Spatial upsampling of sparse head-related transfer function setsby directional equalization - Influence of the spherical sampling scheme
Johannes M. AREND(1)(2)† , Christoph PÖRSCHMANN(1)(1) Institute of Communications Engineering, TH Köln, D-50679 Cologne, Germany,(2) Audio Communication Group, TU Berlin, D-10587 Berlin, Germany† Corresponding author, E-mail: johannes.arend@th-koeln.de

Abstract
Many immersive audio applications rely on a dense set of head-related transfer functions (HRTFs). However,often only measurements on a specific sparse grid are available. To obtain dense HRTF sets from sparse measurements, one common approach is to apply spatial interpolation in the spherical harmonics (SH) domain.
However, the SH representation of sparse HRTF sets is order-limited, leading to spatial aliasing and truncation
errors. In a recent publication, we presented the so-called SUpDEq method (Spatial Upsampling by Directional
Equalization) for spatial upsampling of sparse HRTF sets. The approach is based on a directional equalizationof the sparse set prior to the spherical Fourier transform to remove direction-dependent temporal and spectral
components. This significantly reduces the spatial complexity of the sparse set, allowing for an enhanced interpolation at reduced SH orders. In this study we investigate how different spherical sampling schemes affect theperformance of common SH interpolation and the SUpDEq method. For this, we compare spatially upsampled
HRTF sets originally based on sparse equiangular, Gaussian, Lebedev, and Fliege grids at various SH orders to areference. The influence of the different grids are assessed spectrally, temporally, and with localization models.
Keywords: Head-Related Transfer Functions (HRTFs), Spatial Audio, Spherical Harmonics

1

INTRODUCTION

Human sound source localization is based on binaural cues, i.e. interaural time differences (ITDs) and interaurallevel differenes (ILDs) between both ears, as well as on monaural cues, i.e. spectral distortions of the incoming
sound caused mainly by the listener’s pinna, head, and torso. Head-related transfer functions (HRTFs) containthese binaural and monaural cues and thus describe the sound incidence from a source to both ears [5].
For headphone-based virtual acoustic environments (VAEs), a set of HRTFs is essential. Ideally, such a set
should include individual HRTFs for a large number of directions, typically measured on a sphere around alistener. However, measuring so-called dense sets of individual HRTFs requires special equipment, experiencein handling the equipment and, depending on the measurement approach, can also be time-consuming (see e.g.
[6]). For this reason, it seems appealing to measure only a small number of HRTFs on a sparse sphericalsampling grid with a simplified measurement system, and to apply a specific interpolation or spatial upsampling
method afterwards to generate a dense HRTF set with perhaps thousands of directions.
One popular approach for spatial upsampling is interpolation in the spherical harmonics (SH) domain. For this,an HRTF set captured on a spherical sampling scheme (also simply called spatial grid) is first transformed to the
SH domain applying the spherical Fourier transform (SFT). The resulting spatially continuous representation of
the HRTF set in the SH domain allows for interpolation, i.e. an HRTF for any desired direction can be obtainedby means of the respective inverse spherical Fourier transform (ISFT) [13]. However, the SH representationand interpolation of sparse HRTF sets suffers from so-called sparsity errors, which is a combination of spatial
aliasing and truncation errors [3]. For this reason, various pre- and post-processing methods have been proposed
to reduce the sparsity error and thus to improve SH interpolation of sparse HRTF sets (see e.g. [7]).
Within this scope, we presented the SUpDEq method (SUpDEq - Spatial Upsampling by Directional Equalization) as a pre- and post-processing approach allowing improved SH interpolation of sparse HRTF sets [11].

2643In the respective paper, we examined the performance of the SUpDEq method regarding spectral and temporalfeatures as well as concerning modeled localization performance of reconstructed HRTFs and showed that theapproach clearly outperforms common SH interpolation in terms of these features. However, as the analysis was
based only on the Lebedev sampling scheme [9], the present paper now provides further evaluation investigating
the influence of various (sparse) spherical sampling schemes on the performance of the SUpDEq method. As
the aliasing error strongly depends on the sampling scheme [3], this evaluation is of particular interest to ensure
the general applicability of the proposed upsampling method.
In this paper, we therefore compare spatially upsampled HRTF sets originally based on sparse equiangular,
Gaussian, Lebedev, and Fliege grids [13, Ch. 3][9][8] at various spatial orders N to a dense reference HRTF
set. Similar to the evaluation in [11], we assess the impact of the grids on the spatially upsampled HRTF
sets spectrally, temporally, and by means of localization models. To anticipate some of the results, the analysis
showed that the sampling scheme has very little, if any, influence on the performance of the SUpDEq method.

2

SPHERICAL SAMPLING SCHEMESA set of HRTFs is commonly measured at discrete points on a surrounding sphere according to a sphericalsampling scheme. Such a full-spherical HRTF set can be transformed to the SH domain with the discrete SFT.
The sampling schemes investigated in this paper provide closed-form expressions to calculate SH coefficients,
whereas SH coefficients for arbitrary sampling configurations can be computed by an inversion of the respective
SH matrix [13, Ch. 3]. The latter is however not further discussed here. Given a spherical sampling scheme Lwith a closed-form expression, the spherical HRTF set H(ω, Ωq ) for the left and right ear (indices for left andright are omitted here and in the following for ease of display) can be described in the SH domain by the SH
coefficients hnm (ω) that are computed with the discrete SFT [13, p. 58]
QLbhnm (ω) = ∑ βq H(ω, Ωq )[Ynm (Ωq )]∗ ,

(1)

q=1

with the temporal frequency ω, the QL directions Ωq = {(φ1 , θ1 ), . . . , (φQ , θQ )} at azimuth φ and elevation θ , andthe sampling weights βq depending on the sampling scheme L. The notation (·)∗ denotes complex conjugation
and Ynm are the complex SH functions of order n and degree m. The ISFT can be applied to recover H(ω) atarbitrary angles allowing for SH interpolation [13, p. 17]
N

b
H(ω,
Ω) = ∑n∑ hnm (ω)Ynm (Ω) ,(2)

n=0 m=−nwhere N is the spatial order (also referred to as SH order). As discrete sampling of a function with infinite orderinduces spatial aliasing and truncation errors, the SH coefficients are only error-free up to a specific schemedependent order NL . If the function sampled on the sphere is strictly order-limited, a sampling scheme providingba sufficient order NL results in hnm (ω) = b
hnm (ω). Similar, H(ω) = H(ω)
holds if N is chosen appropriately.
The maximum resolvable order of the sampling scheme NL is generally defined by the number of directions (orsampling points) QL and by the way the sampling points are distributed around the surface of the sphere. This
relationship can be expressed by QL ≥ η(N + 1)2 , with η describing the efficiency of the sampling scheme [12].
The SH order of an HRTF set however increases as the frequency increases, following the relation N ∼ kr, withk the wavenumber and r the radius of a sphere surrounding the head [11][3]. Assuming an average human headradius of r = 8.75 cm, a minimum SH order N = 32 is required to perform a nearly perfect SFT, ISFT, and thus
SH interpolation of HRTFs for frequencies up to 20 kHz.
In research, various schemes have been developed in order to sample the sphere with the highest possible
accuracy and efficiency. A good overview on different sampling approaches in the context of spatial audio canbe found for example in [13, Ch. 3]. For this study, we focused on four different frequently applied schemes,
namely the equiangular, Gaussian, Lebedev, and Fliege grids. The equiangular grids have a uniform distribution2644

of samples along φ and θ , with both angles sampled at 2(N + 1) locations, requiring 4(N + 1)2 samples in total[12]. The Gaussian grids require only 2(N +1)2 samples, as the elevation θ is only sampled at (N +1) locations,
resulting in a nearly-uniform distribution of samples along both angles [12]. However, the equiangular and theGaussian sampling schemes do not provide uniform distributions of sample points on the surface of the sphere.
The Lebedev and Fliege schemes however offer nearly-uniform distribution of samples around the surface ofthe sphere, with the advantage that even less sample points are required to reach a specific SH order. Thus,
the Lebedev grids require approximately 1.3(N + 1)2 samples whereas the Fliege grids only require (N + 1)2
sample points [12][13, Ch. 3]. Figure 1 shows the four introduced grids on the sphere, exemplarily of SH orderN = 7, resulting in 256 points for the equiangular grid (a), 128 points for the Gaussian grid (b), 86 points for
the Lebedev grid (c), and 64 points for the Fliege grid (d).

(a)

(b)

(c)

(d)

Figure 1. Equiangular (a), Gaussian (b), Lebedev (c), and Fliege (d) sampling schemes of SH order N = 7.

3

SPATIAL UPSAMPLING BY DIRECTIONAL EQUALIZATION (SUpDEq)

The following section gives a brief overview of the basic concept behind the SUpDEq method, as illustrated in
the block diagram in Fig. 2. Further details on the implementation and evaluation can be found in [11]. Basically, the approach aims at enhanced SH interpolation and spatial upsampling of sparse HRTF sets. To achievethis, a sparse HRTF set H(ω, ΩS ) measured at S sampling points Ωs = {(φ1 , θ1 ), . . . , (φS , θS )} is equalized directionally by spectral division with an appropriate equalization dataset DEQ (ω, Ωs ) before the SFT:HEQ (ω, Ωs ) = H(ω, Ωs )/DEQ (ω, Ωs ) .

(3)

As a rather good and established approximation of a human head, direction-dependent rigid sphere transfer
functions for an incident plane wave [13, p. 44] are used as the equalization dataset. The spherical headmodel should match the respective human head as best as possible. As a first and easy to implement approach,the radius of the sphere is calculated according to the physical dimensions of the head [1] and the ears are
positioned at φ = ±90◦ and θ = 0◦ on the sphere. Using a spherical head model also has the advantage thatit can be described analytically, which allows the calculation of the rigid sphere transfer functions at high SHorders Nhigh ≥ 32. The directional equalization described in Eq. (3) significantly reduces the spatial complexityof the sparse HRTF set, therefore minimizing the required SH order for the SFT. The reason for the decreaseof the SH order is that on the one hand, the equalization leads to a time-alignment of the HRTFs, similar to are-centering, and on the other hand, direction-dependent influences of the sphere or the head are compensated.
After equalization, the equalized HRTF set HEQ (ω, Ωs ) is transformed to the SH domain with the SFT (Eq. (1))
at a low SH order Nlow according to the maximum resolvable SH order of the spare sampling scheme. Then, anbHRTF,EQ is calculated on a dense sampling grid Ωd = {(φ1 , θ1 ), . . . , (φD , θD )},upsampled (equalized) HRTF set H
with D  S using the ISFT (Eq. (2)). Finally, HRTFs are reconstructed with a subsequent de-equalization by
spectral multiplication with an appropriate de-equalization dataset DDEQ :bDEQ (ω, Ωd ) = H
bEQ (ω, Ωd ) · DDEQ (ω, Ωd ) .
H(4)Again, rigid sphere transfer functions for an incident plane wave can are used as the de-equalization dataset.
In general, the de-equalization recovers energies at higher SH orders that were transformed to lower orders by2645

Equalization

Equalization
DatasetL
SH
Coefficients

Nhigh
SparseHRTF Set

SparseSampling
Grid

De-Equalization
L

R

ISFT
Nhigh

Dense
Sampling
Grid

Equalization

De-Equalization
DatasetR

ISFT

SH
Coefficients

Nhigh

Nhigh

De-Equalization

L

L

R

R

SFT

Dense
HRTF Set

ISFT
Nlow

Nlow

Figure 2. Block diagram of the SUpDEq method. Left panel: A sparse HRTF set is equalized on the corresponding sparse sampling grid and then transformed to the SH domain with N = Nlow . Right panel: The
equalized set is de-equalized on a dense sampling grid, resulting in a dense HRTF set.
bDEQ holds if, in this case, Nlow is sufficient for the SFT
the equalization. Similar as described in Sec. 2, H = H
of HEQ and Nhigh is chosen appropriately. Otherwise, spatial aliasing and truncation errors occur, resulting in
bDEQ . The following section now analyzes the performance of the SUpDEq method with respect to theH≈Hsparse sampling scheme underlying the input HRTF set in comparison to common SH interpolation.

4

INFLUENCE OF THE SAMPLING SCHEME

In our previous publication [11], we investigated the performance of the SUpDEq method for two different
dummy heads, but only for Lebedev grids of different SH orders. To further ensure the general applicability of
the SUpDEq method, the present paper now focuses on the influence of the spherical sampling scheme underlying the sparse input HRTF set. As a reference set, we used HRTFs of a Neumann KU100 dummy head thatwere measured on a Lebedev grid with 2702 sampling points [4]. This reference HRTF set was transformed to
SH domain at N = 35, further referred to as hREF,nm . The various sparse HRTF sets required as input data weregenerated by spatial subsampling of the reference set hREF,nm to the respective sparse equiangular, Gaussian,
Lebedev, or Fliege grids of (limited) SH orders N = 1 − 15 applying the ISFT. Next, these sparse HRTF sets
were spatially upsampled to a dense sampling grid (again the Lebedev grid with 2702 sampling points, further
abbreviated Lebedev2702 ), applying the SUpDEq method as well as (order-limited) SH interpolation without any
pre- or post-processing before or after the SFT/ISFT. The upsampled dense HRTF sets were then again transformed to SH domain at N = 35, resulting in SH coefficients further referred to as hDEQ,nm and hOL,nm , with
DEQ standing for de-equalized and OL for (strictly) order-limited. The de-equalized and order-limited HRTFs,
as hereinafter referred to, were then obtained via the ISFT at the direction required for the respective analysis
method. The optimal radius for the rigid sphere model used for (de-)equalization was calculated based on thedimensions of the Neumann KU100 dummy head [1], leading to r = 9.19 cm.
4.1 Spectral differences
As a first error measure, we analyzed the spectral differences between hREF,nm and hDEQ,nm or hOL,nm as afunction of the SH order N on various test sampling grids with T sampling points Ωt = {(φ1 , θ1 ), . . . , (φT , θT )}.

2646

The frequency-dependent spectral differences per sampling point were calculated in dB as∆g(ω, Ωt ) = 20lg

| HREF (ω, Ωt ) |,| HTEST (ω, Ωt ) |(5)where HREF is the left ear HRTF extracted from hREF,nm and HTEST is the left ear HRTF extracted from hOL,nmor hDEQ,nm at the sampling point Ωt . Then, the absolute value of ∆g(ω, Ωt ) was averaged across all sampling
points Ωt to obtain the frequency-dependent measure ∆G f (ω) (in dB)
n1 Ωt| ∆g(ω, Ωt ) |,∆G f (ω) =nΩt Ω∑t =1(6)

and across ω and Ωt , resulting in a single value ∆G (in dB) describing the spectral differencen∆G =1 1 Ωt nω
∑ | ∆g(ω, Ωt ) | .
nΩt nω Ω∑
t =1 ω=1

(7)

8

8

Magnitude [dB]

Magnitude [dB]

Figure 3 (a) shows the spectral differences ∆G across N over the full audio bandwidth for the four different
sampling schemes applying the SUpDEq method or order-limited interpolation. The test sampling grid Ωt was
the reference Lebedev2702 grid. Independent of the sampling scheme, SUpDEq processing results in about 2 dBless spectral differences than order-limited interpolation. The Fliege scheme however has distinct outliers at
N = 10 and N = 12 for both upsampling methods. Interestingly, exactly at these orders, some of the calculated
weights are negative, which is something Fliege and Maier could not explain [8]. Applying the SFT according
to Eq. (1), the negative weights most probably lead to a phase shift in the complex SH coefficients, certainly
resulting in reconstruction errors when transformed back with the ISFT. Apart from these outliers, the spectral
differences for the four different sampling schemes are pretty similar, indicating that the performance of the
SUpDEq method is independent of the sampling scheme. For order-limited interpolation, the equiangular scheme
leads to slightly higher spectral differences than the other schemes.
Figure 3 (b) illustrates the frequency-dependent spectral differences, exemplarily at N = 7, for the four different sampling schemes applying the SUpDEq method or order-limited interpolation. As before, Ωt was theLebedev2702 grid. It can be seen that the spectral differences are significantly smaller for the SUpDEq methodthan for order-limited interpolation, and furthermore that order-limited interpolation leads to a sharp increasein spectral differences above the spatial aliasing frequency. Regarding the SUpDEq method, the Fliege scheme
performs a little worse than the three other schemes, but overall there is only a marginal influence of the sampling scheme on the performance of the method. Furthermore, the equiangular scheme induces slightly higher
spectral differences than the other schemes when applying order-limited interpolation.

4

4

2

2
0
1

Equiangular
Gaussian
Lebedev
Fliege66510Order N
(a)

15

0

1k

Frequency [Hz]

10k

(b)

Figure 3. Spectral differences in dB (left ear) between reference HRTF set and order-limited (red) or deequalized (blue) HRTF sets for four different sampling schemes (color saturation). The test grid Ωt was always
the Lebedev2702 grid. (a) Spectral differences ∆G across N over the full audio bandwidth. (b) Frequencydependent spectral differences ∆G f (ω) at N = 7.

2647

4.2 Binaural cues
Next, we compared the ILDs and ITDs of the reference HRTF set to those of order-limited or de-equalized
sets, again with respect to different sampling schemes. For this, HRTFs in the horizontal plane (θ = 0◦ ) withan angular spacing of φ = 1◦ were extracted from the reference set hREF,nm and, depending on N, from therespective order-limited or de-equalized set hOL,nm and hDEQ,nm . The broadband ILDs were then calculated as
the ratio between the energy of the left and right ear HRIR (HRIR, the time-domain equivalent of an HRTF).
The ITDs were calculated by means of a threshold-based onset detection on the ten times up-sampled andlow-pass filtered HRIRs (10th order Butterworth low-pass at 3 kHz).
Figure 4 illustrates the calculated ILDs and ITDs of the reference HRTF set as well as of the order-limited
and de-equalized sets, again exemplarily at N = 7. As can be seen in Fig. 4 (a), the ILDs of the de-equalized
HRTFs are in good agreement with the reference and mostly unaffected by the sampling scheme. Overall, theFliege grid shows the most notable deviations, especially at lateral directions. At these directions, also theGaussian scheme provides clear deviations from the reference, whereas the equiangular and Lebedev schemes
show only slight differences over the entire angular range. In contrast, the ILDs of the order-limited HRTFs(see Fig. 4 (b)) differ significantly from the reference. However, there is also only a rather weak influence ofthe sampling scheme. Regarding the ITDs, Fig. 4 (c) and (d) illustrate that there is virtually no influence of thesampling scheme, regardless of the upsampling method. Thus, at N = 7, the ITDs of the de-equalized HRTFs(see Fig. 4 (c)) as well as of the order-limited HRTFs (see Fig. 4 (d)) are in good agreement with the reference.
330°
300°
270°

0°
330°
30°
16 dB
12
300°
60°
8
4
90° 270°
0

240°

120°
210°

180°

0°
330°
30°
16 dB
12
300°
60°
8
4
90° 270°
0

240°

120°
210°

150°

(a)

180°

150°

240°

120°
210°

Equiangular

180°

150°

Gaussian

0°30°
0.8 ms
0.6
60°
0.4
0.2
90°
0

240°

120°
210°

180°

150°

(d)

(c)

(b)
Reference

0°
30°
330°
0.8 ms
0.6
60°
300°
0.4
0.2
90° 270°
0

Lebedev

FliegeFigure 4. ILDs (a), (c) and ITDs (b), (d) in the horizontal plane of the reference (black) HRTFs as well as ofthe order-limited (red) or de-equalized (blue) HRTFs for four sampling schemes (color saturation) at N = 7.

4.3 Localization performance
To conclude the analysis, we compared the localization performance of order-limited and de-equalized HRTFs
with respect to the sampling scheme applying two different auditory models from the Auditory Modeling Toolbox [14]. To assess the localization performance in the median sagittal plane, we used the model from Baumgartner et al. [2], which provides estimates for the polar RMS error (PE) as well as for the quadrant error rate
(QE) based on monaural spectral cues. To evaluate the performance in the horizontal plane, we applied themodel from May et al. [10], which estimates the azimuthal position of a sound source based on binaural cues.
By comparing the intended and the estimated source position, a lateral error (LE) can be calculated. To calculate the error measures, first the performance of hREF,nm , hOL,nm , and hDEQ,nm was determined for each sampling
scheme as a function of N. To estimate median plane localization performance, we used a test sampling grid Ωtwith φ = {0◦ , 180◦ } and −30◦ ≤ θ ≤ 90◦ in steps of 1◦ , and assumed a median listener sensitivity of S = 0.76.
To estimate the horizontal plane localization performance, we applied a test sampling grid with φ = ±90◦ insteps of 5◦ . As final error measures, the absolute polar error difference (in degree)2648

∆PE =| PEREF − PETEST |,(8)

the absolute quadrant error difference (in percent)∆QE =| QEREF − QETEST |,(9)

as well as the absolute lateral error difference (in degree)
∆LE =

1 T
∑ | LEREF (Ωt ) − LETEST (Ωt ) |,
T t=1

(10)

were calculated for each sampling scheme and order N, with the subscripts REF and TEST as defined above.
In the horizontal plane (see Fig. 5 (a)), the order-limited interpolation leads to an error increase at low orders
N ≤ 4. Similar to previous results, the Fliege grid performs worst here, especially at these low orders. In
contrast, the SUpDEq method leads to hardly any increase in lateral error over the entire tested range of N, nomatter which sampling scheme was applied. This shows that even at low orders, upsampling with the SUpDEq
method always results in sufficient binaural cues.
In the median sagittal plane (see Fig. 5 (b) and (c)), order-limited interpolation leads to considerably higher
errors over the entire range of N. Obviously, the high-frequency deviations in order-limited HRTFs badly affects
the monaural spectral cues. Overall, the Fliege and the Lebedev grids seem to perform worse, but it is difficult
to see a clear trend besides a general decrease in error with increasing order N. The SUpDEq method however
amplifies the polar error only slightly at low orders N ≤ 2. Once again, the Fliege grid tends to lead to a higher
increase in error than the other grids, both for the polar error as well as for the quadrant error rate. The othergrids perform more or less the same, with only slight increases in polar error and quadrant error rate at N ≥ 2,
indicating that spectral cues are only marginal impaired.

5
0
1

5

10
Order N(a)15

Abs. QE Diff. [%]

10

15

10

Equiangular
Gaussian
Lebedev
FliegeAbs. PE Diff. [°]Abs. LE Diff. [°]

15

10

5

0
1

5

10
Order N
(b)

15

5
0
1

5

10
Order N

15(c)

Figure 5. Absolute lateral error difference ∆LE (a), polar error difference ∆PE (b), and quadrant error difference∆QE (c) across N for four different sampling schemes (color saturation) applying order-limited interpolation(red) or the SUpDEq method (blue).

5

CONCLUSION

This paper presented further evaluation of the SUpDEq method for spatial upsampling of sparse (individual)
HRTF sets by investigating the influence of the spherical sampling scheme on the performance of the method.
The study compared spatially upsampled HRTF sets originally based on sparse equiangular, Gaussian, Lebedev,
and Fliege grids at various spatial orders N to a dense reference HRTF set, applying the SUpDEq method aswell as common SH interpolation for upsampling. The analysis of spectral features, binaural cues, and localization performance revealed that the influence of the sampling scheme on the results of SUpDEq processing2649

is only marginal. Overall, only the Fliege scheme tended to perform a little worse than the other three tested
schemes. With order-limited interpolation, the sampling scheme affected the examined features slightly stronger.
The results of this study confirm or at least increase the general applicability of the SUpDEq method regarding
the sampling scheme of the input HRTF set. Thus, the SUpDEq method might be applied with any sparse HRTF
set measured on a proper full-spherical sampling grid. However, we only examined sampling schemes providing a closed-form expression in this paper. Therefore, further tests with arbitrary sampling schemes could be
performed, even though it seems that given a reasonable sparse sampling scheme providing a well-conditioned(inverse) SH matrix, the results will be quite similar. Furthermore, listening experiments could be performed toanalyze the perceptual influence of the sampling scheme, although the analysis in this paper suggests that theperceptual influence might be marginal. A Matlab-based implementation of the SUpDEq method is available on
https://github.com/AudioGroupCologne/SUpDEq. The research presented in this paper was funded by theGerman Federal Ministry of Education and Research (BMBF 03FH014IX5-NarDasS).

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