# Ear Centering for Accurate Synthesis of Near-Field Head-Related Transfer Functions +
## 3 Evaluation with Calculated HRTF Datasets
### Conditions  
In practice, HRTF datasets are obtained for a finite number of point sources distributed in spherical grids. The infinite sum in (10) representing the inverse SFT will therefore be truncated up to a maximum order \(N\), which depends on the spherical sampling scheme.  
Among the existing calculated near-field HRTF collections [28; 29], we chose the one in [29] because its data are open and its resolution across distance is dense. Left-ear HRTFs for two individual head models without torso were used in evaluations. The spatial features due to the torso are prominent at the lower frequencies and can extend up to 3 kHz, whereas the features due to the head and pinna span the middle and high frequencies [30]. Although our assessment focuses on head and pinna features, the lower frequencies are also covered; hence, the torso absence does not limit our results. The HRIRs have 512 samples along time, were sampled at 48 kHz, and \(c=344\) m/s. The left-ear positions were extracted from the head models.  
The sound sources were distributed in spherical grids based on subdivisions of the edges of the icosahedron. Icosahedral samplings were chosen because they achieve bounded integration errors distributed across all orders, whereas other samplings aiming at perfect quadrature at low orders yield large errors concentrated in the high orders [27].  
The number of points \(P\) in an icosahedral grid, generated with a subdivision factor \(q\), is calculated as  
\[P=10q^{2}+2. \tag{14}\]  
For almost regular spherical samplings, such as the icosahedral ones, the maximum SFT order \(N_{\text{g}}\) achievable with a grid of \(P\) points is  
\[N_{\text{g}}=\lfloor\sqrt{P}\rfloor-1. \tag{15}\]It ensures reliable synthesis up to a maximum frequency  
\[f_{\text{max}}=\frac{cN_{\text{g}}}{2\pi r_{\text{bound}}}, \tag{16}\]  
where \(r_{\text{bound}}\) is the radius of a sphere fully containing the head.  
Datasets at a distance \(a=100\) cm were used as inputs. Three icosahedral grids with \(P=12,42,252\), correspondingly \(q=1,2,5\), and \(N_{\text{g}}=2,5,14\), were used. The maximum SFT orders to analyze the spherical HRTF datasets were limited by the far-to-near field transitions and the input resolutions as follows:  
\[N=\min(\lceil kr_{\text{bound}}\rceil,N_{\text{g}}). \tag{17}\]  
Here, \(r_{\text{bound}}\) is the same used in (16) and the value \(kr_{\text{bound}}\) indicates the far-to-near field transition.  
Because the ideal DVFs in (9) yield excessive values for higher orders and lower frequencies [13], their action needs to be limited according to  
\[\hat{\mathcal{H}}_{nm}(b,\mathbf{r_{\text{ear}}},\mathbf{r_{\text{ear}}})= \mathcal{W}_{n}(a,b)\mathcal{H}_{nm}(b,\mathbf{r_{\text{ear}}},\mathbf{r_{ \text{ear}}}) \tag{18}\]  
with an order truncation and scaling window defined in [31] as:  
\[\mathcal{W}_{n}(a,b)=\begin{cases}\frac{b}{a}e^{-jk(b-a)},&n\leq N,\\ 0,&\text{otherwise}.\end{cases} \tag{19}\]  
In (16), (17), and (19), \(r_{\text{bound}}\) should ideally be the radius of the smallest sphere containing the head. However, this theoretical limit yielded artifacts due to the discontinuities of the truncation rule \(n\leq\lceil kr_{\text{bound}}\rceil\). To reduce these artifacts, we have empirically chosen \(r_{\text{bound}}=16\) cm as a convenient value for the two individual head models in [29].  
Datasets at distances \(b\), ranging from 20 to 100 cm with 1 cm spacing, were used as target data. For each distance, \(P=642\) directions on an icosahedral grid with \(q=8\) were considered. Datasets were also synthesized for these distributions of points to evaluate three scenarios:  
* No ear centering (NEC);
* Plane-wave ear centering (PEC) using the translation operators in (3) and (12);
* Spherical-wave ear centering (SEC) using the translation operators in (4) and (13).