# A Study of Dielectric Breakdown Along Insulators Surrounding Conductors in
Liquid Argon

S. Lockwitz Corresponding author<EMAIL_ADDRESS>(S. Lockwitz) and H.
Jostlein
Fermi National Accelerator Laboratory  P.O. Box 500  Batavia  IL  60510  USA

###### Abstract

High voltage breakdown in liquid argon is an important concern in the design
of liquid argon time projection chambers, which are often used as neutrino and
dark matter detectors. We have made systematic measurements of breakdown
voltages in liquid argon along insulators surrounding negative rod electrodes
where the breakdown is initiated at the anode. The measurements were performed
in an open cryostat filled with commercial grade liquid argon exposed to air,
and not the ultra-pure argon required for electron drift. While not addressing
all high voltage concerns in liquid argon, these measurements have direct
relevance to the design of high voltage feedthroughs especially for averting
the common problem of flash-over breakdown. The purpose of these tests is to
understand the effects of materials, of breakdown path length, and of surface
topology for this geometry and setup. We have found that the only material-
specific effects are those due to their permittivity. We have found that the
breakdown voltage has no dependence on the length of the exposed insulator. A
model for the breakdown mechanism is presented that can help inform future
designs.

###### keywords:

Liquid Argon; Time Projection Chambers; Dielectric Strength, Electric
Breakdown; High Voltage

##  1 Motivation

Liquid Argon Time-Projection Chambers (LArTPCs) are a popular detector choice
for neutrino and dark matter experiments. This technology relies upon an
electric field typically on the order of 50 kV/m to drift signal electrons
from ionization due to charged particle interactions in the argon. Drift
distances are typically on the scale of less than a few meters. It has become
important to understand parameters related to high voltage use in liquid argon
to design future detectors and understand any technical limitations.

Dielectric breakdown in liquids, particularly transformer oils, has been
actively studied for many decades, mostly to meet the engineering needs of the
power distribution industry. Very little generic research has been done on
dielectric breakdown in liquid argon. Several recent breakdown studies [1, 2,
3, 4] have yielded useful results related to breakdown through the bulk
liquid. This report presents a comparative study of dielectric breakdown in
liquid argon along insulator surfaces surrounding high voltage conductors. The
goal is to understand these breakdowns and eventually find methods to predict
and avoid them.

##  2 Experiment Setup

The experiment setup is shown schematically in Figure 1. We started with
commercially pure liquid argon with contaminants at the parts-per-million
(ppm) level. The tests were performed in an open-to-air cryostat over the
course of weeks. This condition implies that water vapor, oxygen, and nitrogen
diffused into the liquid argon over the course of the measurements. We also
noted an accumulation of dust in the liquid argon, with dust especially
attracted to areas of high electric field. The effects of these contaminants
are unknown. We have not found published data on the effect of contaminants in
the relevant range.

(a)  (b)

Figure 1: Drawings of the test apparatus. The left graphic, (a), shows the
complete test setup including the current-limiting resistor. A detailed
drawing including the test piece and holder is highlighted in (b). The
insulator test piece could be removed and replaced without removing the anode
ring, holder and high voltage feedthrough from the liquid argon. The high
voltage was applied by a high voltage feedthrough placed on top of the test
piece. The ring could be raised and lowered from outside of the cryostat.

The high voltage was supplied by a Glassman LX150N12 power supply [5] able to
provide up to $-150$ kV. The power supply was connected through an oil-
insulated series resistor (75 M$\Omega$) to a high voltage feedthrough
partially submersed in liquid argon. The resistor served to partition the
energy stored in the setup and limit the energy released into the cryostat
during an electrical discharge. The high voltage feedthrough was made of a
stainless steel central conductor surrounded by an ultra high molecular weight
polyethylene (UHMW PE) tube inside an outer stainless steel grounded tube. The
lower edge of the ground tube was encased in a UHMW PE cylinder to prevent
streamer propagation along the feedthrough itself. The lower end of the UHMW
PE protruded beyond the ground tube and its surface was grooved. The inner
conductor extended further out and was machined with a concave end to accept a
sphere.

Each insulator studied started in the form of a 5.1 cm diameter smooth
cylinder, 30.2 cm long with a 2.54 cm diameter center-bored hole along the
length of the material to a depth of 16.5 cm. A 2.54 cm diameter stainless
steel rod with a 3.8 cm diameter sphere welded on its end was inserted into
each test piece as shown in Figure 1. The high voltage feedthrough was placed
on top of the welded sphere during testing. Three samples were additionally
tested with grooved profiles cut into the insulator surfaces.

Each piece was lowered into a grounded holder submerged in liquid argon to
secure the piece during testing. The holder had a grounded stainless steel
ring attached to it with a 5.2 cm inner diameter that fit loosely around the
body of the test piece. The ring thickness flared out from the test piece at a
15 degree angle from an intentionally sharp top edge near the insulator
surface to a 0.5 cm thick ring where it attached to the holder. This top edge,
the knife edge, was designed to create a high electric field to induce
breakdown along the insulator surface. The ring could be raised and lowered
along the test piece by turning a threaded rod that extended outside of the
cryostat.

###  2.1 Experiment Procedure

After insertion into the argon, each test piece was allowed to cool until
there was no visible boiling before testing began. This was done in an effort
to reduce any thermal effects on spark formation.

A LabView [6] program was used to raise the high voltage in 500 V steps every
1.6 s. When the current draw as registered by the supply was above 45 $\mu$A,
the supply would trip setting the voltage to zero. After a trip was
identified, the program would wait one minute and then begin raising the
voltage again. Breakdown values were determined from a log file monitoring the
voltage at sub-second intervals.

While the definition of a tripping spark is specific to our setup, it is the
same for all test pieces. There were occasions where there was an audible
spark that did not trip the supply. These events did not enter our dataset and
the voltage would increase until a tripping spark occurred.

We evaluated the 11 materials listed in Table 1. Tested ring elevations varied
for each sample, but at least three exposed insulator lengths were evaluated
between 3.5-12 cm for each undamaged smooth sample. For a given test piece and
length of exposed insulator above the grounded ring, the number of trips
tested ranged from 42-297 depending on the available time before refilling the
argon. The voltage values of these trips were plotted sequentially to check
for stability, and then histogrammed and fit with a Weibull function [7] as
shown for example in Figure 2. The Weibull function is described elsewhere [8]
as being an appropriate function to describe the variability of breakdown
voltages. The mean and the square root of the variance were taken from the fit
to compare breakdown voltages versus lengths between materials.

| Thermal | Chemical | Chemical | Dielectric | Dielectric
---|---|---|---|---|---
Material | Contraction | Name | Abbreviation | Constant | Strength
| (%) |  |  |  | (kV/mm)
Delrin | 1.2 $\pm$ 0.1 | Polyoxymethylene-homopolymer | Acetal-Homopolymer | 2.7 | 20
POM
Polypropylene |  1.1 $\pm$ 0.09 | Polypropylene | PP | 2.2-2.6 | 30-40
Noryl |  0.5 $\pm$ 0.08 | Polyphenyleneoxide+Polystyrene | PPO + PS | 2.7 | 16-20
Polycarbonate |  0.6 $\pm$ 0.09 | Polycarbonate | PC | 2.9 | 15-67
Polystyrene |  0.5 $\pm$ 0.06 | Polystyrene | PS | 2.4-3.1 | 20
UHMW PE |  1.8 $\pm$ 0.3 | Polyethylene-UHMW | UHMW PE | 2.3 | 28
FR4 |  0.2 $\pm$ 0.07 | Fiberglass-epoxy | G10-FR4 | 4.8 | 20
PTFE |  0.9 $\pm$ 0.09 | Polytetrafluoroethylene | PTFE | 2.0-2.1 | 50-170
Ultem |  0.4 $\pm$ 0.07 | Polyetherimide | PEI | 3.1 | 30
PBT |  0.9 $\pm$ 0.08 | Polyethtylene Terephtalate | PBT | 3.2 | 20
Polyester |  0.9 $\pm$ 0.1 | Polybutylene Terephtalate | PET, PETB | 3 | 17
Table 1: The thermal contraction, dielectric constants, and dielectric
strengths for the materials under test. The thermal contraction was measured
between room temperature and 78 K. The Delrin dielectric constant was
referenced from [9], the FR4 values from [10], all others are from [11].

(a)  (b)

Figure 2: The sequential breakdown voltages for the Ultem test piece with 1.43
cm exposed is shown in (a), and a Weibull functional fit to the resulting
distribution is shown in (b).

##  3 Results

###  3.1 Cold Shrinkage Measurements

Before electrically testing the insulators, their thermal contraction was
measured. The diameter of 5.1 cm rod stock of each material was measured at
room temperature and at liquid nitrogen temperature. The technique was to
submerge each piece in liquid nitrogen until there was no visible boiling.
Each test piece was then removed and measured at three common reference
lengths evaluating the minimum and maximum diameters. The values are reported
in Table 1. These values are useful for understanding the capabilities and
limits of components of LArTPCs, and planning for dimensional change at
cryogenic temperatures.

###  3.2 Electrical Insulator Test Results

The insulator test pieces can be classified into three groups: those that
catastrophically fractured upon submersion in liquid argon, those that broke
after a number of successful dielectric breakdown tests, and those that
underwent the full suite of testing. Here, a successful breakdown test is
defined as a dielectric breakdown near the surface of the insulator that trips
the power supply. For some materials, additional testing of surface profiles
was performed. This testing is discussed in Section 3.3.

####  3.2.1 Group I

Polystyrene fractured upon submersion in liquid argon. The Noryl test piece
made a popping noise as it was lowered into the liquid argon. The piece was
then removed for inspection and burst as it warmed. The contraction of the
solid stock of both materials had been measured without sign of damage. It
appears that introducing the non-contracting center conductor led to too much
strain for the material when exposed to liquid argon temperatures. These
materials are problematic for use in high voltage feedthroughs.

####  3.2.2 Group II

The next group of materials was able to undergo a number of successful
breakdown measurements, but eventually suffered mechanical damage preventing
further testing. This group included polyester, PBT, and PTFE. Polyester and
PBT both suffered catastrophic mechanical failures while testing with 11.43 cm
of material exposed. Polyester failed after 50 breakdown events, and PBT
lasted through 100.

PTFE is often used in cryogenic devices, and initially showed much promise in
this study with average breakdown values among the highest evaluated. However,
while testing with 11.43 cm length exposed, the breakdown voltage suddenly
dropped after about 17 breakdowns. Upon inspection, it was found that a
pinhole had burned through the bulk of the material and a major crack
developed around the diameter nearly severing the piece as seen in Figure 3.
This was the only material that broke through the bulk near the base of the
center conductor. The peak field at the bottom corner of the center conductor
was $\sim$400 kV/cm which is in the range of the published dielectric strength
of the material.

Figure 3: The PTFE test piece after evaluation.

####  3.2.3 Group III

The remaining materials survived mechanically during testing. All had stable
breakdown voltages in time showing no degradation after hundreds of
discharges.

Polypropylene and UHMW PE had the highest average breakdown voltages. UHMW PE
exhibits light brown traces about 0.2-0.5 mm wide and of uniform contrast
along their length on the surface presumably where the spark traveled. UHMW PE
displays this tracing, seen in Figure 4, more clearly than any of the other
materials. These traces show that the breakdowns follow the surface closely
and we found that new traces formed as there were more breakdowns showing that
breakdowns do not preferentially follow existing traces.

(a)

(b)

Figure 4: Traces along the UHMW PE test piece are shown in (a). In the right
of (a), the traces are more dense due to many more electric breakdowns being
evaluated at that elevation. In (b), a glowing path from breakdown along the
surface of the FR4 test piece is shown. The glowing trace goes on a path from
the ring along the right side of the piece up the surface and then radially
inward to the center conductor.

####  3.2.4 Analysis of Group III

We now turn to the measurements made on the materials of Group III: those that
survived the mechanical tests and hundreds of discharges. The average
breakdown voltages from the Weibull fit for the Group III materials are
plotted versus exposed insulator length in Figure 5. A linear fit of the
resulting averages is also shown with the slopes reported in Figure 6. As can
be seen from the fits, there is only a very weak dependence, if any, of the
breakdown voltage on exposed length.

The material’s dielectric constant was the primary driver of the difference in
the field at the ring for a given voltage. An FEA (Finite Element Analysis) of
the setup was performed for different materials with a 0.06 cm gap between the
ring and the insulator surface; an example FEA near the ring is shown in
Figure 7. For a 6 cm exposed insulator on UHMW PE, the peak field at 100 kV
was $470\pm 10$ kV/cm. Shrinking the insulator diameter to the minimum and
maximum of the thermal contractions led to a change of only 1.2% and 1.6% in
peak electric field when keeping the dielectric constant fixed.

Figure 5: Mean breakdown voltages as a function of exposed insulator length
for the materials that survived testing. Figure 6 gives the values of the
slopes, ($\Delta$V)/($\Delta$exposed length). Some configurations were
repeated at different times during the run of the experiment resulting in
multiple data points for a given material’s exposed length.

Figure 6: Slopes from the fit to the breakdown voltage versus exposed
insulator length data. Figure 7: The magnitude of the electric field near the
ring tip from the FEA of UHMW PE with a 6 cm exposed length. Here, 1 V is
applied; the field (V/cm) from a desired voltage can be scaled. The insulator
is on the left; the center conductor is on the left beyond the frame of the
image.

###  3.3 Insulators with Grooved Profiles

In addition to material and exposed insulator length, the surface profile of
an insulator is also a parameter of interest. Ridged or grooved surface
profiles are commonly used in high voltage bushings. For example, the often
seen ceramic bushings used in outdoor power distribution are designed to
prevent flash-over by increasing the creepage path and keeping it dry [12].
Grooves have also been employed in high voltage feedthroughs in liquid argon
[13]. In this section, we present a study of the performance of profiles in
liquid argon.

In UHMW PE, the profiles shown in Figure 8(a) and (b) were prepared to compare
ridge shape and spacing. In FR4, and polycarbonate, only the profile shown in
Figure 8(b) was evaluated.

Figure 8: The two profiles used in testing, and the test apparatus.

(a)  (b)

Figure 9: Summary plots for the mean breakdown voltage values as a function of
exposed insulator length for the UHMW PE samples.

The results from the grooved samples show at most a small effect on breakdown
voltage. The results for the smooth and grooved UHMW PE are shown in Figure 9.
It is worth noting that the traces were observed to follow the surface of the
grooves on the UHMW PE samples.

The remaining grooved materials all suffered damage of some sort. The grooved
polycarbonate test piece developed damaged ridges during testing. Only one
distance at 4.13 cm was evaluated and it is difficult to determine if the
ridges were made weaker during the initial installation of the ring, or if the
ridges were only damaged from electrical discharge. The grooved FR4 piece also
had only one exposed length at 6.67 cm evaluated due to damage. Here, however,
the ridges were not damaged, but rather a crack burned through from the center
conductor outward along fibers of the FR4.

##  4 Discussion

We have evaluated insulator performance by studying dielectric breakdown along
the surface of insulators in liquid argon in an open cryostat. In our setup, a
high voltage conductor was encased in the insulator under test. We have
studied a variety of candidate materials and compared their performance. We
have also evaluated the effect of exposed insulator length on breakdown
voltage.

###  4.1 Dependence of Breakdown Voltage on Material

The average breakdown voltage for a common exposed insulator length of 6 cm is
shown in Figure 10 versus the inverse of the materials’ permittivities.
Materials with high permittivity increase the field near the knife edge on the
ring. We find that the material dependence of breakdown voltage can be
entirely understood as a consequence of the sample’s permittivity and the
resulting electric field near the insulator surface, and does not display any
other material effects.

Figure 10: The breakdown voltage for several materials at an exposed insulator
length of 6 cm versus the inverse permittivity.

###  4.2 Dependence of Breakdown Voltage on Exposed Insulator Length

From our testing, we found little, if any, dependence on breakdown voltage
versus exposed insulator length as shown in Figure 6. This result differs from
broad experience at room temperature in the high voltage community that
generally relates an increase in creepage path to an increase in breakdown
voltage.

In an attempt to understand breakdown behavior in liquid argon, we turned to
the literature on breakdown in oil. A paper by Meek [14] is one of the first
to tackle breakdown in dense media, and provides a narrative for the breakdown
chain of events. This paper states, “The breakdown of a uniform field is
considered to occur by the transition of an electron avalanche proceeding from
cathode to anode into a self-propagating streamer, which develops from anode
to cathode to form a conducting filament between the electrodes.” This model
of an initializing avalanche followed by streamer propagation is still
accepted today.

In our setup, there is a concentrated electric field by design near the sharp
feature at the knife edge of the ring. This field is roughly independent of
ring depth as shown in Figure 11 where different exposed lengths yield a
similar electric field from the ring tip vertically along the insulator
length. The sharp field enables the initialization of a streamer.

Figure 11: The electric field magnitude along the vertical direction from the
ring tip for three exposed insulator lengths. The dominant field near the ring
tip is independent of exposed insulator length.

Once started, the streamer propagates toward the cathode. The streamer grows
by ionization from electrons and photons; therefore it needs energy to
propagate. In Reference [15], a mechanism is described, “The cathode-directed
streamer starts near the anode. It looks like and operates as a thin
conductive needle growing from the anode. The electric field at the tip of the
‘anode needle’ is very high, which stimulates the fast streamer propagation in
the direction of the cathode.”

In our test setup, just as the electric field on the edge of the ring created
a field strong enough to ionize the argon, the highly localized charge in the
streamer head concentrates the electric field from the center conductor
supporting further ionization for streamer growth. The field from the center
conductor is independent of elevation and is the dominating field along the
insulator. This field grows the streamer without any need for additional
fields along the insulator surface. Hence, the voltage required for breakdown
is independent of exposed insulator length.

Our results of breakdown voltages as a function of exposed insulator length
are shown in Figure 5 and the resulting slopes from the fits are summarized in
Figure 6. We see no significant material dependence, and further, the central
values are consistent with zero as one would expect from the model described.

###  4.3 Dependence of Breakdown Voltage on Insulator Profile

A number of arguments have been made to motivate the addition of grooves to
high voltage components in liquid argon. Path length arguments are popular, as
is the suggestion that creating a surface perpendicular or in opposition to
the electric field inhibits the travel of charge. We observed at most a modest
effect from the addition of grooves to our test pieces.

At least part of the efficacy of grooves in our tests is due to the reduction
in field by replacing plastic with argon near the grounded ring. During
testing, we did not know the precise location of the ring tip relative to the
grooves. FEAs of the field at a common cathode voltage for Profile II grooves
were performed for different ring heights relative to the groove. A reduction
in field can be seen in Figure 12 where the ratio of the magnitude of the
field for a grooved sample to that of a smooth sample is shown for a common
input voltage. Similar results were found for polycarbonate and FR4. The
largest effect is seen when the ring is between the middle of a valley of a
groove and the top edge of a peak. Here, the ratio of the field at a common
voltage of the smooth insulator, $E_{s}$, to the field of the grooved
insulator, $E_{G}$, was nearly 0.90 for UHMW PE, 0.84 for polycarbonate, and
0.75 for FR4. These values along with the observed breakdown voltages are
summarized in Table 2. One can scale the modeled peak electric fields by the
observed breakdown voltages, $V_{S}$ and $V_{G}$ for smooth and grooved
samples respectively, to obtain the field at breakdown. If one assumes there
is a similar electric field at breakdown between the geometries, then
$V_{S}E_{S}\simeq V_{G}E_{G}$. If $(V_{S}/V_{G})\leq(E_{G}/E_{S})$, the
increase in breakdown voltage of the grooved samples can be accounted for by
the reduction in field due to the change in the permitivity by removing the
insulator material. For polycarbonate and FR4, the ratios $E_{G}/E_{S}$ are in
rough agreement with the observed average breakdown voltages ratio
$V_{S}/V_{G}$.

(a)  (b)

Figure 12: Ratio of the electric field near the ring for different exposed insulator lengths with respect to the groove profile. Figure (a) is the ratio of the magnitude of the electric field in a line directed radially toward the sample from the ring tip. Figure (b) is the ratio of the magnitude of the electric field in a vertical line from the ring tip. Sample | Grooved Exposed | Grooved | Smooth | FEA | Measured
---|---|---|---|---|---
Length (cm) | $<V_{BD}>$ | $<V_{BD}>$ | $E_{G}/E_{S}$ | $V_{S}/V_{G}$
FR4 | 6.67 | $78.1\pm 14.3$ | 55.9 | 0.75 | 0.72
Polycarbonate | 4.13 | $98.4\pm 13.3$ | 84.6 | 0.84 | 0.86
UHMW PE | 1.11—15.56 |  $106.5\pm 17.9$ — $82.0\pm 11.1$ | 109.9—98.6 | 0.90 | 1—1.2
Table 2: The average breakdown voltage, $V_{BD}$, for smooth and Profile II
grooved samples, and their resulting ratio, $V_{S}/V_{G}$. The smooth value is
from the fitted line to the breakdown voltages as a function of exposed
length. Also included is the minimum ratio of the electric fields vertically
from the ring tip for the samples as computed by an FEA. Here the ring tip is
just below a valley and this is where the maximum effect on the electric field
is seen.

The UHMW PE Profile II sample had a measured $V_{S}/V_{G}>1$ in contrast to
the other samples, including the UHMW PE Profile I sample, and the FEA model.
The precise effect of grooves on performance is not clear, however, they do
not appear to have a large effect on breakdown voltage in this study as shown
in Figure 9(b) where the slopes and errors are reported for the grooved and
smooth UHMW PE. The values are again consistent with zero.

##  5 Conclusion

We have performed a study of electrical breakdown in liquid argon along the
surface of insulators surrounding a conductor with regard to breakdown path
length, insulator material, and insulator surface profile. The measurements
were performed in an open-to-air cryostat initially filled with commercial
grade argon, and the breakdown was initiated at the anode. We identified some
materials that failed mechanically in our test setup. We found the breakdown
voltage does not depend upon the length of the exposed insulator. Material-
specific effects could be explained as due to their permittivities.

The evaluation of surface profiles point to at most a modest effect on
breakdown voltage by adding grooves. Two grooved materials failed mechanically
after one exposed length test. Before failing, the breakdown voltage recorded
was lower when compared to a smooth sample. However, this change in breakdown
voltage could be accounted for by the change in the peak electric field
resulting from the grooved material geometry. A limited study of exposed
length of grooved UHMW PE pointed to little, if any, effect on breakdown
voltage with all of the average voltage gains with length being consistent
with zero.

A practical consequence of our observations (in impure liquid argon) is that
for insulator tubes surrounding high voltage electrodes, the peak surface
electric field is the relevant value. Thus, the permittivity of the materials,
provided they can survive mechanically, is an important parameter of concern.
Adjusting the radial dimension will likely also improve performance. Longer
insulators provide little, if any, improvement in high voltage standoff.
Grooving the insulator surface similarly seems to have little effect beyond
reducing the local field. Improvements must focus on preventing avalanche
initiation, or preventing the avalanche to escape to where it can grow to be a
longer streamer.

##  6 Acknowledgements

The authors wish to thank Brian Rebel and Stephen Pordes for their support,
editorial contributions, and scientific discussions, and Jim Walton and Alan
Hahn for their technical assistance.

Fermilab is operated by Fermi Research Alliance, LLC under Contract No. De-
AC02-07CH11359 with the United States Department of Energy.

## References

  * [1] A. Blatter et al., “Experimental study of electric breakdowns in liquid argon at centimeter scale,” JINST 9 (2014) P04006, arXiv:1401.6693 [physics.ins-det].
  * [2] F. Bay et al., “Evidence of electrical breakdown induced by bubbles in liquid argon,” arXiv:1401.2777 [physics.ins-det].
  * [3] M. Auger et al., “A method to suppress dielectric breakdowns in liquid argon ionization detectors for cathode to ground distances of several millimeters,” JINST 9 (2014) P07023, arXiv:1406.3929 [physics.ins-det].
  * [4] R. Acciarri et al., “Liquid Argon Dielectric Breakdown Studies with the MicroBooNE Purification System,” JINST 9 no. 11, (2014) P11001, arXiv:1408.0264 [physics.ins-det].
  * [5] Glassman High Voltage Inc., PO Box 317, 124 West Main Street, High Bridge, NJ 08829-0317, U.S.A.
  * [6] National Instruments Corp., 11500 N Mopac Expwy, Austin, TX 78759-3504.
  * [7] W. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech 18 (1951) 293–277.
  * [8] W. F. Gauster, “Über Oberflächeneffekte beim elektrischen Durchbruch von Flüssigkeiten,” Ingenieur Archiv 10 (1956) 160.
  * [9] “Design guide–module iii: Delrin acetal resin.”  http://plastics.dupont.com/plastics/pdflit/americas/delrin/230323c.pdf. Page 34, Figure 31.
  * [10] “Fr-4.”  http://en.wikipedia.org/wiki/FR-4.
  * [11] “Professional plastics.”  http://www.professionalplastics.com.
  * [12] L. L. Alston, ed., High-Voltage Technology.  Harwell Post-Graduate Series. Oxford University Press, 1968.  Creepage path is discussed in Section 14.5.1.
  * [13] S. Amerio et al., “Design, construction and tests of the ICARUS T600 detector,” Nucl. Instrum. Methods 527 (2004) 329.
  * [14] J. M. Meek, “A theory of spark discharge,” Phys. Rev. 57 (Apr, 1940) 722–728. http://link.aps.org/doi/10.1103/PhysRev.57.722.
  * [15] A. Fridman, A. Gutsol, and Y. Cho, “Non-thermal atmospheric pressure plasma,” vol. 40 of Advances in Heat Transfer, pp. 1 – 142.  Elsevier, 2007.  http://www.sciencedirect.com/science/article/pii/S0065271707400016.


# Self-Supervised Arbitrary-Scale Point Clouds Upsampling via Implicit Neural
Representation

Wenbo Zhao1,2, Xianming Liu1,2, Zhiwei Zhong1,2, Junjun Jiang1,2, Wei Gao3, Ge
Li3, Xiangyang Ji4
1Harbin Institute of Technology, 2Peng Cheng Laboratory
3Peking University Shenzhen Graduate School, 4Tsinghua University
Corresponding author<EMAIL_ADDRESS>
###### Abstract

Point clouds upsampling is a challenging issue to generate dense and uniform
point clouds from the given sparse input. Most existing methods either take
the end-to-end supervised learning based manner, where large amounts of pairs
of sparse input and dense ground-truth are exploited as supervision
information; or treat up-scaling of different scale factors as independent
tasks, and have to build multiple networks to handle upsampling with varying
factors. In this paper, we propose a novel approach that achieves self-
supervised and magnification-flexible point clouds upsampling simultaneously.
We formulate point clouds upsampling as the task of seeking nearest projection
points on the implicit surface for seed points. To this end, we define two
implicit neural functions to estimate projection direction and distance
respectively, which can be trained by two pretext learning tasks. Experimental
results demonstrate that our self-supervised learning based scheme achieves
competitive or even better performance than supervised learning based state-
of-the-art methods. The source code is publicly available at
https://github.com/xnowbzhao/sapcu.

##  1 Introduction

Point clouds serve as a popular tool to represent 3D data due to their
flexibility and compactness in describing objects/scenes with complex geometry
and topology. They can be easily captured by modern scanning devices, and have
been widely used in many applications, such as autonomous driving, robotics,
etc. However, due to the inherent limitations of 3D sensing technology, raw
point clouds acquired from 3D scanners are usually sparse, occluded and non-
uniform. In many downstream applications, such as surface reconstruction and
understanding, dense point clouds are desired for representing shapes with
richer geometric details. Accordingly, people turn to develop a computational
approach, referred to as point clouds upsampling, which has attracted
extensive attention in both industry and academia [21, 20, 7, 13, 22, 8].

Unlike conventional grid based images, point clouds are irregular and
unordered, which make point clouds upsampling a more challenging task than its
2D image counterpart. The goal of point clouds upsampling is two-fold: 1)
generating a dense point set from the sparse input to provide richer details
of the object; 2) generating a uniform and complete point set to cover the
underlying surface faithfully.

In recent years, deep neural networks based point clouds upsampling approaches
emerge and become popular, which adaptively learn structures from data and
achieve superior performance than traditional methods, such as optimization-
based ones [1, 5, 6]. For instance, Yu et al. [21] propose to learn multi-
level features per point and expand the point set via a multi-branch
convolution unit implicitly in feature space, which is then split to a
multitude of features for reconstruction of an upsampled point set. Wang et
al. [20] propose to progressively train a cascade of patch-based upsampling
networks on different levels of detail. Li et al. [7] apply generative
adversarial network into point clouds upsampling, which constructs an up-down-
up expansion unit in the generator for upsampling point features with error
feedback and self-correction, and formulate a self-attention unit to enhance
the feature integration. To better represent locality and aggregate the point
neighborhood information, Qian et al. [13] propose to use a Graph
Convolutional Network to perform point clouds upsampling.

In summary, the outlined deep learning based methods take a general approach:
first design an upsampling module to expand the number of points in the
feature space, then formulate losses to enforce the output points to be as
close as possible to the ground truth dense points. However, these methods
suffer from the following two limitations:

End-to-End Training. These methods are trained in an end-to-end supervised
learning manner, which requires a large amount of pairs of input sparse and
ground-truth dense point sets as the supervision information. The training
data is constructed by sampling from synthetic models, whose distributions are
inevitably biased from that of the real-scanned data. This would lead the
trained models to have poor generalization ability in real-world applications.
Thus, it is more desirable to develop self-supervised or unsupervised point
cloud upsampling schemes.

Fixed Upsampling Factor. Due to resource constraints, such as display
resolution and transmission bandwidth, the required upsampling factor is
usually various. These existing methods treat upscaling of different scale
factors as independent tasks, which train a specific deep model for a pre-
defined factor and have to build multiple networks to handle upsampling with
varying factors. This manner is clumsy, which increases both model complexity
and training time significantly. Thus, it is more desirable to develop unified
point cloud upsampling schemes that can handle arbitrary scale factor.

Some methods developed very recently [9, 22, 19, 14] investigate the above
limitations and attempt to address them:

  * •

Regarding self-supervised point cloud upsampling, Liu et al. [9] propose the
coarse-to-fine framework, which downsamples the input sparse patches into
sparser ones and then exploits them as pairs of supervision information to
perform end-to-end training. [22] proposes an end-to-end self-supervised
learning manner, in which the loss functions enforce the input sparse point
cloud and the generated dense one to have similar 3D shapes and rendered
images. However, these two methods are still limited to a fixed upsampling
factor.

  * •

Regarding arbitrary-scale upsampling, inspired by the counterpart Meta-SR in
image [4], Ye et al. [19] propose Meta-PU for magnification-flexible point
cloud upsampling, in which the meta-subnetwork is learned to adjust the
weights of the upsampling blocks dynamically. Qian et al. [14] design a neural
network to adaptively learn unified and sorted interpolation weights as well
as the high-order refinements, by analyzing the local geometry of the input
point cloud. However, these two methods still follow the end-to-end supervised
learning manner, which need to construct a large-scale training set including
ground truth dense point sets with scales within a wide range.

In this paper, we propose a novel and powerful point clouds upsampling method
via implicit neural representation, which can achieve self-supervised and
magnification-flexible upsampling simultaneously. Specifically, to get rid of
the requirement of ground truth dense point clouds, we do not directly learn
the mapping between the input sparse and output dense point sets.
Alternatively, inspired by the notion that an implicit surface can be
represented by signed distance function (SDF) [15, 10, 11], we turn to seek
the nearest projection points on the object surface for given seed points
through two implicit neural functions, which are used to estimate projection
direction and distance respectively. The two function can be trained by two
constructed pretext self-supervised learning tasks. In the way, as long as the
seed points are sampled densely and uniformly, we can produce high-resolution
point clouds that are dense, uniform and complete. To guarantee the uniformity
of seeds sampling, we exploit equally-paced 3D voxels to divide the space of
point cloud. Experimental results demonstrate our self-supervised learning
based scheme achieves competitive or even better performance that supervised
learning based state-of-the-art methods. The main contributions of this work
are highlighted as follows:

  * •

To the best of our knowledge, we are the first in the literature to
simultaneously consider self-supervised and arbitrary-scale point clouds
upsampling.

  * •

We formulate point clouds upsampling as the task of seeking nearest projection
points on the implicit surface for seed points, which can be done by two
implicit neural functions trained by pretext tasks. From the generated dense
point clouds, we can achieve arbitrary-scale upsampling by farthest point
sampling.

  * •

Although our method is self-supervised, it produces high-quality dense point
clouds that are uniform and complete, and achieves competitive objective
performance and even better visual performance compared with state-of-the-art
supervised methods.

##  2 Method

Define $\mathcal{X}=\\{{\mathbf{p}}_{i}\\}_{i=1}^{n}\in\mathbb{R}^{n\times 3}$
as the input sparse point cloud. For a desirable scaling factor $r$, we target
to obtain a corresponding dense point cloud
$\mathcal{Y}=\\{{\mathbf{p}}_{i}\\}_{i=1}^{N}\in\mathbb{R}^{N\times 3}$
including $N=\lfloor r\times n\rfloor$ points. ${\mathcal{S}}$ is defined as
the underlying surface of the dense point cloud. The high-resolution point
cloud $\mathcal{Y}$ is required to be dense and uniform, as well as be able to
handle occlusion and noise, i.e., to be complete and clean.

Unlike the existing methods that take the end-to-end training framework, we do
not directly learn the mapping between the input sparse and output dense point
sets, but instead seek the nearest projection point on the object surface for
a given seed point in a self-supervised manner. By densely and uniformly
sampling seed points in the space, we can obtain dense and approximately
uniform projection points, which can describe the underlying surface
faithfully. The proposed self-supervised point clouds upsampling strategy
includes the four steps:

  * •

Seeds Sampling. We represent the geometric space of the point cloud by 3D
voxel grid, from which we choose the centres of voxels that are close to the
implicit surface $\mathcal{S}$ as the seed points.

  * •

Surface Projection. For seed points, we project them to the implicit surface
$\mathcal{S}$ to obtain the projected points, which construct the generated
dense point cloud.

  * •

Outliers Removal. We further remove the projected points that are generated by
far seed points to achieve cleaner point cloud.

  * •

Arbitrary-Scale Point Cloud Generation. To obtain the desired upsampling
factor, we adjust the number of vertices of the generated dense clouds by
farthest point sampling.

In the following, we introduce each step in detail.

###  2.1 Seeds Sampling

To obtain uniformly sampled seed points, given a point cloud, we divide the 3D
space into equally spaced voxels $\\{{\mathbf{V}}_{(x,y,z)}\\}$, where
${\mathbf{V}}_{(0,0,0)}$ represents the voxel locating in the origin of the 3D
Cartesian coordinate system. We define the resolution of a 3D voxel volume as
$l\times l\times l$. The centre of a voxel ${\mathbf{V}}_{(x,y,z)}$ is thus
${\mathbf{c}}_{(x,y,z)}=[x+0.5*l,y+0.5*l,z+0.5*l]$. The centres of voxels are
equally distributed in the space, which serve as good candidates of seed
points. However, we do not use them all, but choose the ones that are closed
to the underlying surface of the point cloud.

A reasonable principle to choose centres is according to their distances to
the surface ${\mathcal{S}}$. We choose a centre ${\mathbf{c}}_{(x,y,z)}$ as
the seed if its distance to the surface within a preset range:
$\mathbb{\text{Dist}}({\mathbf{c}}_{(x,y,z)},{\mathcal{S}})\in[D_{l},D_{u}]$.
The difficulty lies in that we cannot directly compute the distance
$\mathbb{\text{Dist}}({\mathbf{c}}_{(x,y,z)},{\mathcal{S}})$, since the
underlying surface ${\mathcal{S}}$ is unknown. We propose an alternative
strategy to approximate
$\mathbb{\text{Dist}}({\mathbf{c}}_{(x,y,z)},{\mathcal{S}})$. Specifically,
from the input sparse point sets $\mathcal{X}$, we choose $M$ points that are
nearest to ${\mathbf{c}}_{(x,y,z)}$, denoted as
$\\{{\mathbf{p}}_{c,1},{\mathbf{p}}_{c,2},\cdots,{\mathbf{p}}_{c,m},\cdots,{\mathbf{p}}_{c,M}\\}$
that are ordered from near to far. From these points, we can form a set of
triangles
$\\{T_{m}=({\mathbf{p}}_{c,1},{\mathbf{p}}_{c,2},{\mathbf{p}}_{c,m})\\}_{m=3}^{M}$.
We then perform the following approximation:

$\mathbb{\text{Dist}}({\mathbf{c}}_{(x,y,z)},{\mathcal{S}})\approx\min\mathbb{\text{Dist}}({\mathbf{c}}_{(x,y,z)},t),t\in\\{T_{m}\\}_{m=3}^{M}$
(1)

where $t$ represents a point contained in the constructed triangles. Finally,
we obtain the seed points set $C$. By setting appropriate $l$, we can generate
dense and uniformly distributed seed points.

Figure 1: The network architecture of implicit neural function.

###  2.2 Surface Projection

With the sampled seed points, the next step is to seek the their projection
points on the surface, which are the target points of the generated dense
point cloud.

In the field of 3D computer vision and graphics, it is well-known that an
implicit surface can be defined as a signed distance function (SDF) [15, 10,
11]. SDF, when passed the coordinates of a point in space, outputs the closest
distance of this point to the surface, whose sign indicates whether the point
is inside or outside of the surface. Inspired by SDF, we propose the following
feasible approach to estimate the projected point on the surface for a query
seed point.

It is worth noting that, the computation strategies of SDF, such as [15, 10,
11], cannot be directly applied for our purpose. For a 3D query point
${\mathbf{x}}$, SDF outputs: $SDF({\mathbf{x}})=s,s\in\mathbb{R}$. The sign of
$s$ only indicates it is inside or outside of a shape, but does not provide
the direction to the surface. In our method, for a seed point ${\mathbf{c}}\in
C$, we divide the task of estimating projection point into two sub-tasks: 1)
estimating the projection direction ${\mathbf{n}}\in[-1,1]^{3}$; 2) estimating
the projection distance $d\in\mathbb{R}$. Then the coordinate of the
projection point of the seed point ${\mathbf{c}}$ can be obtained as:
${\mathbf{c}}_{p}={\mathbf{c}}+{\mathbf{n}}*d$.

Projection Direction Estimation. We train a multi-layer fully-connected neural
network $f_{n}(\cdot;\Theta_{n})$ for this purpose, which takes the query
point ${\mathbf{c}}$ and the sparse point cloud $\mathcal{X}$ as inputs:

${\mathbf{n}}=f_{n}({\mathbf{c}},\mathcal{X};\Theta_{n})$ (2)

To reduce the computational complexity, we take $k$ nearest points to
${\mathbf{c}}$ in $\mathcal{X}$ instead of the whole $\mathcal{X}$ as input.
We denote this subset of points as
${\mathcal{X}}_{c}=\\{{\mathbf{p}}_{1},\cdots,{\mathbf{p}}_{k}\\}$. Moreover,
to facilitate the inference process of neural networks, we perform
normalization on the point coordinates by setting ${\mathbf{c}}$ as the
origin. In this way, we can simplify the estimation function as:

${\mathbf{n}}=f_{n}(\widehat{\mathcal{X}}_{c};\Theta_{n})$ (3)

where
$\widehat{\mathcal{X}}_{c}=\\{{\mathbf{p}}_{1}-{\mathbf{c}},\cdots,{\mathbf{p}}_{k}-{\mathbf{c}}\\}$.

Projection Distance Estimation. Similarly, for estimating the projection
distance ${\mathbf{d}}$, we also train a multi-layer fully-connected neural
network $f_{d}(\cdot;\Theta_{n})$, which takes the query point ${\mathbf{c}}$,
the subset of nearest points ${\mathcal{X}}_{c}$ and the estimated projection
direction ${\mathbf{n}}$ as inputs:

$d=f_{d}({\mathbf{c}},{\mathcal{X}}_{c},{\mathbf{n}};\Theta_{d})$ (4)

Normalization is also helpful for this network. Different from $f_{n}$, here
the input ${\mathbf{n}}$ involves direction. Therefore, it should perform
normalization on both position and direction, which can be done in two stages:
1) moving ${\mathbf{c}}$ to the origin; 2) applying the rotation matrix
$\mathbf{W}_{r}$ to rotate ${\mathbf{n}}$ to a specific direction
${\mathbf{n}}_{t}$, i.e., ${\mathbf{n}}_{t}=\mathbf{W}_{r}{\mathbf{n}}$. After
normalization, ${\mathcal{X}}_{c}$ becomes
$\widetilde{\mathcal{X}}_{c}=\\{\mathbf{W}_{r}({\mathbf{p}}_{1}-{\mathbf{c}}),\cdots,\mathbf{W}_{r}({\mathbf{p}}_{k}-{\mathbf{c}})\\}$,
which is the only required input for $f_{d}$:

$d=f_{d}(\widetilde{\mathcal{X}}_{c};\Theta_{d})$ (5)

###  2.3 Outliers Removal

In the step of seeds sampling, some points that are actually far away from
${\mathcal{S}}$ may be included into the seed points set $C$ due to the error
in approximation. The normal vector and distance of these points cannot be
well estimated, leading to outliers in the resulting dense point cloud. We
turn to exploit the post-processing procedure to remove them.

Specifically, for a projection point ${\mathbf{c}}_{p}$, we find its $v$
nearest points $\\{{\mathbf{c}}_{p,1},\cdots,{\mathbf{c}}_{p,v}\\}$. We then
compute the average bias between ${\mathbf{c}}_{p}$ and them:

$b_{p}=\frac{1}{v}\sum_{i=1}^{v}\text{Dist}({\mathbf{c}}_{p},{\mathbf{c}}_{p,i})$
(6)

For all projection points, we do in the same way to get $\\{b_{p}\\}$, the
average of which is denoted as $\bar{b}$. We determine a point as outlier if
it satisfies $b_{p}>\lambda\bar{b}$, where $\lambda$ is set as 1.5 in
practical implementation.

###  2.4 Arbitrary-Scale Point Cloud Generation

Note that the above process cannot accurately control the number of vertices
generated. Thus, it is necessary to adjust the number of vertices to achieve
upsampling with the desired scale factor. In our context, we first perform
inverse normalization on the generated point cloud, and then adjust the number
of vertices to $N$ by the farthest point sampling algorithm [12].

Scale |  2$\times$ |  4$\times$
---|---|---
Metric ($10^{-3}$) |  CD $\downarrow$ |  EMD $\downarrow$ |  F-score $\uparrow$ |  mean $\downarrow$ |  std $\downarrow$ |  CD $\downarrow$ |  EMD $\downarrow$ |  F-score $\uparrow$ |  mean $\downarrow$ |  std $\downarrow$
Fixed Scale |  PU-Net [21] | 12.9 (6) | 6.75 (6) | 334 (5) | 4.02 (5) | 5.62 (4) | 11.3 (7) | 7.02 (7) | 462 (7) | 5.05 (7) | 6.81 (6)
MPU [20] | - | - | - | - | - | 10.4 (4) | 5.64 (5) | 527 (4) | 3.61 (4) | 5.50 (3)
PU-GAN [7] | 12.7 (5) | 5.09 (5) | 326 (6) | 4.32 (6) | 6.01 (5) | 10.9 (5) | 6.66 (6) | 484 (6) | 4.66 (6) | 6.56 (5)
PU-GCN [13] | 12.2 (3) | 4.94 (4) | 360 (3) | 3.47 (2) | 5.09 (2) | 10.2 (3) | 5.50 (4) | 537 (3) | 3.35 (2) | 5.01 (2)
PU-DR [8] | 11.0 (1) | 3.55 (1) | 409 (1) | 2.30 (1) | 4.05 (1) | 8.71 (1) | 3.98 (2) | 625 (1) | 2.24 (1) | 3.89 (1)
Arbitrary Scale |  Meta-PU [19] | 12.6 (4) | 3.94 (2) | 339 (4) | 3.77 (4) | 5.41 (3) | 10.9 (5) | 3.56 (1) | 506 (5) | 3.89 (5) | 5.60 (4)
Proposed | 12.1 (2) | 4.93 (3) | 371 (2) | 3.49 (3) | 10.2 (6) | 10.1 (2) | 4.87 (3) | 561 (2) | 3.49 (3) | 9.35 (7)
Scale | $8\times$ | $16\times$
Metric ($10^{-3}$) |  CD $\downarrow$ |  EMD $\downarrow$ |  F-score $\uparrow$ |  mean $\downarrow$ |  std $\downarrow$ |  CD $\downarrow$ |  EMD $\downarrow$ |  F-score $\uparrow$ |  mean $\downarrow$ |  std $\downarrow$
Fixed Scale |  PU-Net [21] | 9.67 (5) | 8.91 (6) | 611 (6) | 4.85 (6) | 6.81 (5) | 9.25 (7) | 10.4 (7) | 632 (7) | 6.04 (7) | 8.04 (6)
MPU [20] | - | - | - | - | - | 8.12 (5) | 7.74 (6) | 727 (5) | 4.01 (6) | 6.11 (5)
PU-GAN [7] | 8.82 (4) | 5.05 (3) | 676 (4) | 3.76 (4) | 5.51 (3) | 7.52 (2) | 6.02 (4) | 770 (3) | 3.14 (2) | 4.92 (2)
PU-GCN [13] | 8.78 (3) | 6.41 (5) | 678 (3) | 3.33 (2) | 5.06 (2) | 7.80 (4) | 7.44 (5) | 749 (4) | 3.39 (4) | 5.11 (3)
PU-DR [8] | 8.34 (1) | 3.95 (1) | 708 (1) | 2.99 (1) | 4.97 (1) | 7.29 (1) | 4.51 (1) | 779 (1) | 2.92 (1) | 4.72 (1)
Arbitrary Scale |  Meta-PU [19] | 9.71 (6) | 4.33 (2) | 625 (5) | 3.95 (5) | 5.68 (4) | 8.96 (6) | 5.62 (2) | 676 (6) | 3.87 (5) | 5.59 (4)
Proposed | 8.70 (2) | 5.53 (4) | 706 (2) | 3.48 (3) | 8.84 (6) | 7.65 (3) | 5.98 (3) | 772 (2) | 3.35 (3) | 8.34 (7)
Table 1: Objective performance comparison with respect to CD, EMD, F-score,
mean and std with state-of-the-art methods. The ranking numbers are also
provided.

##  3 Implicit Neural Networks Training

In this section, we introduce the architecture of implicit neural networks and
the training strategy.

###  3.1 Architectures

The networks $f_{n}$ and $f_{d}$ share the same architecture as shown in
Figure 1, which borrows the idea of encoder-decoder framework [10]. The
network takes the normalized subset of points as input, which are feed into
the encoder to obtain a 2048-dimensional feature vector. Here we employ a
state-of-the-art method DGCNN [17] as the encoder to preserve surface
information in multi-levels. The feature vector is then passed through 4 full-
connected (FC) layers with batch normalization and ReLU, the output
dimensional of which are 1024, 512 and 128 respectively. The output
dimensional of the last FC layer is 3 for the projection direction
${\mathbf{n}}$ and 1 for the the projection distance $d$. Note that the design
of the network is not the main contribution of this paper. We can exploit any
suitable network for our purpose.

###  3.2 Training Data Preparation

To train the two implicit neural functions, we construct two pretext tasks,
for which we prepare training samples that consist of 3D points and the
corresponding ground truth projection direction and distance values. We train
with normalized watertight meshes that are constructed by the TSDF-Fusion
presented in [16, 10] from a subset of the ShapeNet [2] that consists of 13
major categories. Since $f_{n}$ and $f_{d}$ are designed with different
purposes, we prepare different training pairs for them:

  * •

For preparing the training data of $f_{n}$, we firstly generate the seed
points: 50K seed points are randomly selected around the mesh surface by
limiting that the distances between them and the surface are within a preset
range $[D_{l}-\epsilon,D_{u}+\epsilon]$, where $\epsilon$ is introduced to
increase robustness. For a seed point ${\mathbf{c}}$, we find the nearest
point on the mesh and sample 5 points around it, then compute the average
vector ${\mathbf{d}}$ between ${\mathbf{c}}$ and them. In this way, we can
effectively handle the error in mesh reconstruction. The ground truth
$\widehat{\mathbf{n}}$ is finally derived as the normalization of
${\mathbf{d}}$. Secondly, for every 16 seed points, we randomly select 2048
points as the corresponding sparse point cloud $\mathcal{X}$.

  * •

For preparing the training data of $f_{d}$, 50K seed points are randomly
selected around the mesh surface in the same way as $f_{n}$. We then find the
corresponding nearest projection points on the surface. The distance between a
seed point and the projection point is used as the ground truth $\widehat{d}$.
The generation of $\mathcal{X}$ is in the same way as $f_{n}$.

It is worth noting that, although the training data are generated from
watertight meshes, our scheme is capable to handle non-watertight point
clouds, which can be observed in the real-world cases shown in Figure 6.

###  3.3 Training Details

The training of $f_{n}$ and $f_{d}$ is done by minimizing the sum over losses
between the predicted and real direction/length values under the mean squared
error (MSE) loss function.

The training process is conducted on a server with two Tesla V100 GPUs. The
networks are trained for 1200 epochs with a batch size of 64, using the Adam
algorithm. Following [10], the learning rate is set as $10^{-4}$, and the
other hyperparameters of Adam are set as $\beta_{1}=0.9$, $\beta_{2}=0.999$,
$epsilon=10^{-8}$, weight decay $=0$.

##  4 Experiments

In this section, we provide extensive experimental results to demonstrate the
superior performance of our method.

###  4.1 Comparison Study

We compare the proposed self-supervised arbitrary-scale point clouds
upsampling (SSAS) method with several state-of-the-art works, which can be
divided into two categories in term of the scale factor: 1) fixed scale
methods, including PU-Net [21], MPU [20], PU-GAN [7], PU-GCN [13], PU-DR [8];
2) arbitrary scale method, including Meta-PU [19]. Note that these methods are
all supervised learning based. The compared models are trained with the
released codes by their authors, following the default settings in their
papers.

We train all these compared methods following the approach mentioned in [19]
for fair comparison. The test samples are from the dataset adopted by [21,
19]. We non-uniformly sample 2048 points using Poisson disk sampling from 20
test models to form the test set.

###  4.2 Parameters Setting

The side length of voxel $l$ and the range of the distance between seed point
and surface $[D_{l},D_{u}]$ decide the number of seed points. However, the
number is also depended on the shape of input point cloud. To ensure that
enough points are generated, we set $l=0.004$ and
$\left[D_{l},D_{u}\right]=\left[0.011,0.015\right]$. Under this condition, the
minimum number of generated points is 99001 (Chair), which meets the demands
of 16$\times$ or higher scale upsampling. The specific direction
${\mathbf{n}}_{t}$ in normalization can be arbitrarily chosen. We set
${\mathbf{n}}_{t}=\left(1,0,0\right)$ in our experiments. The number of
nearest points $k$ is set as 100 in Section 2.2 and 30 in Section 2.3. The
number of nearest point $M$ for computing
$\mathbb{\text{Dist}}({\mathbf{c}}_{(x,y,z)},{\mathcal{S}})$ affects the
number of outliers and continuity of seed points. We set $M=10$ and further
discuss the effect of different $M$ in ablation study.

Scale | $2\times$ | $4\times$
---|---|---
$p$ ($10^{-2}$) |  0.2% $\downarrow$ |  0.4% $\downarrow$ |  0.6% $\downarrow$ |  0.8% $\downarrow$ |  1.0% $\downarrow$ |  0.2% $\downarrow$ |  0.4% $\downarrow$ |  0.6% $\downarrow$ |  0.8% $\downarrow$ |  1.0% $\downarrow$
Fixed Scale |  PU-Net [21] | 3.05 (6) | 2.33 (6) | 2.03 (6) | 1.86 (6) | 1.75 (6) | 2.72 (7) | 2.19 (7) | 1.97 (7) | 1.84 (7) | 1.76 (7)
MPU [20] | - | - | - | - | - | 2.53 (6) | 2.03 (6) | 1.80 (6) | 1.67 (6) | 1.59 (6)
PU-GAN [7] | 2.46 (3) | 1.86 (3) | 1.62 (4) | 1.50 (4) | 1.43 (4) | 2.45 (4) | 1.94 (5) | 1.73 (5) | 1.62 (5) | 1.56 (5)
PU-GCN [13] | 2.68 (5) | 2.06 (5) | 1.80 (5) | 1.65 (5) | 1.57 (5) | 2.40 (3) | 1.93 (4) | 1.72 (4) | 1.61 (4) | 1.54 (4)
PU-DR [8] | 1.83 (1) | 1.34 (1) | 1.17 (1) | 1.09 (1) | 1.06 (1) | 1.77 (2) | 1.46 (2) | 1.34 (2) | 1.28 (2) | 1.24 (2)
Arbitrary Scale |  Meta-PU [19] | 2.53 (4) | 1.86 (3) | 1.58 (3) | 1.43 (3) | 1.35 (3) | 2.50 (5) | 1.87 (3) | 1.60 (3) | 1.45 (3) | 1.37 (3)
Proposed | 1.96 (2) | 1.50 (2) | 1.33 (2) | 1.25 (2) | 1.21 (2) | 1.72 (1) | 1.40 (1) | 1.27 (1) | 1.21 (1) | 1.19 (1)
Scale | $8\times$ | $16\times$
p ($10^{-2}$) |  0.2% $\downarrow$ |  0.4% $\downarrow$ |  0.6% $\downarrow$ |  0.8% $\downarrow$ |  1.0% $\downarrow$ |  0.2% $\downarrow$ |  0.4% $\downarrow$ |  0.6% $\downarrow$ |  0.8% $\downarrow$ |  1.0% $\downarrow$
Fixed Scale |  PU-Net [21] | 2.64 (5) | 2.21 (6) | 2.02 (6) | 1.91 (6) | 1.84 (6) | 2.86 (6) | 2.42 (6) | 2.22 (7) | 2.10 (7) | 2.02 (7)
MPU [20] | - | - | - | - | - | 2.30 (4) | 1.96 (4) | 1.82 (4) | 1.73 (4) | 1.69 (4)
PU-GAN [7] | 1.72 (2) | 1.47 (3) | 1.38 (3) | 1.34 (3) | 1.32 (3) | 1.79 (3) | 1.57 (3) | 1.48 (3) | 1.44 (3) | 1.41 (3)
PU-GCN [13] | 2.31 (4) | 1.93 (4) | 1.75 (4) | 1.65 (5) | 1.59 (5) | 2.42 (5) | 2.07 (5) | 1.91 (5) | 1.82 (5) | 1.76 (5)
PU-DR [8] | 1.42 (1) | 1.20 (1) | 1.13 (1) | 1.11 (1) | 1.11 (1) | 1.56 (1) | 1.38 (1) | 1.32 (1) | 1.29 (1) | 1.28 (1)
Arbitrary Scale |  Meta-PU [19] | 2.72 (6) | 2.05 (5) | 1.76 (5) | 1.60 (4) | 1.51 (4) | 3.27 (7) | 2.51 (7) | 2.14 (6) | 1.93 (6) | 1.79 (6)
Proposed | 1.72 (2) | 1.45 (2) | 1.34 (2) | 1.29 (2) | 1.27 (2) | 1.75 (2) | 1.51 (2) | 1.41 (2) | 1.36 (2) | 1.33 (2)
Table 2: Uniformity performance comparison with respect to NUC scores. The
ranking numbers are also provided.

(a) (b) (c) (d) (e) (f) (g) (h) (i)

Figure 2: $4\times$ point upsampling results of Chair, Camel and Fandisk. (a)
the input point cloud; (b) the ground truth; (c) to (i) the results of PU-Net
[21] , MPU [20], PU-GAN [7], PU-GCN [13], PU-DR [8], Meta-PU [19] and ours.
Please enlarge the PDF for more details.

###  4.3 Objective Performance Comparison

Objective Evaluation. We employ six popular metrics for objective evaluation:
1) Chamfer Distance (CD) and Earth Mover Distance (EMD) [21]: which evaluate
the similarity between the predicted points and ground truth one in the
Euclidean space. For both metrics, smaller is better. 2) F-score [18]: which
treats upsampling as a classification problem. For this metric, larger is
better. 3) mean and std [21]: which evaluate the distance between the
predicted point cloud and ground truth mesh. For both metrics, smaller is
better. 4) Normalized Uniformity Coefficient (NUC) [21]: which evaluates the
uniformity of points on randomly selected disk with different area percentage
$p=0.2\%,0.4\%,0.6\%,0.8\%,1.0\%$. For this metric, smaller is better.

In Table 1, we offer the comparison results for four scale factors
$\left[2\times,4\times,8\times,16\times\right]$ with respect to CD, EMD,
F-score, mean and std. Surprisingly, it can be found that, although our model
is trained in a self-supervised manner without accessing to the ground-truth,
it achieves competitive performance with those supervised learning based ones
with respect to metrics CD, EMD, F-score and mean. Taking CD for example, our
method is ranked #2, #2, #2 and #3 among seven compared methods for
$\left[2\times,4\times,8\times,16\times\right]$ respectively. Similar results
can be found for F-score, for which our method is ranked #2 for all cases.
Note that our method performs worst with respect to std, which is because that
outliers cannot be completely removed.

Table 2 shows the uniformity evaluation results with respect to NUC. Our
method is ranked #2 for $\left[2\times,8\times,16\times\right]$, just blow PU-
DR [8]. In the case of $4\times$, our method is ranked #1. These results
demonstrate that the proposed method produces dense and uniform point clouds.

Inference Time Cost Comparison. This experimental analysis is conducted on a
server with two 1080Ti GPU. In our inference process, estimation of direction
and length are the most time-consuming steps. According to the experiments,
generating 40000 projected points by our method would cost 46s in average. As
a comparison, the time costs of generating 40000 points ($16\times$) are
342.4s by MPU [20] and 0.228s by Meta-PU [19]. It should be noticed that the
estimation of each point is independently performed and thus can be done in
parallel to speed up significantly.

###  4.4 Subjective Performance Comparison

Input $4\times$ $16\times$ $64\times$

Figure 3: $4\times$, $16\times$ and $64\times$ upsampling results of Moai.

512 points 1024 points 2048 points 4096 points

Figure 4: $4\times$ Upsampling results of Eight with varying size of input.
The first row are the inputs, the second row are the corresponding upsampling
results.

Clean 1% 2%

Figure 5: $4\times$ Upsampling results of Star with different additive
Gaussian noise level.  Figure 6: $8\times$ Upsampling result on a real-world
sample from KITTI. Please enlarge the PDF for more details.

(a) (b) (c)

Figure 7: Ablation on the choice of $M$. (a) input point cloud, (b) $M=3$, (c)
$M=10$.

(a) (b) (c)

Figure 8: Ablation on the outliers removal. (a) input point cloud, (b) without
outliers removal, (c) with outliers removal.

Visual Comparison. Figure 2 illustrates the $4\times$ upsampling results
generated by our method and the compared state-of-the-art methods on three
models Chair, Camel and Fandisk. The results show that our method achieves
better visual performance than other methods. The produced high-resolution
point clouds are dense and uniform, which also have continuous and complete
contours. Specifically, results of the highlighted part of Chair show that our
method succeeds in recovering structure from very few points; results of the
highlighted part of Camel show that our method can handle complex contour.
Furthermore, our method can reconstruct the edge region very well, as
demonstrated in the highlighted part of Fandisk. The above visual comparisons
verify the superiority of our proposed method.

Results on Variable Scales. Figure 3 shows the upsampling results of Moai with
different scale factors.

It can be observed that the contours of all the results are consistent, and
the uniformity of points is well preserved.

Robustness against Varying Sizes of Input. Figure 4 shows the $4\times$
upsampling results of Eight with different sizes of input point sets. Our
method generates consistent outlines regardless of the number of input points.
Figure 5 shows the $4\times$ upsampling results of star with noise level 0%,
1% and 2%. Our scheme also works well on the noisy input while the uniformity
is well preserved. Overall, our method is robust to the input size and noise.

Result on Real-world Sample. We choose one real-world sample from KITTI [3] to
evaluate the generalization capability of our method. In Figure 6, the
upsampling results of three different regions are presented. It can be found
that, even though the input point cloud is sparse and non-uniform, our scheme
can still recover the high-resolution one very well.

###  4.5 Ablation Study

About the choice of $M$. $M$ works in the distance approximation of a seed
point to the underlying surface, which affects the number of outliers and the
continuity of projected points. When $M$ increases, both the continuity and
the number of outliers increase. To show the effect of different $M$, we
provided the 4$\times$ upsampling result of cow with $M=3$ and $M=10$. The
results are shown in Figure 7. It can be observed that, when $M=3$, no outlier
is introduced to the blue box. However, the surface is discontinuous in the
red box. When $M=10$, the surface in the red box becomes continuous, however,
a few amount of outliers is still introduced to the blue box after outliers
removal.

About the necessity of outliers removal. To show the effect of outliers
removal, we provid the 4$\times$ upsampling result of cow with and without
outliers removal. The results are shown in Figure 8. It can be observed that
the outliers removal does not affect the smooth region in the red box, while
it can remove most of the outliers in the blue box.

###  4.6 Limitations

The limitation of our method are two-fold. Firstly, even outliers removal is
performed, there still exist a certain number of outliers. Secondly, our
method cannot precisely control the number of upsampled point set. We have to
first generate a dense one with over-sampled points and then adjust the number
of vertices to the target number by the farthest point sampling algorithm.

##  5 Conclusion

In this paper, we present a novel and effective point clouds upsampling method
via implicit neural representation, which can achieve self-supervised and
arbitrary-scale upsampling simultaneously. We formulate point clouds
upsampling as the task of seeking nearest projection points on the implicit
surface for seed points, which can be done by two implicit neural functions
trained without the ground truth dense point clouds. Extensive experimental
results demonstrate that our method can produce high-quality dense point
clouds that are uniform and complete, and achieves competitive objective
performance and even better visual performance compared with state-of-the-art
supervised methods.

##  6 Acknowledgements

This work was supported by National Key Research and Development Project under
Grant 2019YFE0109600, National Natural Science Foundation of China under
Grants 61922027, 6207115 and 61932022.

## References

  * [1] Marc Alexa, Johannes Behr, Daniel Cohen-Or, Shachar Fleishman, David Levin, and Claudio T. Silva.  Computing and rendering point set surfaces.  IEEE Transactions on visualization and computer graphics, 9(1):3–15, 2003.
  * [2] Angel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, et al.  Shapenet: An information-rich 3d model repository.  arXiv preprint arXiv:1512.03012, 2015.
  * [3] Andreas Geiger, Philip Lenz, and Raquel Urtasun.  Are we ready for autonomous driving? the kitti vision benchmark suite.  In 2012 IEEE conference on computer vision and pattern recognition, pages 3354–3361. IEEE, 2012.
  * [4] Xuecai Hu, Haoyuan Mu, Xiangyu Zhang, Zilei Wang, Tieniu Tan, and Jian Sun.  Meta-SR: A magnification-arbitrary network for super-resolution.  In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 1575–1584, 2019.
  * [5] Hui Huang, Dan Li, Hao Zhang, Uri Ascher, and Daniel Cohen-Or.  Consolidation of unorganized point clouds for surface reconstruction.  ACM transactions on graphics (TOG), 28(5):1–7, 2009.
  * [6] Hui Huang, Shihao Wu, Minglun Gong, Daniel Cohen-Or, Uri Ascher, and Hao Zhang.  Edge-aware point set resampling.  ACM transactions on graphics (TOG), 32(1):1–12, 2013.
  * [7] Ruihui Li, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng.  PU-GAN: A point cloud upsampling adversarial network.  In 2019 IEEE/CVF International Conference on Computer Vision (ICCV), pages 7202–7211, 2019.
  * [8] Ruihui Li, Xianzhi Li, Pheng-Ann Heng, and Chi-Wing Fu.  Point cloud upsampling via disentangled refinement.  In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 344–353, 2021.
  * [9] Xinhai Liu, Xinchen Liu, Zhizhong Han, and Yu-Shen Liu.  SPU-Net: Self-supervised point cloud upsampling by coarse-to-fine reconstruction with self-projection optimization, 2020.
  * [10] Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger.  Occupancy networks: Learning 3d reconstruction in function space.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4460–4470, 2019.
  * [11] Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove.  DeepSDF: Learning continuous signed distance functions for shape representation.  In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 165–174, 2019.
  * [12] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas.  Pointnet++: Deep hierarchical feature learning on point sets in a metric space.  In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
  * [13] Guocheng Qian, Abdulellah Abualshour, Guohao Li, Ali Thabet, and Bernard Ghanem.  PU-GCN: Point cloud upsampling using graph convolutional networks.  In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 11678–11687, 2021.
  * [14] Yue Qian, Junhui Hou, Sam Kwong, and Ying He.  Deep magnification-flexible upsampling over 3d point clouds.  IEEE Transactions on Image Processing, 30:8354–8367, 2021.
  * [15] David Stutz and Andreas Geiger.  Learning 3d shape completion from laser scan data with weak supervision.  In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1955–1964, 2018.
  * [16] David Stutz and Andreas Geiger.  Learning 3d shape completion under weak supervision.  CoRR, abs/1805.07290, 2018.
  * [17] Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E Sarma, Michael M Bronstein, and Justin M Solomon.  Dynamic graph cnn for learning on point clouds.  Acm Transactions On Graphics (tog), 38(5):1–12, 2019.
  * [18] Huikai Wu, Junge Zhang, and Kaiqi Huang.  Point cloud super resolution with adversarial residual graph networks.  arXiv preprint arXiv:1908.02111, 2019.
  * [19] Shuquan Ye, Dongdong Chen, Songfang Han, Ziyu Wan, and Jing Liao.  Meta-PU: An arbitrary-scale upsampling network for point cloud.  IEEE Transactions on Visualization and Computer Graphics, pages 1–1, 2021.
  * [20] Wang Yifan, Shihao Wu, Hui Huang, Daniel Cohen-Or, and Olga Sorkine-Hornung.  Patch-based progressive 3d point set upsampling.  In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 5951–5960, 2019.
  * [21] Lequan Yu, Xianzhi Li, Chi-Wing Fu, Daniel Cohen-Or, and Pheng-Ann Heng.  PU-Net: Point cloud upsampling network.  In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2790–2799, 2018.
  * [22] Yifan Zhao, Le Hui, and Jin Xie.  SSPU-Net: Self-Supervised Point Cloud Upsampling via Differentiable Rendering, page 2214–2223.  Association for Computing Machinery, New York, NY, USA, 2021.