also the spectral reflectance for a plane surface which is shown in Fig. 5.

3.1 Spectral dependencies and appearance interpretation
In order to be able to interpret calculated light quantities as appearance at-
tributes, the above described calculation has to be carried out for each wave-
length of the visible spectrum (380 to 780 nm) separately because the refractive
index is dispersive. Accumulated reflectance spectra for a given viewing direc-
tion were calculated after integrating over all incidence angles. An example of
spectral reflectance for viewing angle 0◦ is shown in Fig. 6. Assuming a certain
illuminant (e.g. D65), integrated reflected (scattered) radiance spectra towards
a given angle of observation can be calculated and interpreted as surface con-
tribution to the overall appearance. Interpretation of the colour under standard
illuminants D65 and A is demonstrated in Fig. 7. The radiance changes with
8 A. Pranovich et al.

Fig. 5. Reflectance for different angles of observation. Calculated by integrating bidi-
rectional reflectances (Fig. 3) over all angles of incidence θi ∈ [−90◦ , 90◦ ] (no cosine
weights). The legend displays the wedge angle. For comparison, we also show the values
for direct specular reflectance of a perfectly planar surface (gray dash-dotted line called
material in the legend). Data has been computed for 550 nm. The discontinuities occur
when the incidence angle reaches ±90◦ .

varying viewing angle. Nevertheless, the redistributed specular reflections repre-
sent mostly a colour of the light source. As can be seen in Fig. 6, the reflectance
is relatively uniform over the visible spectrum.

3.2 Including surface imperfections

Real surfaces contain rough details. In the case of the studied printed wedges,
each individual slope surface is rough, also the edges of the layers are not per-
fectly sharp, but are to a certain degree round. Those two factors lead to sta-
tistically fluctuating angular deviation of scattered radiation, which introduces
the diffusion and blurring of specular reflection. Additionally, the finite size of
specimen and light source induce an angular spread. Furthermore, angular devi-
ations can be observed for light that is incident from outside the plane of interest
depicted in Fig. 1. All these factors can be represented through applying a con-
volution with Gaussian broadening to the computed BRDF. Fig. 8 shows the
resulting scattered light angular distribution after applying convolution repre-
senting additional angular spread of reflection caused by surface roughness.
Surface Discretisation Effects on 3D Printed Surface Appearance 9

Fig. 6. Calculated reflectance spectra for 6 wedges (angles in the legend) using an
“all-to-normal” configuration, where the bidirectional reflectance is integrated over all
angles of incidence with a 0◦ angle of observation (along the normal direction). For
comparison, the insert shows a close-up of Fig. 5 around θo = 0◦ .

Fig. 7. Colour representation of integrated incidence light scattered towards different
viewing angles. Example for the wedges with 10◦ to 60◦ angles and standard illuminants
D65 (left) and A (right). For comparison, colour representation was also calculated for a
plane surface (on the right side of each series). The figure demonstrates the pure colour
effect from the surface, i.e. modified glossy reflections from the discretised surface.
10 A. Pranovich et al.

Fig. 8. BRDF values for wedges with 10◦ (left) and 45◦ (right) wedge angles printed
with PLA after applying a convolution with Gaussian broadening (standard deviation
of 5◦ ). Calculated for 550 nm.

3.3 Comparison with photographic images

As a first attempt at verifying our approach, we compare the computed predic-
tions to real visual impressions. For this purpose, a series of photographs was
prepared. A small point-like LED light source was placed in 50 cm distance from
the specimen. Photographs were taken at different angles at 90 cm distance with
135 mm objective and the following setting: ISO 100, exposure time 1 second
and aperture 5.6. Figure 9 compares theoretical expectations with real appear-
ance. The main quality that varies with changing viewing angle is the brightness
which is casued by the redistribution of specular reflections. Despite of large
scattering between the pictures, the general trend can be observed. Apparently,
the real structures reveal large angular broadening. Remarkably, the 10◦ wedge
has a generally darker appearance than the other ones. Also, the reflectances of
30◦ and 60◦ wedges differ from each other while the model predicts symmetric
behaviour. The unit cells for both configurations have the same geometry, how-
ever the size of the unit cell decreases with the wedge angle. The unit cell of
the 30◦ wedge is approximately 1.7 times bigger than that of the 60◦ one. As a
conclusion, the absolute unit size plays a role in the appearance of the oblique
3D printed surfaces. A possible explanation is the impact of the real (rounded)
edges of the printed layers (cf. Fig. 2). If the unit cell size approaches the cur-
vature radius, those edges will dominate the scattering characteristics. Another
consequence of large angular broadening for structures with smaller unit cells
is the bright intensity scattered towards −80◦ for wedge angles around 40◦ as
seen in the photographs. For example, for the 45◦ wedge, we would expect a
direct reflection toward −90◦ which would not be observable. The brightness at
−80◦ may result from the discussed angular broadening by imperfections like
the round edges.
Surface Discretisation Effects on 3D Printed Surface Appearance 11

Fig. 9. Reflectance towards different angles with angle of incidence 0◦ for 6 different
wedges. Left: computation similar to Fig. 8 after applying convolution with Gaussian
distribution. Right: Gray-scaled photographs of the printed wedges taken at differ-
ent angles. The higher reflectivity lines can be immediately recognized by increased
brightness. Note that no pictures were taken for the 0◦ angle of observation.

4 Summary

In this work, the effects of oblique surfaces created with 3D printing on the
object’s appearance were studied. Due to the limited printing resolution, these
surfaces are discretised according to the surface inclination. Using geometrical
optics, shadowing effects and angular redistribution of specular reflections were
modelled. Geometrical considerations were complemented by calculating wave-
length dependent reflectance. The so calculated reflectances allowed us to predict
the surface contribution to the object appearance, especially for highly absorbing
materials. The approach was used to compute the colour tone of black material
under a certain illuminant for different viewing angles. A series of wedges was 3D
printed using FDM technique and black PLA filament. A comparison of model-
predicted scattering with photographs taken at different viewing angles indicates
reasonable correspondance in particular for those wedges with coarser discreti-
sation. Such imperfections as rounded edges are not discribed by the model but
have strong impact on surfaces with finer discretisation. For future work, we
suggest a study of other aspects such as diffuse scattering by surface microfacets
and subsurface scattering in a translucent print material.

Acknowledgment

Funded by the Horizon 2020 programme of the European Union. Grant #
814158.
12 A. Pranovich et al.

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