1

Multiple spatial spectral components of static skindeformation for predicting macroscopic roughness perception
Qingyu Sun1 , Shogo Okamoto1,2 , Yasuhiro Akiyama3 , and Yoji Yamada1,4

Abstract—A previous study suggested a relationship between the spatial
spectrum of finger pad skin deformation and perception of macroscopicroughness features. This study tested a new hypothesis that macroscopicroughness perception is the result of a weighted linear combination ofmultiple spatial spectral components of skin deformation. Experiments
were conducted by capturing close-up images of finger pad deformation
while the pads were pushed onto specimens with macroscopic features.
Additionally, the roughness perceptions of these specimens were collectedusing a magnitude estimation method. The combination of spectral
components predicted the roughness perception more accurately thanany single spectral component. This suggests that roughness perception
is mediated by multiple Gabor filter-like neural systems with different
spatial periods, such as visual perception.

I. I NTRODUCTION
Roughness is a fundamental property of texture perception [1]–[3]. Many neurophysiological and psychophysical studies have elucidated the perceptual mechanisms of this property. For example,
neurophysiological studies have found that slowly adapting typeI (SAI) units play a significant role in perceiving macroscopic
surface roughness [4]–[6]. SAI units are responsible for sustained
skin deformation and pressure [7], and their neural activity can be afunction of local stress in the skin [8], [9]. In contrast, fast-adapting
units mediate microscopic roughness features [10]–[12]. These units
are sensitive to skin vibrations caused when the finger slides overtextured surfaces. In psychophysics, the relationship between surface
features and roughness perception has been extensively studied. For
example, the inter-dot spaces of dot scales and groove widths ofgrating scales are considered the primary determinants of the perceived roughness [13]–[17]. Humans typically feel the most intenseroughness for grating scales with groove widths in the range of 2.0
to 4.0 mm [11], [14], [15], [18]–[20].
Thus far, neurophysiological studies have mainly focused on how
afferent units respond to roughness features and skin deformation [4]–[6], [9], [21], [22]. Psychophysical studies have primarily focused on
the dependence of roughness perception on roughness features [13]–[17], [23]. Although roughness perception, skin deformation, andafferent activation occur after touching rough surfaces, few studies
have attempted to link skin deformation and perceived roughness
when touching rough surfaces. For example, Taylor and Ledermann
proposed that the penetration of the finger pad skin into the grooves
could be the determinant of the perception of macroscopic roughness [24]. Okamoto and Oishi pointed out that the spatial spectral
components of the surface deformation of finger skin are highly
1 Sun, Okamoto, and Yamada are with the Department of Mechanical
Systems Engineering, Nagoya University, Nagoya, Aichi, Japan.
2 Okamoto is also with the Department of Computer Science, Tokyo
Metropolitan University, Hino, Tokyo, Japan. okamotos@tmu.ac.jp
3 Akiyama is with the Department of Mechanical Engineering and Robotics,
Shinshu University, Ueda, Nagano, Japan.
4 Yamada is also with the National Institute of Technology, Toyota College,
Toyota, Aichi, Japan.
This research was funded by MEXT Kakenhi 20H04263 and 21H05819.
QS and SO are equal contributors.

correlated with the roughness perception of macroscopic grating
scales [25].
In [25], the authors predicted the perceived roughness using asingle spatial spectral component of the finger pad deformationand the outputs of a single Gabor filter, which is a classical basis
function for wavelet transform. They found that components withwavelengths ranging from 2.45 to 4.00 mm exhibited high correlation
with subjective roughness. These wavelength values are close to thoseused in neurophysiological studies [11], [18], [26], where roughness
perception was estimated based on Gabor-filter outputs of spatial
neural activation plots of SAI units. These studies [18], [25]–[27]considered the existence of an optimal wavelength for the Gaborfilter and an optimal spatial frequency for estimating macroscopicroughness perception. Tymms et al. estimated the roughness perception based on the simulated spatial variation of compressive strains
in an elastic model pressed on roughened surfaces and found thatthe optimal spatial distance for computing the spatial variation was
smaller than 2.4 mm [28]1 . In other words, they hypothesized that
there is only one type of Gabor-filter-like neural detector for SAI
activity that is responsible for macroscopic roughness.
This study focused on the question of whether there may be
multiple types of detectors with different sensitive wavelengths andhow the linear combination of these detector outputs may contributeto roughness perception. This is a natural expansion from simple
cells to complex cells in the primary visual cortex, where a simple
cell functions as a Gabor filter, and different simple cells exhibit
sensitivities to different spatial wavelengths. The combination of these
simple cells innervates other neural cells for complex processing ofvisual information [29].
We captured close-up images of the finger pad deformation while
pushing the pads against grating scales with macroscopic features.
Gabor filters of various wavelengths were applied to the images of
surface deformation of the skin to compute the spatial spectrum. The
subjective roughness values for the grating scales were then measured
using magnitude estimation. We established two types of estimators
for the subjective roughness. The first estimator uses a single Gaborfilter output that responds intensely to a specific wavelength value.
The second estimator uses a weighted linear combination of theoutputs of multiple Gabor filters with different wavelengths. These
two types of estimators were compared in terms of their estimation
accuracy. If the latter estimator outperforms the former, it suggeststhat multiple types of Gabor filter-like detectors with sensitivities todifferent wavelengths exist. If the performance of these two types of
estimators are comparable, it suggests that only one type of Gaborfilter-like detector is sensitive to a specific spatial wavelength function
for macroscopic roughness perception.
This study extends the study conducted in [25] where the relationships between the spatial spectral components of static skin deformation and macroscopic-roughness perception were investigated. They
1 They computed the variation of strains between two separate circularregions instead of points. Hence, their definition of distance is different fromthose in other studies.

2(b) Shallow rectangular(a) Rectangular
Groove width

GrooveheightRidge width(c) Circular(d) Triangular
Circular interval（0 or 0.5mm)

Groovewidth

Vertexangle

Diameter(e) Double- or triple sinusoidald(x)x(f) 3D features

20mm

50mm

10mm

Fig. 1. Five types of grating scales: (a) rectangular, (b) shallow rectangular,(c) circular, (d) triangular, and (e) double- or triple-sinusoidal surfaces. (f) 3Dview of grating scales.

concluded that the spectral components with wavelengths ranging
2.45–4.00 mm nearly equally predicted roughness perception. To
determine which component best predicts roughness perception, theyused rectangular, circular, and triangular roughness grating scales intheir experiment. By contrast, we investigated the effect of combining
multiple components, for which more complex roughness scales must
be adopted. Roughness scales with two or three spatial frequency
components were used in this study.
II. METHODOLOGY
The protocol for this experiment was approved by the Institutional
Review Board of the School of Engineering, Nagoya University (#199).
A. Grating roughness scales
Six types of 3D-printed grating scales were used: rectangular,
shallow rectangular, triangular, circular, double-sinusoidal, and triplesinusoidal, as shown in Fig. 1. The printer was Form 3 (Formlabs, MA) with a nominal resolution of 25 µm. After printing
the specimens, we visually and haptically inspected their surfaces
and verified that they had no recognizable microscopic patterns.
The dimensions of the specimens were 50 × 20 × 10 mm. The
rectangular specimens were defined by the groove and ridge widths,
whereas the shallow rectangular specimens also included the groove
height. The groove height of the shallow rectangular specimens was
designed so that the finger skin would reach the bottom of thegrooves. Shallow and deep grating scales lead to different roughness
perceptions, even if they have the same groove width and ridge width.
Shallow macroscopic-roughness specimens elicit a smaller roughness
perception than deep specimens [30]. Hence, we included both deepand shallow rectangular scales to investigate the contribution of themagnitude of skin deformation. Triangular specimens were defined
based on the vertex angle and groove width. Circular specimens were
defined by their diameters and intervals between two neighboring
semicircular shapes. To extend our experiment to a wider variety ofsurfaces, we introduced double- and triple-sinusoidal specimens. The
surfaces of the triple-sinusoidal specimens were defined by the sumof three sinusoidal functions as follows:xxxd(x) = a1 sin(2π ) + a2 sin(2π ) + a3 sin(2π ), (1)λ1λ2λ3where a1,2,3 and λ1,2,3 denote the amplitude and spatial wavelength,
respectively, in millimeters. The double-sinusoidal specimens were
defined by the first two sinusoidal functions. Two or three sinusoidal
components were superimposed with no phase differences. The λvalues were chosen such that they differed by at least 0.5 mm
and were not multiples of each other. Furthermore, we adjusted theamplitudes such that the double- and triple-sinusoidal specimens areapproximately divided into two groups. In one group, the finger pads
could reach the bottom of the grooves, whereas in the other groupthey could not. Some specimens were excluded because they hadlarge peak heights and the finger pads could not contact the surfaces
in multiple separated areas.
These shapes, excluding the double- and triple-sinusoidal and shallow rectangular scales, were also used in [25]. Although most previous psychophysical studies adopted only a single type of scales [14]–[17], [26], we used all six types of grating scales mentioned above.
We originally designed over 60 different specimens and then excluded
those with similar subjective roughness perceptions while maintaininga variety of shapes. The specific parameters of the 37 specimens used
are listed in Table I.
B. Participants
Ten university students (22 ± 2.0 (mean and standard deviation)
years) participated in this experiment after providing written informed
consent. The objectives of this study were not explained to theparticipants before the experiments.
C. Magnitude estimation of roughness perception
Magnitude estimation tasks were conducted to measure participants’ subjective roughness perceptions of the specimens. A circulargrating scale with a diameter of 2 mm and groove width of 0.25 mm
was used as the modulus with a roughness value of 100. All the
specimens were rated relative to the modulus (i.e., if the roughness
of one specimen was perceived to be three times that of the modulus,
its roughness perception was 300). The participants were required toplace their index finger parallel to the specimen and push it with aforce of 300 gf according to an electric scale. This pressing force, i.e.,
300 gf, lies in the range of the values at which roughness perceptionfunctions fairly [31]. Fig. 2 shows the variations in pressing force
in typical trials. The participants identified the roughness magnitudes
after removing their fingers from the stimulus. No sliding motion wasallowed on the specimens. Each specimen was tested three times, andthe specimens were presented in random order. A total of 111 trials(37 specimens × three repetitions) were conducted in randomizedorder for each participant. The participants wore a pair of glasses
whose lenses were blurred by translucent tape, so that they could not
see the detailed shapes of the specimens.

3(a) Raw picture

TABLE IP HYSICAL PARAMETERS OF 37 GRATING SCALES

Shallow
rectangular
Triangular

Circular

Double-sin

Pressing force (gf)

Triple-sin

Groove
width
(mm)
1.0
1.75
1.0
2.25
1.5
2.25
2.25
1.0
1.75
1.0
1.5
1.75
2.75
4.0
0
0.5
0
0
0

a1
(mm)

a2
(mm)

0.2
0.2
1
0.05
0.15
0.35
0.3
0.1
0.1
0.1
0.2
0.2
0.2
0.15
0.15
0.3
0.15
0.25

0.25
0.4
1
0.15
0.35
0.2
0.2
0.3
0.2
0.25
0.2
0.15
0.25
0.2
0.1
0.4
0.35
0.4

Vertex
angle
(◦ )
30
30
30
45
60
-

Circulardiameter
(mm)
1.0
1.75
2.5
3.0
4.0

a3
(mm)

λ1
(mm)

λ2
(mm)

0.2
0.15
0.2
0.2
0.25
0.2
0.1
0.15
0.2
0.4
0.3

1.5
1.5
2
1.5
1.3
1
1.7
1
1.2
1.6
1
1.1
1.3
1.3
1.6
1.2
1.3
1.2

3.5
2
3
2.5
2.1
1.7
3.8
1.6
2.7
2.3
1.6
2.7
2.1
2.4
2.4
1.8
1.9
1.7

Grooveheight
(mm)
3.0
3.0
3.0
3.0
3.0
0.25
0.5
0.25
0.25
-

Symbol
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s

λ3
(mm)

(c) Determined spatial skin central deformation
0.2
0.2
0.1

y (mm)

Rectangular

Ridge
width
(mm)
0.75
0.75
1.0
1.0
1.5
1.0
1.0
0.75
0.75
0
0
0
0
0
0
0
0
0
0

(b) Pixels of finger surface
selected by boundary extraction

0

-5

5
10

0

x (mm)(3R-Viewter-500uv, 3R Solution Ltd., Japan; 500 million pixels) as
shown in Fig. 3a. The deformation curve of the skin surface wasobtained using Photoshop (ver. 2019, Adobe Inc., CA), in which thequick selection tool and stroke functions were used to select the skinborder, as shown in Fig. 3b. The camera zoom was adjusted suchthat the image resolution was 1200 pixels/cm. As shown in Fig. 3c,the binarized images were processed using MATLAB (MATLAB,
2021a, Mathworks, MA, USA). The red area represents the region
extracted by the aforementioned process after removing the linear
trend. The central curve of this region was used to represent thesurface deformation of the skin.
We used a one-dimensional Gabor filter to compute the spatial
spectra from the processed curves of finger deformation. As shown in
Fig. 4a, the one-dimensional Gabor filter is a combination of a cosine
function with a spatial period (λ mm) and exponential attenuation.
g(x) = exp(

Trial 1
Trial 2
Trial 3

−x2
2πx
)cos(
),
2σ 2
λ

200

0
0.5

1

1.5

2
Time (s)

2.5

3

3.5

4

Fig. 2. Example profiles of normal force during magnitude estimation trial.
Profiles of the three trials are shown.

For our subsequent analysis, we used the median values of three
trials for each roughness scale. The values were then shifted such thatthe mean was zero for each individual according to the computational
requirements described in Section II-F.
D. Skin images and spatial spectra
Similar to the magnitude estimation task, participants placed theirindex fingers parallel to the specimens and applied a normal force of
300 gf. A cross-sectional image was captured using a digital camera(2)

where σ denotes the width of the amplitude attenuation and λ−1
denotes the central spatial frequency. σ was set to 1.1 mm as
described in [25], [26], [32]. Fig. 4b shows the extracted central
deformation of the skin, that is, the black border line in Fig. 3c. As
shown in Fig. 4c, for each λ value, the convolution was computed
between the Gabor filter and deformation curve as follows:go (x) = (g ∗ y)(x)0xFig. 3. (a) Original image of an index finger captured from the lateral side.
(b) Extracted finger deformation region. (c) Central curve (black) of surface
deformation region (red).

t
u
v
w
x
y
z
1
2
3
4
5
6
7
8
9
10
11

400

-0.1

−0.2
-0.2

Symbol

2.2
4.3
3.1
3.3
3.4
3.1
3.1
3.9
2.9
3.4
2.8

y

00

(3)

where y(x) is the skin displacement in mm normal to the contact
surface at position x. The maximum absolute output of the Gaborfilter is used as the value of the spatial spectral component for λ.
As shown in Fig. 4d, the spatial spectrum s(λ) is the set of
spatial spectral components for different λ values computed froma single image of skin deformation. We vary the parameter λ from
0.35 to 4.95 mm with an interval of 0.2 mm. There may be no clear
boundary between microscopic and macroscopic roughness, althoughmost researchers agree that the mechanism for roughness perceptionchanges below or above the spatial wavelength ranging from hundredsof µm to 1 mm [14], [33], [34]. Although we used the range of 0.35–
4.95 mm, a slight variation did not affect our final conclusions. For
example, the adoption of a range of 0.10–6.9 mm did not improve
the final estimation accuracy of the subjective roughness.
E. Weight function for the spatial spectra used to predict subjective
roughness
In continuous form, the subjective roughness value r is hypothesized to be the integral of the weight function w(λ) and spatial

4

where S + is the generalized inverse matrix of S. However, due
to collinearity among the explanatory variables (i.e., spectral components), we employed another approach known as partial leastsquares (PLS) regression. PLS is a combination of multiple regression
analyses and supervised principal component analysis (PCA). PLS
uses the scores of the principal components (PCs) as explanatory
variables that are correlated with objective variables (i.e., subjective
roughness values). The PCs are independent of each other; thus, theproblem of collinearity can be circumvented.
Therefore, the computational method of PLS regression was
adopted in this study. The matrix of spatial spectra S and the vectorof subjective roughness r are decomposed into the scores of the uPCs as follows:
u
X
S=
ti pT
(9)
i +E
i=1
u
X

r=

ai ti + e,

(10)

i=1

Fig. 4. Computation of a spatial spectrum. (a) One-dimensional Gabor filter
with spatial period λ. (b) Skin deformation curve y(x). (c) Convolution output
between g(x) and y(x) (in blue). The maximum absolute value (red) is
the spatial spectral component of the spatial period λ. (d) Spatial spectrum
computed from an image of skin deformation for λ values ranging from 0.35
to 4.95 mm.

spectrum s(λ), as follows:
Z 4.95r=s(λ)w(λ) dλ.

(4)where ti ∈ Rn×1 denotes the score of the ith (i = 1, ..., u) PC
pi ∈ R24×1 , and E and e denote the model errors. The number ofPCs, denoted by u, was selected to maximize the prediction accuracy
in a cross-validation method, as discussed in Section III. ai is thecoefficient of ti . Note that ti ⊥ tj (i 6= j) and collinearity are not aconcern in (10).
In contrast to PCA, PLS regression analysis, determines the PCs pi
and their scores ti in a supervised manner using the objective variable
r. ti is computed such that the covariance with r i is maximized asfollows:

0.35

In the discrete form, r is computed from the inner product of thespatial spectrum vector s ∈ R24×1 containing 24 spectral componentsfor λ ranging from 0.35–4.95 mm with increments of 0.20 mm andthe perceptual weight w ∈ R24×1 , as follows:r = sT w.

(5)This formula holds for all trials, that is, all combinations of specimens
and participants, as follows:   Tr1
s1
 ..   .. (6)
 .  =  .  w,
T
rn
sn
where n indicates the number of trials, and si (i = 1, ..., n) denotes
the spatial spectrum vector for the trial i. For all trials, (6) can be
rewritten as follows:r = Sw,(7)
where r ∈ Rn×1 and S ∈ Rn×24 represent the vector of subjective
roughness and the matrix of spatial spectra, respectively. The spectral
components were normalized (z-scored) for individual participantsand λ values. Owing to this normalization, s is dimensionless.
Accordingly, w is also dimensionless. This process is necessary
because the magnitude of skin deformation varies among individuals
owing to the differences in skin hardness.
F. Computation of the weight function via partial least-squares
regression
To solve for w in (7), a multiple regression approach is typicallyadopted and w is computed as
w = S + r,

(8)

ti =

SiST
i ri
,
kS T
i ri k(11)

where k k denotes the L2 norm. Then, as least-squares solutions, pi
and ai are computed as follows:
ST
i ti
kti k
r T tiai = i ,
kti k
pi =

(12)
(13)

where S 1 = S and r 1 = r. S i and r i are the explanatory andobjective values, respectively, after excluding the influence of thePCs up to the i − 1th component:
S i = S i−1 − ti−1 pT
i−1

(14)

r i = r i−1 − ai−1 ti−1 .

(15)

Using the coefficients a and PC vectors p, we calculate the weighting
vector w as:uXw=ai pi .
(16)iIII. R ESULTS
A leave-one-out cross-validation method was used to determine
the number of PCs for PLS regression. For this computation, allparticipants, except for one who was used as the testing dataset, wereused as the training dataset. The test participants were then switchedto calculate the mean estimation performance of the regression
models. R2 was calculated for the estimation of subjective roughness
using the PLS models, and it was the highest (0.636) when two PCs
were used. The R2 value was 0.630 when only the first PC was used.
Therefore, we selected the first two PCs for PLS regression.
Fig. 5a presents the relationship between the reported and estimatedsubjective roughness values for all trials with ten participants when(a) Subjective roughness & estimated subjective roughness by a weighted
spatial spectrum
Rectangular
Shallow rectangular
Triangular
Circular
Double sinusoidal
Triple sinusoidal

Estimated subjective roughness (-)

200
150

100

0.7
𝑟𝑟 = 0.72

0.6
0.5
0.4

0

1

50

2
3
Spatial wavelength 𝜆𝜆 (mm)

4

4.35

5

Fig. 6. Correlation coefficients between single spatial components andreported subjective roughness.

0
TABLE IIM EAN ABSOLUTE ERROR (MAE) OF THE TWO ESTIMATORS BASED ON

−50
−100

DIFFERENT SURFACE SHAPES

Type

−100

−50

0
50
100
Subjective roughness (-)

150

200

(b) Mean reported & estimated subjective roughness among participants by
PLS regression
150

Estimated subjective roughness (-)

Correlation coefficient

5100

Rectangular
Shallow rectangular
Triangular
Circular
Double sinusoidal
Triple sinusoidal

n
10

50

0

−50
−100
−100

g

c
w r

35 z
b 4
s
9
x
2 8
u 11
m
1 t
7
y 6

i

e

f qp

h
a

v

od

l k

−50

0
50
Subjective roughness (-)

j

100

150

Fig. 5. Estimation of subjective roughness. (a) Reports and estimates using
PLS regression. (b) Mean reports and estimates among the participants using
PLS regression.

the first two PCs were incorporated into the weighted linear model.
The correlation coefficient was 0.78 (R2 is 0.61). Table II lists themean absolute errors (MAE) of the two estimators, i.e., single spatial
spectral component and weighted spatial spectrum. The participants’subjective roughness ranged from approximately −100 to 200, andthe overall MAE was 27.77 and 23.53 for the single spatial component and weighted spatial spectrum, respectively. The weighted
spatial spectrum exhibited smaller estimation errors, particularly forthe rectangular, circular, and triple-sinusoidal specimens.
The average correlation coefficient between the reported and estimated subjective roughness values was 0.82 ± 0.023 (mean andstandard error), with maximum and minimum values of 0.91 and
0.69, respectively, when the model was executed for individuals.

Rectangular
Shallow rectangular
Triangular
Circular
Double-sinusoidal
Triple-sinusoidal
Mean

MAE by a single spatial
spectral component
22.76
26.74
40.38
22.47
26.02
28.23
27.77

MAE by weighted
spatial spectrum
17.46
26.08
37.24
12.62
24.84
23.26
23.53

Fig. 5b presents the estimation performance of PLS regression after
averaging the individual responses. The correlation coefficient was
0.88. The triangular specimens tended to be underestimated except
for those with a groove width of 1 mm (denoted as ‘j’). In contrast,
the predictions for rectangular and circular specimens were relatively
accurate. The estimations for double-sin and triple-sin specimens
were also fairly accurate.
Fig. 6 shows the variation in correlation coefficients between the
single spatial spectral component and subjective roughness, which
generally increases to 0.72 at λ = 4.35 mm, although there is a local
peak at 1.15 mm.
We compared the spatial spectrum weighted by the two PCs andthe single spatial spectral component at 4.35 mm in terms of thecorrelation coefficients between the reported and estimated subjective
roughness. The correlation coefficients were calculated for individual
participants and then normalized using the Fisher z-transformation
among all participants. The correlation coefficients obtained using
the weighted spatial spectrum (r = 0.82 ± 0.023, mean ± standard
error) were significantly higher than those obtained using the single
spatial spectral component (r = 0.75 ± 0.023)2 with t = −8.83 andp = 9.97 × 10−6 by a paired t-test.
Even when only the first PC was used in the weighted spatial spectrum, the correlation coefficient between the reported and estimatedsubjective roughness was higher (r = 0.80 ± 0.027) than that of
the single spatial spectral component (t = 5.23, p = 5.40 × 10−4 ).
Therefore, the first PC was sufficient for testing our hypothesis. We
selected two PCs such that the R2 was maximized through crossvalidation, although the contribution of the second component wasminor.
Fig. 7 presents the weight vector w for the spatial spectral components. As in (7), this links the subjective roughness with the spatial
spectra of skin deformation. The peak value lies at λ = 3.15 mm,indicating that macroscopic roughness perception is most sensitive
2 Note that 0.72 in Fig. 6 was calculated using all the samples of allparticipants, whereas 0.75 was the mean of the correlation coefficients among
all individuals.

6

Perceptual weight (-)

1600
1400
1200
1000
𝜆𝜆 = 3.15 mm

800
0
0

1

2

4

3

𝜆𝜆 (mm)5

Fig. 7. Weight vector w acquired by PLS regression with 95% confidence
interval. The confidence interval was computed by the variation of the weights
among the participants.
20

First PC
1

Second PC

Loadings (-)

15
10
5
0
𝜆 =2.95 mm

𝜆 =1.15 mm

−5
0

1

2

𝜆 (mm)

3

4

5

Fig. 8. Loading vectors p for the first two PCs obtained by PLS regression.
The blue curve represents the first PC and the orange curve represents thesecond PC.

to a wavelength of 3.15 mm. However, considering the confidence
interval, a true peak may exist approximately above 2.5 mm.
Fig. 8 shows the vectors p of the first two PCs obtained using PLS
regression. The first PC shows a perceptual peak at 2.95 mm, while
the second PC has a peak at 1.15 mm.
IV. D ISCUSSION
In this study, we used six types of grating scales, in contrast to
previous studies that did not test specimens on different types ofscales (e.g., [14], [15], [17], [26]). As shown in Fig. 5, the roughness
perception for all types of grating scales can be estimated based on
the finger pad skin deformation; however, triangular scales tend to beunderestimated. A potential explanation for this is that the triangularspecimens were not perceived purely as roughness stimuli. They
also included pain stimuli owing to their pointed edges, although noparticipant complained of unacceptable pain during the experiments.
In the magnitude estimation task, participants may have responded toroughness plus pain perception; thus, the reported values were greaterthan those obtained through pure roughness perception.
Although previous studies have considered the contribution ofa single spatial spectral component to macroscopic roughness perception [18], [25]–[27], this study investigated the hypothesis thatmacroscopic roughness perception is mediated by a weighted linear
combination of multiple spectral components. The single spatial
component for λ = 4.35 mm exhibited a correlation coefficient of
0.75 ± 0.023 with the reported roughness values. This correlation
coefficient was significantly lower than that for the linear combinationof spectral components, which was 0.82 ± 0.023. These results
suggest that humans utilize multiple Gabor filter-like detectors, suchas those in the visual system.
This raises the question of what these two PCs indicate. As shownin Fig. 8, each of the two PCs exhibit a peak in a certain spatial

period. The first PC has a peak at 2.95 mm. This value is close
to the results of former studies (2.8 mm in [18], [26]). This value
may be related to the activity of the SAI-unit system. The second PC
exhibits a peak at 1.15 mm. This may be mediated by the fast adaptive(FA) units. In our experimental protocol, we asked the participants
to press their fingertips on the specimens until the electronic scale
showed a force of 300 gf; thus the fingers went through an active
pushing process. Therefore, the subjective roughness reported in our
experiment might have been due to the combined effect of SAI and
FA activities, although FA units were observed as a relatively minor
factor in our setting.
Here, we discuss the wavelength of the spatial component that ismost sensitive to the roughness perception. It is reasonable to assume
that such a wavelength is largely determined by the spatial distributionof SAI units beneath the skin. The density of the SAI units at thefingertip is approximately 70 units per cm2 [35], [36]. Based on
this density and the assumption that SAI units have equal spacing
distances, we performed calculations and found that the average
distance between two neighboring SAI fibers was approximately
1.45 mm. If each SAI unit is considered as a pressure-sensitiveelement, then based on the sampling theorem, the system of SAI units
is suitable for detecting spatial wavelength components of 2.90 mm
or greater. This value is close to those used for the wavelengths of thefilters applied to the spatial activity plots of SAIs: 2.8 mm in [18],[26] and 2.6 mm in [11]. Furthermore, in [25], which is the basis of
our study, the spatial spectral component of 2.66 mm exhibited thelargest correlation coefficient with roughness perception. As shownin Fig. 7, the maximum perceptual weight was λ = 3.15 mm.
These values considerably overlap with the loci of the maximum
subjective roughness reported in [14], [15], [19], [37], where theroughness perception expressed as a function of the spatial periods ofthe grating scales and dot scales exhibited inverted U-shaped curves
with maximum values at approximately 2.0–4.0 mm. In a study using
the spatial spectra of finger pads [25], the spatial components between
2.45 to 4.00 mm exhibited equally high correlation coefficientsfor roughness perception. Furthermore, Connor et al. [18] reported
that Gabor filters with λ = 2.0–4.0 mm exhibited equally good
performance in estimating subjective roughness. The SAI units are
not uniformly distributed beneath the skin, and each unit has several
branches with several Merkel cells at their ends [21]. Additionally,
the densities of the afferent units depend on finger size [38], [39].
Therefore, we cannot expect a unique optimal λ value. Instead, we
speculate that the spatial spectral components for wavelengths ofapproximately 2.0 to 4.0 mm are equally sensitive to the perceptionof macroscopic roughness.
Previous studies have suggested that in the relationship between
tactile acuity and finger size, small fingers lead to dense epidermal
ridges and SAI units [38], [39]. Although individual finger sizes were
not recorded in this study, if a correlation between finger size and themost significant λ value in the perceptual weight vector is identified,then we will have clues on how to determine the most suitable λ value
for different individuals. Moreover, although the finger pressing force
was controlled at approximately 300 gf in our study, humans do notstrictly control their pressing forces for roughness perception in their
daily lives. Our next challenge is to investigate how the pressing force
and its active control can be related to the spatial spectrum of finger
pad deformation in judging roughness.
V. C ONCLUSIONSA previous study investigated the link between the roughness
perception of macroscopic surface features and the spatial spectrum of
finger pad skin deformation [25]. To investigate this link further, we7estimated the roughness perception using a weighted linear combination of multiple spectral components, in contrast to a previous study
that estimated the roughness perception using only individual spectral
components. Close-up images of the finger pads were captured when
the pads were statically pushed into the roughened specimens, andtheir Gabor filter outputs were computed to acquire the spatial spectra.
The spectra were then linked to the subjective roughness values
reported in the magnitude estimation task by using the PLS regression
method. The weighted linear combination of spectral components
exhibited good agreement with the roughness perception with acorrelation coefficient of 0.82 ± 0.023, which is significantly higher
than the correlation coefficient (0.75 ± 0.023) computed from thesingle spectral component. These results support the hypothesis thatroughness perception is mediated by a linear combination of theoutputs of multiple Gabor filter-like detectors for different spatial
wavelengths. These findings can help us further understand theperceptual mechanisms of macroscopic roughness together with theexisting literature on roughness perception.
R EFERENCES[1] W. M. Bergmann Tiest, “Tactual perception of material properties,”
Vision Research, vol. 50, no. 24, pp. 2775–2782, 2010.
[2] S. Okamoto, H. Nagano, and Y. Yamada, “Psychophysical dimensionsof tactile perception of textures,” IEEE Transactions on Haptics, vol. 6,
no. 1, pp. 81–93, 2013.
[3] S. J. Bensmaı̈a, “Texture from touch,” Scholarpedia, vol. 4, no. 8, p.
7956, 2009.
[4] K. Sathian, A. Goodwin, K. John, and I. Darian-Smith, “Perceived
roughness of a grating: correlation with responses of mechanoreceptive
afferents innervating the monkey’s fingerpad,” Journal of Neuroscience,
vol. 9, no. 4, pp. 1273–1279, 1989.
[5] K. O. Johnson and S. S. Hsiao, “Evaluation of the relative roles of slowly
and rapidly adapting afferent fibers in roughness perception,” Canadian
Journal of Physiology and Pharmacology, vol. 72, no. 5, pp. 488–497,
1994.
[6] J. R. Phillips, R. S. Johansson, and K. O. Johnson, “Representation of
braille characters in human nerve fibres,” Experimental Brain Research,
vol. 81, no. 3, pp. 589–592, Aug 1990.
[7] R. S. Johansson and Åke B. Vallbo, “Tactile sensory coding in theglabrous skin of the human hand,” Trends in Neurosciences, pp. 27–
32, 1983.
[8] J. R. Phillips and K. O. Johnson, “Tactile spatial resolution. iii. a continuum mechanics model of skin predicting mechanoreceptor responses
to bars, edges, and gratings,” Journal of neurophysiology, vol. 46, no. 6,
pp. 1204–1225, 1981.
[9] D. R. Lesniak and G. J. Gerling, “Predicting SA-I mechanoreceptor spiketimes with a skin-neuron model,” Mathematical Biosciences, vol. 220,
no. 1, pp. 15–23, 2009.
[10] K. O. Johnson, “The roles and functions of cutaneous mechanoreceptors,” Current Opinion in Neurobiology, vol. 11, no. 4, pp. 455–461,
2001.
[11] D. T. Blake, S. S. Hsiao, and K. O. Johnson, “Neural coding mechanisms
in tactile pattern recognition: the relative contributions of slowly andrapidly adapting mechanoreceptors to perceived roughness,” Journal of
Neuroscience, vol. 17, no. 19, pp. 7480–7489, 1997.
[12] S. J. Lederman, J. M. Loomis, and D. A. Williams, “The role of vibrationin the tactual perception of roughness,” Perception & Psychophysics,
vol. 32, no. 2, pp. 109–116, 1982.
[13] S. J. Lederman and M. M. Taylor, “Fingertip force, surface geometry,
and the perception of roughness by active touch,” Perception & Psychophysics, vol. 12, no. 5, pp. 401–408, Sep 1972.
[14] T. Yoshioka, B. Gibb, A. K. Dorsch, S. S. Hsiao, and K. O. Johnson,“Neural coding mechanisms underlying perceived roughness of finely
textured surfaces,” Journal of Neuroscience, vol. 21, no. 17, pp. 6905–
6916, 2001.
[15] M. A. Lawrence, R. Kitada, R. L. Klatzky, and S. J. Lederman, “Haptic
roughness perception of linear gratings via bare finger or rigid probe,”
Perception, vol. 36, no. 4, pp. 547–557, 2007.
[16] A. Dépeault, E.-M. Meftah, and C. E. Chapman, “Tactile perceptionof roughness: raised-dot spacing, density and disposition,” Experimental
brain research, vol. 197, no. 3, pp. 235–244, 2009.

[17] K. Drewing, “Judged roughness as a function of groove frequency
and groove width in 3d-printed gratings,” in International Conferenceon Human Haptic Sensing and Touch Enabled Computer Applications.
Springer, 2018, pp. 258–269.
[18] C. E. Connor and K. O. Johnson, “Neural coding of tactile texture: comparison of spatial and temporal mechanisms for roughness perception,”
Journal of Neuroscience, vol. 12, no. 9, pp. 3414–3426, 1992.
[19] G. A. Gescheider, S. J. Bolanowski, T. C. Greenfield, and K. E.
Brunette, “Perception of the tactile texture of raised-dot patterns: A
multidimensional analysis,” Somatosensory & Motor Research, vol. 22,
no. 3, pp. 127–140, 2005.
[20] J. Eck, A. L. Kaas, J. L. Mulders, L. Hausfeld, Z. Kourtzi, and R. Goebel,“The effect of task instruction on haptic texture processing: The neural
underpinning of roughness and spatial density perception,” Cerebral
Cortex, vol. 26, no. 1, pp. 384–401, 2014.
[21] D. R. Lesniak, K. L. Marshall, S. A. Wellnitz, B. A. Jenkins, Y. Baba,
M. N. Rasband, G. J. Gerling, and E. A. Lumpkin, “Computation
identifies structural features that govern neuronal firing properties inslowly adapting touch receptors,” eLife, vol. 3, p. e01488, 2014.
[22] H. P. Saal, B. P. Delhaye, B. C. Rayhaun, and S. J. Bensmaı̈a, “Simulating tactile signals from the whole hand with millisecond precision,”
Proceedings of the National Academy of Sciences, vol. 114, no. 28, pp.
E5693–E5702, 2017.
[23] S. Kuroki, M. Sawayama, and S. Nishida, “The roles of lower-andhigher-order surface statistics in tactile texture perception,” Journal of
Neurophysiology, 2021.
[24] M. Taylor and S. J. Lederman, “Tactile roughness of grooved surfaces:
A model and the effect of friction,” Perception & Psychophysics, vol. 17,
no. 1, pp. 23–36, 1975.
[25] S. Okamoto and A. Oishi, “Relationship between spatial variations
in static skin deformation and perceived roughness of macroscopic
surfaces,” IEEE Transactions on Haptics, vol. 13, no. 1, pp. 66–72,
2020.
[26] A. I. Weber, H. P. Saal, J. D. Lieber, J.-W. Cheng, L. R. Manfredi, J. F.
Dammann, and S. J. Bensmaı̈a, “Spatial and temporal codes mediate
the tactile perception of natural textures,” Proceedings of the National
Academy of Sciences, vol. 110, no. 42, pp. 17 107–17 112, 2013.
[27] J. D. Lieber, X. Xia, A. I. Weber, and S. J. Bensmaı̈a, “The neuralcode for tactile roughness in the somatosensory nerves,” Journal of
Neurophysiology, vol. 118, no. 6, pp. 3107–3117, 2017.
[28] C. Tymms, E. P. Gardner, and D. Zorin, “A quantitative perceptual modelfor tactile roughness,” ACM Transactions on Graphics, vol. 37, no. 5,
pp. 1–14, 2018.
[29] T. Lindeberg, “Normative theory of visual receptive fields,” Heliyon,
vol. 7, no. 1, p. e05897, 2021.
[30] A. Sutu, E.-M. Meftah, and C. E. Chapman, “Physical determinantsof the shape of the psychophysical curve relating tactile roughness
to raised-dot spacing: implications for neuronal coding of roughness,”
Journal of Neurophysiology, vol. 109, no. 5, pp. 1403–1415, 2013.
[31] S. J. Lederman, “Tactual roughness perception: spatial and temporal
determinants.” Canadian Journal of Psychology, vol. 37, no. 4, p. 498,
1983.
[32] C. E. Connor, S. S. Hsiao, J. R. Phillips, and K. O. Johnson, “Tactile
roughness: neural codes that account for psychophysical magnitude
estimates,” Journal of Neuroscience, vol. 10, no. 12, pp. 3823–3836,
1990.
[33] S. J. Lederman, “Tactile roughness of grooved surfaces: The touchingprocess and effects of macro-and microsurface structure,” Perception &Psychophysics, vol. 16, no. 2, pp. 385–395, 1974.
[34] M. Hollins, F. Lorenz, and D. Harper, “Somatosensory coding ofroughness: the effect of texture adaptation in direct and indirect touch,”
Journal of Neuroscience, vol. 26, no. 20, pp. 5582–5588, 2006.
[35] G. Corniani and H. P. Saal, “Tactile innervation densities across thewhole body,” Journal of Neurophysiology, vol. 124, no. 4, pp. 1229–
1240, 2020.
[36] R. S. Johansson, “Tactile sensibility in the human hand: receptive fieldcharacteristics of mechanoreceptive units in the glabrous skin area.”
Journal of Physiology, vol. 281, no. 1, pp. 101–125, 1978.
[37] C. E. Chapman, F. Tremblay, W. Jiang, L. Belingard, and E.-M. Meftah,“Central neural mechanisms contributing to the perception of tactile
roughness,” Behavioural Brain Research, vol. 135, no. 1, pp. 225 – 233,
2002.
[38] R. M. Peters, E. Hackeman, and D. Goldreich, “Diminutive digits discern
delicate details: Fingertip size and the sex difference in tactile spatial
acuity,” Journal of Neuroscience, vol. 29, no. 50, pp. 15 756–15 761,
2009.

8

[39] C. Tymms, D. Zorin, and E. P. Gardner, “Tactile perception of theroughness of 3d-printed textures,” Journal of Neurophysiology, vol. 119,no. 3, pp. 862–876, 2018.

Qingyu SUN received a B.S. degree in engineeringfrom Tongji University, China in 2020 and has beena master’s student at Nagoya University, Japan, since2020. His research interests include haptics andperception.

Shogo OKAMOTO received a B.S. degree in engineering from Kobe University in 2005 and M.S.
and Ph.D. degrees in information sciences in 2007
and 2010, respectively, from the Graduate School ofInformation Sciences, Tohoku University. He was an
assistant and associate professor in the Departmentof Mechanical Systems, Nagoya University till 2015
and 2021, respectively. Since 2021, he has beenwith the Department of Computer Sciences, Tokyo
Metropolitan University, as an associate professor.

Yasuhiro AKIYAMA received his B.E. degree inengineering from Tokyo Institute of Technology,
Tokyo, Japan in 2006, and his M.S. and Ph.D.
degrees in engineering from the University of Tokyo,
Tokyo in 2008 and 2011, respectively. Until 2016,he was an assistant professor at Nagoya University,
Japan. Since 2022, he has been an associate professor in the Department of Mechanical Engineeringand Mechatronics, Shinshu University, Japan. His
main research interests include mechanical safety,
human–robot interaction, and biomechanics.

Yoji YAMADA received his Ph.D. from the Tokyo
Institute of Technology, Japan in 1990. He has
been an associate professor at Toyota Technological
Institute, Nagoya, Japan since 1983. In 2004, hejoined AIST as the leader of the Safety Intelligence
Research Group at the Intelligent Systems ResearchInstitute. In 2008, he became a professor at the Graduate School of Engineering at Nagoya University.
Since 2022, he has been the president of the National
Institute of Technology, Toyota College.