A Magnitude-Based Parametric Model Predicting the Audibility of HRTF Variation

Abstract

This work proposes a parametric model for just noticeable differences of unilateral differences in head-related transfer functions (HRTFs). For seven generic magnitude-based distance metrics, common trends in their response to inter-individual and intra-individual HRTF differences are analyzed, identifying metric subgroups with pseudo-orthogonal behavior. On the basis of three representative metrics, a three-alternative forced-choice experiment is conducted, and the acquired discrimination probabilities are set in relation with distance metrics via different modeling approaches. A linear model, with coefficients based on principal component analysis and three distance metrics as input, yields the best performance, compared to a simple multi-linear regression approach or to principal component analysis–based models of higher complexity.

Full text

PAPERS Freely available online
S. Doma, C. A. Ermert and J. Fels, “A Magnitude-Based
Parametric Model Predicting the Audibility of HRTF Variation”
J. Audio Eng. Soc., vol. 71, no. 4, pp. 155–172, (2023 April).
DOI: https://doi.org/10.17743/jaes.2022.0080
A Magnitude-Based Parametric Model Predicting
the Audibility of HRTF Variation
SHAIMAA DOMA, AES Student Member
(sdo@akustik.rwth-aachen.de)
, COSIMA A. ERMERT,
(cer@akustik.rwth-aachen.de)
AND JANINA FELS
(jfe@akustik.rwth-aachen.de)
Institute for Hearing Technology and Acoustics, RWTH Aachen University, Aachen, Germany
This work proposes a parametric model for just noticeable differences of unilateral differ-
ences in head-related transfer functions (HRTFs). For seven generic magnitude-based distance
metrics, common trends in their response to inter-individual and intra-individual HRTF dif-
ferences are analyzed, identifying metric subgroups with pseudo-orthogonal behavior. On the
basis of three representative metrics, a three-alternative forced-choice experiment is conducted,
and the acquired discrimination probabilities are set in relation with distance metrics via dif-
ferent modeling approaches. A linear model, with coefficients based on principal component
analysis and three distance metrics as input, yields the best performance, compared to a simple
multi-linear regression approach or to principal component analysis–based models of higher
complexity.
0 INTRODUCTION
Recent advances in audio playback technology have
made high-quality sound accessible to the masses. A va-
riety of applications, such as video games or home theater
systems, benefit from the proper design and equalization
of loudspeaker systems and headphones, allowing for bet-
ter immersion in the spatial acoustic environment [1]. With
this development—and with the increasing awareness and
expectations of users—spatial reproduction is further opti-
mized and tailored to the individual listener. Individualized
head-related transfer functions (HRTFs) approximate the
listener-specific sound directivity, which provides listeners
with their learned cues for auditory localization of spatial
sources. However, the available approximation methods are
accompanied by a potential deterioration in perceived qual-
ity.
For a better understanding of the impact on quality, a
holistic approach is required to relate auditory perception
to a wide range of possible spectral errors, e.g., a loss of
spectral or spatial detail, the presence of non-individual
spatial cues, changes in sound level and/or coloration, or
a combination of these (or other) factors. Several studies
have examined the detectability of manually applied spec-
tral changes, e.g., these changes included peaks and notches
of different form and depth introduced to the spectra [2],
a stepwise variation of inter-aural time differences (ITDs)
[3, 4], or different degrees of notch smoothing applied to
HRTFs [5].
However, to properly predict the audibility and percep-
tual effects of not only such controlled HRTF differences,
but also arbitrary spectral errors, a generalized metric is
needed. For example, in the context of developing HRTF
individualization methods and spatial interpolation tech-
niques, or of direct comparison between HRTF acquisi-
tions methods and setups, the nature of underlying spectral
deviations is not specifically known a priori. Studies per-
forming comparisons of this kind generally follow one of
the following approaches:
In the first approach, different levels of “pseudo”-
arbitrary variations are defined in relation to a specific pa-
rameter. This can, for example, be a property of a filter
used for HRTF reconstruction. In that case, the detection
thresholds acquired in a listening experiment describe a
required resolution of the parameter [6]. Although this in-
formation is relevant for the specific application, it is not
possible to generalize for arbitrary HRTF differences, un-
less these differences can be approximately described as a
filter degradation. Therefore, the second—more flexible—
approach draws conclusions based on a selection of distance
metrics from literature. These metrics may, on their part, be
more or less suitable for capturing certain differences.
In the context of localization errors, metrics have been
analyzed regarding how they scale with a given angular
error [7]. Numerous binaural models have further integrated
machine learning [8] and concepts of auditory cognition,
quantifying the likelihood of misinterpreting a stimulus as
coming from a deviant sound incidence direction [9–11].
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DOMA ET AL. PAPERS
Whereas spatial infidelities largely impact perceived
playback quality, timbral infidelities have been reported
to be even more detrimental [12], and these timbral dif-
ferences would not be accurately represented by models
targeting localization cues. Moreover, the spectral proper-
ties affecting the different perceptual attributes are closely
linked. For example, the incidence angle directly influences
sound color (as demonstrated in the directional bands theory
[13]), entailing differences of up to 10 dB in narrow-band
loudness sensitivity [14]. Given this complexity, it becomes
evident that HRTF accuracy cannot be described on a uni-
dimensional scale. An overarching model that scales with
HRTF differences should incorporate multiple models and
retain a multi-dimensional nature in its output.
The perceptual validation of such a model proves to be
an intricate task. Especially higher levels of HRTF errors
would need to be correctly mapped by the listeners to spe-
cific perceptual attributes. This requires them to have a
proper (and common) understanding of appropriate vocab-
ulary [15]. For example, in [16], the authors make do with
the general term of “discoloration” of stimuli without ad-
dressing more detailed timbral descriptors. In the experi-
ment, participants grade different degrees of discoloration.
A model then receives two-channel spatial signals and pre-
dicts the degree of binaural discoloration as the mean of
two separately calculated monaural values. Although the
model integrates both ear channels, it omits a representa-
tion of binaural interaction and, more specifically, does not
identify separate contributions of monaural and binaural
cue differences. In another study on the degree of timbral
and localization changes [17], ITDs are included, in addi-
tion to simple summation of the two separately calculated
monaural errors.
The mentioned studies focus on error grading, while not
explicitly identifying just noticeable differences (JNDs) for
the perceptual descriptors. In fact, the question of mere
detectability is rather detached from such definitions. Al-
though it can be assumed that the different aspects of per-
ceptual dissimilarity contribute to discrimination of stim-
uli, they may only be consciously identifiable for supra-
threshold stimuli. Thus, they need not be considered in the
error range close to the discrimination threshold.
Instead, a “generalized” JND would serve as a “worst-
case” limit for permissible spectral deviations. Below this
minimum required resolution, further HRTF optimization
would no longer be of value. This is the goal of the present
study: to examine the JND for HRTF dissimilarity given
generic differences, i.e., no artificial degradation of the
filters. A previous attempt for predicting discrimination
thresholds for generic stimuli was done in [18] in the con-
text of piano signals played on different instruments. Ex-
perimental results from a three-alternative forced choice
(3AFC) discrimination task in noise provided thresholds in
the form of SNR values. A single-channel (monaural) and
physiologically motivated dissimilarity model was devel-
oped; its output achieved “moderate to high correlation”
with the experimental data.
In the present work, the authors decided against apply-
ing a similar approach. The binaural nature of HRTF sets
brings about an inherent complexity to the design of a suit-
able listening test paradigm. Possible interactions between
the background noise and the already complex spatial and
timbral percepts may affect the experimental results. More-
over, binaural unmasking effects [19] would need to be ac-
counted for, based on the decision to use either inter-aurally
identical or uncorrelated noise signals.
On this account, the present study targets a direct com-
parison of stimuli convolved with free-field HRTFs. In the
range of small errors, spectral dissimilarity is regarded as
binary information with a probability of occurrence for the
two states: perceptually “similar” and “dissimilar.” These
probabilities are to be modeled, as a first step, for unilat-
eral HRTF variation, i.e., only one of the two ear signals
being varied. This limitation to single-channel JND com-
ponents serves a later extension to a binaural JND model,
in which monaural and binaural contributions are identified
separately.
In the error range where the JND is to be expected, a
modeling of higher-level cognitive representation of au-
ditory percepts is not essential. Instead, seven signal-near
metrics are used, with the goal of predicting the probabil-
ity of detection for different kinds of spectral deviations.
As shown in previous work by the authors [20], a prelim-
inary assessment of distance metric interrelation indicates
a variation in correlation patterns depending on the type
of compared datasets. Here, a selection of seven metrics
(not identical to the previous work) is analyzed numeri-
cally through both correlation and factor analysis, identi-
fying common trends in their reactions to spectral errors.
A thus-selected subset of metrics is employed in the de-
sign of a JND experiment for unilateral spectral deviations,
which provides perceptual data for modeling the audibil-
ity of errors. Different modeling approaches and their per-
formance are contrasted, upon which an optimal model
is identified.
The paper is structured as follows: SEC. 1 introduces the
magnitude-based distance metrics and the HRTF datasets
employed for metric analysis. SEC. 2 follows an objective
evaluation of distance metric interrelation, with the goal
of defining pseudo-orthogonal subsets of metrics. The lat-
ter observations find application in SEC. 3, in which the
paradigm for a JND listening experiment is designed and
first experimental results are presented. These results form
the basis for subsequent modeling of the discrimination
probabilities in SEC. 4. Finally, SEC. 5 contextualizes the
findings, followed by a brief summary and an outlook in
SEC.6.
1 MATERIALS
1.1 Distance Metrics
HRTFs provide information on incident sound arriving
at both ears for a broad range of frequencies and directions.
Due to high dimensionality, the task of identifying differ-
ences between HRTF sets is not straight-forward. Various
distance metrics enable a direct comparison of two HRTF
sets, summarizing the differences by means of a scalar
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quantity or vector. In the present work, metrics with the
following key aspects were selected:
- They quantify unilateral differences, i.e., they con-
trast data of a single ear in two HRTF sets through
single-channel calculations.
- They focus on magnitude deviations, disregarding
phase information. All calculations are therefore per-
formed in the frequency domain, rather than on im-
pulse responses.
- They provide scalar dissimilarity values in the direc-
tional domain, i.e., for each direction kdefined by
elevation angle θand azimuth angle ϕ.
Four of the presented metrics incorporate psychoacoustic
concepts to provide a closer relation to perception.
A well-established method for digital filter comparison
is the Mean Squared Error (MSE). In the context of HRTFs
[7], it is computed for each direction kby averaging the
squared spectral difference over frequency bins i∈[1, N]
as
MSE(k)=1
N
N

i=1
[HRTF1(k,i)−HRTF2(k,i)]2.(1)
Here, each frequency is equally weighted in the difference
measure. Because of the non-linear nature of the frequency
resolution of the human ear, errors in high frequencies may
be overvalued in this metric, compared to low-frequency
differences. The critical band model provides a mathemat-
ical description of the frequency processing in the auditory
system [21]. As proposed in [7], this property can be incor-
porated into the metric by means of a frequency dependent
weighting factor α(i), leading to the Critical-Band (CB)
MSE:
CB(k)=1
N
N

i=1
(α(i)[HRTF1(k,i)−HRTF2(k,i)])2.(2)
Analogous to the auditory frequency resolution, the weight-
ing factor decreases for higher frequencies and is computed
as
α(i)=1
α0fCB(i),(3)
with fCB(i) being the critical bandwidth for each fre-
quency bin iand the normalization value α0defined as
α0=
N

i=1
1
fCB(i).(4)
Because both the MSE and CB rely on absolute dif-
ferences, they will naturally yield larger dissimilarity val-
ues for directions of larger magnitude, e.g., for ipsilateral
incidence. However, human perception of magnitude dif-
ferences is not linear but approximately logarithmic [21].
As a consequence, differences at low magnitudes, e.g., at
notches, can have a higher perceptual impact than would
be captured on a linear scale. The MSE and CB may offer
no adequate representation of such perceptual subtleties.
Therefore, to counteract the influence of absolute magni-
tudes, a variation of the metrics is introduced: the logarith-
mic squared errors
MSElog(k)=1
N
N

i=1
log10HRTF1(k,i)
HRTF2(k,i)2
(5)
and
CBlog(k)=1
N
N

i=1
α(i)log
10HRTF1(k,i)
HRTF2(k,i)2
.(6)
Another metric performing squared error calculations is the
Mel Frequency Cepstral Distortion (MFCD) [22]. Here, the
HRTF spectrum is divided into NB=24 Mel bands by
means of a gammatone filter-bank. Mel bands offer a linear
scale that links physical frequencies to perceived pitch [21].
So-called Mel-frequency cepstral coefficients (MFCCs) are
obtained by performing a discrete cosine transform on the
energy within each Mel band [23]. Similarly to the MSE,
the MFCD is defined as
MFCD(k)=1
NB
NB

n=1
[MFCC1(k,n) −MFCC2(k,n)]2.
(7)
The variance of logarithmic magnitude differences is eval-
uated in the Inter-Subject Spectral Difference (ISSD), as
introduced in [24]. This distance metric is defined for fre-
quency bins up to 13 kHz. Originally, the metric uses di-
rectional transfer functions (DTFs), which are computed
by omitting the information common to all directions of
an HRTF set. This, however, makes the spectrum DTF(k)
dependent on the HRTF spectra of all directions. As the cur-
rent work targets a direct comparison of individual spectra,
the original HRTF spectra are here employed instead (cf.
[25]), yielding
ISSD(k)=σ2
i20 ·log10HRTF1(k,i)
HRTF2(k,i),(8)
with σ2
ibeing the variance over frequency and f(i)<13 kHz.
Finally, the Loudness Level Spectral Error (LLSE) is de-
signed to capture coloration differences between HRTF.
Based on [17], a pink noise signal is convolved with the
two HRTF spectra. Subsequently, the loudness levels LL
of the resulting signals are computed in phon for each of
the m=1...NEequivalent rectangular bandwidth filters
[21], respectively. An error value for each direction is then
obtained by evaluating the variance σ2
mof the difference in
loudness levels:
LLSE(k)=σ2
m(LL1(m,k)−LL2(m,k)).(9)
Because the computational procedures of the distance mea-
sures differ substantially from each other, it can be expected
that different “types” of dissimilarity will be evaluated de-
pending on the metric in use. For example, incorporating the
frequency selectivity of the human auditory system (e.g.,
CB, CBlog, MFCD, and LLSE) avoids over-representation
of errors in high frequencies. A brief overview of metric
properties is summarized in Table 1.
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DOMA ET AL. PAPERS
Table 1. Overview of the selected distance metrics (in
alphabetical order) and their main features.
Abbrev. Arithmetic operation Psychoacoustic model Scale
CB Squared mean Critical band weights Linear
CBlog Squared mean Critical band weights Log
ISSD Variance ··· Log
LLSE Variance ERB filters Log
MFCD Squared mean Mel filters Log
MSE Squared mean ··· Linear
MSElog Squared mean ··· Log
ERB, equivalent rectangular bandwidth.
1.2 HRTF Data Sets
The analysis of the distance metrics is performed on the
basis of three HRTF datasets, in the following termed: the
“measured,” “idealPCA,” and “anthroPCA” datasets.
The “measured” set was directly extracted from the ITA
HRTF database [26], in which acoustically acquired HRTFs
at a resolution of 5◦×5◦as well as anthropometric fea-
tures are provided for 47 individuals. For the “idealPCA”
dataset, the “measured” HRTFs were approximated using
Principal Component Analysis (PCA) reconstruction [27].
In this individualization approach, HRTF spectra are de-
picted as a weighted sum of Principal Components (PCs),
with “ideal” weights provided for HRTF spectra originally
used as input for the analysis. With increasing number of
PCs, the discrepancy between reconstructed and original
HRTF spectra in terms of ISSD error has been reported to
reach a saturation level at 15 PCs, with no further improve-
ment achieved by higher complexity [28]. In this work,
however, the reconstruction is performed with 23 PCs, as
the sufficiency of 15 PCs has not been validated with other
distance metrics. The “anthroPCA” dataset was generated
by approximating the ideal weights using a linear combi-
nation of six anthropometric features: “h,” “w,” “du,” “df,”
“d6,” and “d8,” according to definitions in [26].
Note that these approximated HRTF spectra initially do
not include phase information. All distance metrics received
only magnitude spectra as input. For use in the perceptual
experiment, additional phase spectra were reconstructed
(cf. SEC. 3.2.3).
The simulated datasets (“ideal-” and “anthroPCA”) were
mainly chosen for this paper to enable the generation of
both small and large error values. When comparing mea-
sured HRTFs from different individuals, distance metrics
oftentimes yield large error values. This is not desirable
here, given the aim of examining JND thresholds. Small
spectral deviations would allow for providing both sub-
threshold and supra-threshold stimuli in an experimental
setting and would further lead to a more precise JND. In
fact, the inclusion of less detailed spectra with, accordingly,
less prominent differences, facilitated the stimulus selection
for the subjective evaluation of distance metrics (see SEC.
3). Another motivation behind this selection of HRTF sets
was to avoid the controlled manipulation or “degradation”
[29] of spectra.
This should allow for more degrees of freedom and, thus,
presumably more realistic variations in the HRTF spectra.
An alternative to generating small errors by approximation
would be to compare HRTF sets obtained from repeated
measurements [30] of the same individual. This approach
was, however, dismissed due to the extensive measurement
effort required.
In the following, a distinction is made between inter-
individual and intra-individual comparisons. In literature,
intra-individual differences typically refer to the compari-
son of left and right HRTFs from a single dataset. In con-
trast, the term is here used to describe the comparison of
left-ear HRTF spectra of different datasets (“measured,”
“idealPCA,” and “anthroPCA”) belonging to the same indi-
vidual. For inter-individual comparison, equilateral HRTF
spectra of different individuals are compared. When per-
forming inter-individual comparisons, the left-ear HRTF of
ID1 from the ITA HRTF database was used as a reference.
Because of excessive data (2,304 directions per member),
data from half of the database (24 subjects) were deemed
sufficient for the present work.
2 NUMERICAL EVALUATION
The following sections observe the reaction of distance
metrics to inter-individual and intra-individual differences
between HRTFs. A special emphasis is laid on datasets eli-
gible for an evaluation of audibility, see SEC. 3. Early infor-
mal listening led to the conclusion that comparisons involv-
ing the “measured” dataset were for the most part clearly
distinguishable and therefore not suited for the planned
JND experiment. The following analysis therefore focuses
mainly on inter-individual comparisons within the “ide-
alPCA” and “anthroPCA” datasets, respectively, and intra-
individual comparisons between the two. To provide an an-
chor point for metric values, inter-individual comparisons
within the “measured” dataset are also included.
2.1 Relative Value Ranges
Fig. 1 (top) visualizes the metric values for the four
presented cases, i.e., inter-individual comparisons within
the “measured,” “idealPCA,” and “anthroPCA” datasets
as well as intra-individual comparison between the recon-
structed HRTFs of ID 1 in the “idealPCA” and “anthroPCA”
datasets, respectively. For better visibility of trends, each
distance metric is normalized by its 97th percentile, with
normalization factors calculated collectively over all four
cases.
Inter-individual comparisons, i.e., cases in Figs. 1(a)–
1(c), show most prominent reactions for metrics MSElog,
MFCD, and LLSE. The ISSD metric is furthermore strongly
affected by the detailed non-individual cues between “mea-
sured” HRTFs of different individuals. Especially cases in
Figs. 1(a) and 1(b) dominate the normalization factors for
these metrics, leading to very small values in the inter-
individual comparison case of Fig. 1(d). On the other hand,
the case in Fig. 1(d), which contrasts the “anthroPCA” and
“idealPCA” datasets, dominates the value range for metrics
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Fig. 1. Top: Relative value ranges of the distance metrics, normalized to their respective 97th percentile. Bottom: Bivariate Pearson
correlation of log-transformed data points. The three cases (a)–(c) represent inter-individual comparisons between HRTF spectra of ID 1
and IDs 2–20 within the same dataset (“measured,” “idealPCA,” and “anthroPCA”), respectively. Case (d) represents the intra-individual
comparison between the “anthroPCA” and “idealPCA” datasets, limited to HRTF spectra of ID 1.
MSE, CB, and CBlog. Smallest metric reactions are found
for the case in Fig. 1(c), in which substantial loss of de-
tail in reconstruction attenuates the differences between the
non-individual cues being approximated.
2.2 Interrelation Analysis
Besides examining common trends in value ranges, mu-
tual information and possible pseudo-orthogonal behavior
of the distance metrics was investigated based on both intra-
individual and inter-individual comparisons. The goal was
to reduce the available metrics to a subset that would likely
capture a variety of HRTF differences with little redun-
dancy. The analysis focused on directional data, with each
value contrasting a pair of HRTF spectra in a certain direc-
tion, rather than a whole set of spatial transfer functions.
Calculations were performed using R (version 4.0.2) and
RStudio (version 1.3.1073).
2.2.1 Correlation Analysis
Bivariate Pearson correlation [31] was first calculated to
determine strong pairwise correlations and trends common
to the different comparison cases. Given the lower zero
boundary of all metrics and the presence of outliers towards
high metric values, the data initially exhibited rather strong
skewness, requiring a logarithmic transform for its reversal
before conducting the correlation analysis.
The acquired correlation coefficients between distance
metrics are displayed in Fig. 1 (bottom) for the three inter-
individual and one intra-individual comparison set. Due
to the large amount of data points, all correlations are
significant (p<0.001). The strongest and most consis-
tent correlation is found between the metrics MSE and CB
(r>0.90), which otherwise (especially MSE) show the least
correlation to the remaining metrics. Second highest is the
correlation between MFCD and CBlog. Further moderate to
strong correlations show a slight case dependency. Most
pronounced is the contrast between the inter-individual
cases and the one intra-individual case in Fig. 1(d). Vari-
ance in the correlation pattern becomes more prominent
when observing ipsilateral and contralateral sound inci-
dence directions separately, yet this differentiation is not
further pursued at this point.
2.2.2 Factor Analysis
To gain further insight on how the metrics could be
grouped, an Exploratory Factor Analysis (EFA) [32] was
conducted as follows: The metric values were first log-
transformed, since the analysis again relies on correlation
matrices. Then, the datasets were tested on eligibility for
the EFA by applying Bartlett’s Test of Sphericity [33] and
the Kaiser-Meyer-Olkin (KMO) criterion [34] as a measure
of sampling adequacy. For subsets of data complying with
these requirements, EFA could then be performed using a
suitable number of factors as determined by the parallel
analysis method [35]. Principal axis factoring was chosen,
as it does not require strict multi-variate normality [36] and
is preferred for exploratory analysis. Varimax rotation was
applied to maximize “extreme” factor loadings.
With the seven distance metrics as a starting point,
Bartlett’s test of sphericity confirmed the presence of re-
dundancy in the data (p<0.01). The KMO criterion yielded
“mediocre” to good values [34], ranging between 0.64 and
0.88 for most metrics. The sole exceptions lay in the MSE
and CB metrics, with values as low as 0.41 to 0.51. The
KMO is known to penalize pairwise grouping of variables
in favor of larger groupings. According to common practice,
it should lead to omission of the two concerned variables
from the analysis, since they would “lead to erroneous in-
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DOMA ET AL. PAPERS
terpretation” [37]. However, as the study does not intend
to create an ideal FA model but to detect tendencies for
pseudo-orthogonal behavior, FA is run both before and af-
ter elimination of MSE and CB.
The analysis attempting to use nFac =3 factors (as sug-
gested by parallel analysis) results in invalid models with
negative specific variances for CB in all three datasets
(“ultra-Heywood case”). Using nFac =2, two models remain
invalid, whereas the case “anthroPCA” vs. “anthroPCA”
successfully maps MSE and CB to Factor 1 (F1), and the
remaining metrics to Factor 2 (F2). Indeed, the previously
presented correlation results support the supposition that
these two metrics would likely dominate the loadings for
one factor and that they present a meaningful contrast to
the other metrics.
After elimination of metrics MSE and CB from the
FA, improved KMO values can be observed (range: [0.72,
0.88]). Again, parallel analysis suggests nFac =3forthe
three datasets, and all analyses lead to valid models. The
following common trends are found: MFCD and CBlog have
highest loadings for F1, ISSD and LLSE for F2, and MSElog
for F3. For all three models, variable complexity is maximal
for MSElog. This metric is explained by 2.1 to 2.5 factors,
indicating that it is not completely detached from F1 and
F2 and thereby not totally separate from the other metrics.
2.3 Pseudo-Orthogonal Metric Subsets
Combining the results from the previous sections, in-
sights on similarities in metric behavior can be gained.
Firstly, the relative value ranges showed complementary
behavior between groups of metrics: MSElog, MFCD, and
LLSE seem to react most strongly to inter-individual dif-
ferences, whereas MSE, CB, and CBlog are affected by
intra-individual errors. The latter case is also reacted to by
MFCD, if not as strongly as to the first.
Marked correlations could be found, e.g., between MSE
and CB. This was to be expected, since both rely on bin-wise
squared mean calculations on a linear scale. Correlation
between ISSD and LLSE can be similarly attributed to
the frequency domain variance calculations performed in
both. The logarithmic Mel band energy in the calculation
of MFCC coefficients may explain the strong correlation
between the MFCD and the two metrics MSElog and CBlog,
in contrast to their linear counterparts.
Combining these observations with the metric allocations
in EFA, three groups can be identified:
- I: MSE and CB.
- II: MFCD and CBlog.
- III: ISSD and LLSE.
Evidently, these groups do not claim total similarity of the
metrics included, but a tendency for a common response to
spectral differences. A group thereby cannot be reduced to
a single metric that explains all of its variance. For the cur-
rent application, however, choosing a representative metric
per group is of benefit, as a smaller number of metrics
can be more easily considered for stimulus selection (see
SEC. 3.2.2). In search of a generalized metric model that
can handle different kinds of spectral alterations, a vari-
ety of these alterations should be included in a perceptual
evaluation of the model. Stimuli to which predominantly
one subgroups responds—and not the others—are likely to
represent a specific category of errors.
On this account, the three groups are reduced to the
following representative subset: MSE, MFCD, and ISSD.
These three metrics are integrated in the experiment design,
which will be described in the next section.
3 PERCEPTUAL EVALUATION
In contrast to previous studies that link distance metrics
or binaural model output to physical quantities (e.g., an-
gular displacement of HRTF incidence directions [7]), the
present study aims to derive a direct connection between
distance metrics and auditory distinguishability. The fol-
lowing sections describe a listening experiment paradigm
developed for this purpose. The approach used for han-
dling experimental output data as well as first findings on
the relation between individual metrics and probabilities
of stimulus discrimination are presented. These observa-
tions will serve as a foundation for modeling JNDs in the
subsequent section.
3.1 Approach for Signal Presentation
The sought JND needs to be captured in an appropriately
designed experimental setting. Multiple possibilities can be
considered regarding the way the signals are presented and
varied:
- Monaural presentation (with unilateral signal varia-
tion).
- Diotic presentation with bilateral signal variation.
- Binaural presentation with bilateral signal variation.
- Binaural presentation with unilateral signal varia-
tion.
Requirements for the present study as well as resulting
potential drawbacks of the listed approaches are discussed
in the following.
For single-channel filters, audibility assessment for spec-
tral changes is a comparatively straightforward task. Unlike
these simple filters, the information present in HRTFs and
the auditory percepts evoked by them require special con-
siderations. Not only are changes in coloration and/or level
possible, but also a spatial difference may be perceived be-
tween two pairs of HRTFs, i.e., a relocated virtual source,
an altered source width and/or degree of externalization.
It can be assumed that conscious recognition of the afore-
mentioned percepts is mainly possible for supra-threshold
(i.e., more prominent) differences. Close to the threshold,
the affected perceptual aspects may not be identifiable in-
dependently. On this account, the authors do not seek sepa-
rate JNDs for each perceptual attribute, but instead, a global
JND, to which the perceptual attributes contribute collec-
tively.
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In this context, retaining bilateral stimulation (with plau-
sible left and right ear HRTFs) is essential to account for the
spatial component to the JND. A simple monaural playback,
in which the HRTF spectra to be compared are successively
presented to only one ear, with no playback at the averted
ear, would therefore not be sufficient.
In the diotic case, this issue is not present. In spite of un-
naturally symmetric monaural cues and the absence of bin-
aural cue information, a spatial percept may be evoked, with
a virtual source being perceived in the median plane. This
scenario has the advantage that the signal variation (i.e.,
the switch between HRTF spectra) is identical at both ears.
Similarly to the monaural case, it can therefore be attempted
to describe the difference between the two HRTF spectra
by means of a single set of distance metrics. Nonetheless,
previous research about the perception of diotic playback
leads to question its suitability in the present case:
Numerous studies have investigated a so-called “diotic
advantage” [38] compared to monaural presentation, identi-
fying a lower audibility threshold [39], lower speech recep-
tion thresholds in speech-shaped noise [40], lower detection
thresholds for both amplitude and frequency modulation
[19], and better distinguishability of speech processing al-
gorithms in the context of quality assessment [41], among
others. To explain this advantage, several hypotheses were
put forward, that likely contribute to this effect. One theory
justifies the improvement by the presence of “two inde-
pendent observations” [19], which intrinsically leads to a
higher probability of detection. Another theory attributes
the effect to a different neural representation in absence of
contralateral stimulation [42, 43]. A heightened “general
attention” was further theorized in presence of bilateral
stimulation [38].
Applying these theories leads to the following conclu-
sions: The latter hypotheses relating to overall lowered
sensitivity upon unilateral stimulation give further reasons
against monaural playback and in favor of diotic (or bin-
aural) presentation. However, the first theory implies, from
a signal-theoretical perspective, that diotic playback would
exaggerate the audibility of differences, since the (same)
variation would be presented at both ears.
This leads to the third proposed approach: binaural pre-
sentation with both ear signals being varied individually.
Because it is not intended to artificially “degrade” spec-
tra according to specific parameters, this approach implies
swapping two generic pairs of HRTFs within each exper-
imental trial. Evidently, this is the most realistic scenario,
which, however, introduces further complexity to the prob-
lem. The subjective percept (i.e., the ability to tell filters
apart) would be influenced by two filters changing simul-
taneously (and, most importantly, in a non-identical way).
These changes would need to be described via two different
sets of distance metric values, which would flow into the
selection of stimulus pairs for the experiment, as well as
into later perceptual modeling.
In addition, changes in both ear signals would (unless
closely monitored) introduce variations to binaural cues
(i.e., to the direct difference values between the left and right
ear). Certainly, the essential contribution of these binaural
effects to audibility is not to be disputed. However, due to
the described complexity, a separate evaluation of monaural
and binaural cue differences would be of benefit as a first
step.
Studying the contribution of monaural cue differences is
enabled by the final suggested approach, which is applied
in the present work: HRTF variation is introduced at only
one ear, while an unchanged signal is maintained at the
opposing ear, thus ensuring plausible binaural presentation.
Simultaneously, inter-aural differences are controlled for.
This is done, on the one hand, by applying identical ITD
phases to the two pairs of magnitude spectra. On the other
hand, the transitions in inter-aural cross-correlation (IACC)
between HRTF pairs is minimized to be below the known
JND for inter-aural noise cross-correlation [44], cf. SEC.
3.2.3.
Clearly, the selected approach is atypical. However, con-
sidering the drawbacks and limitations of the other ap-
proaches, it can be considered a first step towards deriving
a more complex model for bilateral HRTF variations. Alto-
gether, the arguments show this novel approach to be better
suited for the given purpose.
3.2 Experimental Design
The goal of the present study is to derive a parametric
model predicting audibility of unilateral HRTF differences.
The input to such a model is a pair of slightly different
HRTF spectra meant for one ear—which are to be varied
during an experiment—while the other ear spectrum is kept
constant. (See SEC. 3.2.3 for a detailed description of the
stimuli.)
Accordingly, the database of feasible HRTFs needed to
be limited to pairs in which the error detection thresh-
old was to be expected. On this account, the “measured”
HRTFs were discarded from the experiment, as they were,
for the most part, too clearly distinguishable in both inter-
individual and intra-individual comparisons. This lead to
three cases, cf. Figs. 1(b)–1(d), which were integrated into
the experiment design. For simplicity, the pool of HRTFs
was further tightened to inter-individual comparisons be-
tween database IDs 1 and 2, and intra-individual compar-
isons for ID 1. Because only audibility was of relevance—
and not the accuracy of source representation as, e.g., in the
context of a localization task—the use of generic HRTFs
from the HRTF database was deemed sufficient, irrespec-
tive of their similarity to the participants’ own HRTF sets.
3.2.1 Paradigm
Audibility of a difference between two stimuli was ex-
amined in a classic three-alternative forced choice (3AFC)
task. The positions of a target stimulus and two reference
stimuli were balanced among the three buttons. Participants
were asked to choose the one stimulus different from the
other two.
As previously motivated, binaural playback was main-
tained in the experiment and the signal for only one ear
was varied per stimulus pair, respectively. The sound inci-
dence directions for the experiment were selected for signal
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DOMA ET AL. PAPERS
Fig. 2. Experimental conditions “Ear” דSide.” The microphones
indicate the ear channel at which the signal is varied within a
trial. The loudspeakers indicate the hemisphere of sound incidence
relative to the ear at which the signal is varied.
variation at the left ear. These directions were additionally
mirrored along the median plane, thereby allowing for a
complementary set of trials with identical signal variations
offered at the right ear. This guaranteed equal representa-
tion of both ears in the study. Supposing a potential dif-
ference in metric behavior for ipsilateral and contralateral
incidence directions, this distinction was further considered
by including virtual sound sources in both hemispheres, see
Fig. 2.
In total, the study was composed of the following inde-
pendent variables:
- Modality: cases in Figs. 1(b)–1(d).
- Side: ipsilateral/contralateral.
- Ear: L(eft) (optimized direction)/R(ight) (mirrored
direction).
For each of the 12 conditions (modality ×side ×ear),
six incidence directions were selected (see next section),
leading to 72 stimulus pairs per block. Using a within-
subject design, each participant was presented with three
blocks (i.e., three repetitions in total), containing identical
yet differently ordered (Latin-square balanced) stimuli.
3.2.2 Stimulus Selection
Within the selected conditions, stimuli close to the
threshold of error detection needed to be chosen. As previ-
ously discussed, the individual metrics may not be sufficient
to describe the audibility of differences. Yet, from a numer-
ical perspective, they are indicators for the presence of a
spectral deviation and can be assumed to scale with the
strength of said deviation. Accordingly, the stimuli were
selected to have rather low metric values.
Since the stimuli should additionally exploit the pseudo-
orthogonal behavior of metric subgroups (as derived in SEC.
2.2), the three representative metrics served as a basis: MSE,
MFCD, and ISSD. Metric values for the limited dataset (ID
1/ID 2) are visualized in Fig. 3(a) collectively for cases Figs.
3(b)–3(d). As before, the values are normalized by their re-
spective 97th percentiles. In accordance with Fig. 1, visibly
little to no common trends are present between the three
Fig. 3. Selection of stimulus pairs from HRTFs of ID1 and ID2
on the basis of small values of the three pre-selected metrics
ISSD, MFCD, and MSE (a) and trying to achieve the best possible
distribution in azimuth (b). The latter condition could be only
partly satisfied while fulfilling the first.
metrics. The red crosses mark the directions used for the
experiment. These points were statistically selected within
empirically defined value ranges for the three metrics.
Besides the small metric values, it was aimed to pro-
vide a proper distribution of incidence directions within
the respective hemispheres, particularly regarding the az-
imuth angle. Fig. 3(b) displays the HRTF incidence an-
gles for the selected stimuli. Despite multiple iterations
and some manual adaptations to the empirical value range,
not all angular ranges were compatible with the constraint
of small metric values. Particularly on the rear contralateral
side (between 200◦and 290◦), no suitable stimuli could
be identified for either of the comparison cases. Similarly,
for frontal ipsilateral directions (azimuth angles ≤80◦), the
intra-individual comparison case (triangular marker) also
came with increased metric values, leading to exclusion of
these directions. The shown directions were used as they
are for trials with signal variation on the left ear, and were
mirrored along the median plane for trials with right ear
signal variation.
3.2.3 Stimulus Preparation
The binaural signals for each experimental trial consisted
of a triple pulse train of pink noise convolved with two pairs
of HRTF spectra. The pulses each had a duration of 200 ms,
with raised cosine ramps of 25 ms for fade-in and fade-out,
and were separated by 150 ms of silence, producing to a
total stimulus duration of 900 ms.
The preparation of HRTFs comprised multiple steps, as
visualized in Fig. 4. First, HRTF magnitude spectra were re-
constructed from PCs, as described in SEC. 1.2. As they only
contained absolute values, a suitable phase was computed:
A minimum-phase spectrum [45] accounted for phase con-
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Fig. 4. Stimulus preparation for the JND experiment. Suitable
phase spectra were calculated for the HRTF magnitude spectra
reconstructed from PCs. The signal for the averted ear, which
would not be varied between a pair of stimuli, was selected based
on a minimization of the transition in IACC.
tributions of monaural cues, as contained in the magnitude
spectra; An ITD phase component was subsequently cal-
culated according to the analytical ellipsoidal model [46],
with head dimension of ID 1 of the database as input; Fi-
nally, and to ensure causality of the filters, a time offset of
1ms was applied. The now complex-valued spectra could
then be convolved with the noise pulses.
Up to this point, stimuli were considered from a monaural
perspective. As left and right ear signals were combined for
spatial audio playback, it was important to ensure that bin-
aural effects would not interfere with the solely unilateral
differences captured by the metrics in use. As previously
noted, the signal of only one ear was varied between the
HRTF pairs. This implicates that the other ear signal was
kept constant and had to be selected among two available
stimuli. Here, IACC should be considered. A transition in
IACC may produce an audible difference between binau-
ral signals, even when separate monaural signals might not
be distinguishable. In [44], the JND for inter-aural noise
cross-correlation was reported as r2=0.4 for a reference
stimulus of r2=0. A worst-case (i.e., easiest discernibility)
JND of r2=0.04 was found for a reference of r2=1.
For the HRTF spectra selected in the present study, mean
and standard deviations of reference IACCs amounted to
μ±σ=0.827 ±0.135. As the values were rather close to
1, a sensitivity to transitions in IACC close to (though not
as bad as) the reported worst-case JND could be expected.
For the selected HRTFs, the absolute transitions |r2| when
replacing a left-ear spectrum by that of another HRTF set
amounted to 0.049 ±0.035, which, in fact, was danger-
ously close to the JNDs. Therefore, a minimization of the
difference in IACC values between the two binaural signal
pairs was applied. This served as a selection criterion for
the right-ear signal, which would be employed as the fixed
ear stimulus (either for right ear playback, or (after mirror-
ing) as a left ear signal). Thus optimized IACC transitions
amounted to μ±σ=0.032 ±0.025, with all but five stim-
uli lying below the JND curve reported in [44] (Fig. 4, curve
denoting 75% correct in their 2AFC paradigm). Thereby,
it was verified that the IACC transitions should, for the
most part, play no part in distinguishing between presented
stimuli.
3.2.4 Playback
The experiment was conducted in a custom-made hearing
booth (length ×width ×height =2.1 m ×2.1 m ×2m).
For playback, Sennheiser HD650 headphones were used,
applying individual headphone equalization [47]. Level cal-
ibration was performed using an artificial head with IEC711
coupler (HMS III, HEAD Acoustics, Herzogenrath, Ger-
many), a conditioning amplifier (Type 2690-A, NEXUS,
Hottinger Br¨
uel & Kjær GmbH, Darmstadt, Germany), and
a soundcard (RME Fireface UC, Audio AG, Haimhausen,
Germany), setting the level to ≈60 dB for frontal and a
maximum of 66 dB for lateral incidence.
3.2.5 Participants
A total of 19 participants (six female, 13 male), aged
23−35 years (μ±σ=27.3 ±2.8), took part in the ex-
periment. All, except for four, had prior experience with
spatial audio and similar experiments. Pure-tone audiome-
try (up to 16 kHz) ensured normal hearing and sensitivity
to changes in HRTF cues in the higher frequency range.
All participants provided informed consent and received no
compensation for their participation.
3.3 Experimental Results
As mentioned above, the participants received each stim-
ulus once per block, leading to three repetitions in total.
Thus, calculating separate probability values for each par-
ticipant would produce values discretized to pi,s∈{1
/
32
/
3,1}
for individual iand stimulus s∈[1,72], respectively. In a
3AFC paradigm, the guessing rate is 1
/
3. Accordingly, the
range between 1
/
3and 1 would therefore be very sparsely
sampled, making it particularly difficult to fit the values
to the sigmoid shape of a psychometric function. On this
account, answers from all participants instead flow into a
single data point per stimulus. Each point is defined by the
overall probability of recognition in the 3AFC task (calcu-
lated based on 19 ×3=57 trials) and the seven metric
values corresponding to a stimulus pair.
3.3.1 Distribution of Discrimination Probabilities
The percentage of correct responses for each stimulus,
respectively, is displayed in Fig. 5. The tested conditions can
be differentiated as follows: the vertical panels indicate the
three different comparison sets (modalities); color coding
denotes ipsilateral (petrol) and contralateral (gray) sound
incidence; and the marker symbols correspond to the ear
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DOMA ET AL. PAPERS
Fig. 5. Percentage of correct answers (over all participant re-
sponses) for each of the 72 stimuli in the 3AFC experiment. Data
points are split after the different conditions (“Modality” דSide”
דEar”), see Fig. 2.
signal that was varied in the trial, with “x” for the left and
“o” for the right ear.
Collectively, the values range between 31.6% and 89.5%
(μ±σ=56.1 ±13.9%). For few individual conditions,
especially the inter-individual comparison within the “an-
throPCA” dataset (middle panel), the maximum probabil-
ity barely reaches the 66.2% threshold point, which cor-
responds to the 50% chance of audibility outside of the
experimental setting. (Note that the target point along the
psychometric function is debated in literature. Here, the
middle point is selected, as suggested in [48], while as-
suming a guessing rate of γ=1
/
3and a lapse rate of λ=
0.01.)
Most of the shown stimuli are independent from each
other, i.e., a direct link between the stimuli of different
conditions does not exist. The only exception lies in the
differentiation between left and right ear playback, in which
the binaural stimuli were presented once in their original
form and once after mirroring along the median plane. On
this account, it is feasible to point out a slight tendency
towards lower detectability of spectral alterations in the
right ear signal (circles), compared to the left ear (crosses).
In contrast, a possible impact of the other independent
variables cannot be evaluated directly, since the observa-
tions are based on different stimuli per condition. The
(partly empirical) choice of these stimuli, however, was
of varying difficulty: for some conditions, finding inaudi-
ble stimulus pairs was a challenge, whereas for others, it
was more demanding to find audible ones. This variation,
although not leading to analytical conclusions, is in accor-
dance with findings on the value range of distance metrics,
which varied greatly for the different comparison cases, cf.
Fig. 1.
3.3.2 Psychometric Representation
Given the partly limited ranges of the acquired audibil-
ity values, cf. Fig. 5, modeling a separate psychometric
function for each condition, respectively, was not always
feasible. Moreover, only few distance metrics showed a
monotonic rise with increasing detectability rate of the cor-
responding stimuli. A “good” and “bad” example can be ob-
Fig. 6. Relation between stimulus distinguishability in the 3AFC
task and normalized metrics for inter-individual comparisons of
HRTF spectra from the “idealPC” dataset. The maximum of each
metric over the 72 stimuli is used as a normalization factor. Marker
shapes correspond to the four conditions (“Ear” דSide”). Here,
CBlog exemplifies cases less suitable for fitting to a psychometric
function, compared to MSE.
served in Fig. 6 for inter-individual comparison within the
“idealPCA” dataset. Normalized metric values for MSE and
CBlog are displayed on the xaxis, the percentage of correct
answers in the 3AFC task on the yaxis. The marker shapes
represent different conditions (“Side” דEar”). (Note that
the cases of left and right ear signal variation for the same
“Side” variable are only mirrored stimuli and correspond to
the same metric values. Therefore, squares and triangle (or
circles and crosses) are always vertically aligned in pairs.)
Collectively, no general trend of increasing audibility
with rising CBlog can be observed, whereas MSE data points
seem to be a much better fit for reconstructing a psycho-
metric function. The latter case was only true for very few
conditions and metrics. This made most cases unsuited for
a modeling approach based on gathered psychometric func-
tion fitting parameters, such as the slope or spread [49].
4 MODELING
In this paper, audibility of spectral deviations is to be
modeled as a function of a set of numerical distance metrics.
The choice of a suitable model type must be based on ob-
servations of the interaction behavior of the metrics and the
collected perceptual information. The experimental results
demonstrated that, e.g., an univariate regression model, re-
lying on a single metric, would be insufficient to describe
audibility. More complex model types are therefore sug-
gested, embedding information from a varying number of
distance metrics. The following sections present two differ-
ent approaches and contrast their performance.
4.1 Multi-Linear Regression
A multi-linear regression (MLR) approach models the
probabilities of recognition as a linear combination of a
selection of Ndistance metrics.
pcorrect =c0+
N

i=1
ci·Xi,(10)
with cidenoting linear coefficients and Xithemetricval-
ues. To maintain a consistent range for coefficients, metric
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values can be normalized prior to summation, replacing Xi
by
ˆ
Xi=Xi−μx,i
σx,i
.(11)
The mean μiand standard deviation σiare calculated for
metric iof dataset X, which is available for model training.
For an arbitrary input dataset Y, vector ˆ
Xin Eq. (11) can
be substituted by the normalized dataset ˆ
Y, while retaining
the values μxand σxof the training data.
4.2 Principal Component Regression
A linear model requires the variables to be uncorrelated,
thus avoiding collinearity issues. This prerequisite is only
partially satisfied by the distance metrics. Therefore, as a
second model type, a PCA-based approach is evaluated.
Here, multi-dimensional data are projected onto a set of
orthogonal coordinate axes (PCs) that capture the variance
of input variables. In matrix representation, the normalized
input data ˆ
X, cf. Eq. (11), is expressed as a weighted sum
of PCs:
ˆ
X≈Wx·Vx.(12)
Weight matrix Wx(also termed ”score”) and coordinates
Vxare estimated based on a model training dataset X.For
an arbitrary input dataset Y, corresponding scores can be
calculated as
Wy≈ˆ
Y·V−1
x=ˆ
Y·VT
x,(13)
with ˆ
Ydenoting the input data after normalization with μx
and σx. (Note that the equality of the inverse and trans-
pose of Vxholds due to the orthonormality property of
the matrix.) Applied to the present issue, a PCA model can
provide a set of virtual metrics that are orthogonal by defini-
tion. Corresponding weights describe the reaction of these
”metrics” for each data point, i.e., for each stimulus pair,
and can therefore be treated similarly to the normalized dis-
tance metric values in SEC. 4.1. Recognition probabilities
are then expressed as
pcorrect =c0+
NPC

i=1
ci·Wy,i,(14)
where Wy,idenotes a column vector ifrom the estimated
score matrix.
The PC regression can be considered as a hierarchical
linear modeling approach. Although it is more complex than
the MLR model, it allows for integrating information from
many metrics, and benefits—rather than suffers—from the
information common to them.
4.3 Psychometric Adjustment
The presented models define the slope of a psychometric
function, predicting the discrimination rate (or percent cor-
rect) in a 3AFC paradigm relative to a few selected distance
metrics or to the score of ”virtual” distance metrics (PCs).
The probabilities pcorrect, as reconstructed in Eq. (10) or
(14), initially have no lower and upper limits, as opposed to
typical data acquired in a 3AFC experiment. The common
psychometric function possesses, besides the quasi-linear
slope, a lower asymptote due to guessing rate γ=1
/
3.It
further possesses an upper asymptote due to lapse rate λ,
often assumed around 0.01 to minimize slope bias [50].
A hard cut is therefore introduced to the model output at
33.3% and 99%, respectively, leading to
plimited =min{99 %,max{33.3%,pcorrect}}.(15)
If pdetect is introduced as the general sensitivity, i.e., the
probability of detection outside of the experimental setting,
the percentage of correct answers given in the experiment
can be expressed as
plimited =γ+(1 −λ−γ)·pdetect.(16)
Accordingly, the sensitivity values are calculated as
pdetect =plimited −γ
(1 −λ−γ).(17)
After this transformation, the probability pdetect is assumed
to be no longer related to the experimental design. The
50% value corresponds to the sought audibility thresh-
old. The chance level of pcorrect =33.3 % is indicated by
pdetect =0%.
4.4 Model Quality Measures
A common measure for quality of approximation is the
root MSE (RMSE). In the present case, true and estimated
discrimination probabilities for a given set of NSt stimuli
are contrasted as
RMSE =


1
NSt
NSt

i=1
[ptrue −pest]2.(18)
Here, a distinction is made between a training and a test
dataset. RMSEtrain is used to measure how well the model
approximates the input data, i.e., the probabilities of dis-
crimination acquired in the listening experiment. RMSEtest
is used to examine the suitability of the model for data that
were not included when the model was created.
Moreover, the chosen metrics should explain as much
as possible of the variance of the audibility values. The
goodness-of-fit of the produced model is captured by the
adjusted coefficient of determination (R2), which describes
the ratio of explained variance relative to the total variance
of the dependent variable. Additionally, the pvalues of each
explanatory variable indicate their respective significance
for the model.
Finally, Pearson correlation coefficients rtest allow for
better comparability to related studies that do not utilize
linear regression. Here, they assess a linear relationship
between true and estimated discrimination probabilities for
the test data stimuli.
4.5 Model Selection
The following subsections present different variants of
the two modeling approaches (MLR and PCA) and an anal-
ysis of their performance. On the basis of the discussion,
it is possible to narrow down the variants to a set of best-
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DOMA ET AL. PAPERS
performing parameters in terms of model quality and com-
plexity.
4.5.1 Parameter Variation
The output of the models depends on several factors:
- The number of distance metrics: All model types
are examined for nDist =1–7 input metrics. In the
MLR models, the metrics flow directly into the lin-
ear combination. In PCA-based models, only nDist
metrics flow into the calculation of PCs.
- The number of linear modeling coefficients: For the
MLR approach, this number equals nDist, because
the model directly creates a linear combination of
the (centered and scaled) distance metric values. In
contrast, for the PCA approach, the number of lin-
ear coefficients corresponds to the number of PCs,
which, on their part, are calculated based on nDist
different metrics. (Note that these two variables are
varied independently.) Because the first few PCs ac-
count for a majority of the explained variance, only
up to two linear modeling coefficients (i.e., two PCs)
are considered sufficient for examining the perfor-
mance of PCA-based models.
- The specific selection of metrics: Depending on the
number of metrics to be selected, a large variety of
combinations is possible. Instead of examining the
model for discrete subsets of metrics, model quality
is first assessed on the basis of a statistical selec-
tion of nDist metrics and 500 iterations per model
type. The following two cases are considered: first,
in the “random” case, nDist arbitrary variables are
selected from the seven available metrics; second,
in the “fixated” case, the three pseudo-orthogonal
metrics MSE, ISSD, and MFCD (as employed for
stimulus selection) are prioritized. This means that
for nDist ≥3, these three metrics are always included,
with an additional (nDist −3) random selections from
the remaining metrics. For nDist <3, only a subset
from MSE, ISSD, and MFCD is selected in the “fix-
ated” case.
Furthermore, the choice of training data influences the
resulting model. Furthermore, the (dis-)similarity between
test and training data may affect the apparent model quality.
For this reason, the selection of the training and test stimuli
was randomized in a first step, which allowed for a proper
assessment of the other factors. Iteratively, 54 stimuli (3/4)
were selected for model creation, and the remaining 18
stimuli were used for subsequent testing. This approach is
similar to that of k-fold cross-validation [51] with k=4,
yet with multiple runs and omitting the final averaging step
over all folds.
4.5.2 Evaluation of Performance
Different combinations of model parameters and their
effect on model quality measures are visualized in Fig. 7.
The spread depicted in the different box plots results from
Fig. 7. Performance evaluation for the different model types as
a function of nDist and for 500 iterations. A fully randomized
selection of metrics shows deteriorated performance, compared
to a prioritization (fixation) of the three metrics ISSD, MSE, and
MFCD. Saturation is reached around three to five input metrics to
the model.
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500 iterations, both in terms of randomized metric subset
and training/ test stimuli selection. Only for a “fixated”
metric choice and nDist =3, the data spread solely reflects
variability due to stimuli selection.
On a descriptive level, with increasing number of dis-
tance metrics flowing into the model, the quality measures
show improvement, i.e., lower RMSE values for training
and test data, higher adjusted R2and higher Pearson corre-
lation coefficients rtest. The quality measures further show
asymptotic behavior and, depending on the model type, al-
ready reach saturation for as low as nDist ≈3−5 distance
metrics. For MLR and PCA models, respectively, asymp-
totes are located at 12.71% and 13.97% for RMSEtrain,at
14.75% and 14.44% for RMSEtest, at 0.58 and 0.54 for the
adjusted R2, and at 0.72 and 0.73 for rtest.
Upon closer inspection, a marked influence of the model
parameters is noticeable. A prioritization of the ISSD, MSE,
and MFCD metrics in the “fixated” case enhances perfor-
mance, compared to a fully randomized choice of metric
subsets. Best overall performance is achieved by the MLR
model applying the three pre-selected metrics (orange box
plots), with RMSEtrain and RMSEtest as small as (μ±σ)=
(12.85 ±0.5)% and (13.83 ±1.6)%, respectively. It further
shows the highest adjusted R2of (0.6 ±0.04) and highest
Pearson correlation coefficients rtest of (0.76 ±0.08) (with
pvalues as low as (0.002 ±0.007) for rtest).
As can be derived from the figure, it is not beneficial to
include more than three metrics; The performance measures
even show a tendency for model deterioration with higher
model complexity. A similar stagnation and deterioration
is observed for the other two cases with “fixated” metric
selection [PCA models with one PC (petrol) or two PCs
(gray)].
Each additionally included distance metric necessitates
more computations. For the ensuing complexity to be jus-
tified, each metric contribution needs to be significant. In
the sense of purely linear models, each coefficient (or ex-
planatory variable) should contribute significantly to the
model (p<αwith a significance level of, e.g., α=0.01).
Fig. 8 displays the maximum pvalues over all explanatory
variables. For linear models (yellow and orange), an in-
crease in maximum pvalues with rising number of distance
metrics can be observed. With as low as nDist =3, the sig-
nificance level is by far exceeded. These cases are therefore
eliminated, including the (until now) best-performing MLR
model with three distance metrics as input.
In the context of PCA models, nDist only affects the com-
position of PCs. The depicted pvalues refer to significance
of the “virtual” metrics, which are linearly combined to
estimate audibility. A rise from one to two PCs leads to
exceeding the significance level, irrespective of metric se-
lection (see purple and gray boxplots). Thus, models based
on two PCs are also disqualified.
After these exclusions, and based on the model quality
measures and significance values of the remaining model
types, the PCA model with one PC and the three pre-
selected input metrics (ISSD, MSE, MFCD) can be identi-
fied as most promising, with μ±σ=(13.88 ±0.65)% for
RMSEtrain, (14.31 ±1.91)% for RMSEtest , (0.54 ±0.04)
Fig. 8. Maximum pvalues among the explanatory variables in
the linear models (top) and pvalues for test data correlation coef-
ficients rtest (bottom). Exceeding the level α=0.01 in the top plot
indicates an insignificant contribution of the metrics to the model
output. Thus unnecessarily complex models are eliminated, in-
cluding the best-performing three-metric MLR model according
to Fig. 7. In the bottom plot, correlation values are increasingly
significant with rising number of metrics.
for the adjusted R2, and (0.75 ±0.09) for rtest (with pvalues
(0.003 ±0.015) for rtest).
It should be noted that the adjusted R2here only serves
a relative comparison between the model types. Because
of lack of a reference for the expected value range (to our
knowledge), it is refrained from drawing conclusions re-
garding overall model effectiveness based on this value.
Similarly, the acquired RMSE values are difficult to com-
pare to other JND studies. In most cases, the 50% thresholds
are evaluated, defining a minimum discriminable resolu-
tion of a specific variable, e.g., loudness. Here, contrarily,
the unit of the model output is in percent. A more useful
measure for effectiveness is therefore provided by the cor-
relation values. A mean of 0.75 can be considered moderate
to strong and further lies in a similar range to that achieved
for some conditions in [18].
4.5.3 Specific Model Implementation
The cross-validation in the previous section already
proved the superiority of a model type with one PC, that
explicitly dictates the three metrics to use (ISSD, MSE,
and MFCD). Now, the model can be re-trained using the
whole data sample to obtain a final numerical model. For
the finalized model, the now scalar quality measures yield
14.02% for RMSEtrain and 0.55 for the adjusted R2.Both
values are very close to the mean data obtained in cross-
validation for this model type. Pearson correlation between
the experimental and predicted discrimination probabilities
yields 0.75 (p<0.001). (Note that this value is not directly
comparable to rtest, because all stimuli flow into training
the final model, with no stimuli left for testing.)
J. Audio Eng. Soc., Vol. 71, No. 4, 2023 April 167

DOMA ET AL. PAPERS
For a pair of stimuli, the probability of a correct answer
in the 3AFC task ( ˆ
pcorrect in percent) is reconstructed as
ˆ
pcorrect =c0+c1·w1,(19)
with linear coefficients c0=57.6023 (intercept) and
c1=7.7673. Weight (score) w1corresponds to PC1and is
acquired by running PCA on the complete training dataset
X (i.e., three metric values for each of the 72 stimuli).
For a given pair of stimuli, w1is estimated as the inner
product
w1=PC1,ˆ
d=
⎛
⎝
0.6671
0.6314
0.378
⎞
⎠,⎛
⎝
ˆ
issd
ˆ
mse
ˆ
mfcd
⎞
⎠(20)
with the centered and scaled metric vector ˆ
dcalculated as
ˆ
d=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎛
⎝
issd
mse
mfcd
⎞
⎠−⎛
⎝
7.2916
0.1615
0.9294
⎞
⎠
 
μX
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎛
⎝
0.189
5.2127
1.4271
⎞
⎠
 
1
σX
.(21)
Here, μXand σXrepresent the mean and standard deviation
of the training data for each of the metrics. The Hadamard
product operator indicates element-wise multiplication.
Finally, the predicted values ˆ
pcorrect are transformed using
Eqs. (15) and (17), yielding values independent from the
3AFC paradigm.
5 DISCUSSION
5.1 Integration and Reproduction of Variance
The numerical evaluation of the distance metrics and
their interrelation behavior demonstrated the presence of
mutual information. Although the correlation patterns fea-
tured common trends, especially a contrast between met-
ric behavior for inter-individual and intra-individual HRTF
comparisons could be observed. The pseudo-orthogonal
metric subsets, created on the basis of correlation and factor
analysis results, were used for the choice of stimuli for the
listening experiment, and were later shown to represent an
efficient choice of input metrics for the parametric model
of error audibility.
The evaluation of different modeling approaches re-
vealed that a model based on three metrics (ISSD, MSE,
and MFCD) and a single PC provides a good balance be-
tween model complexity and performance. Clearly, a more
complex approach could provide a more detailed descrip-
tion of the stimuli present. Eliminating all but three metrics
from the model implicitly neglects some aspects of spectral
deviation. Furthermore, a limitation to a single PC captures
only 59.6% of the explained variance of metric behavior.
As shown, however, the potentially added variance infor-
mation of the second PC (29.8%) does not help represent
the variance of perceptual data in the model output. Similar
observations hold true for an increased number of met-
rics, which only led to insignificant contributions in both
the PCA and MLR model types. The variance of percep-
tual data can thereby not be fully explained by the given
metrics. This could indicate the need for further metrics
that capture other aspects of spectral dissimilarity, possi-
bly including also phase information. (Note that, although
minimum-phase spectra were reconstructed for the HRTF
magnitudes before convolution with the noise pulse train,
phase differences were not directly evaluated by the em-
ployed metrics.)
5.2 Model Applicability
In the following sections, the generalized validity of the
model is discussed, reviewing the meaningfulness of a uni-
lateral model in practice, as well as the applicability on
arbitrary HRTF data.
5.2.1 Relevance of Unilateral Modeling
The concept and motivation for choosing an approach of
binaural presentation with unilateral signal variation were
outlined in SEC. 3.1. Still, the question of usability and
informative value of the model must be addressed.
In the conducted experiment, only unilateral signal vari-
ations were presented. In practice, however, a direct com-
parison of HRTFs very likely entails changes to both ear
signals. In order to apply the model in practice, it needs
to be extended to a bilateral variation model. For the de-
velopment of such a model, the presented unilateral JND
approach would be of value in two ways: On the one hand,
it could serve a selection of binaural stimuli covering com-
binations of different degrees of similarity. As indicated
in SEC. 3.2, the task of finding near-threshold stimuli al-
ready proved quite hard for the single-channel variation.
Attempting a similar approach for binaural variation, with
the pure distance metric values as a starting point, would be
very restrictive. The present model would facilitate stimu-
lus selection and thereby allow for the inclusion of more
conditions.
On the other hand, the current model output could serve
as direct input for the new, more complex model. The latter
would then predict distinguishability using appropriate bin-
aural weighting, in addition to integrating binaural effects,
e.g., the IACC transitions that have been controlled for in
the present work. It was decided against the inclusion of
a bilateral extension in the present study, as it would have
exceeded the intended scope of this paper.
5.2.2 Validity for Generic Datasets
As previously noted, only a subset of available HRTFs
were used for the listening experiment. This included the
elimination of “measured” HRTF set, for which an empiri-
cal selection of near-threshold stimuli proved to be difficult.
This raises the question whether the derived model is ap-
plicable to HRTF datasets other than those used for the
stimuli.
To examine this, the model was applied to the four HRTF
comparison cases, as introduced in SEC. 2. The different
stages of the model output are visualized in Fig. 9. The
direct output of the linear model is represented by pcorrect.
Introducing a cut-off to the model output yields plimited,in
which the unrealistic probabilities above 100% are elimi-
168 J. Audio Eng. Soc., Vol. 71, No. 4, 2023 April

PAPERS MAGNITUDE-BASED MODEL FOR AUDIBILITY OF HRTF VARIATION
0
50
100
150
Fig. 9. Modeled probability of discrimination for the four com-
parison cases. A distinction is made between the percentage of
correct answers in the 3AFC paradigm as a direct output of the
PCA model (pcorrect), with introduced cutoff for guessing and lapse
rate (plimited), and transformed to a general probability of detection
(pdetect).
nated, whereas the median values remain unchanged. The
gray boxplots (pdetect) show the actual predicted probabili-
ties of detection (for a non-experimental setting).
The inter-individual comparison Figs. 9(a) and 9(b) show
the highest probabilities, with a median and interquartile
range of median (IQR) =98.3 ([65.5, 100])% and 89.2
([58.8, 100])%, respectively. This observation is in accor-
dance with the difficulties found in selecting suitable stimuli
for the experiment, which lead to the total exclusion of the
“measured” dataset (a) and to an unequal azimuthal dis-
tribution of stimulus incidence directions in Fig. 9(b), cf.
Fig. 6. Approximately half of the stimulus pairs in the intra-
individual comparison dataset in Fig. 9(d) lies below the
threshold for audibility, with median (IQR) =52.9 ([37.4,
78.4]), The same goes for most of the dataset in Fig. 9(c),
with median (IQR) =25.4 ([19.9, 35.3]). A comparison
of pcorrect of Figs. 9(b)–9(d) with the experimental data in
Fig. 5 indicates that the rather empirical selection of stimuli
is representative of the datasets.
Informal listening by the authors showed that the model
is successful in finding HRTF pairs below the JND for
the “measured” dataset—a task that had been particularly
difficult when relying on separate minimization of the three
metrics. It can therefore be assumed that the application
of the model is not strictly limited to the reconstructed
HRTFs used for the experiment. Though the “measured”
data played a part in metric selection, it could be argued that
the analysis already covered a variety of HRTF variations.
Still, a validation based on other datasets (e.g., from other
HRTF databases) would be useful.
5.3 Limitations
The derived model is based on the available perceptual
data. Here, the selection of stimuli close to the JND for the
listening experiment had a major influence on modeling
possibilities, restricting the model to the slope of the psy-
chometric function. By including stimuli that are further
above or below the threshold, it would have been possible
to model the full sigmoid shape. An increased number of
stimuli would further enable a distinction between condi-
tions, with the cost of a more time-consuming listening ex-
periment. Separate models created for, e.g., ipsilateral and
contralateral incidence directions, would take into account
the influence of HRTF magnitude and hemisphere-specific
cues on distance metric behavior.
As in former JND studies for psychoacoustic properties,
the loudness level of stimuli might affect the sensitivity to
spectral changes. Especially in low-level spectral compo-
nents, changes (e.g., in notch quality of HRTFs) may go
undetected if below the auditory threshold. Although the
choice of 60–65 dB was deemed sufficient to assess the
concept of the model in a first step, a potential level depen-
dence should still be examined in future work.
Further perceptual validation is required to ensure the
applicability of the model to other types of spectral alter-
ations, possibly not represented in the used datasets. How-
ever, it should be noted that, e.g., HRTFs acquired in differ-
ent measurement or simulation setups likely possess large
differences [52]. They might feature too clearly audible
differences and thus have to be excluded from the JND as-
sessment. Supra-threshold stimulus pairs of this kind would
derive more benefit from a model predicting the type or the
markedness of a perceptual difference, rather than its mere
presence.
6 CONCLUSION
In this study, seven magnitude-based distance metrics
for HRTFs were used on measured HRTFs from the ITA
HRTF database and on approximations thereof, represent-
ing two levels of spectral detail loss. Metric behavior for
the different inter-individual and intra-individual compari-
son cases was analyzed using correlation and factor analy-
sis. Although the interrelation patterns of the metrics could
be partly attributed to the arithmetic operations involved in
their calculation, metric variance was not fully explained
by the common factors. For the purpose of the study, the
metrics ISSD, MSE, and MFCD were selected, represent-
ing three subgroups with a tendency for related metric re-
sponses within each.
A listening test paradigm was designed for the assess-
ment of perceptibility of unilateral differences of HRTFs.
The choice of test stimuli was optimized, selecting HRTF
pairs to which the three metrics did not respond equally.
Furthermore, a minimization of the influence of inter-aural
cross-correlation on the detection of dissimilarities between
stimuli was considered. The experiment provided detection
probabilities around the perception threshold, allowing for
modeling the slope of a psychometric function to describe
these probabilities.
A multi-linear regression approach and a linear model
based on the score of principal components were examined
regarding their performance. Different complexities of the
model in terms of the number of linear coefficients and the
number of integrated metrics were contrasted. A trade-off
was made between model accuracy and complexity, choos-
ing to model the variance of the three pre-selected models
metrics ISSD, MSE, and MFCD using one principal com-
J. Audio Eng. Soc., Vol. 71, No. 4, 2023 April 169

DOMA ET AL. PAPERS
ponent. The weighting score of this PC served as a “virtual”
metric, approximating detection probabilities in a univariate
linear model. It was shown that more complex approaches,
e.g., integrating more than three distance metrics, did not
improve the goodness-of-fit of the model and partly even
caused a deterioration of performance.
Observing the residual variance of the perceptual data,
which could not be modeled by increasing model com-
plexity, the question arises whether other types of metrics
should be considered. This could include, e.g., phase-based
calculations or variants of the employed metrics, evaluat-
ing spectral deviations within specific frequency ranges.
Nonetheless, the model will remain subject to the reser-
vation of inter-individual variation, as responses can (and
will) certainly deviate from the modeled mean threshold.
For better applicability, future work should extend the
results to a binaural model with simultaneous variation of
both ear signals. This bilateral approach could either make
use of the direct output of the present single-channel model
or follow similar methodology in deriving a model from a
subset of metrics. For the acquisition of perceptual data on
detection of bilateral variation, the unilateral model could
further help in the selection of near-threshold stimulus pairs.
It should be noted that the employed pulsed noise sig-
nals, combined with free-field conditions, emphasize the
audibility of spectral differences, leading to comparatively
strict JND values. For more natural situations, involving
in-door playback of more common sounds, the sensitivity
to changes is expected to be substantially lower. Further
validation with binaural room impulse responses and, e.g.,
speech or music signals could provide insight on the accu-
racy level of HRTF representation required in practice.
7 ACKNOWLEDGMENT
This research was funded by the German Research Foun-
dation (DFG, project no. 402811912, Individual Binaural
Synthesis of Virtual Acoustic Scenes). The authors would
like to thank Nat´
alie Broˇ
zov´
a for implementing and con-
ducting the listening experiment. Many thanks to Hark
Braren, Lukas Vollmer, Dr. Manuj Yadav, and Dr. Ming
Yang for the fruitful discussions on modeling and evalua-
tion, as well as to the experiment participants for their time.
The authors would furthermore like to express their grati-
tude to the four anonymous reviewers, whose remarks have
significantly improved the quality of this paper.
Declaration of Conflicting Interests
Part of this work has been previously presented at the 48th
Annual Conference on Acoustics, DAGA 2022, Stuttgart,
Germany. The authors take complete responsibility for the
integrity of the data and the accuracy of the data analysis.
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THE AUTHORS
Shaimaa Doma Cosima A. Ermert Janina Fels
Shaimaa Doma is a research assistant at the Institute
for Hearing Technology and Acoustics (IHTA), RWTH
Aachen University, where she started her Ph.D. in 2019.
She received her M.Sc. in Electrical Engineering and In-
formation Technology from RWTH Aachen University in
2018, with a specialization in Biomedical Engineering. In
2017, she was an intern at the Advanced Bionics European
Research Center, Hanover, Germany.
•
Cosima A. Ermert received her M.Sc. in Electrical Engi-
neering and Information Technology from RWTH Aachen
University in 2020. After an internship in the clinical re-
search group at Oticon A/S in Denmark, she started work-
ing as a research assistant at the Institute for Hearing Tech-
nology and Acoustics (IHTA) at the RWTH Aachen Uni-
versity. In her research, she focuses on distance metrics for
HRTF comparison and cognitive research in virtual envi-
ronments.
•
Janina Fels is a full professor and director of the Chair
and Institute for Hearing Technology and Acoustics at
RWTH Aachen University, Germany, since 2020. In 2012–
2020, she was Professor for Medical Acoustics at RWTH
Aachen University, Germany. She studied electrical en-
gineering (diploma 2002) at RWTH Aachen University,
Germany, where she also received her Ph.D. from the In-
stitute of Technical Acoustics in 2008. In 2009, she was
a post-doc at the Center for Applied Hearing Research
(CAHR) at the Technical University of Denmark (DTU)
and Widex, Denmark. From 2012–2015, she was a visit-
ing scientist at the Institute of Neuroscience and Medicine,
Structural and Functional Organization of the Brain (INM-
1) at Forschungszentrum J¨
ulich, Germany. In 2013, she was
awarded the Lothar Cremer Prize by the German Acoustics
Society for her innovative and pioneering work in the field
of binaural technology and medical acoustics. In 2014, she
was appointed to the Young College of the North Rhine-
Westphalian Academy of Sciences and Arts. In 2020, she
was elected as a Review Board Member for Acoustics for
the German Research Foundation (DFG).
172 J. Audio Eng. Soc., Vol. 71, No. 4, 2023 April