# Strong Increase in Ultrasound Attenuation Below Tc in Sr2RuO4: Possible
Evidence for Domains

Sayak Ghosh  Laboratory of Atomic and Solid State Physics, Cornell University,
Ithaca, NY 14853, USA  Thomas G. Kiely  Laboratory of Atomic and Solid State
Physics, Cornell University, Ithaca, NY 14853, USA  Arkady Shekhter  National
High Magnetic Field Laboratory, Florida State University, Tallahassee, FL
32310, USA  F. Jerzembeck  Max Planck Institute for Chemical Physics of
Solids, Dresden, Germany  N. Kikugawa  National Institute for Materials
Science, Tsukuba, Ibaraki 305-0003, Japan  Dmitry A. Sokolov  Max Planck
Institute for Chemical Physics of Solids, Dresden, Germany  A. P. Mackenzie
Max Planck Institute for Chemical Physics of Solids, Dresden, Germany  SUPA,
School of Physics and Astronomy, University of St Andrews, North Haugh, St
Andrews KY16 9SS, UK  B. J. Ramshaw<EMAIL_ADDRESS>Laboratory of
Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA

Recent experiments suggest that the superconducting order parameter of Sr2RuO4
has two components. A two-component order parameter has multiple degrees of
freedom in the superconducting state that can result in low-energy collective
modes or the formation of domain walls—a possibility that would explain a
number of experimental observations including the smallness of the time
reversal symmetry breaking signal at $T_{\mathrm{c}}$ and telegraph noise in
critical current experiments. We perform ultrasound attenuation measurements
across the superconducting transition of Sr2RuO4 using resonant ultrasound
spectroscopy (RUS). We find that the attenuation for compressional sound
increases by a factor of seven immediately below $T_{\mathrm{c}}$, in sharp
contrast with what is found in both conventional ($s$-wave) and
high-$T_{\mathrm{c}}$ ($d$-wave) superconductors. We find our observations to
be most consistent with the presence of domain walls between different
configurations of the superconducting state. The fact that we observe an
increase in sound attenuation for compressional strains, and not for shear
strains, suggests an inhomogeneous superconducting state formed of two
distinct, accidentally-degenerate superconducting order parameters that are
not related to each other by symmetry. Whatever the mechanism, a factor of
seven increase in sound attenuation is a singular characteristic with which
any potential theory of the superconductivity in Sr2RuO4 must be reconciled.

## Introduction

One firm, if perhaps counter-intuitive, prediction of Bardeen, Cooper, and
Schrieffer (BCS) theory is the contrasting behavior of the nuclear spin
relaxation rate, $1/T_{1}$, and the ultrasonic attenuation, $\alpha$ Bardeen
_et al._ (1957). Upon cooling from the normal state to the superconducting
(SC) state, one might expect both $1/T_{1}$ and $\alpha$ to decrease as both
processes involve the scattering of normal quasiparticles. In the SC state,
however, Cooper pairing produces correlations between quasiparticles of
opposite spin and momentum. These correlations produce “coherence factors”
that add constructively for nuclear relaxation and produce a peak—the Hebel-
Slichter peak—in $1/T_{1}$ immediately below $T_{\mathrm{c}}$ Hebel and
Slichter (1957). In contrast, the coherence factors add destructively for
sound attenuation and there is an immediate drop in $\alpha$ below
$T_{\mathrm{c}}$ Morse  _et al._ (1959). These experiments provided some of
the strongest early evidence for the validity of BCS theory Bardeen  _et al._
(1957), and the drop in sound attenuation below $T_{\mathrm{c}}$ was
subsequently confirmed in many elemental superconductors Levy (1963);
Leibowitz (1964); Claiborne and Einspruch (1965); Fossheim (1967).

It came as a surprise, then, when peaks in the sound attenuation were
discovered below $T_{\mathrm{c}}$ in two heavy-fermion superconductors: UPt3
and UBe13 Batlogg  _et al._ (1985); Golding  _et al._ (1985); Müller  _et al._
(1986). Specifically, peaks were observed in the longitudinal sound
attenuation—when the sound propagation vector $\mathbf{q}$ is parallel to the
sound polarization $\mathbf{u}$: ($\mathbf{q}\parallel\mathbf{u}$). Transverse
sound attenuation ($\mathbf{q}\perp\mathbf{u}$), on the other hand, showed no
peak below $T_{\mathrm{c}}$ but instead decreased with power law dependencies
on $T$ that were ultimately understood in terms of the presence of nodes in
the SC gap Moreno and Coleman (1996). Various theoretical proposals were put
forward to understand the peaks in the longitudinal sound attenuation,
including collective modes, domain-wall friction, and coherence-factors Miyake
and Varma (1986); Monien  _et al._ (1987); Joynt  _et al._ (1986); Coffey
(1987), but the particular mechanisms for UPt3 and UBe13 were never pinned
down (see Sigrist and Ueda (1991) for a review). What is clear, however, is
that a peak in sound attenuation below $T_{\mathrm{c}}$ is not a prediction of
BCS theory and surely indicates unconventional superconductivity.

Figure 1:  Measuring ultrasonic attenuation with resonant ultrasound
spectroscopy. (a) The Sr2RuO4 unit cell under a deformation corresponding to
the longitudinal strain $\epsilon_{xx}$, associated with the elastic constant
$c_{11}$. This mode is a superposition of pure compression
$\epsilon_{xx}+\epsilon_{yy}$ and pure shear $\epsilon_{xx}-\epsilon_{yy}$,
associated with the elastic constants $(c_{11}+c_{12})/2$ and
$(c_{11}-c_{12})/2$, respectively. (b) Resonant ultrasound spectrum of Sr2RuO4
between 2.2-2.8 MHz. $X(\omega)$ and $Y(\omega)$ are the real and the
imaginary parts of the response. The boxed resonance is shown in detail in
(c). (c) Zoom-in on the resonance near 2.34 MHz. The center of the resonance
and the linewidth are indicated. Inset shows the same resonance plotted in
complex plane and fit to a circle—$z_{c}$ denotes the center of the circle.

The superconductivity of Sr2RuO4 has many unconventional aspects, including
time reversal symmetry (TRS) breaking Luke  _et al._ (1998); Xia  _et al._
(2006); Kidwingira  _et al._ (2006), the presence of nodal quasiparticles
Lupien  _et al._ (2001); Hassinger  _et al._ (2017); Sharma  _et al._ (2020),
and two components in its SC order parameter Ghosh  _et al._ (2021); Benhabib
_et al._ (2021). These observations have led to various recent theoretical
proposals for the SC state in Sr2RuO4 Ramires and Sigrist (2019); Rømer  _et
al._ (2019); Røising  _et al._ (2019); Scaffidi (2020); Suh  _et al._ (2020);
Kivelson  _et al._ (2020); Willa  _et al._ (2021), requiring further
experimental inputs to differentiate between them. Not only should the
coherence factors differ for Sr2RuO4 compared to the $s-$wave BCS case, but
there is the possibility of low-energy collective modes Chung  _et al._ (2012)
and domain-wall motion Yuan  _et al._ (2021), all of which could be observable
in the ultrasonic attenuation.

Prior ultrasonic attenuation measurements on Sr2RuO4 reported a power-law
temperature dependence of the transverse sound attenuation, interpreted as
evidence for nodes in the gap Lupien  _et al._ (2001), but found no other
unconventional behaviour. This may be in part due to the specific ultrasound
technique employed: pulse-echo ultrasound. While pulse-echo can measure a pure
shear-strain response in the transverse configuration, the in-plane
longitudinal configuration is a combination of both compression strain and
shear strain in a tetragonal crystal like Sr2RuO4 Brugger (1965).
Specifically, the L100 mode measures the elastic constant $c_{11}$, which is a
mixture of pure compression, $(c_{11}+c_{12})/2$, and pure shear,
$(c_{11}-c_{12})/2$ (see Figure 1(a)). Shear and compression strains couple to
physical processes in fundamentally different ways and thus effects that
couple exclusively to compressional sound may have been missed in previous
measurements.

## Experiment

We have measured the ultrasonic attenuation of Sr2RuO4 across $T_{\mathrm{c}}$
using an ultrasound technique distinct from pulse-echo ultrasound—resonant
ultrasound spectroscopy (RUS). RUS allows us to obtain the attenuation in all
the independent symmetry channels in a single experiment (i.e. for all 5
irreducible representations of strain in Sr2RuO4). See Ghosh  _et al._ (2021)
for details of our custom built low-temperature RUS apparatus. The high-
quality Sr2RuO4 crystal used in this experiment was grown by the floating zone
method—more details about the sample growth can be found in Bobowski  _et al._
(2019). A single crystal was precision-cut along the [110], [1$\bar{1}$0] and
[001] directions and polished to the dimensions 1.50 mm $\times$ 1.60 mm
$\times$ 1.44 mm, with 1.44 mm along the tetragonal $c$ axis. The sample
quality was characterized by heat capacity and AC susceptibility measurements,
as reported in Ghosh  _et al._ (2021). The SC $T_{\mathrm{c}}$ measured by
these techniques—approximately 1.43 K—agrees well with the $T_{\mathrm{c}}$
seen in our RUS experiment, indicating that the sample underwent uniform
cooling during the experiment.

RUS measures the mechanical resonances of a three-dimensional solid. The
frequencies of these resonances depend on the elastic moduli, density, and
geometry of the sample, while the widths of these resonances are determined by
the ultrasonic attenuation Ramshaw  _et al._ (2015); Ghosh  _et al._ (2020).
Because each resonance mode is a superposition of multiple kinds of strain,
the attenuation in all strain channels can be extracted by measuring a
sufficient number of resonances—typically 2 or 3 times the number of unique
strains (of which there are 5 for Sr2RuO4).

A typical RUS spectrum from our Sr2RuO4 sample is shown in Figure 1(b) (see
methods for details of the measurement.) Each resonance can be modeled as the
response $Z(\omega)$ of a damped harmonic oscillator driven at frequency
$\omega$ (see Figure 1(c)),

$Z(\omega)=X(\omega)+iY(\omega)=Ae^{i\phi}/((\omega-\omega_{0})+i\Gamma)$ (1)

where $X$ and $Y$ are the real and imaginary parts of the response, and $A$,
$\Gamma$, and $\phi$ are the amplitude, linewidth, and phase, respectively.
The real and imaginary parts of the response form a circle in the complex
plane. The response is measured at a set of frequencies that space the data
points evenly around this circle: this is the most efficient way to precisely
determine the resonant frequency $\omega_{0}$ and the linewidth $\Gamma$ in a
finite time (see Shekhter  _et al._ (2013) for details of the fitting
procedure). We plot the temperature dependence of the linewidth of all our
experimentally measured resonances through $T_{\mathrm{c}}$ in the SI. For
comparison, the attenuation $\alpha$ measured in conventional pulse-echo
ultrasound is related to the resonance linewidth via $\alpha=\Gamma/v$, where
$v$ is the sound velocity.

## Results

When the sound wavelength, $\lambda=\frac{2\pi}{q}$, is much longer than the
electronic mean free path $l$, i.e. when $ql\ll 1$, the electron-phonon system
is said to be in the ‘hydrodynamic’ limit Khan and Allen (1987) (this is
different than the hydrodynamic limit of electron transport). Given that the
best Sr2RuO4 has a mean free path that is at most of order a couple of
microns, and that our experimental wavelengths are of the order of 1 mm, we
are well within the hydrodynamic limit. In this regime, we can express the
linewidth $\Gamma$ of a resonance $\omega_{0}$ as,

$\frac{\Gamma}{\omega_{0}^{2}}=\frac{1}{2}\sum_{j}\alpha_{j}\frac{\eta_{j}}{c_{j}},$
(2)

where $\eta_{j}$ and $c_{j}$ are the independent components of the viscosity
and elastic moduli tensors, respectively (see SI for details). The $\alpha$
coefficients define the composition of a resonance, such that
$\alpha_{j}=\partial(\ln\omega_{0}^{2})/\partial(\ln c_{j})$ and
$\sum_{j}\alpha_{j}=1$Ramshaw  _et al._ (2015).

We measured the linewidths of 17 resonances and resolved them into the
independent components of the viscosity tensor. The tetragonal symmetry of
Sr2RuO4 dictates that there are only six independent components, arising from
the five irreducible representations (irreps) of strain in $D_{4h}$ plus one
component arising from coupling between the two distinct compression strains
Ghosh  _et al._ (2021). The six symmetry-resolved components of viscosity in
Sr2RuO4 are plotted in Figure 2.

The shear viscosity $(\eta_{11}-\eta_{12})/2$ decreases below $T_{\mathrm{c}}$
in a manner similar to what is observed in conventional superconductors Morse
_et al._ (1959); Levy (1963). We find that $(\eta_{11}-\eta_{12})/2$ is much
larger than the other two shear viscosities, which is consistent with previous
pulse-echo ultrasound experiments Lupien  _et al._ (2001); Lupien (2002). On
converting attenuation to viscosity, we find relatively good agreement (within
a factor of 2) between the resonant ultrasound and pulse-echo measurements of
$(\eta_{11}-\eta_{12})/2$. This is particularly non-trivial given that sound
attenuation scales as frequency squared in the hydrodynamic regime and the
pulse-echo ultrasound measurements were performed at frequencies roughly two
orders of magnitude higher than those used in the RUS measurements. The much
larger magnitude of $(\eta_{11}-\eta_{12})/2$, in comparison to $\eta_{66}$,
may be due to the fact that the $\epsilon_{xx}-\epsilon_{yy}$ strain is
associated with pushing the $\gamma$ Fermi surface pocket toward the van Hove
singularity Barber  _et al._ (2019). The small values of $\eta_{44}$ and
$\eta_{66}$ are comparable to the experimental noise and any changes at
$T_{\mathrm{c}}$ are too small to resolve at these low frequencies.

In contrast with the rather conventional shear viscosities, the three
compressional viscosities each exhibit a strong increase below
$T_{\mathrm{c}}$. For in-plane compression—the strain that should couple
strongest to the rather two-dimensional superconductivity of Sr2RuO4—the
increase is by more than a factor of seven. After peaking just below
$T_{\mathrm{c}}$, the attenuation slowly decreases as the temperature is
lowered. The large increase below $T_{\mathrm{c}}$ was not observed in
previous longitudinal sound attenuation measurements made by pulse-echo
ultrasound Lupien  _et al._ (2001); Lupien (2002). Longitudinal sound is a
mixture of pure shear and pure compression, as shown in Figure 1(a). At the
frequencies where pulse-echo ultrasound is measured—of order 100 MHz—the shear
viscosity $(\eta_{11}-\eta_{12})/2$ is at least one order of magnitude larger
than the compression viscosity Lupien (2002) and thus completely dominates the
sound attenuation. The relative time-scales between the dynamics of the
attenuation mechanism and the measurement frequency may also play a role—we
will return to this idea later on in the discussion.

Figure 2:  Symmetry-resolved sound attenuation in Sr2RuO4. (a) Compressional
and (b) shear viscosities through $T_{\mathrm{c}}$. The irreducible strain
corresponding to each viscosity is shown—$\eta_{13}$ arises due to coupling
between the two $A_{1g}$ strains. An increase in the compressional viscosities
is seen immediately below $T_{\mathrm{c}}$, while no such feature is seen in
the shear viscosities.

## Analysis

Figure 3:  Comparison of different mechanisms for sound attenuation in the
superconducting state. (a) Attenuation in the superconducting state
$\alpha_{S}$ (normalized by the normal state attenuation $\alpha_{N}$) for an
isotropic $s$-wave gap and a $d_{x^{2}-y^{2}}$ gap, calculated within the BCS
framework. (b) $\alpha_{S}/\alpha_{N}$ for a time reversal symmetry breaking
gap below $T_{\mathrm{c}}$. A peak is seen at high enough frequencies
($\sim$THz) but not at our experimental frequencies ($\sim$MHz). (c)
Attenuation peak at different frequencies due to pair-breaking effects in a
$d_{x^{2}-y^{2}}$ gap. The inset shows the plot at our experimental frequency
in detail—a tiny peak is seen about 0.01 nK below $T_{\mathrm{c}}$. (d)
Normalized attenuation in the $A_{1g}$ channels of Sr2RuO4 through
$T_{\mathrm{c}}$, fit to the attenuation expected from domain wall motion
below $T_{\mathrm{c}}$. The fit works well only close to $T_{\mathrm{c}}$,
probably because it does not include other temperature-dependent effects (see
text for details).

We now analyze possible mechanisms that could give rise to such an increase in
sound attenuation below $T_{\mathrm{c}}$. First, we calculate sound
attenuation within a BCS-like framework, accounting for the differences in
coherence factors that occur for various unconventional SC order parameters.
We find that a peak can indeed arise under certain circumstances but not under
our experimental conditions. Second, we consider phonon-induced Cooper pair
breaking in the SC state that does lead to a sound attenuation peak just below
$T_{\mathrm{c}}$, but which is inaccessibly narrow in our experiment. Finally,
we show that a simple model of sound attenuation due to the formation of SC
domains best matches the experimental data.

First we examine the possibility of increased sound attenuation due to
coherent scattering in the SC state. Sound attenuation and nuclear spin
relaxation in an $s$-wave superconductor are proportional to the coherence
factors $F_{\pm}=(1\pm\Delta_{0}^{2}/E_{k}E_{k^{\prime}})$, where $\Delta_{0}$
is the uniform $s$-wave gap and $E_{k}$ is the Bogoliubov quasiparticle
dispersion Bardeen  _et al._ (1957). Scattering off of a nucleus flips the
spin of the quasiparticle and the resultant coherence factor is $F_{+}$, where
the $+$ sign produces the Hebel-Slichter peak below $T_{\mathrm{c}}$.
Scattering off of a phonon, on the other hand, does not flip quasiparticle
spin and the resultant coherence factor is $F_{-}$, producing a sharp drop in
sound attenuation below $T_{\mathrm{c}}$. In general, the coherence factors
depend on the structure of the superconducting gap, motivating the idea that
an unconventional superconducting OP might produce a peak in the sound
attenuation. Calculating within the BCS framework, we find that attenuation
for a $d_{x^{2}-y^{2}}$ gap decays slowly compared to the isotropic $s$-wave
gap, but does not exhibit a peak (Figure 3(a), see SI for details of the
calculation). This slow decrease can be attributed to the presence of nodes in
the $d_{x^{2}-y^{2}}$ gap Won and Maki (1994); Vekhter  _et al._ (1999). For a
TRS breaking gap such as $p_{x}+ip_{y}$ or $d_{xz}+id_{yz}$, a Hebel-Slichter-
like peak appears below $T_{\mathrm{c}}$ if sufficiently large-angle
scattering is allowed (Figure 3(b)). Scattering at these large
wavevectors—essentially scattering across the Fermi surface—would require
ultrasound with nanometer wavelengths. This regime is only accessible at THz
frequencies, whereas our experiment operates in the MHz range. Hence we rule
out coherent scattering as the mechanism of increased compressional sound
attenuation below $T_{\mathrm{c}}$.

Next we consider how phonon-induced Cooper pair breaking may give rise to a
sound attenuation peak, similar to what has been observed in superfluid
3He-$B$ below $T_{\mathrm{c}}$ Adenwalla  _et al._ (1989). Pair-breaking in
BCS superconductors requires a minimum energy of 2$\Delta_{0}$, where
$\Delta_{0}$ is the gap magnitude. This energy scale is generally much higher
than typical ultrasound energies. For example, the maximum gap magnitude in
Sr2RuO4 is 2$\Delta\sim$ 0.65 meV Firmo  _et al._ (2013), which would require
a frequency of approximately 1 THz to break the Cooper pairs. However, the
pair-breaking energy is lowered for a gap with nodes, such as
$d_{x^{2}-y^{2}}$. In particular, since the gap goes to zero at
$T_{\mathrm{c}}$, it may be small enough near $T_{\mathrm{c}}$ such that pair-
breaking is possible at a few MHz. Our calculations for a $d_{x^{2}-y^{2}}$
gap, however, show that $\sim$10 GHz frequencies are required to produce an
experimentally discernible peak (Figure 3(c)). At our experimental
frequencies, the peak is only visible within 0.01 nK of $T_{\mathrm{c}}$. For
a fully gapped superconductor, like the TRS breaking state $p_{x}+ip_{y}$, the
peak will be even smaller. This clearly rules out pair-breaking as the origin
of the increased sound attenuation.

Finally, we consider the formation of SC domains. When different
configurations of a SC order parameter are degenerate, such as $p_{x}+ip_{y}$
and $p_{x}-ip_{y}$, domains of each configuration will form separated by
domain walls. These domain walls can oscillate about their equilibrium
positions when sound propagates through the sample Joynt  _et al._ (1986).
Sigrist and Ueda (1991) derive an expression for the sound attenuation
coefficient, $\alpha$, from domain wall motion of the form

$\alpha\left(\omega,T\right)\propto\frac{\omega^{2}}{\omega^{2}+\omega_{DW}^{2}}\rho_{s}^{2},$
(3)

where $\rho_{s}$ is the superfluid density (proportional to the square of the
superconducting gap), $\omega$ is the angular frequency of the sound wave, and
$\omega_{DW}$ is the lowest vibrational frequency of the domain wall. While
the full functional form of $\rho_{s}$ and $\omega_{DW}$ are unknown, near
$T_{\mathrm{c}}$ they can be written within a Ginzburg-Landau (GL) formalism
as $\rho_{s}\propto|T-T_{c}|$ and $\omega_{DW}\propto|T-T_{c}|^{3/2}$ which
gives the explicit temperature dependence of the above equation as

$\alpha\left(\omega,T\right)\propto\frac{\omega^{2}}{\omega^{2}+\omega_{1}^{2}\left|T/T_{\rm
c}-1\right|^{3}}\left|T/T_{\rm c}-1\right|^{2},$ (4)

where $\omega_{1}$ is the domain wall frequency as $T\rightarrow 0$. We fit
all three attenuation channels to Equation 4 and extract $\omega_{1}=500\pm
25$ MHz (Figure 3(d)). As the temperature approaches $T_{\mathrm{c}}$ from
below the domain wall frequency decreases to zero, producing a peak in the
attenuation when the experimental frequency is approximately equal to the
domain wall frequency. This may partially explain why no peak was observed in
previous pulse-echo ultrasound measurements at $\sim$100 MHz Lupien  _et al._
(2001) (as mentioned above, there is also the issue that the attenuation for
longitudinal sound along the [100] direction is almost entirely dominated by
$(\eta_{11}-\eta_{12})/2$).) The fit of Equation 4 deviates substantially from
the data for $T/T_{\rm c}\lesssim 0.95$: this is to be expected, as the GL
theory is only valid near $T_{\mathrm{c}}$ Sigrist and Ueda (1991).
Nevertheless, Equation 4 captures the correct shape of the rapid increase in
attenuation below $T_{\mathrm{c}}$ in all three compression channels, using
the same value of $\omega_{1}$ for all three fits. The extracted frequency
scale of $\omega_{1}\approx 500$ MHz is also reasonable: studies of sound
attenuation in nickel at MHz frequencies show similar magnitudes of increase
in the magnetically ordered state when domains are present Leonard  _et al._
(1962). We note that the results of Josephson interferometry measurements have
previously been interpreted as evidence for SC domains in Sr2RuO4 Kidwingira
_et al._ (2006).

## Discussion

The factor of seven increase we find in the in-plane compressional viscosity
is without precedent in a superconductor. For comparison, longitudinal
attenuation increases by 50% below $T_{\mathrm{c}}$ in UPt3 Müller  _et al._
(1986), and by a bit more than a factor of two in UBe13 Golding  _et al._
(1985). There is also a qualitative difference between the increase in Sr2RuO4
and the increase seen in the heavy fermion superconductors: the attenuation
peaks sharply below $T_{\mathrm{c}}$ in both UPt3 and UBe13, with a peak width
of approximately 10% of $T_{\mathrm{c}}$. The compressional attenuation in
Sr2RuO4, by contrast, decreases by only about 10% over the same relative
temperature range. This suggests that something highly unconventional occurs
in the SC state of Sr2RuO4, leading to a large increase in sound attenuation
that is not confined to temperatures near $T_{\mathrm{c}}$. The mechanism we
find most consistent with the data is domain wall motion.

Assuming that we have established the likely origin of the increase in sound
attenuation, we consider its implications for the superconductivity of
Sr2RuO4. The formation of domains requires a two-component order parameter
(OP), either symmetry-enforced or accidental, reaffirming the conclusions of
recent ultrasound studies of the elastic moduli and the sound velocity Ghosh
_et al._ (2021); Benhabib  _et al._ (2021). We can learn more about which
particular OPs are consistent with our experiment by considering which
symmetry channels show the increase in attenuation. Domains attenuate
ultrasound when the application of strain raises or lowers the condensation
energy of one domain in comparison to a neighboring domain. A simple example
is the “nematic” superconducting state proposed by Benhabib  _et al._ (2021),
which is a $d-$wave OP of the $E_{g}$ representation, transforming as
$\left\\{d_{xz},d_{yz}\right\\}$. Under $(\epsilon_{xx}-\epsilon_{yy})$
strain, domains of the $d_{xz}$ configuration will be favored over the
$d_{yz}$ configuration (depending on the sign of the strain). This will cause
some domains to grow and others to shrink, attenuating sound through the
mechanism proposed by Sigrist and Ueda (1991). We find no increase in
$\left(\eta_{11}-\eta_{12}\right)/2$ below $T_{\mathrm{c}}$, suggesting that a
$\left\\{d_{xz},d_{yz}\right\\}$ OP cannot explain the increase in
compressional sound attenuation.

More generally, the lack of increase in attenuation in any of the shear
channels implies that that the SC state of Sr2RuO4 does not break rotational
symmetry. Domains that are related to each other by time reversal symmetry can
also be ruled out: there is no strain that can lift the degeneracy between,
for example, a $p_{x}+ip_{y}$ domain and a $p_{x}-ip_{y}$ domain. The observed
increase in sound attenuation under compressional strain is therefore quite
unusual: as Sigrist and Ueda (1991) point out, compressional strains can never
lift the degeneracy between domains that are related by any symmetry, since
compressional strains do not break the point group symmetry of the lattice.
Instead, attenuation in the compressional channel requires domains that couple
differently to compressional strain, which in turn requires domains that are
accidentally degenerate. Examples that are consistent with both NMR Pustogow
_et al._ (2019) and ultrasound Ghosh  _et al._ (2021); Benhabib  _et al._
(2021) include $\\{d_{x^{2}-y^{2}},g_{xy(x^{2}-y^{2})}\\}$ Kivelson  _et al._
(2020); Willa  _et al._ (2021); Clepkens  _et al._ (2021) and $\\{s,d_{xy}\\}$
Rømer  _et al._ (2021). Then, for example, domains of $d_{x^{2}-y^{2}}$ will
couple differently to compressional strain than domains of
$g_{xy(x^{2}-y^{2})}$, leading to the growth of one domain type and an
increase in compressional sound attenuation below $T_{\mathrm{c}}$. Shear
strain, meanwhile, does not change the condensation energy of any single-
component order parameter (e.g. $s$, $d_{xy}$, $d_{x^{2}-y^{2}}$, or
$g_{xy(x^{2}-y^{2})}$) to first order in strain, which means that the lack of
increase in shear attenuation below $T_{\mathrm{c}}$ is also consistent with
an accidentally-degenerate OP. This is also consistent with the lack of a cusp
in $T_{\mathrm{c}}$ under applied shear strain Hicks  _et al._ (2014); Watson
_et al._ (2018).

Recent theoretical work Yuan  _et al._ (2021) has shown that domain walls
between $d_{x^{2}-y^{2}}$ and $g_{xy(x^{2}-y^{2})}$ OPs may provide a route to
explain the observation of half-quantum vortices in Sr2RuO4 without a spin-
triplet order parameter Jang  _et al._ (2011)—a result that is otherwise
inconsistent with the singlet pairing suggested by NMR Pustogow  _et al._
(2019). Willa  _et al._ (2021), followed by Yuan  _et al._ (2021), have shown
that domains between such states stabilize a TRS-breaking $d_{x^{2}-y^{2}}\pm
ig_{xy(x^{2}-y^{2})}$ state near the domain wall. This would naturally explain
why probes of TRS breaking, such as the Kerr effect and $\mu$SR Xia  _et al._
(2006); Grinenko  _et al._ (2021), see such a small effect at $T_{\mathrm{c}}$
in Sr2RuO4.

One significant challenge for the two-component order parameter scenario is
that, whether accidentally degenerate or not, a two component order parameter
should generically produce two superconducting $T_{\mathrm{c}}$s. The lack of
a heat capacity signature from an expected second transition under uniaxial
strain Li  _et al._ (2021) can only be explained if the second, TRS-breaking
transition is particularly weak—a result that might be consistent with the
TRS-breaking state appearing only along domain walls. Finally, it is worth
noting that there are other mechanisms of ultrasonic attenuation that we have
not explored here, including collective modes and gapless excitations such as
edge currents that might appear along domain walls even if the domains are
related by symmetry. Future ultrasound experiments under applied static strain
and magnetic fields are warranted as certain types of domain walls can couple
to these fields, thereby affecting the sound attenuation through
$T_{\mathrm{c}}$.

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## Acknowledgments

B. J. R. and S. G. acknowledge support for building the experiment, collecting
and analyzing the data, and writing the manuscript from the Office of Basic
Energy Sciences of the United States Department of Energy under award no. DE-
SC0020143. B.J.R. and S.G. acknowledge support from the Cornell Center for
Materials Research with funding from the Materials Research Science and
Engineering Centers program of the National Science Foundation (cooperative
agreement no. DMR-1719875). T. G. K. acknowledge support from the National
Science Foundation under grant no. PHY-2110250. N. K. acknowledges support
from Japan Society for the Promotion of Science (JSPS) KAKENHI (Nos.
JP17H06136, JP18K04715, and 21H01033)) and Japan Science and Technology Agency
Mirai Program (JPMJMI18A3) in Japan.

## Supplemental Material

### Viscosity from RUS Measurements

RUS measures the mechanical resonances of a solid, which are determined by the
elastic constants and the density of the material along with its dimensions.
Each resonance $\omega_{0}$ is a superposition of the various irreducible
strains, and therefore is a function of the independent elastic moduli
$c_{j}$. In a tetragonal system like Sr2RuO4, there are six such moduli
($j=1,2,...,6$) such that

$\omega_{0}^{2}=\mathfrak{F}(c_{j},\rho,l_{k}),$ (5)

where $\rho$ is the density and $l_{k}$ are the dimensions of the sample.
Sound attenuation in the solid leads to these frequencies having a finite
linewidth $\Gamma$, giving them a characteristic Lorentzian shape (as
discussed in the main). Within linear response, sound attenuation is related
to the strain rate through the viscosity tensor Moreno and Coleman (1996),
which has the same symmetries as the elastic moduli tensor. To relate the
experimentally measured linewidths to the irreducible viscosities, we replace
$\omega_{0}\rightarrow\omega_{0}+i\Gamma$ and $c_{j}\rightarrow
c_{j}+i\omega_{0}\eta_{j}$ in Equation 5,

$\displaystyle(\omega_{0}+i\Gamma)^{2}=\mathfrak{F}(c_{j}+i\omega_{0}\eta_{j},\rho,l_{k})$
(6) $\displaystyle\implies$
$\displaystyle\omega_{0}^{2}+2i\omega_{0}\Gamma\approx\mathfrak{F}(c_{j},\rho,l_{k})+\sum_{j}\frac{\partial\mathfrak{F}}{\partial
c_{j}}\cdot i\omega_{0}\eta_{j}$ $\displaystyle\implies$
$\displaystyle\Gamma=\frac{1}{2}\sum_{j}\frac{\partial\mathfrak{F}}{\partial
c_{j}}\cdot\eta_{j}=\frac{1}{2}\omega_{0}^{2}\sum_{j}\alpha_{j}\frac{\eta_{j}}{c_{j}},$

where $\alpha_{j}=\partial(\ln\omega_{0}^{2})/\partial(\ln c_{j})$ and
$\sum_{j}\alpha_{j}=1$ Ramshaw  _et al._ (2015). The fit procedure outlined in
Ramshaw  _et al._ (2015) gives us the $\alpha_{j}$ coefficients for all our
experimentally measured resonances. Knowing these coefficients, we can
calculate the six independent viscosities as a function of temperature by
measuring the temperature evolution of sufficiently many resonance linewidths
(typically 2-3 times the number of independent viscosities). Note that
Equation 6 is true in the weak attenuation limit ($\Gamma\ll\omega_{0}$),
which is easily satisfied in our experiments ($\Gamma/\omega_{0}\sim 10^{-4}$
for all our measured resonances, see Figure 4).

Figure 4: RUS attenuation data. (a) Temperature evolution of normalized
resonance linewidth of 18 resonances of Sr2RuO4 through $T_{\mathrm{c}}$, with
panels (a) and (b) each showing 9 resonances. These 18 resonances were used to
calculate the six independent components of the viscosity tensor through
$T_{\mathrm{c}}$.

### Quasiparticle Scattering in the Superconducting State

The conventional method for describing ultrasound attenuation assumes that
sound waves attenuate by scattering quasiparticles. In particular, we assume
that the attenuation rate is proportional to the scattering rate induced by
the sound wave. The effect of the superconducting transition on scattering was
introduced by BCS Bardeen  _et al._ (1957) and is described pedagogically by
Tinkham Tinkham (1996) and Schrieffer Schrieffer (1964). We assume that the
induced scattering can be described by an interaction Hamiltonian

$\mathcal{H}_{\rm
int}=\sum_{k,k^{\prime},\sigma}M_{k,k^{\prime}}c^{\dagger}_{k,\sigma}c_{k^{\prime},\sigma}$
(7)

where $M_{k,k^{\prime}}$ is symmetric under time reversal (TR). This is
crucial: as an $s$-wave superconductor consists of time-reversed fermionic
pairs, the symmetry of the interaction under TR has drastic effects on the
scattering properties. In particular, an interaction Hamiltonian that is even
under TR will result in destructive interference between Bogoliubov
quasiparticles, and vice versa.

The scattering rate can be computed by Fermi’s golden rule. Following Tinkham
Tinkham (1996), we take $M_{k,k^{\prime}}=Me^{i\theta_{k,k^{\prime}}}$ to have
a constant modulus. This is a drastic approximation that allows for
arbitrarily wide-angle scattering. While it has little effect on the $s$-wave
calculation, we will have to remedy this approximation for the $p$-wave gap
function. We perform the calculations in 2D for each of the Fermi surfaces of
${\rm Sr}_{2}{\rm RuO}_{4}$. The band structure adds a Jacobian factor to the
integrand, $J_{i}(\phi)$, where $\phi$ is the azimuthal angle in momentum
space and $i\in\\{\alpha,\beta,\gamma\\}$ for the $\alpha$, $\beta$ and
$\gamma$ bands, respectively. The general integral is of the form

$\Gamma_{s}(\omega,T)=|M|^{2}\int_{0}^{\infty}dE~{}\int\frac{d\phi_{1}d\phi_{2}}{(2\pi)^{2}}J_{i}(\phi_{1})J_{i}(\phi_{2})n_{s}(E,\phi_{1})n_{s}(E+\hbar\omega,\phi_{2})\times\\\
\big{(}f(E)-f(E+\hbar\omega)\big{)}\big{(}F_{\gamma^{\dagger}\gamma}(E,\phi_{1},\phi_{2})+F_{\gamma\gamma}(E,\phi_{1},\phi_{2})\big{)}$
(8)

where $n_{s}(E,\phi)$ is the angle-resolved density of states, obeying
$\int\frac{d\phi}{2\pi}n_{s}(E,\phi)=N_{s}(E)$, $f(E)$ is the Fermi factor,
and $F(E,\phi_{1},\phi_{2})$ is the coherence factor. The coherence factors
arise due to interference between quasiparticles. They are derived by
rewriting the interaction Hamiltonian in terms of Bogoliubov quasiparticles:

$\mathcal{H}_{\rm
int}=M\sum_{k,k^{\prime}}F_{\gamma^{\dagger}\gamma}(k,k^{\prime})\gamma^{\dagger}_{k,\sigma}\gamma_{k^{\prime},\sigma}+M\sum_{k,k^{\prime}}F_{\gamma\gamma}(k,k^{\prime})\gamma_{k,\sigma}\gamma_{k^{\prime},\sigma}+h.c.$
(9)

Linearizing the functions $F(k,k^{\prime})$ about the Fermi surface and
evaluating the energy delta function gives us the functions
$F(E,\phi,\phi^{\prime})$.

The coherence factors depend on the structure of the gap function in spin and
momentum space. We will not treat the general case here. For the case of a
$p_{x}+ip_{y}$ gap function, the coherence factors are both of the form

$F_{\pm}(E,\phi,\phi^{\prime})=1\pm\frac{\Delta_{0}^{2}}{E(E+\hbar\omega)}\cos(\phi-\phi^{\prime}).$
(10)

In Figure 3 we plot the results for the $(p_{x}+ip_{y})\hat{z}$ gap function,
for which $F_{\gamma^{\dagger}\gamma}=F_{\gamma\gamma}=F_{+}$. As we are
considering a 2D scattering problem, the gap function has a constant modulus
in momentum space, so the angle-resolved density of states is equal to the
density of states:

$n_{s}(E,\phi)=N_{s}(E)=\frac{1}{\sqrt{E^{2}-\Delta_{0}^{2}}}.$ (11)

Collecting terms, we find that the scattering rate is given by

$\Gamma_{x+iy}(\omega,T)=2|M|^{2}\int_{0}^{\infty}dE~{}\int\frac{d\phi_{1}d\phi_{2}}{(2\pi)^{2}}J_{i}(\phi_{1})J_{i}(\phi_{2})\frac{1}{\sqrt{(E^{2}-\Delta_{0}^{2})((E+\hbar\omega)^{2}-\Delta_{0}^{2})}}\times\\\
\big{(}f(E)-f(E+\hbar\omega)\big{)}\bigg{(}1-\frac{\Delta_{0}^{2}}{E(E+\hbar\omega)}\cos(\phi_{1}-\phi_{2})\bigg{)}.$
(12)

This is essentially equivalent to the s-wave scattering problem except for the
factor of $\cos(\phi_{1}-\phi_{2})$. Importantly, one can immediately see that
the integral over $J_{i}(\phi_{1})J_{i}(\phi_{2})\cos(\phi_{1}-\phi_{2})$ will
vanish due to the $\mathbb{C}_{4}$ symmetry of the band structure. This means
that the coherence factors are effectively equal to 1, as if the Bogoliubov
quasiparticles do not interfere with one another. It is this lack of
interference that leads to the peak shown in panel (b) of Fig. 3.

Note that the vanishing (average) interference between quasiparticles depends
on integrating the relative angle, $\phi_{1}-\phi_{2}$, over the full period
of the cosine function. This involves wide-angle scattering events, where the
particles being scattered sit on opposite sides of the Fermi surface. Such
scattering events would require that the sound wave have a momentum ${\bf
q}={\bf k_{1}}-{\bf k_{2}}$ that can be as large as $2k_{F}$. Experimentally,
however, the sound waves have a frequency $q/2k_{F}\sim 10^{-7}$. We therefore
propose a phenomenological method for confining the relative angle
$|\phi_{1}-\phi_{2}|\lesssim\eta$ by including the Gaussian factor
$G(\phi_{1}-\phi_{2})$ in the integrand, where

$G(x)=\frac{1}{\eta\sqrt{2\pi}}e^{-(x/\eta)^{2}/2}.$ (13)

As the relative angle on a circular Fermi surface is given by
$\phi_{1}-\phi_{2}=\arcsin(q/2k_{F})$, we convert the parameter $\eta$ to an
equivalent sound frequency using $\omega_{\eta}=2v_{s}k_{F}\sin(\eta)$. For
large $\omega_{\eta}$, we recover the non-interfering results. For
$\omega_{\eta}$ on the order of the experimental frequency, however,
$\cos(\phi_{1}-\phi_{2})\approx 1$ and we recover the standard s-wave
calculation with a pronounced dip across $T_{c}$. This corresponds to strong
destructive interference between quasiparticles. These curves are compared in
panel (b) of Fig. 3.

The coherence factors for a $d_{x^{2}-y^{2}}$ gap function are of the form

$F_{\pm}(E,\phi,\phi^{\prime})=1\pm\frac{\Delta_{0}^{2}}{E(E+\hbar\omega)}|\cos(2\phi)||\cos(2\phi^{\prime})|$
(14)

and the angle-resolved density of states is
$n_{s}(E,\phi)=|E|/\sqrt{E^{2}-\Delta_{0}^{2}\cos^{2}(2\phi)}$. Inserting
these directly into Equation 8, one obtains

$\Gamma_{x^{2}-y^{2}}(\omega,T)=2|M|^{2}\int_{0}^{\infty}dE~{}\int\frac{d\phi_{1}d\phi_{2}}{(2\pi)^{2}}J_{i}(\phi_{1})J_{i}(\phi_{2})\big{(}f(E)-f(E+\hbar\omega)\big{)}\times\\\
\frac{1}{\sqrt{(E^{2}-\Delta_{0}^{2}\cos^{2}(2\phi_{1}))((E+\hbar\omega)^{2}-\Delta_{0}^{2}\cos^{2}(2\phi_{2}))}}\bigg{(}1-\frac{\Delta_{0}^{2}}{E(E+\hbar\omega)}|\cos(2\phi_{1})||\cos(2\phi_{2})|\bigg{)}.$
(15)

We do not include the phenomenological Gaussian factor for the results in
panel (a) of Fig. 3 because there is no erroneous peak, so we do not expect
any qualitative change in the results. In principle, however, the structure of
the $d$-wave gap in momentum space means that a quantitative calculation of
the scattering rate should remove wide-angle scattering events. All theory
curves in panels (a), (b) and (c) of Fig. 3 were computed for the $\beta$
band. Figure 5 shows the normalized scattering rate for the $\alpha$, $\beta$,
and $\gamma$ bands.

Figure 5: Scattering rate for ${\rm Sr}_{2}{\rm RuO}_{4}$ bands. Comparison of
the scattering rate with a $d_{x^{2}-y^{2}}$ gap function for each of the
three ${\rm Sr}_{2}{\rm RuO}_{4}$ Fermi surfaces. The electron-like $\gamma$
and $\beta$ bands show nearly identical behavior while the hole-like $\alpha$
band differs significantly at low temperatures. We find that these
discrepancies do not change the qualitative behavior across $T_{c}$. Note that
the $\beta$ curve is reproduced in panel (a) of Fig. 3 in the main text.

The $\alpha$ band is hole-like, and its scattering rate differs qualitatively
from that of the electron-like bands. We note that this effect appears most
pronounced at low temperatures and that it does not change the behavior across
$T_{c}$. The $s$-wave curve, also plotted in panel (a) of Fig. 3, has
coherence factors $1\pm\frac{\Delta_{0}^{2}}{E(E+\hbar\omega)}$ and a density
of states $n_{s}(E)=(E^{2}-\Delta_{0}^{2})^{-1/2}$. The $s$-wave singlet
problem is covered extensively in Refs. Bardeen  _et al._ (1957); Tinkham
(1996); Schrieffer (1964) so we do not review it here.

### Cooper Pair-breaking below $T_{\mathrm{c}}$

In Equation 8, we neglected a set of quadratic Bogoliubov terms that are
referred to as pair-breaking terms. These terms are neglected because the
domain of the integral formally vanishes when the driving frequency
$\omega<2\Delta(T)$ where $\Delta(T)$ is the temperature-dependent s-wave gap.
For gap functions with nodes, however, the frequency domain does not vanish
explicitly. Crucially, the pair breaking coherence factors have the opposite
sign of the dominant coherence factors discussed above. This means that an
interaction that leads to destructive interference between quasiparticles (and
thus a sharp drop in the scattering rate across $T_{c}$) will exhibit
constructive interference in its pair-breaking terms. Thus, pair breaking
terms will induce a peak where there is otherwise none.

Pair breaking terms can be identified immediately by the energy delta
function. For example, in going from the general statement of Fermi’s golden
rule to Equation 8, we evaluated the energy delta function
$\delta(E-E^{\prime}-\hbar\omega)$ assuming that $E,E^{\prime}>0$. This is the
natural choice in the $\omega\to 0$ limit. For $\omega>0$, however, there are
also solutions to the delta function where $E<0$ ($E^{\prime}>0$). These
contributions are what we will explicitly include now. Note that even with
nodes in the gap function, the domain of this integral formally vanishes as
$\omega\to 0$.

For the d-wave gap, normal contributions are given by Equation 15. Pair
breaking contributions differ in the domain of the $E$ integral, the sign of
the coherence factor, and in the product of Fermi factors:

$\Gamma^{PB}_{x^{2}-y^{2}}(\omega,T)=2|M|^{2}\int_{0}^{\omega}dE~{}\int\frac{d\phi_{1}d\phi_{2}}{(2\pi)^{2}}J_{i}(\phi_{1})J_{i}(\phi_{2})\big{(}1-f(E)-f(E+\hbar\omega)\big{)}\times\\\
\frac{1}{\sqrt{(E^{2}-\Delta_{0}^{2}\cos^{2}(2\phi_{1}))((E+\hbar\omega)^{2}-\Delta_{0}^{2}\cos^{2}(2\phi_{2}))}}\bigg{(}1+\frac{\Delta_{0}^{2}}{E(E+\hbar\omega)}|\cos(2\phi_{1})||\cos(2\phi_{2})|\bigg{)}.$
(16)

We sum these two terms, normalized by the scattering rate in the normal state,
for two values of $\omega$ in panel (c) of Fig. 3. An interpretation of these
results is covered in the main text.

Figure 6: Sound attenuation from order parameter modes. Normalized
$(\eta_{11}+\eta_{12})/2$ in Sr2RuO4 fit to two different models of increased
sound attenuation below $T_{\mathrm{c}}$. The green curve is a fit to Equation
17, which models sound attenuation due to OP modes. The red curve is a fit to
Equation 18, which models the sound attenuation arising from domain wall
motion. Near $T_{\mathrm{c}}$, the red curve clearly fits the experimental
data better than the green one.

### Attenuation Due to Order Parameter Relaxation

The formation of the SC order parameter below $T_{\mathrm{c}}$ can lead to
relaxational dynamics as the OP interacts with the strain. Within a Landau
theory, the relaxation timescale diverges as $\left|T/T_{\rm c}-1\right|^{-1}$
close to $T_{\mathrm{c}}$. Unlike the resonant sound absorption arising from
domains, OP relaxation can cause non-resonant absorption of ultrasound and
lead to a broad peak in sound attenuation below $T_{\mathrm{c}}$ Sigrist
(2002). We fit our measured $(\eta_{11}+\eta_{12})/2$ to the attenuation
expression derived by Sigrist (2002),

$\alpha(\omega,T)\propto\frac{\omega^{2}\tau}{1+\omega^{2}\tau^{2}}\sim\frac{\omega^{2}\tau_{0}/\left|T/T_{\rm
c}-1\right|}{1+\omega^{2}\tau_{0}^{2}/\left|T/T_{\rm c}-1\right|^{2}}$ (17)

as shown in Figure 6. However we find that this expression does not capture
the sharp increase in attenuation below $T_{\mathrm{c}}$, which the expression
for attenuation from domain wall motion given in Sigrist and Ueda (1991) does,

$\alpha\left(\omega,T\right)\propto\frac{\omega^{2}}{\omega^{2}+\omega_{1}^{2}\left|T/T_{\rm
c}-1\right|^{3}}\left|T/T_{\rm c}-1\right|^{2}.$ (18)

In fact, the sharp peak-like behavior of attenuation right below
$T_{\mathrm{c}}$, which is already present in the raw data (for example, 2495
kHz and 2573 kHz in Figure 4), points strongly to a resonant absorption
mechanism compared to a non-resonant one.


11institutetext: Department of Electrical and Electronics Engineering,
Yeditepe University, İstanbul 34755, Turkey.
<EMAIL_ADDRESS>

# Parametric Curve Segment Extraction by Support Regions

Cem Ünsalan

###### Abstract

We introduce a method to extract curve segments in parametric form from the
image directly using the Laplacian of Gaussian (LoG) filter response. Our
segmentation gives convex and concave curves. To do so, we form curve support
regions by grouping pixels of the thresholded filter response. Then, we model
each support region boundary by Fourier series and extract the corresponding
parametric curve segment.

##  1 Introduction

High level computer vision tasks, such as object recognition and scene
understanding requires primitives to be extracted from the image. One such
primitive is the edge information. State of the art edge detectors proposed in
the literature can be found in a standard image processing book such as [2].
One of the best known edge detectors is Canny’s method [1].

As the edge pixels are detected, they can be represented as curves. For high
level processing, these curves need to be segmented and represented in a
standardized form. The literature is vast on curve segmentation [3, 4, 5, 6,
7]. Most of these methods require the candidate curve segment pixels to be
extracted first, either by edge detection or a similar preprocessing. Then,
these pixels are grouped by a given criteria to form the corresponding curve
segments.

These curve segments should also be represented in parametric form. One way of
parametric representation is polygonal approximation. Some of the existing
work on this problem is given in references [8, 9, 10]. Most of these methods
require curve segments to be extracted first by a preprocessing step. Some of
these methods are also based on the area information, such that the contours
in the binary image is represented in terms of polygons. However, these
methods require a thresholding step to be applied to grayscale images.

In this study, we extract the trigonometric polynomial representation of curve
segments in the image by a method combining edge pixel grouping and curve
segmentation. Our method is similar to straight line extraction using line
support regions introduced by Burns  _et. al._ [11]. They grouped pixels of
contiguous gradient directions and formed support regions. Then they obtained
straight line segments from these. This method also combines steps in
parametric curve segment extraction. However, the final segments extracted are
only straight lines and the method is not suitable for a general curve segment
extraction. Another drawback of their method is its computational cost because
of the pixel grouping step.

To overcome these problems, we extend the idea to represent the general
parametric curve segments by dividing them into convex and concave sections.
Hence, our method can handle curved and straight line segments in the image.
We group pixels to form curve support regions by the Laplacian of Gaussian
(LoG) filter response (by extending its usage beyond zero crossing edge
detection). Then, we obtain the corresponding parametric curve segment by
fitting a Fourier series to its boundary. Next, we explore our curve
segmentation and parametrization method step by step, starting with
mathematical derivation followed by experimental results.

##  2 Curve Support Region (CSR) Extraction

Zeros of the second derivative of a one dimensional function indicate its
inflection points. Therefore, zeros of the second derivative can be used to
distinguish convex and concave parts of the function. Haralick and Shapiro
[12] point out that “The isotropic generalization of the second order
derivative to two dimensions is the Laplacian”. Based on Haralick and
Shapiro’s definition, the Laplacian should have the same property
(partitioning the convex and concave parts) for two dimensional functions
(such as the brightness function of an image). In this study, we give examples
showing that this is indeed the case. Therefore, this property will lead to
direct curve segmentation, partitioned into convex and concave parts, from the
image at hand.

###  2.1 An Example in Continuous Domain

We next give an example in continuous domain. We model an intensity change (in
an image brightness function) having a sinusoidal shape, consisting of convex
and concave segments. This example gives insight into the mathematical
background of the proposed method. As we are working in continuous domain, we
obtain curve segments mathematically.

Let a curve on a continuous image brightness function be:

$y-\sin(x)+\rho=0$ (1)

where $\rho$ is a constant. To generalize the curve in 2D, let’s add a
$\theta$ radian rotation.

$\begin{bmatrix}y_{r}\\\ x_{r}\\\
\end{bmatrix}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\\\ \end{bmatrix}\times\begin{bmatrix}y\\\ x\\\
\end{bmatrix}=\begin{bmatrix}\alpha&-\beta\\\ \beta&\alpha\\\
\end{bmatrix}\times\begin{bmatrix}y\\\ x\\\ \end{bmatrix}$ (2)

where $\alpha=\cos(\theta)$ and $\beta=\sin(\theta)$. Our rotated curve
becomes

$y_{r}-\sin(x_{r})+\rho=0$ (3)

Now, let’s define an image brightness function $B(x,y)$ as

$B(x,y)=b_{1}+\Delta bu(y_{r}-\sin(x_{r})+\rho)$ (4)

where $b_{1}$ is a brightness value, $\Delta b$ is the contrast, and
$u(\cdot)$ is the unit step function. The brightness function becomes

$B(x,y)=b_{1}+\Delta bu(\alpha y-\beta x-\sin(\beta y+\alpha x)+\rho)$ (5)

The Laplacian over $B(x,y)$ is calculated as

$\nabla^{2}B=\frac{\partial^{2}B}{\partial
x^{2}}+\frac{\partial^{2}B}{\partial y^{2}}$ (6)

where the second order partial derivatives are

$\begin{split}\frac{\partial^{2}B}{\partial x^{2}}=&\Delta
b[\alpha^{2}\sin(\beta y+\alpha x)\delta(\alpha y-\beta x-\sin(\beta y+\alpha
x)+\rho)\\\ &+(-\beta-\alpha\cos(\beta y+\alpha x))^{2}\delta^{\prime}(\alpha
y-\beta x-\sin(\beta y+\alpha x)+\rho)]\end{split}$ (7)

$\begin{split}\frac{\partial^{2}B}{\partial y^{2}}=&\Delta
b[\beta^{2}\sin(\beta y+\alpha x)\delta(\alpha y-\beta x-\sin(\beta y+\alpha
x)+\rho)\\\ &+(\alpha-\beta\cos(\beta y+\alpha x))^{2}\delta^{\prime}(\alpha
y-\beta x-\sin(\beta y+\alpha x+\rho)]\end{split}$ (8)

We obtain

$\begin{split}\nabla^{2}B=&\Delta b[\sin(\beta y+\alpha x)\delta(\alpha
y-\beta x-\sin(\beta y+\alpha x)+\rho)\\\ &+(1+\cos^{2}(\beta y+\alpha
x))\delta^{\prime}(\alpha y-\beta x-\sin(\beta y+\alpha x)+\rho)]\end{split}$
(9)

Here, we take $\delta(t)$ to be nonzero only at $t=0$ and $\delta^{\prime}(t)$
to be nonzero only as $t\rightarrow 0^{+}$ (with negative response) and
$t\rightarrow 0^{-}$ (with positive response). This is consistent with the
zero crossing edge detection in computer vision, where we have a zero response
on the edge and nonzero responses (with different polarities) on the two sides
of the edge.

If we want to localize edges in this brightness function, we should equate

$\delta(\alpha y-\beta x-\sin(\beta y+\alpha x)+\rho)=1$ (10)

This is satisfied only for

$\alpha y-\beta x-\sin(\beta y+\alpha x)+\rho=0$ (11)

which is the expanded form of the rotated curve in Eqn. 3. Therefore, the
$\delta(\cdot)$ response in the Laplacian of $B(x,y)$ gives the edge locations
mathematically.

The magnitude of $\delta(\cdot)$ on the curve is $\sin(x_{r})$. This function
is zero only for $x_{r}=n\pi$, where $n$ is any integer number. On the curve,
these $x_{r}$ values lead to $y_{r}=-\rho$. Therefore, points $(n\pi,-\rho)$
correspond to zeros of the Laplacian of the brightness function in rotated
coordinates. These points are the actual inflection points of the rotated
curve. Hence, finding zeros of the Laplacian actually corresponds to finding
the inflection points of the 2D function.

More important to us is the response of $\delta^{\prime}(\cdot)$ around zero.
Let’s consider the positive response obtained from $\delta^{\prime}(0^{-})$.
In order to obtain the nonzero response for this function, we must have

$\alpha y-\beta x-\sin(\beta y+\alpha x)+\rho=\lim_{\epsilon\rightarrow
0^{-}}\epsilon$ (12)

This equation indicates that the curve can be located from the
$\delta^{\prime}(0^{-})$ response in $\epsilon$ distance from the actual
location as in Eqn. 11.

The magnitude of the positive response of $\delta^{\prime}(\cdot)$ is
$1+cos^{2}(x_{r})$. Thresholding this function around two divides the response
into subparts separated by points $(n\pi,-\rho)$. These are the inflection
points on the rotated curve. Therefore, these subparts correspond to separate
convex and concave curve segments. Similar to this example, obtaining the
nonzero $\delta^{\prime}(\cdot)$ response and thresholding it will give the
segmented curves in continuous domain functions.

We simulate the brightness function, $B(x,y)$ in Fig. 1 (a) with a $\pi/3$
radians rotation and $\rho=0.2$. The curve segments are given in Fig. 1 (b).
As can be seen, the Laplacian performs the same operation in two dimensions as
does the second derivative in one dimension. Therefore, we can extract curve
segments directly by using the $\delta^{\prime}(\cdot)$ part of the Laplacian
of a continuous domain function.

(a) $B(x,y)$ (b) Curve segments

Figure 1: A curve segmentation example in continuous domain

###  2.2 An Example in Discrete Domain

To apply the above idea on digital images, we need to obtain the discrete
version of the Laplacian operator with a possible smoothing applied on the
image. Marr and Hildreth [13] introduced the rotation invariant Laplacian of
Gaussian (LoG) filter for a similar purpose. They used the Laplacian of a
smoothed image to extract edges in a zero crossing sense. In this method, the
Laplacian operator performs the partial second derivative calculations and the
Gaussian filter performs smoothing on the image. By the property of
convolution, both smoothing and Laplacian operations can be represented as one
filtering operation, hence the LoG filter:

$f(x,y)=\frac{1}{\pi\sigma^{4}}\left[\frac{x^{2}+y^{2}}{2\sigma^{2}}-1\right]\exp(-\frac{x^{2}+y^{2}}{2\sigma^{2}})$
(13)

where $\sigma$ is the scale parameter (for smoothing) of the LoG filter. The
scale parameter leads to a natural scale space property for curve extraction
and segmentation.

We next apply the LoG filter to obtain the positive and negative responses
similar to the $\delta^{\prime}(\cdot)$ response above. We label pixels having
responses higher than a threshold. Connected components analysis on these
pixels leads to curve support regions of the image. We give an example of this
procedure on an image containing a rectangle shape in Fig. 2 (a).

(a) Rectangle (b) LoG response (c) CSR extracted (d) Curve segments

Figure 2: Curve support regions and curve segments over the rectangle image

In Fig. 2 (b) the LoG filter response has three levels as: the zero level
(gray), high level (bright), and the low level (black). To form curve support
regions, we threshold this response and take only the high valued pixels
(bright pixels in Fig. 2 (b)). Curve support regions for the rectangle image
are given in Fig. 2 (c). As can be seen, curve support regions indicate four
separate curve segments for this shape. We will consider obtaining the final
parametric representation of the curve segments from these support regions
next.

##  3 Parametric Curve Segment Extraction

To extract the final parametric curve segment from the corresponding support
region, we represent the outer boundary of each support region using its
Fourier descriptors. In a previous study, we applied a similar strategy to
extract straight line segments from the line support regions [14]. Here, we
extend its usage to a more general case, curve extraction from curve support
regions.

We represent the outer boundary of each support region using its Fourier
descriptors. Let a complex periodic function, $u(t)=x(t)+jy(t)=u(t+rT)$,
$j=\sqrt{-1}$ for any integer values of $t$ and $r$, represent the outer
boundary of the support region. $T$ is the total number of points in the
contour. The complex periodic contour can be approximated by a Fourier series
as [15]:

$\hat{u}(t)=\sum_{n=0}^{T-1}U_{n}e^{j\frac{2\pi nt}{T}}$ (14)

$U_{n}=\frac{1}{T}\sum_{t=0}^{T-1}u(t)e^{-j\frac{2\pi nt}{T}}$ (15)

The curvature of the boundary representation leads to the parametric curve
segment extraction from the corresponding support region. The curvature is a
differential geometric entity giving a measure of how rapidly the curve
deviates from the tangent line [16]. We find the curvature of the approximated
boundary ($\hat{u}(t)=\hat{x}(t)+j\hat{y}(t)$) as:

$K(t)=\frac{\frac{d\hat{x}(t)}{dt}\frac{d^{2}\hat{y}(t)}{dt^{2}}-\frac{d\hat{y}(t)}{dt}\frac{d^{2}\hat{x}(t)}{dt^{2}}}{\left((\frac{d\hat{x}(t)}{dt})^{2}+(\frac{d\hat{y}(t)}{dt})^{2}\right)^{3/2}}$
(16)

Given the support region construction and approximation above, the extremal
points of this curvature correspond to the endpoints of the parametric curve
segment (on the boundary) to be extracted. There is no algebraic solution to
obtain the roots of Eqn. 16 directly; we solve it numerically.

Assume that we obtain two extremum points $P_{1},P_{2}$, such that each
correspond to an endpoint of the curve segment. We divide the approximated
boundary into two parts $x_{u}+jy_{u}$ and $x_{l}+jy_{l}$ based on these
extremum points. These parts have the same parameter such that
$k=[0,P_{2}-P_{1})$ and 0 corresponds to $P_{1}$, $P_{2}-P_{1}$ corresponds to
$P_{2}$ on the boundary. Therefore, they have the same increment direction.
Our final parametric curve segment, $s(k)$ for $k=[0,P_{2}-P_{1})$ is:

$s(k)=\frac{1}{2}\left[\left(x_{u}(k)+x_{l}(k)\right)+j\left(y_{u}(k)+y_{l}(k)\right)\right]$
(17)

Depending on the complexity of the scene and the LoG scale parameter selected,
some curve support regions may merge. This will lead to more than two extremum
points in the curvature function. For such cases, we first obtain the centroid
on the support region. Then, we obtain all parametric curve segments
separately by using each extremum of the curvature and the centroid with the
previous setup as if we have two extremum points.

##  4 Experimental Results

We test our parametric curve segment extraction method on various images.
First, we extract parametric curve segments from black and white images. They
give insight on the applicability of our method to the shape information only.
Then, we test our method on grayscale images with diverse characteristics, to
emphasize its general applicability.

###  4.1 Experiments on Black and White Images

We start with the rectangle image in Fig. 2 (a). We give the parametric curve
segments in Fig. 2 (d). As can be seen, our method extracts four parametric
curve segments from the rectangle shape as desired. In this and the following
examples, each curve segment is labeled by dots at their endpoints.

We apply the method on two more black and white images. The first image
contains an ‘S’ shape. The second image contains characters ‘A, B, C, D, E’.
We specifically chose these black and white images to observe the performance
of our method on different shape characteristics. Furthermore, to test the
robustness of our method, we applied a rotation of $\pi/6$ radians, an affine
transformation as in Eqn. 18, and an affine transformation followed by a
rotation to our images.

$\begin{bmatrix}y^{\prime}\\\ x^{\prime}\\\
\end{bmatrix}=\begin{bmatrix}1&0\\\ 0.5&1\\\
\end{bmatrix}\times\begin{bmatrix}y\\\ x\\\ \end{bmatrix}$ (18)

For the ‘S’ shape, extracted parametric curve segments are given in Fig. 3.
The method works fairly well on the original image. Obtained curve support
regions are correct and consistent through rotations and translations. In
rotated and transformed images, there are extra segments, each being a valid
curve segment. They can be merged by a postprocessing step.

Figure 3: Parametric curve segments obtained from the ‘S’ shape in the
original, rotated, affine transformed, affine transformed and rotated versions
(from left to right respectively)

As for the characters ‘A, B, C, D, E’, extracted curve segments (with the
above variations) are given in Fig. 4. In this example, ‘B’ has a support
region containing both convex and concave parts because of a large filter
scale selection. Since the filter scale is large, such small changes can not
be detected in this character. Similarly, for ‘E’ lines inside the character
are combined as one curve. This also occurs from the large filter scale
selection, such that the corners of these lines are merged. Similar to the
previous example, extra segmentations can be merged. Besides, all curve
segments are extracted from these characters with success.

Figure 4: Parametric curve segments obtained from characters in the original,
rotated, affine transformed, affine transformed and rotated versions (from
left to right respectively)

We can comment on the black and white image performance of our method as
follows. Since we have a rotation invariant filter, ideally there should be no
change on the support regions extracted after rotation. However, we have extra
segmentations in the rotated shapes possibly due to the (discrete) rotation
operation. As for the affine transformed and affine transformed and rotated
images, we obtain the desired support regions with some extra segments. As we
mentioned previously, these extra segments can be merged easily.

###  4.2 Experiments on Grayscale Images

Next, we consider the performance of our parametric curve segment extraction
method on grayscale images. The first three of these images (Rice, Puffin, and
Mandrill), given in the first row of Fig. 5, have relatively low content. Our
method locates most of the curve segments (in parametric form) in the low
content images with success. In This image set, the Mandrill image deserves a
special consideration. It contains heavy textured regions as well as a general
shape. By selecting a large filter scale, we can detect curve segments (in
parametric form) corresponding to the general characteristics of the Mandrill
image among the fine details. Unfortunately, a large scale causes relatively
large shift in the location of the detected curve segments. This also causes
some small curve segments to have convex and concave parts. However, the
overall performance is satisfactory.

Figure 5: Parametric curve segment extraction from grayscale images.

The last three grayscale images (Lena, Boat, and Urban), given in the third
row of Fig. 5, have high content. As can be seen, for these images our method
works fairly well. Since there are many curve segments in these images, we did
not put heavy dots at the end of each curve segment.

##  5 Conclusions

We introduce a method to extract curve segments (represented in parametric
form) directly from the LoG filter response of the image. Hence, our method
combines three basic operations to one by a simple filtering and grouping.
Therefore, the proposed method is potentially faster and simpler than the
existing curve extraction, segmentation, and parametric representation
methods. Although the LoG filtering causes a small shift in the location of
curves extracted, the general shape of the object is not affected. Therefore,
this method is suitable for most of the recognition and computer vision
applications. The experiments support the usefulness of the proposed method in
extracting curve segments both on black and white and grayscale images having
diverse characteristics. For some shapes, the proposed method may give extra
segments. However, since we have the parametric representation of these
segments at hand, they can be merged by a postprocessing step.

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