Magnitude Modelling of HRTF Using Principal Component Analysis Applied to Complex Values

Abstract

Principal components analysis (PCA) is frequently used for modelling the magnitude of the headrelated transfer functions (HRTFs). Assuming that the HRTFs are minimum phase systems, the phase is obtained from the Hilbert transform of the log-magnitude. In recent years, the PCA applied to HRTFs is also used to model individual HRTFs relating the PCA weights with anthropometric measurements of the head, torso and pinnae. The HRTF log-magnitude is the most used format of input data to the PCA, but it has been shown that if the input data is HRTF linear magnitude, the cumulative variance converges faster, and the mean square error (MSE) is smaller. This study demonstrates that PCA applied directly on HRTF complex values is even better than the two formats mentioned above, that is, the MSE is the smallest and the cumulative variance converges faster after the 8th principal component. Different objective experiments around all the median plane put in evidence the differences which, although small, seem to be perceptually detectable. To elucidate this point, psychoacoustic discrimination tests are done between measured and reconstructed HRTFs from the three types of input data mentioned, in the median plane between −45• and +90• .

Individualization of head-related transfer functions using principal component analysis and anthropometric dimensions

“…Assessing all available data sets, the 90% threshold of the variances is reached for 13 components. In comparison, the study of Ramos et al 24 reached this threshold within the first eight components. It could not be clarified whether the higher number of necessary components is forced by the phase, for instance by a delay, or the number of observations p. The complex-valued reconstructions using six or twelve components show large deviations in Fig.…”

“…For logarithmic magnitudes or a complex spectrum, 12 components are necessary to reach this threshold. 24 As an example, the reconstructed magnitudes of six directions of a randomly chosen HRTF data set are shown in Fig. 2.…”

“…The required number of components is often motivated by the threshold where the cumulative variance reaches 90%. 12,17,24 Dependent on the number of observations of the input, the number of required components to fulfill this limit varies between five and six. 17,24 This corresponds with the database, which reaches the threshold under consideration of six components.…”

“…22 The Principal Component Analysis (PCA) 13,14,17,23,24 in combination with a regression analysis provides one opportunity to calculate individualized HRTFs. 10 The PCA is a statistical approach 15 and has the advantage that it reduces the complexity of a data set while containing the majority of the information.…”

Ensemble Learning Based HRTF Personalization Using Anthropometric Features

“…For example, the active sensory tuning (AST) was used to synthesize HRTFs for the target person from a generic HTRF by adjusting parameters of poles and zeros of the transfer functions through several optimization steps [8][9]. The principal component analysis (PCA) and regression analysis were combined for estimating HRTFs [10][11] [12][13] [14]. In recent years, with deep learning widely used in almost all research fields, deep neural networks (DNNs) were proposed to estimate HRIRs using anthropometric features [15], and even to generate HRTFs from the image of the pinna [16].…”