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1) Problem description

I am trying to implement a 3D audio simulator in Python. I am using the HUTUBS dataset as HRIR database (more informations here: https://depositonce.tu-berlin.de/handle/11303/9429), in particular I am using the simulated one, which has a higher resolution than the measured dataset.

Since I need continous movement of sound sources, I am interpolating the available HRIR using the VBAP technique proposed by Pulkki (http://lib.tkk.fi/Diss/2001/isbn9512255324/article1.pdf), in particular I am referring to this paper solution: https://www.ime.usp.br/~mqz/TwoApproachesForHRTFInterpolation.pdf , where the authors suggest to interpolate the available HRIR exploiting the VBAP technique.

So far I have implemented all the code and it works fine to a certain extent, but I am not able to achieve constant gain amplitude using the proposed interpolation technique. When I use the interpolation technique, I get like "jumps" of volume (I will show an audio example later)

2) My implementation

Using the previous HRIR database, I am going to compute the 3 closest HRIR to the requested position using the following function:

 def compute_three_closest_positions(self, azimuth_angle, elevation): requested_position = np.array([azimuth_angle, elevation, 0]) # computing the absolute difference between the requested angles and the available one in the dataset result = abs(self.sourcePositions - requested_position) result = np.delete(result, 2, 1) result = result.sum(axis=1) # returning index of the requested IR indexes = result.argsort()[:3] interpolated_positions = self.sourcePositions[indexes] print(interpolated_positions) self.real_azimuth_angles = interpolated_positions[:,0] self.real_elevations = interpolated_positions[:,1] print('printing azimuth angles: ',self.real_azimuth_angles) print('printing elevation angles: ',self.real_elevations) return indexes, interpolated_positions, requested_position

This function returns the index in the database which contains the 3 closest HRIR to the desider point (along with the positions of those points and the requested position). self.sourcePosition is computed by means of self.sourcePositions = self.HRIR_SOFA_file.getVariableValue('SourcePosition') which leverages on the pysofaconventions library to read SOFA datasets.

Once I get the desired HRIRs, I compute the VBAP gains using its formula:

def compute_gains(self, indexes, interpolated_positions, requested_position): interpolated_positions_cartesian = self.polar2cart(1.47, interpolated_positions[:,0], interpolated_positions[:,1]) requested_position_cartesian = self.polar2cart(1.47, requested_position[0], requested_position[1], array = False) inverse_interpolated_positions_cartesian = np.linalg.inv(interpolated_positions_cartesian) g = np.matmul(requested_position_cartesian,inverse_interpolated_positions_cartesian) g_normalized = (np.sqrt(1.47)*g)/np.sqrt(g[0][0]**2 + g[0][1]**2 + g[0][2]**2) return g_normalized

Where self.polar2cart is a simple support function that converts the input data from spherical to cartesian coordinates and the constant value 1.47 has been used because the dataset has HRIRs measured at a distance of 1.47 meters.

The above two functions have been used inside a function which returns the final interpolated HRIR:

 def get_interpolated_IR(self,azimuth_angle, elevation, distance): indexes, interpolated_positions, requested_position = self.compute_three_closest_positions(azimuth_angle, elevation) gains = self.compute_gains(indexes, interpolated_positions, requested_position) aux_IR = np.array((self.IR_dictionary[indexes, 0, :], self.IR_dictionary[indexes, 1, :])) aux_IR = np.moveaxis(aux_IR, 1,0) gains = np.squeeze(gains,axis=0) weighted_IR = 0 for i in range(0,3): weighted_IR =+ gains[i] * aux_IR[i,:,:] return weighted_IR

Where self.IR_dictionary has been computed by means of self.IR_dictionary = self.HRIR_SOFA_file.getDataIR()

Finally, I use my weighted_IR to filter some audio data. The code about the audio processing, reproduction ecc... is omitted since it works fine.

3) Problems and examples

If I use the standard HRIR dataset (so no interpolations via VBAP) using the single closest HRIR to the desired position, this is what I get (use headphones): https://drive.google.com/file/d/1ywMwmHMKlmnG0MnjEnrmztxmuiZDxDjO/view?usp=sharing As you might hear, the 3D audio simulation works quite decently, even though you can hear a kind of "step" behaviour between each spatial position.

On the other hand, if I use the above mentioned interpolation method, this is the result I get (the audio movement is slower, to make more evident the volume jumps): https://drive.google.com/file/d/1F_9IdfiWgeHQNYSYrdqMYC1SQ9q_U7QE/view?usp=sharing

4) Known possible bugs

I noticed that in my approach, computing the 3 closest points to the desired source position, sometimes leads to a triangle that doesn't contain the source position, thus obtaining negative gain factors. An example is the following:

Requested point: azimut, elevation, distance [200 0 1.47] Three closest points: azimut, elevation, distance [[199.69547046 0. 1.47 ] [199.40203659 5.61508214 1.47 ] [199.40203659 -5.61508214 1.47 ]] gain matrix: [[ 2.03778511 -0.52140831 -0.52140831]] g_normalized: [[1.1400898290447332 -0.29171491366465313 -0.2917149136646515]]

NB: I get "jumping" volumes also for valid gains (all positives) like in the following example:

Requested position: [190 0 0] Three closest points: [[191.83033621 0. 1.47 ] [188.02560265 2.34839855 1.47 ] [188.02560265 -2.34839855 1.47 ]] gain matrix: [[0.51921064 0.24087231 0.24087231]] g_normalized: [[1.013732209139594 0.47029087177669177 0.47029087177668905]]

5) question

Which could be the cause of jumping gains and non constant volume? (beside the above already mentioned bug) I get that behaviour also for "valid" triplets of positions.

NB: My data has the following shape (i added a visualization to simplify the understanding of my problem)

Sphere

And here is an example of the data matrix with the available positions in the dataset (i can't provide the full matrix due to characters limitations in the question):

Source Positions [[ 0. 90. 1.47 ] [ 0. 85.2073787 1.47 ] [ 0. 78.16966379 1.47 ] [ 0. 70.30452954 1.47 ] [ 0. 62.0367492 1.47 ] [ 0. 53.56289304 1.47 ] [ 0. 45. 1.47 ] [ 0. 36.43710696 1.47 ] [ 0. 27.9632508 1.47 ] [ 0. 19.69547046 1.47 ] [ 0. 11.83033621 1.47 ] [ 0. 4.7926213 1.47 ] [ 0. 0. 1.47 ] [ 0. -4.7926213 1.47 ] [ 0. -11.83033621 1.47 ] [ 0. -19.69547046 1.47 ] [ 0. -27.9632508 1.47 ] [ 0. -36.43710696 1.47 ] [ 0. -45. 1.47 ] [ 0. -53.56289304 1.47 ] [ 0. -62.0367492 1.47 ] [ 0. -70.30452954 1.47 ] [ 0. -78.16966379 1.47 ] [ 0. -85.2073787 1.47 ] [ 0. -90. 1.47 ] [ 1.64008341 -1.6394119 1.47 ] [ 1.64008341 1.6394119 1.47 ] [ 2.37160039 8.01881788 1.47 ] [ 2.37160039 -8.01881788 1.47 ] [ 2.80356493 -15.58649429 1.47 ] [ 2.80356493 15.58649429 1.47 ] [ 3.16999007 23.70233802 1.47 ] [ 3.16999007 -23.70233802 1.47 ] [ 3.56208248 -32.09871039 1.47 ] [ 3.56208248 32.09871039 1.47 ] [ 4.04606896 40.63141594 1.47 ] [ 4.04606896 -40.63141594 1.47 ] [ 4.1063771 -4.09587122 1.47 ] [ 4.1063771 4.09587122 1.47 ] [ 4.70089981 49.2024397 1.47 ] [ 4.70089981 -49.2024397 1.47 ] [ 4.7926213 0. 1.47 ] [ 5.18003348 11.34988901 1.47 ] [ 5.18003348 -11.34988901 1.47 ] [ 5.65662228 -57.72525378 1.47 ] [ 5.65662228 57.72525378 1.47 ] [ 5.950799 19.30523849 1.47 ] [ 5.950799 -19.30523849 1.47 ] [ 6.69117127 -27.61716167 1.47 ] [ 6.69117127 27.61716167 1.47 ] [ 6.99679324 6.9451987 1.47 ] [ 6.99679324 -6.9451987 1.47 ] [ 7.17914676 -66.09875172 1.47 ] [ 7.17914676 66.09875172 1.47 ] [ 7.54685777 36.10960472 1.47 ] [ 7.54685777 -36.10960472 1.47 ] [ 8.02560265 -2.34839855 1.47 ] [ 8.02560265 2.34839855 1.47 ] [ 8.31015293 14.75097316 1.47 ] [ 8.31015293 -14.75097316 1.47 ] [ 8.65428668 -44.67195851 1.47 ] [ 8.65428668 44.67195851 1.47 ] [ 9.4113917 22.9850347 1.47 ] [ 9.4113917 -22.9850347 1.47 ] [ 9.94527716 -74.16952628 1.47 ] [ 9.94527716 74.16952628 1.47 ] [ 10.16924521 10.01274933 1.47 ] [ 10.16924521 -10.01274933 1.47 ] [ 10.20644567 -53.21545998 1.47 ] [ 10.20644567 53.21545998 1.47 ] [ 10.57849119 31.43870375 1.47 ] [ 10.57849119 -31.43870375 1.47 ] [ 11.39521763 -5.07846224 1.47 ] [ 11.39521763 5.07846224 1.47 ] [ 11.71781242 18.18752703 1.47 ] [ 11.71781242 -18.18752703 1.47 ] [ 11.83033621 0. 1.47 ] [ 12.00910817 39.98781129 1.47 ] [ 12.00910817 -39.98781129 1.47 ] [ 12.55572647 -61.64591517 1.47 ] [ 12.55572647 61.64591517 1.47 ] [ 13.18808541 26.61516878 1.47 ] [ 13.18808541 -26.61516878 1.47 ] [ 13.57223424 -13.20668987 1.47 ] [ 13.57223424 13.20668987 1.47 ] [ 13.93222636 48.53955038 1.47 ] [ 13.93222636 -48.53955038 1.47 ] [ 14.86973294 -35.15709506 1.47 ] [ 14.86973294 35.15709506 1.47 ] [ 14.9005703 8.03442269 1.47 ] [ 14.9005703 -8.03442269 1.47 ] [ 15.39564168 -21.62918492 1.47 ] [ 15.39564168 21.62918492 1.47 ] [ 15.6042627 2.70038857 1.47 ] [ 15.6042627 -2.70038857 1.47 ] [ 16.36946156 -81.6400267 1.47 ] [ 16.36946156 81.6400267 1.47 ] [ 16.48669352 69.83188274 1.47 ] [ 16.48669352 -69.83188274 1.47 ] [ 16.7150029 -57.00288364 1.47 ] [ 16.7150029 57.00288364 1.47 ] [ 17.02691737 43.71636296 1.47 ] [ 17.02691737 -43.71636296 1.47 ] [ 17.19395061 -16.46805812 1.47 ] [ 17.19395061 16.46805812 1.47 ] [ 17.30592586 30.17930126 1.47 ] [ 17.30592586 -30.17930126 1.47 ] [ 18.5483898 -11.124716 1.47 ] [ 18.5483898 11.124716 1.47 ] [ 19.35894951 25.04791476 1.47 ] [ 19.35894951 -25.04791476 1.47 ] [ 19.40203659 -5.61508214 1.47 ] [ 19.40203659 5.61508214 1.47 ] [ 19.62236001 38.75484136 1.47 ] [ 19.62236001 -38.75484136 1.47 ] [ 19.69547046 0. 1.47 ] [ 20.01992939 52.2039386 1.47 ] [ 20.01992939 -52.2039386 1.47 ] [ 21.04482476 -19.7531228 1.47 ] [ 21.04482476 19.7531228 1.47 ] [ 21.08234374 65.25985975 1.47 ] [ 21.08234374 -65.25985975 1.47 ] [ 21.80747487 -33.65623199 1.47 ] [ 21.80747487 33.65623199 1.47 ] [ 22.35617628 14.28763204 1.47 ] [ 22.35617628 -14.28763204 1.47 ] [ 22.7019775 -47.26870019 1.47 ] [ 22.7019775 47.26870019 1.47 ] [ 23.26547643 8.65819735 1.47 ] [ 23.26547643 -8.65819735 1.47 ] [ 23.63943111 -28.41582338 1.47 ] [ 23.63943111 28.41582338 1.47 ] [ 23.7346488 2.9023498 1.47 ] [ 23.7346488 -2.9023498 1.47 ] [ 24.21803858 -77.53803241 1.47 ] [ 24.21803858 77.53803241 1.47 ] [ 24.47980904 60.51175385 1.47 ] [ 24.47980904 -60.51175385 1.47 ] [ 24.91227418 -42.20657626 1.47 ] [ 24.91227418 42.20657626 1.47 ] [ 25.15024972 23.02517483 1.47 ] [ 25.15024972 -23.02517483 1.47 ] [ 26.35068855 -17.47624816 1.47 ] [ 26.35068855 17.47624816 1.47 ] [ 26.75062552 37.01937754 1.47 ] [ 26.75062552 -37.01937754 1.47 ] [ 27.0913713 -55.62120718 1.47 ] [ 27.0913713 55.62120718 1.47 ] [ 27.23340936 11.76937527 1.47 ] [ 27.23340936 -11.76937527 1.47 ] [ 27.77832208 -5.92590316 1.47 ] [ 27.77832208 5.92590316 1.47 ] [ 27.9632508 0. 1.47 ] [ 28.28221843 -31.70311174 1.47 ] [ 28.28221843 31.70311174 1.47 ] [ 28.76376549 73.11464641 1.47 ] [ 28.76376549 -73.11464641 1.47 ] [ 29.1592042 -50.60695206 1.47 ] [ 29.1592042 50.60695206 1.47 ] [ 29.54728944 26.24983039 1.47 ] [ 29.54728944 -26.24983039 1.47 ] [ 30.56674657 -20.65061551 1.47 ] [ 30.56674657 20.65061551 1.47 ] [ 30.83270788 45.47781292 1.47 ] [ 30.83270788 -45.47781292 1.47 ] [ 31.34578223 -14.9011977 1.47 ] [ 31.34578223 14.9011977 1.47 ] [ 31.72271753 68.47240153 1.47 ] [ 31.72271753 -68.47240153 1.47 ] [ 31.87747186 -9.01147133 1.47 ] [ 31.87747186 9.01147133 1.47 ] [ 32.14860743 3.0170113 1.47 ] [ 32.14860743 -3.0170113 1.47 ] [ 32.20717627 -40.23544245 1.47 ] [ 32.20717627 40.23544245 1.47 ] [ 33.34422421 34.87601987 1.47 ] [ 33.34422421 -34.87601987 1.47 ] [ 33.80374677 -63.6670032 1.47 ] [ 33.80374677 63.6670032 1.47 ] [ 34.28301486 29.39174041 1.47 ] [ 34.28301486 -29.39174041 1.47 ] [ 35.04658869 -23.77296238 1.47 ] [ 35.04658869 23.77296238 1.47 ] [ 35.34895957 58.73011406 1.47 ] [ 35.34895957 -58.73011406 1.47 ] [ 35.64559338 -18.01217841 1.47 ] [ 35.64559338 18.01217841 1.47 ] [ 36.08099358 12.11111736 1.47 ] [ 36.08099358 -12.11111736 1.47 ] [ 36.3472473 -6.09089013 1.47 ] [ 36.3472473 6.09089013 1.47 ] [ 36.43710696 0. 1.47 ] [ 36.54240134 -53.67929106 1.47 ] [ 36.54240134 53.67929106 1.47 ] [ 37.49104029 48.52285063 1.47 ] [ 37.49104029 -48.52285063 1.47 ] [ 38.26049473 -43.26244646 1.47 ] [ 38.26049473 43.26244646 1.47 ] [ 38.89235441 37.89458326 1.47 ] [ 38.89235441 -37.89458326 1.47 ] [ 39.41317128 -32.41179615 1.47 ] [ 39.41317128 32.41179615 1.47 ] [ 39.83934649 26.80417916 1.47 ] [ 39.83934649 -26.80417916 1.47 ] [ 40.17988842 -21.06217068 1.47 ] [ 40.17988842 21.06217068 1.47 ] [ 40.4381312 15.18178714 1.47 ] [ 40.4381312 -15.18178714 1.47 ] [ 40.61322257 -9.17306752 1.47 ] [ 40.61322257 9.17306752 1.47 ] [ 40.70208965 3.06953652 1.47 ] [ 40.70208965 -3.06953652 1.47 ] [ 45. -87.68120488 1.47 ] [ 45. -84.20260831 1.47 ] [ 45. -80.15364767 1.47 ] [ 45. -75.76582943 1.47 ] [ 45. -71.14963815 1.47 ] [ 45. -66.36539941 1.47 ] [ 45. -61.44781755 1.47 ] [ 45. -56.41666981 1.47 ] [ 45. -51.28206575 1.47 ] [ 45. -46.04723346 1.47 ] [ 45. -40.71010299 1.47 ] [ 45. -35.26438968 1.47 ] [ 45. -29.70075831 1.47 ] [ 45. -24.00879372 1.47 ] [ 45. -18.18080994 1.47 ] [ 45. -12.21860297 1.47 ] [ 45. -6.14282693 1.47 ] [ 45. 0. 1.47 ] [ 45. 6.14282693 1.47 ] [ 45. 12.21860297 1.47 ] [ 45. 18.18080994 1.47 ] [ 45. 24.00879372 1.47 ] [ 45. 29.70075831 1.47 ] [ 45. 35.26438968 1.47 ] [ 45. 40.71010299 1.47 ] [ 45. 46.04723346 1.47 ] [ 45. 51.28206575 1.47 ] [ 45. 56.41666981 1.47 ] [ 45. 61.44781755 1.47 ] [ 45. 66.36539941 1.47 ] [ 45. 71.14963815 1.47 ] [ 45. 75.76582943 1.47 ] [ 45. 80.15364767 1.47 ] [ 45. 84.20260831 1.47 ] [ 45. 87.68120488 1.47 ] [ 49.29791035 3.06953652 1.47 ] [ 49.29791035 -3.06953652 1.47 ] [ 49.38677743 -9.17306752 1.47 ] [ 49.38677743 9.17306752 1.47 ] [ 49.5618688 15.18178714 1.47 ] [ 49.5618688 -15.18178714 1.47 ] [ 49.82011158 -21.06217068 1.47 ] [ 49.82011158 21.06217068 1.47 ] [ 50.16065351 26.80417916 1.47 ] [ 50.16065351 -26.80417916 1.47 ] [ 50.58682872 -32.41179615 1.47 ] [ 50.58682872 32.41179615 1.47 ] [ 51.10764559 37.89458326 1.47 ] [ 51.10764559 -37.89458326 1.47 ] [ 51.73950527 -43.26244646 1.47 ] [ 51.73950527 43.26244646 1.47 ] [ 52.50895971 48.52285063 1.47 ] [ 52.50895971 -48.52285063 1.47 ] [ 53.45759866 -53.67929106 1.47 ] [ 53.45759866 53.67929106 1.47 ] [ 53.56289304 0. 1.47 ] [ 53.6527527 -6.09089013 1.47 ] [ 53.6527527 6.09089013 1.47 ] [ 53.91900642 12.11111736 1.47 ] [ 53.91900642 -12.11111736 1.47 ] [ 54.35440662 -18.01217841 1.47 ] [ 54.35440662 18.01217841 1.47 ] [ 54.65104043 58.73011406 1.47 ] [ 54.65104043 -58.73011406 1.47 ] [ 54.95341131 -23.77296238 1.47 ] [ 54.95341131 23.77296238 1.47 ] [ 55.71698514 29.39174041 1.47 ] [ 55.71698514 -29.39174041 1.47 ] [ 56.19625323 -63.6670032 1.47 ] [ 56.19625323 63.6670032 1.47 ] [ 56.65577579 34.87601987 1.47 ] [ 56.65577579 -34.87601987 1.47 ] [ 57.79282373 -40.23544245 1.47 ] [ 57.79282373 40.23544245 1.47 ] [ 57.85139257 3.0170113 1.47 ] [ 57.85139257 -3.0170113 1.47 ] [ 58.12252814 -9.01147133 1.47 ] [ 58.12252814 9.01147133 1.47 ] [ 58.27728247 68.47240153 1.47 ] [ 58.27728247 -68.47240153 1.47 ] [ 58.65421777 -14.9011977 1.47 ] [ 58.65421777 14.9011977 1.47 ] [ 59.16729212 45.47781292 1.47 ] [ 59.16729212 -45.47781292 1.47 ] [ 59.43325343 -20.65061551 1.47 ] [ 59.43325343 20.65061551 1.47 ] [ 60.45271056 26.24983039 1.47 ] [ 60.45271056 -26.24983039 1.47 ] [ 60.8407958 -50.60695206 1.47 ] [ 60.8407958 50.60695206 1.47 ] [ 61.23623451 73.11464641 1.47 ] [ 61.23623451 -73.11464641 1.47 ] [ 61.71778157 -31.70311174 1.47 ] [ 61.71778157 31.70311174 1.47 ] [ 62.0367492 0. 1.47 ] [ 62.22167792 -5.92590316 1.47 ] [ 62.22167792 5.92590316 1.47 ] [ 62.76659064 11.76937527 1.47 ] [ 62.76659064 -11.76937527 1.47 ] [ 62.9086287 -55.62120718 1.47 ] [ 62.9086287 55.62120718 1.47 ] [ 63.24937448 37.01937754 1.47 ] [ 63.24937448 -37.01937754 1.47 ] [ 63.64931145 -17.47624816 1.47 ] [ 63.64931145 17.47624816 1.47 ] [ 64.84975028 23.02517483 1.47 ] [ 64.84975028 -23.02517483 1.47 ] [ 65.08772582 -42.20657626 1.47 ] [ 65.08772582 42.20657626 1.47 ] [ 65.52019096 60.51175385 1.47 ] [ 65.52019096 -60.51175385 1.47 ] [ 65.78196142 -77.53803241 1.47 ] [ 65.78196142 77.53803241 1.47 ] [ 66.2653512 2.9023498 1.47 ] [ 66.2653512 -2.9023498 1.47 ] [ 66.36056889 -28.41582338 1.47 ] [ 66.36056889 28.41582338 1.47 ] [ 66.73452357 8.65819735 1.47 ] [ 66.73452357 -8.65819735 1.47 ] [ 67.2980225 -47.26870019 1.47 ] [ 67.2980225 47.26870019 1.47 ] [ 67.64382372 14.28763204 1.47 ] [ 67.64382372 -14.28763204 1.47 ] [ 68.19252513 -33.65623199 1.47 ] [ 68.19252513 33.65623199 1.47 ] [ 68.91765626 65.25985975 1.47 ] [ 68.91765626 -65.25985975 1.47 ] [ 68.95517524 -19.7531228 1.47 ] [ 68.95517524 19.7531228 1.47 ] [ 69.98007061 52.2039386 1.47 ] [ 69.98007061 -52.2039386 1.47 ] [ 70.30452954 0. 1.47 ] [ 70.37763999 38.75484136 1.47 ] [ 70.37763999 -38.75484136 1.47 ] [ 70.59796341 -5.61508214 1.47 ] [ 70.59796341 5.61508214 1.47 ] [ 70.64105049 25.04791476 1.47 ] [ 70.64105049 -25.04791476 1.47 ] [ 71.4516102 -11.124716 1.47 ] [ 71.4516102 11.124716 1.47 ] [ 72.69407414 30.17930126 1.47 ] [ 72.69407414 -30.17930126 1.47 ] [ 72.80604939 -16.46805812 1.47 ] [ 72.80604939 16.46805812 1.47 ] [ 72.97308263 43.71636296 1.47 ] [ 72.97308263 -43.71636296 1.47 ] [ 73.2849971 -57.00288364 1.47 ] [ 73.2849971 57.00288364 1.47 ] [ 73.51330648 69.83188274 1.47 ] [ 73.51330648 -69.83188274 1.47 ] [ 73.63053844 -81.6400267 1.47 ] [ 73.63053844 81.6400267 1.47 ] [ 74.3957373 2.70038857 1.47 ] [ 74.3957373 -2.70038857 1.47 ] [ 74.60435832 -21.62918492 1.47 ] [ 74.60435832 21.62918492 1.47 ] [ 75.0994297 8.03442269 1.47 ] [ 75.0994297 -8.03442269 1.47 ] [ 75.13026706 -35.15709506 1.47 ] [ 75.13026706 35.15709506 1.47 ] [ 76.06777364 48.53955038 1.47 ] [ 76.06777364 -48.53955038 1.47 ] [ 76.42776576 -13.20668987 1.47 ] [ 76.42776576 13.20668987 1.47 ] [ 76.81191459 26.61516878 1.47 ] [ 76.81191459 -26.61516878 1.47 ] [ 77.44427353 -61.64591517 1.47 ] [ 77.44427353 61.64591517 1.47 ] [ 77.99089183 39.98781129 1.47 ] [ 77.99089183 -39.98781129 1.47 ] [ 78.16966379 0. 1.47 ] [ 78.28218758 18.18752703 1.47 ] [ 78.28218758 -18.18752703 1.47 ] [ 78.60478237 -5.07846224 1.47 ] [ 78.60478237 5.07846224 1.47 ] [ 79.42150881 31.43870375 1.47 ] [ 79.42150881 -31.43870375 1.47 ] [ 79.79355433 -53.21545998 1.47 ] [ 79.79355433 53.21545998 1.47 ] [ 79.83075479 10.01274933 1.47 ] [ 79.83075479 -10.01274933 1.47 ] [ 80.05472284 -74.16952628 1.47 ] [ 80.05472284 74.16952628 1.47 ] [ 80.5886083 22.9850347 1.47 ] [ 80.5886083 -22.9850347 1.47 ] [ 81.34571332 -44.67195851 1.47 ] [ 81.34571332 44.67195851 1.47 ] [ 81.68984707 14.75097316 1.47 ] [ 81.68984707 -14.75097316 1.47 ] [ 81.97439735 -2.34839855 1.47 ] [ 81.97439735 2.34839855 1.47 ] [ 82.45314223 36.10960472 1.47 ] [ 82.45314223 -36.10960472 1.47 ] [ 82.82085324 -66.09875172 1.47 ] [ 82.82085324 66.09875172 1.47 ] [ 83.00320676 6.9451987 1.47 ] [ 83.00320676 -6.9451987 1.47 ] [ 83.30882873 -27.61716167 1.47 ] [ 83.30882873 27.61716167 1.47 ] [ 84.049201 19.30523849 1.47 ] [ 84.049201 -19.30523849 1.47 ] [ 84.34337772 -57.72525378 1.47 ] [ 84.34337772 57.72525378 1.47 ] [ 84.81996652 11.34988901 1.47 ] [ 84.81996652 -11.34988901 1.47 ] [ 85.2073787 0. 1.47 ] [ 85.29910019 49.2024397 1.47 ] [ 85.29910019 -49.2024397 1.47 ] [ 85.8936229 -4.09587122 1.47 ] [ 85.8936229 4.09587122 1.47 ] [ 85.95393104 40.63141594 1.47 ] [ 85.95393104 -40.63141594 1.47 ] [ 86.43791752 -32.09871039 1.47 ] [ 86.43791752 32.09871039 1.47 ] [ 86.83000993 23.70233802 1.47 ] [ 86.83000993 -23.70233802 1.47 ] [ 87.19643507 -15.58649429 1.47 ] [ 87.19643507 15.58649429 1.47 ] [ 87.62839961 8.01881788 1.47 ] [ 87.62839961 -8.01881788 1.47 ] [ 88.35991659 -1.6394119 1.47 ] [ 88.35991659 1.6394119 1.47 ] [ 90. 85.2073787 1.47 ] [ 90. 78.16966379 1.47 ] [ 90. 70.30452954 1.47 ] [ 90. 62.0367492 1.47 ] [ 90. 53.56289304 1.47 ] [ 90. 45. 1.47 ] [ 90. 36.43710696 1.47 ] [ 90. 27.9632508 1.47 ] [ 90. 19.69547046 1.47 ] [ 90. 11.83033621 1.47 ] [ 90. 4.7926213 1.47 ] [ 90. 0. 1.47 ] [ 90. 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-19.30523849 1.47 ] [ 96.69117127 -27.61716167 1.47 ] [ 96.69117127 27.61716167 1.47 ] [ 96.99679324 6.9451987 1.47 ] [ 96.99679324 -6.9451987 1.47 ] [ 97.17914676 -66.09875172 1.47 ] [ 97.17914676 66.09875172 1.47 ] [ 97.54685777 36.10960472 1.47 ] ecc...
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For your known bug:

Since your azimuth-elevation grid is uniformly distributed, finding the closest triangle containing the target direction is not difficult.

Say the target direction is $(\varphi, \vartheta)$. Let $\varphi = [\varphi_1, \varphi_2, ..., \varphi_M]$ and $\vartheta=[\vartheta_1, \vartheta_2, ..., \vartheta_N]$ be ascending sorted arrays of azimuths and elevations, respectively. You can use binary search to find the closest azimuth-elevation pair $(\varphi_m, \vartheta_n)$ satisfying $\varphi_m\leq\varphi$ and $\vartheta_n\leq\vartheta$, and the other two points should be $(\varphi_{m+1}, \vartheta_n)$ and $(\varphi_m, \vartheta_{n+1})$. Note the azimuth wraping at 360$^\circ$.

Other things to note:

As you are interpolating HRIR, you should calculate the weighted average of the minimum-phase parts of HRIRs. The detailed steps are: remove the delays of HRIRs, interpolate, add new delays calculated by specific interpolation method.

Why should you perform interpolation on the minimum-phase part, think of two HRIRs which are ideal delta functions.

$h_1(t) = A_1\delta(t_1)$ and $h_2(t) = A_2\delta(t_2)$, where $A_1$ and $A_2$ are respectively amplitude of HRIRs and $t_1$ is generally not equal to $t_2$. You will get two pulses if you just sum $h_1(t)$ and $h_2(t)$ up with different weights, which is obviously not the result we want.

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  • $\begingroup$ Are you assuming to order the azimuths and elevations indipendently? Because (maybe i'm wrong) if i do this way, i could end up with a couple of (azimuth,elevation) which doesnt exist in my dataset. I edited my question with an example of the available positions data. I am sorry but I didn't understand the second point, what do you mean by 'also found in the first case'? $\endgroup$ Commented Jan 14, 2021 at 9:50
  • $\begingroup$ @MattiaSurricchio Yes my approach to find triangles only works for mesh grid of azimuth-elevation pairs, so it doesn't work for your database. Maybe you can search for a mature VBAP code, or choose another database. I think most databases have uniformly distributed source positions, because it is easy to do the measurement. And the second point is saying that as seen from the waveform of your first audio example without interpolation, the jumping gains are also observed. $\endgroup$ Commented Jan 14, 2021 at 12:25
  • $\begingroup$ Did you plot the waveform of the audio file? I actually didn't, but from the only listening perspective, it seemed that the volume of my audio file in the different spatial positions is more or less constant, there are no huge jumps in the perceived audio volume. Do you have any other database you could suggest me? The ones that i know (KEMAR, CIPIC ecc...) they still have the same problem i mentioned, you have fixed couples of (azimuth,elevation). Maybe i'm doing something wrong, but ordering azimuth and elevations indipendetly, could lead to the same problem i've already mentioned $\endgroup$ Commented Jan 14, 2021 at 12:48
  • $\begingroup$ If you are referring to those spikes, as you might notice they're almost periodical, they are probably the consequence of my real time processing. I'm basically taking n samples from real time audio and then i convolve them with the selected HRIR. $\endgroup$ Commented Jan 14, 2021 at 13:11
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    $\begingroup$ @MattiaSurricchio A decently working HRTF processing will not produce these peaks, unless your input signals contain these peaks periodically. Otherwise, there is something wrong with your framing operation. BTW, the database I use is Priceton's 3D3A database, which has 72 azimuths: [0°, 5°, 10°, …, 355°] and 9 elevations: [–57°, –30°, –15°, 0°, 15°, 30°, 45°, 60°, 75°] (72 × 9 = 648 positions in total). $\endgroup$ Commented Jan 14, 2021 at 13:23
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This is a long post with a lot of info, I just and give some high level feedback

  1. Loose the 1.47 scale factor and just do everything on the unit sphere. If absolute distance/gain matters, just scale your whole HRIR set by 1.47/1.
  2. If you do a three point interpolation, the three points should form a triangle which encloses the point that you want. These are NOT necessarily the three closest point (depending on how your grid looks like) and your result seem to indicate cases where this is not the enclosing triangle.
  3. Gain normalization is tricky: at low frequencies HRIR add in amplitude and not in energy, i.e. they are highly correlated.

Here is what I would do

  1. Normalize HRIR data base to the unit sphere.
  2. Find the enclosing triangle.
  3. If the target point is "very close" to one side of the triangle, I would just do a two point interpolation
  4. Calculate the gains as the inverse of the distance of each vertex (triangle or line) to the target point
  5. Normalize the gains so that the sum of the gains (NOT the square of the gains) is unity. You have a fairly dense grid of HRIRs so you will see way more correlated summing than uncorrelated summing.

An alternative approach would be a 4 point interpolation. Do two 2-point azimuth interpolations at the two nearest elevations and then another 2-point elevation between the results of azimuth interpolations. That has the advantage that you can line up the interaural delays for different azimuth angles before interpolating. So you interpolate the interaural delays and the spectral features separately.

A completely different approach for this type of thing as spherical harmonics. You can represent the entire data set through a series of spherical harmonics and just do a weighted sum of the harmonics at the target location.

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  • $\begingroup$ So I removed the 1.47 constant (thus working on the unit circle) and used the suggested gain normalization, but the problem still holds. Your 4th point suggest to compute the gains in that way, is it wrong to use the VBAP formula? Furthermore, I updated the question with a visualization of the data. I expected to get triangle meshes by computing the 3 nearest points, but sometimes it seems that it doesn't work. Do you have an idea about why? Given the data "structure" it seemed reasonable to compute the three nearest point and expect a triangle as result $\endgroup$ Commented Jan 13, 2021 at 19:49
  • $\begingroup$ I'm not familiar with VBAP so I can't help. However, it looks like some of the gains are negative, which makes no sense, so something must be wrong there, $\endgroup$ Commented Jan 13, 2021 at 21:24
  • $\begingroup$ Yes, they are negative for some triplets because the desired point is not contained inside the triangle created by the 3 closest point, i mentioned that in the "known bugs" part $\endgroup$ Commented Jan 14, 2021 at 9:23

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