Audio Engineering Society

Convention Paper 9892Presented at the 143rd Convention

2017 October 18–21, New York, NY, USAThis convention paper was selected based on a submitted abstract and 750-word precis that have been peer reviewed by at least two qualified anonymous reviewers. The complete manuscript was not peer reviewed. This convention paper has been reproduced from the author’s advance manuscript without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents. This paper is available in the AES E-Library (http://www.aes.org/e-lib), all rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.

A method for efficiently calculating head-related transferfunctions directly from head scan point cloudsRahulram Sridhar and Edgar Y. Choueiri

3D Audio and Applied Acoustics Laboratory, Princeton University, Princeton, NJ, 08544, USA

Correspondence should be addressed to Rahulram Sridhar (rahulram@princeton.edu)

ABSTRACTA method is developed for efficiently calculating head-related transfer functions (HRTFs) directly from head scanpoint clouds of a subject using a database of HRTFs, and corresponding head scans, of many subjects. Consumerapplications require HRTFs be estimated accurately and efficiently, but existing methods do not simultaneouslymeet these requirements. The presented method uses efficient matrix multiplications to compute HRTFs fromspherical harmonic representations of head scan point clouds that may be obtained from consumer-grade cameras.The method was applied to a database of only 23 subjects, and while calculated interaural time difference errors arefound to be above estimated perceptual thresholds for some spatial directions, HRTF spectral distortions up to 6kHz fall below perceptual thresholds for most directions.

1 INTRODUCTION

One of the challenges associated with spatial soundreproduction using synthesized binaural signals is ac-quisition of a listener’s head-related transfer functions(HRTFs) accurately, efficiently, and conveniently. Sev-eral methods exist for estimating individualized HRTFs,since direct measurement in an anechoic chamber isinfeasible for commercial implementation [1, Ch. 3].For a comprehensive review, see Blauert [1], Xie [2],and references therein.

Amongst the methods, those that use 3D morphologicalscans of a listener to estimate HRTFs can be accurate,but suffer from computational inefficiency and imple-mentation complexity. The most common approach

to estimate HRTFs from morphological scans is by us-ing the boundary element method to numerically solvethe wave equation on a meshed surface of the scan [2,Ch. 4]. This method is limited by its high computa-tional cost and need for an accurate 3D mesh, whichrequires careful capture and extensive post-processingof raw scan data [3, 4]. Time-domain techniques suchas the finite-difference time-domain method [5] andadaptive rectangular decomposition [6] have also beenused. However, these techniques also either suffer fromcomputational inefficiency [4, Table I], or estimationinaccuracy (see, for example, Fig. 2 in Meshram et al.[6] where important spectral features are sometimesnot captured, even at low frequencies).

Tao et al. [7] use 11th-degree spherical harmonics torepresent the shape of the KEMAR dummy head and

Sridhar and Choueiri Calculating HRTFs from head scans

a boundary element method to compute HRTFs forfrequencies below 3 kHz. This method is prone toinefficiency at higher frequencies, and still requires ameshed scan. More recently, Politis et al. [8] assignthe same set of measured interaural time differences(ITDs) to listeners with similar head shapes that areidentified by one of three types of spherical transfor-mations of meshed scans. However, while this methodcan be extended to obtain HRTF estimates, it remainsinaccurate for listeners who do not have head shapesthat are similar to those of other listeners.

To address these issues, we present a method that al-lows computation of HRTFs directly from sphericalharmonic representations of head scan point clouds thatmay be conveniently obtained from consumer-gradecameras. Our method does not require sophisticatedmesh generation techniques and is computationally ef-ficient, requiring only matrix multiplications.

In the rest of the paper, we formulate our method inSec. 2, apply and validate it in Sec. 3, and conclude inSec. 4.

2 METHOD FORMULATION

2.1 Definitions and conventions

To represent ITDs, HRTF magnitudes, and head scanpoint clouds, we use real-valued spherical harmonicsof degree n � 0 and order m 2 [�n,n] given by

Y mn (q ,f) = N|m|

n P|m|n (sinf)⇥

(cosmq for m � 0,

sin |m|q for m < 0,

where P|m|n is the associated Legendre polynomial of

degree n and order |m|, and Nmn is a normalization term

given by

Nmn = (�1)m

s(2n+1)(2�dm)

4p

(n�m)!(n+m)!

,

where dm is the Kronecker delta. Furthermore, q 2[0�,360�) denotes azimuth, and f 2 [�90�,90�] el-evation, following the coordinate systems describedin AES69-2015 [9], and illustrated in Fig. 1. Theradial distance, r, from the origin is given by r =p

x2 + y2 + z2.

y

x

z

q

f

Fig. 1: Coordinate systems used here, and specifiedby AES69-2015 [9].

.

2.2 Spherical harmonic representation offunctions sampled on a sphere

Given the column vector,

s = [S(q1,f1),S(q2,f2), · · · ,S(qV ,fV )]T ,

of V spatial-samples of a band-limited function, S, de-fined on the surface of a sphere, and the vector ofspherical harmonics,

ymn = [Y m

n (q1,f1),Y mn (q2,f2), · · · ,Y m

n (qV ,fV )]T , (1)

we can determine the projection, sP, of s onto the col-umn space of Y := [y0

0,y�11 , · · · ,yn

n] as

sP = Y�YT Y

��1 YT s = Yc, (2)

where c is a vector of so-called spherical harmoniccoefficients, and YT denotes the transpose of Y. sP isthe spherical harmonic representation of s.

2.2.1 HRTF magnitude and ITD

Romigh et al. [10] show that, compared to complex-valued HRTFs or their absolute magnitudes, HRTFmagnitudes in dB are most efficiently represented us-ing spherical harmonics. Therefore, in this work, weuse Eq. (2) with s consisting of HRTF magnitudes indB for each frequency, and n = nH , where nH is the

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 2 of 9

Sridhar and Choueiri Calculating HRTFs from head scans

largest spherical harmonic degree used for representingthese magnitudes, to compute c = cHl/r [k], where thesubscript is used to identify either the left- or right-earHRTF magnitude. k 2

⇥0,Nf �1

⇤is the frequency-

index with Nf being the number of frequency samplesin each HRTF magnitude spectrum, and is proportionalto frequency, f , given by f = kFs/Nf , where Fs is thesampling rate. A similar calculation is performed byreplacing HRTF magnitudes with measured ITD valuesto compute c = cI .

2.2.2 Head scan point clouds

To represent head scan point clouds in terms of spher-ical harmonics, we choose s to be the vector of Eu-clidean distances, r, of each of the points (each associ-ated with a unique spatial direction) from the coordi-nate system origin. We then use a similar procedure, asdescribed in Sec. 2.2.1, to compute the vector, cS, ofpoint cloud spherical harmonic coefficients.

2.3 Computation of HRTFs from scans

Let u = 1,2, . . . ,U identify each of U subjects, andlet c(u) denote the vector c associated with subject u.Define matrices DS, DHl/r [k], and DI of spherical har-monic coefficients such that

DS =hc(1)

S , . . . ,c(U)S

iT,

DHl/r [k] =h⇣

c(1)Hl/r

[k]⌘

, . . . ,⇣

c(U)Hl/r

[k]⌘iT

,

DI =hc(1)

I , . . . ,c(U)I

iT,

where cS, cHl/r [k], and cI for all U subjects are obtainedas described in Secs. 2.2.1 and 2.2.2.

We seek a matrix XHl/r [k] whose columns are the pro-jections of the corresponding columns of DHl/r [k] onthe column space of DS, and a column vector XI whichis the projection of DI , also on this space. XHl/r [k] isgiven by

XHl/r [k] =�DT

S DS��1 DT

S DHl/r [k], (3)

and XI may be computed similarly using DI instead ofDHl/r [k].

The matrix XHl/r [k] and vector XI may then be used tocompute cHl/r [k] and cI , from cS, using the relations

cHl/r [k] = XTHl/r

[k]cS and cI = XTI cS, (4)

respectively, for any subject.

Finally, HRTF magnitudes for a set of directions(qi,fi), i = 1,2, . . . ,V may then be computed usingEq. (2) with c = cHl/r [k] from Eq. (4), and with n = nH .The resulting sP is the vector of desired magnitudes ata given frequency. These calculations may be repeatedfor each frequency to compute complete HRTF mag-nitude spectra. ITDs may be computed similarly byusing cI instead of cHl/r [k] and setting n = nI . CompleteHRTFs may be derived by determining the minimum-phase HRIRs from the HRTF magnitude spectra, andintroducing the appropriate computed ITDs.

2.3.1 Comparison with anthropometry-basedregression methods

Like other anthropometry-based methods, we assumerelationships exist between a listener’s HRTFs and an-thropometric features. However, we further assume thatthese relationships may be found by representing bothHRTF and anthropometric data using the same kindof basis functions (spherical harmonics in this case).This assumption distinguishes our method from otheranthropometry-based methods (see Nishino et al. [11],for example), where such relationships are sought be-tween measured anthropometric features represented inEuclidean space with naive Cartesian basis, and HRTFfeatures represented in some arbitrarily chosen, typi-cally principal component, space. Since our method,like other anthropometry-based methods, is data-driven,we assume that convergence to the aforementioned re-lationships will be achieved with a sufficiently largeamount of data. However, unlike our method, thosethat represent HRTF features on a principal compo-nent space, for example, rely on convergence not onlyto these relationships, but also to a set of basis func-tions, since the principal component space is typicallya function of the data used to develop the method. Fur-thermore, while previous anthropometry-based HRTFmethods are based on the difficult task of first identify-ing and measuring specific anthropometric features ofa subject [2, Ch. 7], our method, in comparison, doesnot require such features to be explicitly identified.

3 APPLICATION AND VALIDATION

3.1 Data acquisition and pre-processing

To apply and validate our method, we use the mea-sured HRTFs and head scans of 25 subjects from the

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 3 of 9

Sridhar and Choueiri Calculating HRTFs from head scans

RIEC1 [12] database. For each subject, we normal-ize HRTFs such that the largest magnitude at 468.75Hz, considering HRTFs for all spatial directions, is0 dB. Following Romigh et al. [10], we make allHRIRs minimum-phase and truncate each to a dura-tion of 5.8 ms. Following Andreopoulou and Katz [13],we compute ITDs by low-pass filtering the original,non-minimum-phase HRIRs using a 3 kHz cut-off fre-quency, identifying left- and right-HRIR onsets usinga 30% threshold2, and subtracting right-HRIR onsetsfrom corresponding left-HRIR onsets.

We resample all head scan point cloud data to contain5000 to 6000 roughly evenly distributed points, andalign each scan such that the y-axis (see Fig. 1) is alsoapproximately the interaural axis, with the coordinatesystem origin located approximately halfway betweenthe entrances to the two ear canals. We do not closeany holes or do any processing on the scan besidesalignment, basic noise reduction (outlier removal), andresampling.

3.2 Application

To apply our method, we compute XHl/r [k] and XI using23 “training” subjects. Due to the limited training data,we are restricted to nS 3 to ensure that the matrix DSis not “fat” (i.e. one with fewer rows than columns). Wechoose nS = 2 to compute HRTF magnitudes since wefound this produces the best results for the amount oftraining data available. We choose nS = 1 to computeITD because this roughly corresponds to an ellipsoidal-approximation to the subject’s head shape, and simplegeometrical methods of the head have previously beensuccessfully used to compute ITD [14, 15].

Figure 2 shows average RMS error, xl [k], computedhere in dB as a function of nH and f . xl [k] is given by

xl [k] =1U

U

Âu=1

s1V

V

Âv=1

���x (u)vl [k]

���2, (5)

where

x

(u)vl [k] =

������20 log10

���H(u)vl [k]

������H(u)

vl [k]���

������(6)

1http://www.riec.tohoku.ac.jp/pub/hrtf/

index.html

2Although Andreopoulou and Katz [13] recommend using 3.16%,we use 30% because many of the HRIRs contain undesirable pre-responses that result in incorrectly identified onsets when using3.16%.

12

3

4

= 6

1 3 5 7 9 11 13 15Frequency, f (kHz)

02468

101214

Deg

ree,

nH

xl [k]

Fig. 2: Contour plot of xl [k], in dB, as a function of nHand f .

denotes absolute spectral distortion in dB, with |·| de-noting absolute value. H(u)

vl [k] and H(u)vl [k] denote, re-

spectively, the measured and projected, left-ear HRTFsfor subject u and direction v, and RMS averaging isperformed over U = 25 subjects and all V = 865 spatialdirections.

We see that xl [k] for nH = 6 is similar to RMS errorsshown in Fig. 5 published by Romigh et al. [10] fora “truncation order” of four, the minimum for imper-ceptible spherical harmonic HRTF representations [10].Therefore, we choose nH = 6 for computing HRTFmagnitudes, and also to make inferences about the per-ceptibility of errors induced by our method.

Figure 3 shows a plot of average absolute ITD error, e ,as a function of nI when averaging over all subjects, andeither over all 865 directions or just the 72 directionson the horizontal plane (i.e., f = 0�). e is given by

e =1

UV

U

Âu=1

V

Âv=1

e

(u)v , (7)

wheree

(u)v =

���t(u)v � t

(u)v

��� , (8)

is the absolute ITD error, and t

(u)v and t

(u)v denote, re-

spectively, the measured and projected ITDs for subjectu and direction v.

We see that e decreases significantly between nI = 0and nI = 1, whereas beyond nI = 3, the decrease is slow.Therefore, we choose nI = 3 for computing ITDs.

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 4 of 9

Sridhar and Choueiri Calculating HRTFs from head scans

0 2 4 6 8 10 12 14Degree, nI

0

80

160

240

320

400

ITD

err

or,

All directionsHorizontal plane

e(µ

s)

Fig. 3: Plot of e as a function of nI , averaged overall 25 subjects, and either over all 865 spatialdirections (solid line), or just the 72 directionson the horizontal plane (dashed line). Error bars(too small to be visible) represent standard errorof the mean when averaging over subjects.

3.3 Validation

We validate our method using two “test” subjects fromthe database whose data were not used to computeXHl/r [k] and XI .

To evaluate the accuracy of the computed HRTF mag-nitudes for these subjects, we compute, for the left-earHRTF magnitudes of each subject, log-weighted aver-age spectral distortion, x

(u)vl , in dB, given by

x

(u)vl =

K2

Âk=K1

c[k] ·x (u)vl [k]

K2

Âk=K1

c[k]

, (9)

where the weights, c[k], are given by

c[k] = logk +0.5k�0.5

.

x

(u)vl [k] is computed as shown in Eq. (6) with H(u)

vl [k]now denoting the HRTFs computed using our method.K1 and K2 are frequency indices corresponding tolower- and upper-frequency bounds over which aver-aging is performed. We average over each of threefrequency bands given by: (i) 0 < f < 2 kHz, (ii)2 < f < 8 kHz, and (iii) 8 < f < 16 kHz. We also

compute RMS values of x

(u)vl , averaging over all spatial

directions, in each frequency band, and compare thesevalues with the RMS error, xl [k], shown in Fig. 2, toapproximately determine the perceptibility of x

(u)vl .

To evaluate the accuracy of computed ITDs, we com-pute e

(u)v as a function of q and f using Eq. (8),

with t

(u)v now denoting the ITDs computed using our

method. To determine the perceptibility of e

(u)v , we use

a threshold ITD error of approximately 30 µs, whichwe compute using the Woodworth and Schlosberg for-mula [16] for estimating ITD for a spherical-head withradius 0.0875 m, using a localization blur angle of3.2� [17, Table 2.1]. Although this 30 µs threshold isonly applicable for broadband white noise stimuli, andmay not be applicable to all spatial directions, we useit as an approximate threshold for the perceptibility ofe

(u)v .

Figure 4 shows matrix plots of x

(u)vl as a function of q

and f for each of the test subjects. Also shown on theseplots are RMS values of x

(u)vl for each frequency band,

when averaging over all spatial directions. We see thatx

(u)vl < 2 dB for all spatial directions when averaging

over 0 < f < 2 kHz, and x

(u)vl > 4 dB between 2 and 8

kHz for f < 30� and |q �270�|< 30� for both subjects.Similar calculations (not shown here) for the right-earHRTFs result in similar errors for |q �90�|< 30�. Thissuggests that the current implementation of our methodis not accurate for computing contralateral HRTFs forsome spatial directions in this frequency range. Fur-thermore, the similarity in results across subjects forf < 8 kHz indicates that the performance of our methodfor these frequencies is reliable, and we expect similarperformance for different test subjects. Additional cal-culations (not shown here) made by varying the choiceof training and test subjects confirm this observation.Finally, when averaging over 8 < f < 16 kHz, x

(u)vl > 7

dB for most spatial directions. This shows that thecurrent implementation of our method is not accuratewhen computing HRTF spectral features in this fre-quency range. This is discussed further in Sec. 4.

From a perceptual standpoint, comparing the RMSvalues of x

(u)vl to those plotted in Fig. 2 for nH = 6, we

see that, for the range 0 < f < 8 kHz, the RMS errorsin the computed HRTFs are similar to those for sixth-degree spherical harmonic representations of measuredHRTFs. Based on the discussion in Sec. 3.2, we may

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 5 of 9

Sridhar and Choueiri Calculating HRTFs from head scans

Test Subject 1

0 90 180 270 355-30

0

30

60

90Test Subject 2

0 f

2 kHz

0 90 180 270 355-30

0

30

60

90

0

0.7

1.4

2.1

2.8

0 90 180 270 355-30

0

30

60

90

Elev

atio

n,

(deg

.) 2 f

8 kHz

0 90 180 270 355-30

0

30

60

90

0

2

4

6

8

0 90 180 270 355Azimuth, (deg.)

-30

0

30

60

90 8 f

16 kHz

0 90 180 270 355-30

0

30

60

90

1

4

7

10

13

x

(u)vl , dB

x

(u)vl , dB

x

(u)vl , dB

RMS: 0.74 dB RMS: 0.71 dB

RMS: 2.34 dB RMS: 2.33 dB

RMS: 6.85 dB RMS: 5.09 dB

Fig. 4: Matrix plots of x

(u)vl as a function of q and f for each of the test subjects. Note that values of x

(u)vl for

elevations f = �30� and f = 90� are not shown because sufficient data points for different q do not existat these elevations.

conclude that any errors introduced by our method forf < 8 kHz are likely imperceptible.

Figure 5 shows matrix plots of x

(u)vl [k] as a function of

f on the median plane (i.e. q = 0� and q = 180�) foreach of the test subjects. For both subjects, x

(u)vl [k] < 3

dB for most f and for f < 6 kHz. Also, for subject 1,x

(u)vl [k] ranges from 9 to 15 dB for 7 < f < 8 kHz and

0� < f < 60�, suggesting that the current implemen-tation of our method is unable to accurately capturespectral features caused by the pinnae, since these fea-tures are typically found in this frequency range [18].Figure 6, which shows measured and computed HRTFmagnitude spectra, in dB, for f = 30� and f = �10�on the median plane of subject 1 and 2, respectively,

further illustrates this. We explain this discrepancy bynoting that a second-degree spherical harmonic rep-resentation of the head scan is insufficient to captureshape variations in the subject’s pinnae that might causethese spectral features in the HRTFs.

Finally, Fig. 7 shows a matrix plot of e

(u)v as a func-

tion of q and f for each of the test subjects. Wesee that e

(u)v < 30 µs (the approximate perceptibility

threshold computed earlier) for most f near the medianplane, except f � 30� behind (i.e. q = 180�) subject 1.Additionally, 60 < e

(u)v < 110 µs for |q �90�| < 30�

and |q �270�| < 30�, thus exceeding the perceptiblethreshold significantly. One possible explanation forthese errors is that a large number of measured HRIRs

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 6 of 9

Sridhar and Choueiri Calculating HRTFs from head scans

Test Subject 1

0 2 4 6 8 -30

0

30

60

90

60

30

0

-30

Elev

atio

n,

(deg

.)

Test Subject 2Front (

= 0 °)Back (

= 180 °)

0 2 4 6 8 Frequency, f (kHz)

-30

0

30

60

90

60

30

0

-30

0

3

6

9

12

15

18x

(u)vl [k], dB

Fig. 5: Matrix plots of x

(u)vl [k] as a function of f on the median plane for each of the test subjects.

102 103 104

Frequency, f (Hz)

-35

-25

-15

-5

5

15

Mag

nitu

de (d

B)

Test Subject 1, = 30°

MeasuredModeled

102 103 104

Frequency, f (Hz)

-35

-25

-15

-5

5

15

Mag

nitu

de (d

B)

Test Subject 2, = -10°

Fig. 6: Line plots of left-ear HRTF magnitude spectra in dB for f = 30� and f =�10� on the median plane for testsubjects 1 (left) and 2 (right), respectively. Note that these plots are for the worst case, which correspondsto the region of highest error shown in Fig. 5.

from the database contained artifacts prior to the mainimpulse, and the thresholding method used here forestimating measured ITD, as described in Sec. 3.1, isnot robust to such artifacts. This, however, needs fur-ther investigation. Currently, our method is unable toestimate ITDs accurately for some spatial directions,

particularly those corresponding to large q and f .

4 CONCLUSIONS

We presented a method for computing HRTFs from low-resolution head scan point cloud data of a subject by

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 7 of 9

Sridhar and Choueiri Calculating HRTFs from head scans

Test Subject 1

0 90 180 270 355-30

0

30

60

90

Elev

atio

n,

(deg

.)

Test Subject 2

0 90 180 270 355Azimuth, (deg.)

-30

0

30

60

90

0

30

60

90

120e

(u)v , µs

Fig. 7: Matrix plot of e

(u)v as a function of q and f for each of the test subjects. Note that values of e

(u)v for

elevations f = �30� and f = 90� are not shown because sufficient data points for different q do not existat these elevations.

first projecting the point cloud data onto a vector spacespanned by spherical harmonics, and then mapping theresulting spherical harmonic coefficients to ones thatrepresent the subject’s HRTFs in the same space. Wedefined this mapping as the projection of spherical har-monic coefficients computed from measured HRTFs fora set of “training” subjects, onto the column space of amatrix with columns containing the spherical harmoniccoefficients computed from the head scan point clouddata of these subjects (see Sec. 2.3 for details). Sepa-rate mappings were derived for computing a subject’sITD in µs and HRTF magnitude spectra in dB.

We then presented an implementation of our methodusing 23 training subjects to derive the mappings andvalidated the method using 2 “test” subjects. We val-idated our method in terms of spectral distortion androot-mean-square (RMS) errors of HRTF magnitudes,and also in terms of absolute ITD errors. Both thesemetrics were related to perceptibility criteria to ap-proximately determine the perceptibility of the errorsinduced by our method.

We show that our method, when implemented usingonly 23 training subjects, is able to reliably compute

HRTF magnitudes up to approximately 6 kHz, and thaterrors in this range may be imperceptible for most spa-tial directions. This shows that a second-degree spher-ical harmonic representation head scan point cloudsmay be sufficient to capture the necessary head shapeinformation that influences HRTF magnitude spectraup to 6 kHz. We also conclude that for f > 6 kHz, thecurrent implementation of our method does not per-form reliably, and any estimation errors that may not beperceptible (for instance, see, in Fig. (5), x

(u)vl [k] < 3 dB

between 6 to 8 kHz for median plane HRTFs for sub-ject 2) are only due to chance. Higher-degree sphericalharmonic representations of head scans are likely nec-essary to improve performance of the method for thesefrequencies. This, in turn, requires a larger trainingdata set.

Finally, we note that the above results are achievedusing unmeshed head scans with no more than 6000data points, and with minimal pre-processing involvingonly resampling, basic noise reduction, and alignment.We also note that HRTF computations using our methodinvolves only a few, computationally efficient matrixmultiplications. Our method is, therefore, well-suitedfor use in consumer spatial audio applications.

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 8 of 9

Sridhar and Choueiri Calculating HRTFs from head scans

Acknowledgements

We wish to thank Joseph Tylka for reviewing themanuscript and for some of the MATLAB code used togenerate the results shown here. We also wish to thankthe anonymous reviewers for their comments on theprécis submission related to this paper. This work wassponsored by the Sony Corporation of America.

References

[1] Blauert, J., The technology of binaural listening,Springer, 2013.

[2] Xie, B., Head-related transfer function and vir-tual auditory display, J. Ross Publishing, 2013.

[3] Katz, B. F., “Boundary element method calcula-tion of individual head-related transfer function.I. Rigid model calculation,” The Journal of theAcoustical Society of America, 110(5), pp. 2440–2448, 2001.

[4] Gumerov, N. A., O’Donovan, A. E., Duraiswami,R., and Zotkin, D. N., “Computation of the head-related transfer function via the fast multipole ac-celerated boundary element method and its spher-ical harmonic representation,” The Journal of theAcoustical Society of America, 127(1), pp. 370–386, 2010.

[5] Xiao, T. and Huo Liu, Q., “Finite difference com-putation of head-related transfer function for hu-man hearing,” The Journal of the Acoustical Soci-ety of America, 113(5), pp. 2434–2441, 2003.

[6] Meshram, A., Mehra, R., and Manocha, D., “Effi-cient HRTF computation using adaptive rectangu-lar decomposition,” in Audio Engineering SocietyConference: 55th International Conference: Spa-tial Audio, Audio Engineering Society, 2014.

[7] Tao, Y., Tew, A. I., and Porter, S. J., “A studyon head-shape simplification using spherical har-monics for HRTF computation at low frequencies,”Journal of the Audio Engineering Society, 51(9),pp. 799–805, 2003.

[8] Politis, A., Thomas, M. R., Gamper, H., and Ta-shev, I. J., “Applications of 3D spherical trans-forms to personalization of head-related transferfunctions,” in Acoustics, Speech and Signal Pro-cessing (ICASSP), 2016 IEEE International Con-ference on, pp. 306–310, IEEE, 2016.

[9] AES69-2015, “AES69-2015: AES standard forfile exchange - Spatial acoustic data file format,”2015.

[10] Romigh, G. D., Brungart, D. S., Stern, R. M., andSimpson, B. D., “Efficient real spherical harmonicrepresentation of head-related transfer functions,”IEEE Journal of Selected Topics in Signal Pro-cessing, 9(5), pp. 921–930, 2015.

[11] Nishino, T., Inoue, N., Takeda, K., and Itakura,F., “Estimation of HRTFs on the horizontal planeusing physical features,” Applied Acoustics, 68(8),pp. 897–908, 2007.

[12] Watanabe, K., Iwaya, Y., Suzuki, Y., Takane, S.,and Sato, S., “Dataset of head-related transferfunctions measured with a circular loudspeakerarray,” Acoustical science and technology, 35(3),pp. 159–165, 2014.

[13] Andreopoulou, A. and Katz, B. F., “Identifyinga perceptually relevant estimation method of theinter-aural time delay,” The Journal of the Acous-tical Society of America, 141(5), pp. 3635–3635,2017.

[14] Sridhar, R. and Choueiri, E., “Capturing the eleva-tion dependence of interaural time difference withan extension of the spherical-head model,” in Au-dio Engineering Society Convention 139, AudioEngineering Society, 2015.

[15] Duda, R. O., Avendano, C., and Algazi, V. R.,“An adaptable ellipsoidal head model for the in-teraural time difference,” in Acoustics, Speech,and Signal Processing, 1999. Proceedings., 1999IEEE International Conference on, volume 2, pp.965–968, IEEE, 1999.

[16] Woodworth, R. S. and Schlosberg, H., Experimen-tal psychology, Holt, 1954.

[17] Blauert, J., Spatial hearing: the psychophysics ofhuman sound localization, MIT press, 1997.

[18] Hebrank, J. and Wright, D., “Spectral cues usedin the localization of sound sources on the medianplane,” The Journal of the Acoustical Society ofAmerica, 56(6), pp. 1829–1834, 1974.

AES 143rd Convention, New York, NY, USA, 2017 October 18–21Page 9 of 9