# Effect of A-site ionic radius on metamagnetic transition in charge ordered
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ compounds

Sanjib Banik<EMAIL_ADDRESS>CMP Division, Saha Institute of Nuclear
Physics, HBNI, 1/AF-Bidhannagar, Kolkata 700 064, India  Kalpataru Pradhan
<EMAIL_ADDRESS>CMP Division, Saha Institute of Nuclear Physics,
HBNI, 1/AF-Bidhannagar, Kolkata 700 064, India  I. Das
<EMAIL_ADDRESS>CMP Division, Saha Institute of Nuclear Physics, HBNI,
1/AF-Bidhannagar, Kolkata 700 064, India

###### Abstract

We investigate the ultra-sharp jump in the isothermal magnetization and the
resistivity in the polycrystalline $Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$
$(y=0,0.1,0.2,0.25,0.3,0.5)$ compounds. The critical field $(H_{cr})$,
required for the ultra-sharp jump, decreases with increase of ‘Sr’
concentration, i.e. with increase of average A-site ionic radius $\langle
r_{A}\rangle$. The magnetotransport data indicate that the phase separation
increases with the increase of $\langle r_{A}\rangle$, i.e. with $y$. The
dependency of $H_{cr}$ with magnetic field sweep rate reveals that the ultra-
sharp jump from antiferromagnetic (AFM) state to the ferromagnetic (FM) state
is of martensitic in nature. Our two-band double exchange model Hamiltonian
calculations show that the ‘Sr’ doping induces the ferromagnetic clusters in
the antiferromagnetic insulating phase and in turn reduces the critical field.
In the end we present a phenomenological picture obtained from our combined
experimental and theoretical study.

##  I Introduction

Recently, materials exhibiting field induced metamagnetic phase transition
between two energetically competing phases have attracted a lot of attention
due to its complex nature Mahendiran ; Autret ; Fisher ; Hardy1 ; KrishnaM .
The occurrence of metamagnetic transition is perceptible by the sharp jump in
isothermal magnetization. It is well established that this transition is
independent to the microstructure and actually related to the intrinsic nature
of the materialsc̃iteOuyang. Many extensive studies on this magnetic field
induced metamagnetic transition have been carried out over the last decades
Roy ; Velez ; Choi ; Danjoh . Examples of such materials studied include
certain phase separated manganites, inter-metallic alloys such as
$Nd_{5}Ge_{3}$, $Gd_{5}Ge_{4}$, $CeF_{2}$, etc. and some phase separated well
known multiferroic $Eu_{1-x}Y_{x}MnO_{3}$ systems. The appearance of the
magnetization steps is found to be sensitive to the cooling magnetic field as
well as on the magnetic field sweep rate Mahendiran ; Hardy1 . Hardy et al
Hardy2  have shown that, in $Pr_{0.5}Ca_{0.5}Mn_{1-x}Ga_{x}O_{3}$, the
spontaneous magnetization jump occurs in the time evolution of magnetization
for a fixed temperature and magnetic field. Wu et al. Wu  have also observed
the same phenomenon in manganite thin films. The observation of magnetization
(and resistivity) steps in $(La_{0.5}Nd_{0.5})_{1.2}Sr_{1.8}Mn_{2}O_{7}$ is
also reported by Liao et al Liao . But, the origin of these metamagnetic
transition is still a matter of investigation. Very different kind of
mechanisms have been proposed, such as field dependent orbital ordering in
$Pr_{0.5}Ca_{0.5}Mn_{0.95}Co_{0.05}O_{3}$ Mahendiran , spin quantum transition
in $Pr_{5/8}Ca_{3/8}MnO_{3}$ Cao , spin reorientation in $FeRh$ thin films
Bordel , geometrical frustration in garnets Tsui , spin flop transition in
$Ca_{3}CoMnO_{6}$ Flint  and burst like growth of the ferromagnetic fraction
in the phase separation picture. According to the most of the authors the
origin of the magnetization steps is of martensitic in nature Hardy1 ; Hardy2
; Hardy3 . In spite of having lot of study to analyze metamagnetism in various
systems the origin through detailed analysis has been rarely addressed.

In manganites the metamagnetic transition is usually observed in low bandwidth
charge ordered systems. Therefore, in our investigation, to understand the
origin of metamagnetic transition we have chosen $Sm_{0.5}Ca_{0.5}MnO_{3}$
(SCMO) as parent compound, which is one of the lowest bandwidth and robust
charge ordered system. It needs 470 kOe magnetic field at 4 K for the
metamagnetic transition Tokura . As the electronic bandwidth depends on the
average A-site ionic radius $\langle r_{A}\rangle$ Moritomo ; Mathieu , our
study by changing the average A-site ionic radius will give us the lead to
figure out the origin of the metamagnetic transition. To obtain materials with
different electronic bandwidth, we replace ‘Ca’ ions in
$Sm_{0.5}Ca_{0.5}MnO_{3}$ compound by ‘Sr’ ions. This ‘Sr’ doping undoubtedly
increases the $e_{g}$ electronic bandwidth as ‘Sr’ has higher ionic radius as
compared with ‘Ca’. By varying the concentration of ’Sr’ doping we prepared a
set of samples with different $\langle r_{A}\rangle$ and analyzed their
structural, magnetic and electrical transport properties. We observe that the
critical field decreases with the increase of $\langle r_{A}\rangle$. We
explain this by taking the induced ferromagnetic clusters with ’Sr’ doping in
to account, which act as the nucleation centers for the metamagnetism. In
addition, we performed spin-fermion Monte Carlo calculations using double
exchange model Hamiltonian to support our experimental results.

Figure 1:  Room temperature XRD data with its corresponding profile fitted
data for the compounds $Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$
(${y=0,0.1,0.2,0.25,0.3,0.5}$)

##  II Sample preparation and characterization

All the bulk polycrystalline compounds $Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$
(${y=0-0.5}$) have been prepared by the well known sol-gel method with
$Sm_{2}O_{3}$, $CaCO_{3}$, $Sr(NO_{3})_{2}$ and $MnO_{2}$ as the starting
chemicals of purity $99.9\%$. In order to prepare the bulk samples decomposed
gel has been pelletized and heated at $1300^{0}C$ for 36 hours.

The single phase nature of the samples have been characterized from room
temperature x-ray diffraction (XRD) measurements by using Rigaku-TTRAX-III
with 9 kW rotating anode copper source of wavelength $\lambda=1.54\AA$.
Magnetic measurements have been performed using quantum design SQUID-VSM. The
transport and magnetotransport measurements have been carried out on bar
shaped samples in longitudinal geometry by four probe method using Cryogenic
setup.

##  III Results and discussion

###  III.1 Structural characterization

The room temperature XRD study (Fig. 1) display the single phase nature of all
the bulk polycrystalline compounds. The crystal structure information has been
obtained from Rietveld refinement of the XRD data using FULLPROF software
which shows that all the samples crystallize in orthorhombic structure with
‘Pnma’ space group. The extracted lattice parameter and the average A-site
ionic radius $\langle r_{A}\rangle$ are calculated from shanon effective ionic
radii for different $y$ (see Table. 1). $\langle r_{A}\rangle$ increases
gradually with ‘Sr’ concentrations because of its larger ionic radius as
compared with ‘Ca’.

Table 1: The lattice parameters and average A-site ionic radii for the samples $Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ (${y=0,0.1,0.2,0.25,0.3,0.5}$)  y | a ($\AA$) | b ($\AA$) | c ($\AA$) |  $\langle r_{A}\rangle$ ($\AA$)
---|---|---|---|---
0 | 5.423 | 5.370 | 7.582 | 1.156
0.1 | 5.415 | 5.381 | 7.593 | 1.169
0.2 | 5.410 | 5.401 | 7.626 | 1.182
0.25 | 5.404 | 5.413 | 7.629 | 1.188
0.3 | 5.410 | 5.416 | 7.634 | 1.195
0.5 | 5.441 | 5.425 | 7.660 | 1.221

We estimate the orthorhombic distortion [defined as
${\delta=\frac{a+b-c/\sqrt{2}}{a+b+c/\sqrt{2}}}$] from the lattice parameters.
The variation of $\delta$ and the unit cell volume with $\langle r_{A}\rangle$
are plotted in Fig. 2. Initially distortion decreases rapidly up to $y=0.2$
with increase of $\langle r_{A}\rangle$ and then increases very slowly with
$\langle r_{A}\rangle$. On the other hand, unit cell volume increases steadily
with increase of $\langle r_{A}\rangle$.

Figure 2: Evolution of orthorhombic distortion ($\delta$) and unit cell volume
with $\langle r_{A}\rangle$.

###  III.2 Magnetotransport and magnetization study

The increase of average A-site ionic radius and reduction of orthorhombic
distortion greatly influences the transport and magnetotransport properties as
the bandwidth of $e_{g}$ electrons in manganites is directly proportional to
the $\langle r_{A}\rangle$. The temperature dependence of resistivity
${[\rho(T)]}$ in absence of any external magnetic field has been performed for
all the samples. The measurements were done during warming cycle after cooling
the samples in zero magnetic field. The evolution of ${\rho(T)}$ with
different ‘Sr’ concentrations ($y$) (Fig. 3) shows that the samples are
insulating down to the measurable resistance limit.

Figure 3:  Temperature dependence of reduced resistivity of the samples
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ (y = 0, 0.1, 0.2, 0.25, 0.5) in absence of
external magnetic field. Inset (A) shows the $d[ln(\rho)]/dT^{-1}$ vs. T plot
for the samples and inset (B) shows the variation of activation energy with
$\langle r_{A}\rangle$.

There is a relative suppression of resistivity value with increase of ‘Sr’
concentrations at low temperatures. Moreover, the ‘Sr’ substitution also
decreases the charge ordering temperature ($T_{CO}$) from 260 K for ${y=0}$ to
210 K for ${y=0.5}$ [see inset (A) of Fig. 3] and increases the fragility of
the CO state. The reduction of resistivity with ‘y’ as well as softening of CO
state is due to the increase of bandwidth by increasing A-site ionic radius.
For further investigation, we analyze the high temperature (${T>T_{CO}}$)
resistivity data with the help of small polaron hopping model (SPH). It is
known that in manganites the electrical resistivity in paramagnetic region is
mainly governed by the polaronic activation. According to the SPH model Banik
the expression of resistivity is $\rho=\rho_{0}Texp(E_{A}/k_{B}T)$ where
$E_{A}$ is the polaronic activation energy. From the fitting of the high
temperature (${T>260K}$) resistivity data, the activation energy $E_{A}$ for
‘Sr’ doped samples has been calculated and its evolution with ‘y’ is shown in
the inset (B) of Fig. 3. The activation energy reduces with ‘y’ as expected
(bandwidth decreases the activation energy). Previously, it was shown that the
increase of $\langle r_{A}\rangle$ converts the charge ordered state to
electronically phase-separated state Gutierrez ; Rao ; Kumar1 ; Shankar  and
results a spontaneous metal insulator transition. Though in our case, there is
no spontaneous metal insulator transition, but the reduction of resistivity
and softening of CO state with ‘y’ points towards a phase coexistence scenario
at larger $y$.

Figure 4:  Temperature dependence of resistivity for the samples
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ (y=0, 0.1, 0.2, 0.25, 0.5) in presence of
90 kOe external magnetic field.

To have a clear vision about this phase coexistence scenario we perform the
temperature variation of resistivity in presence of 90 kOe magnetic field (see
Fig. 4). There is almost no effect of 90 kOe field on the resistivity of
${y=0}$ sample as expected (critical field for robust charge ordered material
$Sm_{0.5}Ca_{0.5}MnO_{3}$ is 470 kOe at 4K). With substitution of ‘Sr’ in
place of ‘Ca’ a huge suppression of resistivity is observed on application of
90 kOe magnetic field and shows an insulator to metal transition. Moreover,
with increasing ‘Sr’ concentration the curve around the $T_{MI}$ gets broaden,
which signifies the enhancement of phase coexistence. Thus, it can be firmly
said that the ‘Sr’ doping weaken the robustness of CO state and introduces the
phase separation.

Manganite systems being strongly correlated in nature, the phenomena of phase
separation should also be reflected in magnetization data. In this regard,
magnetization as a function of temperature [M(T)] has been measured in the
field cooled warming (FCW) protocol in presence of 100 Oe magnetic field for
(${y=0,0.1,0.2,0.25,0.5}$). The evolution of M(T) for these samples are shown
in Fig. 5. At low temperature (${T<50K}$) with ‘Sr’ doping the magnetization
increases. The value of magnetization increases from ${0.004\mu_{B}}$ to
${0.303\mu_{B}}$ at ${25K}$ with increasing ‘Sr’ concentration from ${y=0.25}$
to ${y=0.5}$. This result verifies the enhancement of ferromagnetic phase
fraction with ‘Sr’ doping. At the same time ‘Sr’ substitution decreases the
charge ordering temperature ($T_{CO}$) as well as the anti-ferromagnetic
ordering temperature (${T_{N}}$) (see the inset of Fig. 5). For ${y=0.5}$
composition there is no signature of $T_{N}$. This is due to the existence of
tri-critical point around ${100K}$ Hiraka .

Figure 5:  Evolution of magnetization with temperature, measured in FCW
protocol in presence of 1 kOe magnetic field for the samples
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ (y=0, 0.1, 0.2, 0.25, 0.5). Inset shows
the temperature derivative of the corresponding magnetization of the samples.
Figure 6:  Temperature dependence of inverse dc susceptibility (H/M) data,
measured in presence of 100 Oe magnetic field for the samples
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ (y = 0, 0.25, 5).

For further investigation, we analyze the high temperature (${T>260K}$)
inverse dc susceptibility (H/M) versus temperature data at 100 Oe magnetic
field with the Curie-Weiss law $\chi=C/(T-\theta_{CW})$ where
$C=\mu_{eff}^{2}/3k_{B}$ [$\mu_{eff}$ and $\theta_{CW}$ are the effective
paramagnetic moment in Bohr magneton and paramagnetic curie temperature,
respectively]. The variation of dc susceptibility (H/M) with temperature for
${y=0.1,0.25,0.5}$ samples and their corresponding Curie-Weiss fitted data are
presented in Fig. 6. From the fitting, paramagnetic curie temperature comes
out to be 110 K, 122 K and 145 K for the samples with ‘Sr’ concentrations
${y=0.1,0.25}$ and ${0.5}$, respectively. Furthermore, the enhancement of the
effective paramagnetic moment ($\mu_{eff}=6.16\mu_{B}$ for ${y=0.1}$ to
$\mu_{eff}=6.76\mu_{B}$ for ${y=0.5}$) has also been observed. This increase
of $\theta_{CW}$ and $\mu_{eff}$ clearly indicates the enhancement of
ferromagnetic interactions with ‘Sr’ doping. Here another point needs to
mention is that the values of the effective moments are larger than the
theoretical calculated value of ${4.42\mu_{B}}$. It indicates the presence of
ferromagnetic clusters in the high temperature region (${T>260K}$). These
clusters behaves as an individual paramagnetic entity which contains more than
one Mn ions Banikrsc . This is why $\mu_{eff}$ value [= ${4.32\mu_{B}}$] for
${y=0}$ sample (without any ferromagnetic clusters) is close to the
theoretical expected value. These systematic result further implies that ‘Sr’
doping induces the ferromagnetic clusters in the host antiferromagnetic phase.

Figure 7:  Isothermal magnetization of the samples
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ (y = 0, 0.1, 0.2, 0.25, 5) at 2 K.

###  III.3 Metamagnetic transition

To see the effect of magnetic field in these phase separated state, we measure
the field dependence of magnetization at ${2K}$ for these samples as presented
in Fig. 7. For the samples without any ‘Sr’ concentrations (i.e. ${y=0}$) and
with small ‘Sr’ concentrations (i.e. ${y=0.1}$) we find a linear increase of
magnetization with magnetic field up to the 70 kOe field due to the strong CO-
AFM phase in parent compound SCMO. With further increase of ‘Sr’ concentration
i.e. to ${y=0.2}$, a sharp metamagnetic transition is observed at 55 kOe (the
magnetization increases from ${0.75\mu_{B}}$ to ${3\mu_{B}}$). So at this
point the system converts completely from a CO antiferromagnetic (CO-AFM)
phase to a ferromagnetic (FM) phase. The descending branch of the M-H curve
remains almost flat down to 10 kOe field which indicates the irreversible
nature of the field induced CO-AFM to FM transformation. With further
decreasing of the field from 10 kOe we find that the magnetization rapidly
decreases. On the other hand, with further increase of ‘Sr’ concentrations
requirement of critical field for the metamagnetic transition decreases to 47
kOe for ${y=0.25}$ and to 20 kOe for ${y=0.5}$. Although in ${y=0.5}$, the
initial increase of magnetization is like a soft ferromagnet, which indicates
the dominance of ferromagnetic phases as compared to the antiferromagnetic
phases in the sample. Previously, the same kind of sharp metamagnetic
transition has been observed in Mn site doped CO manganites, for instance in
$Pr_{0.5}Ca_{0.5}Mn_{1-x}M_{x}O_{3}$. According to Raveau et al. Mahendiran
the occurrence of this step like behavior in Mn site doped manganites is
because of the presence of the short range ordered ferromagnetic regions in
the CO-AFM region. Here, the existence of ferromagnetic clusters and their
growth with ‘Sr’ doping is also observed from the analysis of dc
susceptibility data. These ferromagnetic clusters play the role of nucleation
centers in the metamagnetic transitions (from CO-AFM to FM phase) and
sharpness of steps indicates the jerky growth of these FM clusters. The
downward shifting of the critical fields with increasing ‘Sr’ concentrations
is possibly because of the increasing number of ferromagnetic clusters.

Figure 8:  Magnetic field dependence of resistance of the samples
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ at 2 K.

Next, we measure the resistance with variation of magnetic fields at 2 K for
our samples to correlate the shifting of the critical field as seen in
magnetization data. Here in the sample ${y=0.1}$, metamagnetic transition is
observed at 89.7 kOe which was not visible in magnetization because of the
instrumental limitation. In ${y=0}$ sample there is almost no effect of
magnetic field (not shown in the figure). In the subsequent ‘Sr’ doped samples
i.e. for ${y=0.2}$ metamagnetic transition occurs at 58.8 kOe [and 48.5 kOe
for ${y=0.25}$]. The slight higher values of the critical field determined
from R-H compared with M-H is possibly due to different average sweep rate of
the field. For example for ${y=0.25}$ sample, in M-H sweep rate was 100 Oe/sec
and in R-H it was 24 Oe/Sec. The dependence of critical field on sweep rate is
a signature of martensitic like transition as smaller sweep rate assist the
progressive accommodation of the martensitic strain which push the instability
towards higher magnetic field as discussed in our earlier work Sanjibnpg .

###  III.4 Theoretical study

In order to further explain the experimental results we perform spin-fermion
Monte Carlo calculations. Our prime motive is to analyze the role of ‘Sr’
disorder on the metamagnetic transition. In SCMO, ’Sr’ doping is simulated by
adding quenched disorder to a well studied model Hamiltonian for manganites in
the large Hund’s coupling limit $(J_{H}\rightarrow\propto)$ Dagotto1 ;
Pradhanprl . Our model Hamiltonian:

$\displaystyle H$ $\displaystyle=$ $\displaystyle-\sum_{\langle
ij\rangle\sigma}^{\alpha\beta}t_{\alpha\beta}^{ij}c^{\dagger}_{i\alpha\sigma}c^{~{}}_{j\beta\sigma}~{}-J_{H}\sum_{i}{\bf
S}_{i}.{\mbox{\boldmath$\sigma$}}_{i}$ $\displaystyle~{}~{}+J\sum_{\langle
ij\rangle}{\bf S}_{i}.{\bf S}_{j}-\lambda\sum_{i}{\bf
Q}_{i}.{\mbox{\boldmath$\tau$}}_{i}+{K\over 2}\sum_{i}{\bf Q}_{i}^{2}$

where $t_{\alpha\beta}$ is the hopping amplitude between nearest neighbor
$e_{g}$ electrons with orbitals ($\alpha$, $\beta$) and
$\sigma\,(=\uparrow,\downarrow)$. $\alpha$ (and $\beta$) denotes
$d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$ Mn-orbitals. $J_{H}$ (Hund’s
coupling) is between Mn $t_{2g}$ spin ${\bf S_{i}}$ and $e_{g}$ electron spin
$\sigma_{i}$ at site $i$, whereas J is the superexchange interaction between
neighboring Mn $t_{2g}$ spins. $\lambda$ is the electron-phonon interaction
between $e_{g}$ electrons and the Jahn-Teller phonon modes $(\textbf{Q}_{i})$
in the adiabatic limit. We treat $(\textbf{S}_{i})$ and $(\textbf{Q}_{i})$ as
classical Dagotto2  variables. $K$ (stiffness of Jahn-Teller modes) and $|{\bf
S}_{i}|$ are set to be 1. For more details please see Ref.30

We incorporated the effect of ’Sr’ disorder by adding
$\sum{\epsilon_{i}n_{i}}$ term to the Hamiltonian. One generally add
$\sum_{i}\epsilon_{i}n_{i}$ to the Hamiltonian such that average
$\overline{\epsilon}_{i}=0$ to model A-site disorder in manganites with two
A-type elements Tokura ; Anamitra  [example: for $Sm_{0.5}Ca_{0.5}MnO_{3}$
(SCMO) or $Sm_{0.5}Sr_{0.5}MnO_{3}$ (SSMO) like samples]. This is done by
choosing $\epsilon_{i}$ at each site from $P(\epsilon_{i})={1\over
2}\delta(\epsilon_{i}-\Delta_{H})+{1\over 2}\delta(\epsilon_{i}+\Delta_{H})$
distribution. In $Sm_{0.5}Ca_{0.25}Sr_{0.25}MnO_{3}$ (SCSMO), with three
A-type elements, ${Sr^{2+}}$ ions (with larger ionic radius than $Sm^{3+}$ and
$Ca^{2+}$ ions) occupy one fourth of the A-sites randomly. So by neglecting
the ionic mismatch between Sm and Ca elements we model Sr and Sm-Ca disorder
by adding $\sum_{i}\epsilon_{i}n_{i}$ at each Mn site picked from the
distribution $P(\epsilon_{i})={1\over
4}\delta(\epsilon_{i}-\Delta_{Q})+{3\over 4}\delta(\epsilon_{i}+\Delta_{Q})$
Sanjibnpg ; Sanjibarxiv . Both $\Delta_{H}$ and $\Delta_{Q}$ are quenched
disorder potentials. $\Delta_{H}$ (H stand for half) and $\Delta_{Q}$ (Q stand
for quartern) are binary disorder with ratio 50-50 and 25-75, respectively. In
an external magnetic field we add a Zeeman coupling term $-\sum_{i}{\bf
h}\cdot{\bf S}_{i}$ in our model Hamiltonian. We measure all the parameters in
terms of the hopping energy $t$. The estimated value of $t$ in manganites is
0.2 eV Dagotto1 .

We use an exact diagonalisation scheme for the $e_{g}$ electrons using
different background $t_{2g}$ spins and phonon mode $(\textbf{Q})$
configurations. Background configurations were chosen using travelling cluster
approximation (TCA) based spin-fermion Monte Carlo technique to access the
large system size $(24\times 24$ lattice) Kumar ; Pradhanprl . In different
magnetic field our measured magnetization is thermally averaged over ten
different disorder samples in addition to the thermal averages during the
Monte Carlo sweeps. Over 10,000 Monte Carlo sweeps were performed to
thermalize the system.

First we start our calculations using $\lambda$ = 1.65 and $J$ = 0.1 that
reproduces the correct magnetic phase (CE-CO-OO-I phase) at low temperatures
for electron density $n$ = $1-x$ = 0.5 Pradhanprb . At T = 0.005 the
magnetization (M) remains very small even for $\Delta_{Q}$ = 0.3 Sanjibnpg .
In Figs. 9(a)–(d) we show the magnetization vs. field ($h$) curve for the
clean system (without any disorder) using dotted lines. The metamagnetic
transition is at $h$ = 0.11. For $\Delta_{Q}$ = 0.1 the metamagnetic
transition remains sharp, but the critical field $h_{cr}$ decreases to 0.09 as
shown in Fig. 9(a). For $\Delta_{H}$ = 0.1, the critical field for the
magnetic transition remains the same to that of $\Delta_{Q}$ = 0.1. So, for
$\Delta_{Q}$ = $\Delta_{H}$ = 0.1 the system behaves more or less like clean
system. Also the inset of Fig. 9(d) shows that the magnetic transition for
$\Delta_{Q}$ = 0.1 remains sharp even for higher temperature (T = 0.01) unlike
SCSMO experiments Sanjibnpg . Therefore we believe that the disorder strength
is larger that $\Delta_{Q}$ = 0.1 in SCSMO. For $\Delta_{H}$ = 0.2 and 0.3 the
magnetic transition is continuous with the field and qualitatively agrees with
previous work Anamitra . But the correct way to model ’Sr’ disorder in SCSMO
is to add disorder that is $\Delta_{Q}$ type. It is apparently clear that the
critical field for metamagnetic transition decreases to $h$ = 0.08 for
$\Delta_{H}$ = 0.2 and 0.3. For $\Delta_{Q}$ = 0.2, the sharpness of magnetic
transition vanishes at T = 0.01, which qualitatively matches with the trend of
SCSMO experimental data Sanjibnpg .

Figure 9:  Magnetization vs. field using $\lambda$= 1.65 and $J$ = 0.1.
(a)–(c) $M-h$ phase diagrams using different $\Delta$ (both $\Delta_{Q}$ and
$\Delta_{H}$) values at temperature T = 0.005. Dotted line (circle symbols) in
(a) [also in (c)–(d)] is for clean system ($\Delta$ = 0). (d) $M-h$ phase
diagram for $\Delta_{Q}$ =0.2 for T = 0.005 and T = 0.01. Inset of (d) shows
the same using $\Delta_{Q}$ =0.1.  Figure 10:  Magnetization vs. field ($M-h$)
for three sets of parameters (mimicking SCMO, SCSMO and SSMO samples). (a)
$\Delta$ =0.2 and (b) $\Delta$ =0.3. Dashed lines in both (a) and (b) are for
$\Delta$ = 0. Please see the text for details.

Next, we move to analyze the M-h curve for the series (SCMO-like, SCSMO-like
and SSMO-like) of materials as prepared in the experiments. We know that SSMO
(SCMO) has larger (smaller) bandwidth than SCSMO. We incorporate the bandwidth
variation by changing $\lambda$ (and $J$) values in our model Hamiltonian.
Smaller $\lambda$ (and $J$) implies larger bandwidth or vice versa. For
clarity we treat SCMO as a clean system (due to the small mismatch between Sm
and Ca ionic radii) and use $\lambda$= 1.73, $J$ = 0.105 for SCMO-like
materials. For SSMO-like materials we set $\Delta_{H}$ = 0.2, $\lambda$= 1.57,
$J$ = 0.095. For SCSMO we choose $\Delta_{Q}$ = 0.2, $\lambda$= 1.65, $J$ =
0.1 as discussed above. Fig. 10(a) shows that our results qualitatively agree
with the experiments. We denote $\Delta$ = 0.2 (0.3) [i.e. $\Delta_{Q}$ = 0.2
(0.3) for SCSMO-like material and $\Delta_{H}$ = 0.2 (0.3) for SSMO like
material] for brevity. Magnetization of SCMO-like material remains small even
for $h=0.12$. For SCSMO-like material the metamagnetic transition is at
critical field $h_{cr}=0.08$. For SSMO we find sizeable magnetization even at
$h=0.0$ and the metamagnetic transition is at $h=0.06$ (smaller than that of
SCSMO). For clarity and completeness we also plot the two curves using
$\Delta_{Q}$= 0.3 and $\Delta_{H}$= 0.3 (along with the clean SCMO) in Fig.
10(b). The trend of M-H curves from SCMO-like to SSMO-like materials remains
qualitatively same for both set of parameters.

Figure 11:  A schematic picture to describe the reduction of critical magnetic
field for the metamagnetic transition from $y$ = 0 to $y$ =0.5.

So the experimental scenario can be explained by a simple phenomenological
picture obtained from our experimental and theoretical study (see Fig. 11).
According to this picture ‘Sr’ doping (in place of ‘Ca’) in SCMO induces
ferromagnetic clusters. The area and number of ferromagnetic cluster increases
gradually with ‘Sr’ doping and at ${y=0.5}$ area of each ferromagnetic cluster
phase is maximum. These clusters act as the nucleation center that give rise
to martensitic like transformation and convert the CO-AFM phase to a FM phase.
With increasing the bandwidth (by increasing $y$) the phase separation
increases, i.e. the strength as well as number of these nucleation center
increases and as a result the critical field decreases with $y$.

##  IV Conclusions

In summary, we have investigated the metamagnetic properties of the
$Sm_{0.5}(Ca_{0.5-y}Sr_{y})MnO_{3}$ compounds through isothermal magnetization
and resistivity measurements. The presence of ultra-sharp jump at low
temperature ($T<5K$) in both isothermal resistivity and magnetization is due
to the strong spin and charge coupling in the systems. The sweep rate
dependence of the critical field ($H_{cr}$) indicates the martensitic
scenario. ‘Sr’ doping (in place of ‘Ca’) in SCMO induces ferromagnetic
clusters and the volume of these ferromagnetic clusters increases in the CO-
AFM materials with average A-site ionic radius $\langle r_{A}\rangle$ as
perceived from the magnetotransport and magnetization data. Our model
Hamiltonian calculations also confirm this scenario. These ferromagnetic
clusters act as the nucleation center in the CO-AFM background for the burst
like growth from CO-AFM to FM phase at critical field. The FM fraction
increases with $\langle r_{A}\rangle$ and as result the $H_{cr}$ decreases
with ’Sr’ doping.

##  V Acknowledgements

The work was supported by Department of Atomic Energy (DAE), Govt. of India.

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# Weakly Supervised Semantic Segmentation for Large-Scale Point Cloud

Yachao Zhang1, Zonghao Li1, Yuan Xie2 111Corresponding Author , Yanyun Qu1 ∗,
Cuihua Li1, Tao Mei3

###### Abstract

Existing methods for large-scale point cloud semantic segmentation require
expensive, tedious and error-prone manual point-wise annotations. Intuitively,
weakly supervised training is a direct solution to reduce the cost of
labeling. However, for weakly supervised large-scale point cloud semantic
segmentation, too few annotations will inevitably lead to ineffective learning
of network. We propose an effective weakly supervised method containing two
components to solve the above problem. Firstly, we construct a pretext task,
i.e., point cloud colorization, with a self-supervised learning to transfer
the learned prior knowledge from a large amount of unlabeled point cloud to a
weakly supervised network. In this way, the representation capability of the
weakly supervised network can be improved by the guidance from a heterogeneous
task. Besides, to generate pseudo label for unlabeled data, a sparse label
propagation mechanism is proposed with the help of generated class prototypes,
which is used to measure the classification confidence of unlabeled point. Our
method is evaluated on large-scale point cloud datasets with different
scenarios including indoor and outdoor. The experimental results show the
large gain against existing weakly supervised and comparable results to fully
supervised methods222Code based on mindspore: https://github.com/dmcv-
ecnu/MindSpore_ModelZoo/tree/main/WS3_MindSpore.

## Introduction

3D scene understanding is required in numerous applications, in particular
robotics, autonomous driving and virtual reality. Large-scale point cloud
semantic segmentation as a fundamental task attracts more and more attention.
The success of deep neural networks in point cloud semantic segmentation is
attribute to their ability to scale up with more well-labeled training data
(Qi et al. 2017a, b; Zhang, Hua, and Yeung 2019; Li, Chen, and Hee Lee 2018;
Wang et al. 2019; Wu, Qi, and Fuxin 2019; Yang et al. 2019; Hu et al. 2020).

Fully supervised point cloud semantic segmentation methods need expensive,
tedious and error-prone manual point-wise annotations. One direct solution is
to achieve effective segmentation via weakly supervised learning with
annotating partial points or semantic category which is still in its infancy.
Xu and Lee proposed a weakly supervised approach (Xu and Lee 2020) whose
results are close to the previous fully supervised performance with $10\times$
fewer labeled points. This approach only deal with an instance (ShapeNetPart)
or a block ($1m\times 1m$ on S3DIS) of point cloud in a small scale.

Figure 1: Visualize the results of semantic segmentation, the
misclassification points are signed in red. From left to right, the columns
show the results with one labeled point, 1% labeled points, 10% labeled points
and the ground truth.

For large-scale point clouds, the network cannot effectively learn feature
representation with a few labeled points. We consider the two strategies to
solve this problem: 1) What prior knowledge can be transferred to the
segmentation network for improving feature representation in weakly supervised
learning? 2) How are the labels of given points propagated to unlabeled points
effectively?

Firstly, transfer learning provides a promising way and is relatively
successful in 2D vision tasks, such as image classification, segmentation, and
so on. But point cloud tasks lack a well-annotated and category-extensive
dataset like ImageNet (Deng et al. 2009) for pre-training. We consider to use
self-supervised learning for knowledge transfer. However, how to use enormous
amounts of unlabeled point cloud to generate labels by itself and learn a
semantically related representation for subsequent tasks is very challenging.
Secondly, propagating pseudo labels to unlabeled points is a common method for
weakly supervised and semi-supervised task to learn effectively. Usually, this
requires constructing a fully-connected graph with all points. However, for
large-scale point clouds ($\sim 10^{6}$ points), the fully-connected graph is
unachievable due to large GPU memory consumption.

To address the above difficulties, we present an effective weakly supervised
method for large-scale point cloud semantic segmentation. Firstly, we notice
that points of the same semantic class have similar color distribution, and
point cloud with color is essentially free. We choose point cloud colorization
as a pretext task for transfer learning. Specifically, we use color space
transformation to construct a self-supervised network for color space
completion. We further introduce a local perceptual regularization term to
enhance the local representation which is consistent with the goal of
segmentation task. As a result, it can learn a prior-based initialization
distribution. After that, we use the pre-trained knowledge to fine-tune weakly
semantic segmentation network. Our learning scheme allows to transfer the
knowledge from the self-supervised task to the weakly supervised task and
improves the effectiveness of feature representation. Moreover, we propose a
sparse label propagation method induced by class prototype which can gradually
propagate pseudo labels to unlabeled points. It is computational friendly that
can be adapted to large-scale tasks.

Our contributions can be summarized as follows:

  * •

A weakly supervised segmentation method is proposed for large-scale point
cloud which only needs very small number of labeled points.

  * •

We adopt the heterogeneous transfer learning method and construct a self-
supervised pretext task by point cloud colorization. It can learn a prior
distribution and be generalized well to segmentation tasks.

  * •

We propose an efficient sparse label propagation method which can propagate
labels to unlabeled points. It expands the supervision information and has low
computational complexity.

  * •

Extensive experimental results demonstrate that our method achieves the
comparable to or even exceeds fully supervised competitors.

## Related Work

Semantic segmentation for large-scale point cloud. PointNet and PointNet++ (Qi
et al. 2017a, b) are pioneering approaches for point clouds. While recent
works have shown promising results on small point clouds, most of them cannot
directly scale up to large-scale point clouds ($\sim 10^{6}$ points) due to
high computational and memory costs (Hu et al. 2020).

SPG (Landrieu and Simonovsky 2018) processes the large-scale point clouds
through a graph convolution-based method. FCPN (Rethage et al. 2018)
preprocesses large-scale point clouds into voxels. However, both the graph
partitioning and voxelization are computationally expensive. Recently, RandLA-
Net (Hu et al. 2020) provides an efficient and lightweight neural architecture
for large-scale point cloud semantic segmentation. The state-of-the-art
methods mentioned above are fully supervised that require a large number of
point clouds with dense point-wise annotation. Obviously, this point-wise
annotation is labor-intensive and time-consuming.

Self-supervised learning on point cloud. Self-supervised learning springs up
in computer vision recently. It aims to learn good representations from
unlabeled visual data, reducing or even eliminating the need for the costly
collection of manual labels (Newell and Deng 2020). Recent self-supervised
learning makes great successes in feature representation which achieves
comparable or outperforms results produced by supervised pre-training
(Bachman, Hjelm, and Buchwalter 2019; He et al. 2020).

Unlike 2D images, self-supervised learning in point cloud is rarely used.
Previous works on unsupervised 3D representation learning (Achlioptas et al.
2018; Gadelha, Wang, and Maji 2018; Hassani and Haley 2019; Li, Chen, and Hee
Lee 2018; Sauder and Sievers 2019; Yang et al. 2018) mainly focuse on
representing an instance (ShapeNet (Chang et al. 2015)). It is difficult to
directly apply the feature representation learned on an instance to real-word
large-scale point cloud tasks due to the large domain gaps (Xie et al. 2020).

PointContrast (Xie et al. 2020) concerns on contrastive embeddings and
proposed a pre-training task for 3D point cloud understanding. The core of
this method is that different views of point cloud should be mapped to similar
embeddings for matched points. It achieves promising results on fully-
supervised downstream tasks. Xu and Lee (Xu and Lee 2020) proposed a Siamese
self-supervision branch by augmenting the training sample with a random in-
plane rotation and flipping, and then made the original and augmented point-
wise predictions be consistent. This method (Xu and Lee 2020) treats self-
supervision as a branch of multi-task and only the current training samples
are used. Therefore, it cannot make full use of other massive point clouds to
learn representations with strong generalization.

Weakly Supervised Point Cloud Semantic Segmentation. There are few researches
on weakly supervised point cloud semantic segmentation. Following the weakly
supervised manner called incomplete supervision in (Zhou 2018), the recent
work (Xu and Lee 2020) utilizes $10\times$ fewer labeled points to achieve
comparable performance to fully supervised method in small scale part
segmentation and small blocks ( $\sim 10^{3}$ points) semantic segmentation.
MPRM (Wei et al. 2020) introduces a multi-path region mining module to
generate pseudo point-level label from a classification network. The
classification network is trained by the subcloud-level weakly labels. For the
scene-level as an input, the performance will degrade. However, up to now,
none of them can be generalized to large-scale problem well.

## Proposed Method

### Overview

Figure 2: The framework of our method consists of three parts: i) Self-
supervised pretext task learns a prior knowledge. ii) The prior knowledge is
used to fine-tune the weakly-supervised semantic segmentation network. iii)
Sparse label propagation generates pseudo label for unlabeled data to improves
the effectiveness of weakly-supervised task.

Our method aims to exploit the knowledge transfer and label propagation to
solve the problem of unstable and poor representation produced by the network
under weakly supervised large-scale point cloud segmentation. We propose an
effective weakly supervised large-scale method and depict the overview
framework in Figure 2.

We take point cloud colorization as a self-supervised pretext task to learn a
prior-based initialization distribution. A local perceptual regularization is
proposed to learn the contextual information. Then, we use pre-trained
parameters of encoder to initialize the weakly-supervised network for
improving the effectiveness of feature presentation.

Furthermore, we leverage labeled points to directly supervise the network and
fine-tune network parameters. We also introduce a non-parametric label
propagation method for weakly supervised semantic segmentation. Some unlabeled
points are assigned pseudo labels through the similarity between class
prototypes (Li et al. 2020; Qiu, Yao, and Mei 2017) and embeddings of
unlabeled points. Therefore, more supervised information is introduced to
improve effectiveness of training. Considering of computational and memory
efficiency for large-scale point cloud, we choose RandLA-Net (Hu et al. 2020)
as the backbone, which is an efficient and lightweight neural architecture for
large-scale point clouds semantic segmentation. In the following, we describe
the self-supervised pretext task and the sparse label propagation method.

### Self-Supervised Pretext Task

Colorization provides a powerful supervisory signal unlike training from
scratch in 2D vision task. Training data are easy to collect, so any point
cloud with color can be used as training data. Due to the progress of point
cloud acquisition equipment, we have access to enormous amounts of unlabeled
point cloud data with color information. We investigate and implement self-
supervised learning on point cloud colorization which is treated as a pretext
task. Point colorization aims to guide the self-supervised model to learn the
feature representation.

It is recognized that the $\mathbf{Lab}$ color space favor for perceptual
distance (Zhang, Isola, and Efros 2016; Larsson, Maire, and Shakhnarovich
2017). Thus, we perform point cloud colorization by $\textbf{a},\textbf{b}$
completion in this colorspace. Given the lightness channel L, the network
predicts the a and b color channels and the local Gaussian distribution for
each point. Notably, the value in channel ${L}$ is replicated to three times
of each point to keep the same dimension as the input of segmentation task.
Therefore, the input point cloud
$X^{s}=[x_{1},x_{2},...,x_{N^{s}}]\in\mathcal{R}^{N^{s}\times 6}$ consists of
$N^{s}$ 3D points with the xyz coordinates and three L. $N^{s}$ is the number
of points in one point cloud.

Moreover, we implement RandLA-Net on the self-supervised task by modifying the
final output layer. That is, the output of the network is a $6$-dimension
vector which contains the predicted $\hat{a}$, $\hat{b}$ and the corresponding
local mean and variance.

#### Loss of Self-Supervised Task

The loss is inherited from standard regression problem, which minimizes
$\mathcal{L}_{1}$ error between the prediction and ground truth. Given a point
cloud with coordinates and triplicate lightness values, the self-supervised
pretext task learns a mapping
$\hat{Y}=\mathcal{F}(X^{s};\Theta),\hat{Y}=\\{\hat{a},\hat{b},\hat{\mu}^{a},\hat{\sigma}^{a},\hat{\mu}^{b},\hat{\sigma}^{b}\\}$,
where $\hat{a}$, $\hat{b}$, $\hat{\mu}$ and $\hat{\sigma}$ denote predicted a,
b and corresponding local mean and variance, respectively. The loss of self-
supervised task can be formulated as:

$\mathcal{L}_{ab}=\frac{1}{2N^{s}}\sum^{N^{s}}_{i=1}(||a_{i}-\hat{a}_{i}||_{1}+||b_{i}-\hat{b}_{i}||_{1}).$
(1)

Figure 3: The framework of sparse label propagation. $\odot$ is Hadamard
product and Gather denotes the operator of getting elements by index.

In addition, to learn the local color distribution of every point, we
introduce a local perceptual regularization term. If the network can predict
the color distribution (mean and variance) of the neighbors, it can embed the
local information which is consistent with the segmentation task using local
features for weakly supervised semantic segmentation. Given a point $x_{i}$ as
the centroid, the local neighbor $\mathcal{N}(x_{i})$ is calculated by KNN
according to the Euclidean distance. The ground truth $\mu_{i}^{a}$ and
$\sigma_{i}^{a}$ of $a$ channel can be obtained by:

$\mu_{i}^{a}=\frac{1}{K}\sum_{j=1}^{K}a_{j},\forall{x_{j}}\in\mathcal{N}(x_{i}),$
(2)
$\sigma_{i}^{a}=\sqrt{\frac{1}{K}\sum_{j=1}^{K}(a_{j}-\mu_{i}^{a})^{2}+\varepsilon},\forall{x_{j}}\in\mathcal{N}(x_{i}),$
(3)

where $\varepsilon$ is a very small constant. $\mu_{i}^{b},\sigma_{i}^{b}$ can
be obtained in the same way. We formulize the local perceptual regularization
term as:

$\begin{split}\mathcal{L}_{local}=\frac{1}{4N^{s}}\sum^{N^{s}}_{i=1}(||&\mu_{i}^{a}-\hat{\mu_{i}}^{a}||_{1}+||\sigma_{i}^{a}-\hat{\sigma_{i}}^{a}||_{1}+\\\
||&\mu_{i}^{b}-\hat{\mu_{i}}^{b}||_{1}+||\sigma_{i}^{b}-\hat{\sigma_{i}}^{b}||_{1}).\end{split}$
(4)

The total loss $\mathcal{L}_{p}$ of self-supervised pretext task can be
expressed as:

$\mathcal{L}_{p}=\mathcal{L}_{ab}+\mathcal{L}_{local}.$ (5)

#### Discussion

Why is the knowledge learned from the pretext task beneficial for semantic
segmentation?

  * •

Pretext task learns similar feature distributions as the semantic segmentation
task. Objects in the same category usually have similar color distribution,
for example the vegetation is typically green, and the road is black. The
surface color texture of the scene provides ample cues for many categories.

  * •

Pretext task embeds the local feature representation. We introduce a local
perceptual regularization term to constrain the local color distribution to be
consistent with the original distribution. Thus, it allows the network to
embed more local information. Therefore, it may enhance the embedding of local
features for semantic segmentation task.

### Sparse Label Propagation

Segmentation performance degrades significantly with few labeled points. The
main reason is that supervisory information provided by few labeled points can
not be propagated well to unlabeled points. Therefore, we use the labeled
points to assign pseudo labels for unlabeled points, and further provide
additional supervised information to improve the representation of the weakly
supervised network.

In order to achieve this goal, the following items require to be taken into
account: 1) Computational complexity is not high and memory recourse is not
large. Large-scale point clouds usually contain $\sim 10^{6}$ points, if using
all the points as nodes to construct a fully-connected graph, it will consume
a lot of memory and computing resources. 2) The anchor points should be
sparse. Some ambiguous points should not be given labels to train the network.
3) The propagated label should be soft. The propagated labels should be
related to their similarity, and the higher of the similarity, the more
similar the label should be.

We design a sparse label propagation method. The overall framework is shown in
Figure 3. It consists of three parts: class prototype generation, class
assignment matrix construction, sparse pseudo-label generation.

Class prototype generation. In the last two layers of the network, we output
embedding
$\mathbf{Z}=[z_{1}^{l},z_{2}^{l},...,z_{M}^{l};z_{1}^{u},z_{2}^{u},...,z_{N}^{u}]\in\mathcal{R}^{(M+N)\times
d}$ and the corresponding prediction
$\mathbf{Y}=[y_{1}^{l},y_{2}^{l},...,y_{M}^{l};y_{1}^{u},y_{2}^{u},...,y_{N}^{u}]\in\mathcal{R}^{(M+N)\times
C}$ which comes from $M$ labeled points and $N$ unlabeled points. We use
$\mathbf{Z^{l}}$, $\mathbf{Z^{u}}$ to represent the embedding of labeled and
unlabeled points, respectively. Firstly we generate $C$ prototypes to
represent the $C$ classes according to the labeled points. Specifically, we
simply take the mean of the labeled point embeddings $\mathbf{Z^{l}}$ for each
class. For class $c$, the prototype $\rho_{c}$ is given by:

$\rho_{c}=\frac{1}{\left|\mathcal{I}_{c}\right|}\sum_{z^{l}_{i}\in\mathcal{I}_{c}}z^{l}_{i},$
(6)

where $\mathcal{I}_{c}$ denotes the embedding sets of class $c$,
$c=\\{1,2,...,C\\}$ for labeled points.

Class assignment matrix construction. We leverage embedding $\mathbf{Z^{u}}$
of unlabeled points and the class prototypes to construct a similarity matrix
$\mathcal{W}\in\mathcal{R}^{N\times C}$ by:

$\mathcal{W}=exp(-\frac{||z_{i}^{u}-\rho_{c}||^{2}}{\sigma}),$ (7)

where $\sigma$ is a hyper-parameter. Each column of $\mathcal{W}$ represents
the similarity between the unlabeled point and the class prototypes. We use
$soft$-$max$ to convert the similarity into a class assignment probability $S$
as:

$S=\mathcal{P}(i\mapsto
c)=\frac{exp(\mathcal{W})}{\sum_{c=1}^{C}exp(\mathcal{W})}.$ (8)

Sparse pseudo label generation. There are some points with low similarity to
each category. These points are not suitable to provide supervisory
information to train the network. Specifically, for each class, according to
the class assignment matrix $S$, we select the $top$-$K$ unlabeled points and
get the mask $M^{k}\in\\{0,1\\}^{N\times C}$, where ${m_{ic}^{k}=1}$ means the
embedding of $i$-$th$ point is the first $K$ points similar to the class $c$.
$N$ is the number of unlabeled points. This is a label expansion method with a
balanced number of categories. It can alleviate the category imbalance to a
certain extent.

As an unlabeled point may belong to multiple categories, we choose the most
similar category and generate a binary mask. According to $M^{k}$, we get the
point mask $M^{pt}\in\\{0,1\\}^{N}$. $m^{pt}_{i}=1$ denotes that the $i_{th}$
point is assigned a pseudo labels. We can get the sparse pseudo label
$Y^{p}\in\mathcal{R}^{N\times C}$ by:

$Y^{p}=M^{pt}\odot S,$ (9)

where $\odot$ denotes Hadamard product. $Y^{p}$ is the form of soft one-hot.
The label propagation loss $\mathcal{L}_{sp}$ can be formulated as:

$\mathcal{L}_{sp}=-\frac{1}{||M^{pt}||_{1}}\sum_{i=1}^{N}m^{pt}_{i}\sum_{c=1}^{C}{y}^{p}_{ic}\log
y^{u}_{ic},$ (10)

where $y^{u}_{ic}$ is the probability that the unlabeled point $i$ is
classified as category $c$, and ${||\cdot||_{1}}$ denotes $L1$ norm.

Compared with the traditional fully-connected graph label propagation method,
our method is computation efficient. The complexity of our method is
$\mathcal{O}(NCd)$, while the method of fully-connected graph is
$\mathcal{O}((N+M)^{2}d)$. $C$ is the number of category and $d$ denotes the
dimension of $\mathbf{Z^{u}}$. The magnitude of $C$ is $\sim 10^{1}$, which is
much smaller than $N(\sim 10^{6}$).

### Loss of Weakly Supervised Task

The loss of weakly supervised semantic segmentation includes two terms:
segmentation loss and label propagation loss.

$\mathcal{L}_{total}=\mathcal{L}_{seg}+\lambda\mathcal{L}_{sp},$ (11)

We utilize a softmax cross-entropy loss on the labeled points. For labeled
points, the segmentation loss is formulated as:

$\mathcal{L}_{seg}=-\frac{1}{M}\sum_{i=1}^{M}\sum_{c=1}^{C}y_{ic}\log\frac{\exp\left(y^{l}_{ic}\right)}{\sum_{c=1}^{C}\exp\left(y^{l}_{ic}\right)},$
(12)

where $y_{ic}$ is the ground truth of labeled point $i$. $M$ denotes the
number of labeled points.

In the early stage of training, the embedding is unreliable. Class prototypes
generated by embedding can not represent classes well. Thus, the label
propagation loss should not be introduced to optimize network parameters. As
the embedding gets better, the weight of the pseudo loss should increase.
Therefore, we introduce a non-linear parameter $\lambda$ to balance the two
losses. In this work, $\lambda$ is formulated as:

$\lambda=\left\\{\begin{aligned} &0,&epoch<30\\\
&e^{\frac{epoch}{max\\_epoch}-1},&otherwise\\\ \end{aligned}\right.$ (13)

where $epoch$ and $max\\_epoch$ denote the current training epoch and the
total epochs, respectively.

## Experiments and Analysis

In this section, we firstly introduce the experimental settings. Then, we
evaluate the weakly semantic segmentation performance on indoor and outdoor
large-scale point clouds, respectively. Furthermore, we perform ablation study
to evaluate the importance of the main components.

### Experiment Settings

#### Dataset Setting for Self-supervised Task

Previous works on unsupervised 3D representation learning (Achlioptas et al.
2018; Gadelha, Wang, and Maji 2018; Hassani and Haley 2019; Li, Chen, and Hee
Lee 2018; Sauder and Sievers 2019; Yang et al. 2018) mainly focused on
ShapeNet (Chang et al. 2015) which is a dataset of single-object CAD point
cloud models. However, the real-world large-scale point cloud usually contains
multiple models of different categories. Pre-training on ShapeNet have poor
scalability because of a large domain gap. We choose ScanNet (Dai et al. 2017)
as the pre-training dataset which is a big real-world point cloud dataset
containing 2.5 million views in more than 1500 indoor scans.

To train self-supervised pretext task, we convert the RGB space to the Lab
space and split $6$ channels corresponding to the coordinate $(x,y,z)$ and
color $(L,a,b)$ into the given channels $(x,y,z,L)$ and the prediction
channels $(a,b)$.

#### Dataset Setting for Weakly Supervised Segmentation

In order to evaluate the performance of our network on weakly semantic
segmentation tasks, we experiment on the indoor scene dataset S3DIS (Armeni et
al. 2016) and ScanNetv2 (Dai et al. 2017) and outdoor dataset Semantic3D
(Hackel et al. 2017). Note that our method is pre-trained on ScanNet dataset.
However, the pre-training task does not use the semantic labels.

#### Implementation Details

Here weakly supervised settings are studied. i) 1 point label ($1pt$), we
assume there is only one point within each category labeled with ground-truth
for each scene. ii) ($x$%) denote $x$ percentage points with ground-truth. We
set $x=\\{1,10\\}$ in our experiments. iii) The cost of labeling $1\%$ points
manually is much higher than $1pt$. We consider a cheap way: super-point
($SPT$), which annotates a local area instead of one point. In all settings,
the annotated points are randomly selected.

Our pre-training and weakly semantic segmentation network are built on the
efficient backbone of RandLA-Net. The two networks can be trained built upon
TensorFlow with a single NVIDIA Titan RTX. We use Adam optimizer with default
parameters. The number of neighboring points $K$ is $16$. Both the self-
supervised pretext task and weakly semantic segmentation network are trained
80 epochs. We calculate the mean Intersect over Union (mIoU) to evaluate the
performance.

Setting | Method | Area5 | 6-fold
---|---|---|---
Fully | PointNet [’17] | 41 | 47.6
DGCNN [’19] | - | 56.1
RSNet [’18] | 56.5 | -
PointCNN [’18] | 57.3 | 65.4
ShellNet [’19] | - | 66.8
RandLA-Net [’20] | 62.8* | 70.0
$1pt$ ($0.2\%$) |  $\Pi$ Model [’16] | 44.3 | -
MT [’17] | 44.4 | -
Xu [’20] | 44.5 | -
$0.2\%$ | Ours | 56.4 | -
$1pt$ ($0.03\%$) | Baseline | 40.4 | -
Ours | 45.8 | -
$1\%$ | Ours | 61.8 | 65.9
$10\%$ |  $\Pi$ Model [’16] | 46.3 | -
MT [’17] | 47.9 | -
Xu [’20] | 48.0 | -
Ours | 64.0 | 68.1
Table 1: Comparisons of performance on S3DIS (Armeni et al. 2016) (mIoU %).
Area5 denotes evaluation on Area 5. 6-fold is cross evaluation of 6 areas. The
superscript * indicates the result evaluated by the official code.

### Evaluation on S3DIS

We conduct the comparison experiments with state-of-the-art weakly supervised
(Weakly) and fully supervised (Fully) on the indoor detaset S3DIS. The former
contains: $\Pi$ Model (Laine and Aila 2016), MT (Tarvainen and Valpola 2017)
and Xu (Xu and Lee 2020). The latter includes: PointNet (Qi et al. 2017a),
DGCNN (Wang et al. 2019), RSNet (Huang, Wang, and Neumann 2018), PointCNN (Li
et al. 2018), ShellNet (Zhang, Hua, and Yeung 2019), RandLA-Net (Hu et al.
2020).

The results are presented in Table 1. Overall, we observe the consistent
improvement in the performance of segmentation with more labeled points. Our
$1pt$ denotes only one labeled point for each category in the entire rooms
instead of small blocks (e.g., $1\times 1$ meter). Therefore, our $1pt$
represents that total labeled points are about 0.03%. While the total labeled
points of (Xu and Lee 2020) are about 0.2%. Ours still achieves better
performance against Xu (Xu and Lee 2020) at $1pt$ setting. Under $1\%$
setting, we achieve comparable performance against the fully supervised
method, RandLA-Net, and even exceed the previously fully supervised methods by
a large margin. At $10\%$ setting, ours is overwhelmingly superior to the
previous weakly supervised methods and even achieves $1.2\%$ improvement
against RandLA-Net (Hu et al. 2020) on Area 5. These results show the
effectiveness of our method.

The qualitative results are shown in Figure 1 on S3DIS dataset under $1pt$,
$1\%$ and $10\%$ settings. It can be seen that at 1% setting, our method can
correctly classify except for challenging boundaries.

### Evaluation on ScanNetv2

We compare our method with five fully supervised methods: PointNet++ (Qi et
al. 2017b), SPLATNet (Su et al. 2018), PointCNN (Li et al. 2018), KPConv
(Thomas et al. 2019), and a weakly supervised method: MPRM (Wei et al. 2020).
Table 2 shows the comparison results on test set. From $SPT$ to $10\%$
setting, ours achieves significant gain and make a breakthrough margin
compared with fully supervised method PointCNN. At $SPT$ setting, ours gains
7.9% against MPRM which is trained with subcloud-level semantic labels
(subcloud). We cannot directly and quantitatively evaluate the annotated labor
of $SPT$ and MPRM. Compared to MPRM, ours needs to draw an extra little region
at random. But it reduces the labor of dividing into subclouds and repeatedly
annotating semantic category for each subcloud.

Setting | Method |  mIoU
---|---|---
Fully | PointNet++ [’17] |  33.9
SPLATNet [’18] |  39.3
PointCNN [’18] |  45.8
KPConv [’19] |  68.4
Weakly | MPRM (subcloud) [’20] |  41.1
Ours ($SPT$) |  49.0
Ours ($1\%$) |  51.1
Ours ($10\%$) |  52.0
Table 2: Quantitative results on ScanNetv2 (mIoU %).

### Evaluation on Semantic3D

For outdoor point cloud semantic segmentation, we compare ours against five
state-of-the-art fully supervised methods: SnapNet (Boulch, Le Saux, and
Audebert 2017), SEGCloud (Tchapmi et al. 2017), ShellNet (Zhang, Hua, and
Yeung 2019), KPConv (Thomas et al. 2019) and RandLA-Net (Hu et al. 2020). The
results of online testing are summarized in Table 3.

Clearly, the more labeled points, the better the semantic segmentation
performance is. With 1% labeled points, we achieve 72.6% mIoU and outperforms
fully supervised ShellNet (Zhang, Hua, and Yeung 2019) by the gain of over
3.3%. With $10\%$ labeled points, our method is close to the current state-of-
the-art RandLA-Net. Therefore, our method is also effective in outdoor
dataset.

Setting | Method | mIoU |  OA
---|---|---|---
Fully | SnapNet [’17] | 59.1 |  88.6
SEGCloud [’17] | 61.3 |  88.1
ShellNet [’19] | 69.3 |  93.2
KPConv [’19] | 74.6 |  92.9
RandLA-Net [’20] | 77.4 |  94.8
Weakly | Ours ($1\%$) | 72.6 |  93.7
Ours ($10\%$) | 73.3 |  94.0
Table 3: Quantitative results on Semantic3D (reduced-8) (Hackel et al. 2017).
’OA’ denotes overall accuracy.

### Ablation Study

We conducted ablation studies to evaluate the importance of the main
components: self-supervised pre-training task and sparse label propagation.
Scratch denotes random initialization. Pre-training represents that using the
parameters of self-supervised pretext task for initialization. $\lambda=0$
represent without sparse label propagation. $\lambda=1$ and ‘nonlinear’ denote
a constant $1$ and a nonlinear way weighting the sparse label propagation,
respectively.

Self-supervised pre-training or training from scratch.

Setting |  | $1pt$ | $SPT$ | $1\%$ | $10\%$
---|---|---|---|---|---
Scratch | $\lambda=0$ | 40.4 | 55.8 | 58.6 | 60.6
Pre-training | $\lambda=0$ | 45.4 | 57.5 | 60.1 | 61.3
$\lambda=1$ | - | - | - | -
nonlinear | 45.8 | 58.2 | 61.8 | 64.0
Table 4: Comparisons of different components on S3DIS Area 5 (Armeni et al.
2016) (mIoU %).

From the comparison of the first and second rows of Table 4, the pre-training
makes the performance better, while training from scratch drops in performance
for 5.0%, 1.7%, 1.5%, 0.7%. We hypothesize that our pre-training is effective
because it can transfer knowledge learned from supervised tasks to subsequent
tasks. This encourages the subsequent task to learn a more robust feature
distribution. We can still find that the fewer annotation points, the greater
the effect of self-supervision. We infer that fewer labeled points cannot
learn a good feature representation. Therefore, it needs more additional
knowledge to promote feature representation learning.

Sparse label propagation. We further analyse the effects of sparse label
propagation. Comparing the second and fourth rows of Table 4, our nonlinear
obtains 0.4%, 0.7%, 1.7%, and 2.7% improvement over $\lambda=0$, respectively.
The more labeled points, the better performance of the semantic segmentation
achieves. Because more labeled points can learn better representations and
produce accurate class prototypes. When $\lambda$ is a constant $1$, the
network training collapses. Because the initial training (the first 30 epochs)
cannot learn a good feature representation. It is unreasonable to use these
features for label propagation. In the process of feature learning gradually
getting better, a larger weight is gradually introduced to the loss term.

Effectiveness and efficiency. We conduct a pilot study to understand the
effectiveness and show the mIoU curve of our method and baseline on validation
set. From Figure 4, our method converges faster and has a higher stable value
which demonstrates our method is more effective for weakly supervised large-
scale point cloud segmentation.

Like RandLA-Net (Hu et al. 2020), we feed the entire scene to the network. We
list the training time of each epoch (RandLA-Net: 267s and Ours: 280s) and
total test time (RandLA-Net: 115s and Ours: 116s) on Area-5 of S3DIS dataset.
As the sparse label propagation is introduced in the training phase, training
time of per-epoch increases by 13s. But their test time are almost the same.

(a) On S3DIS (b) On Semantic3D

Figure 4: mIoU curve of our method and baseline on the validation set.

We also visualize the results of colorization in Figure 5 on S3DIS. Although
our point cloud colorization focuses on the representation learning, it can be
seen that the colorization is close to real-world color.

Figure 5: Visualize the results of point cloud colorization.

## Conclusion

We present a weakly supervised semantic segmentation method for large-scale
point cloud. With the help of a self-supervised pretext task and sparse label
propagation, our method significantly outperforms the weakly supervised and
almost reaches the accuracy of fully-supervised methods on three challenging
large-scale point cloud dataset. The experimental results show self-supervised
knowledge transfer is an effective way to improve weak supervised performance
and our method make a breakthrough margin against the compared weakly
supervised method. Furthermore, the more the labeled points, the better
performance our method achieves, which is consistent with our expectation. In
the future, we expect future work to explore more effective self-supervised
knowledge representation for weakly or semi-supervised point cloud tasks.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
under Grant 61876161, Grant 61772524, the National Key Research and
Development Program of China No.2020AAA0108301, and Natural Science Foundation
of Shanghai No.20ZR1417700. CAAl-Huawei Mind-Spore Open Fund.

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