## 4 Evaluation Experiment
### Experiment 1: Evaluation of Sound Image Localization Accuracy
Hence, we set \(\omega_{\text{L}}=6283\) rad/s and \(\omega_{\text{H}}=12,\)566 rad/s with consideration of the crossover. These values are related to 1000 and 2000 Hz in terms of frequency. From hereafter, \(E_{\Omega}(\omega,\theta,\phi)\) is called the error function. 3. Estimation of the direction of reconstructed 3-D sound image. The direction of the reconstructed 3-D sound image can be estimated by obtaining \((\hat{\theta}_{\Omega},\hat{\phi}_{\Omega})\) of which the error function \(E_{\Omega}(\omega,\theta,\phi)\) is the smallest value. Here, the directions of the sound image were estimated for each recorded sound as: \[(\hat{\theta}_{\Omega},\hat{\phi}_{\Omega})=\underset{0\leq\theta\leq 180^{ \circ}}{\arg\min}\left|\int_{0}^{\omega_{\text{max}}}E_{\Omega}(\omega,\theta, \phi)d\omega\right|,\] (26) where \(\omega_{\text{max}}\) is the highest angular frequency of the recorded sound. In this experiment, the frequency of the sound is up to 8000 Hz and \(\omega_{\text{max}}=50265\) rad/s. 4. Estimation of the position of the reconstructed 3-D sound image. Finally, the position of the reconstructed 3-D sound image was estimated by using the estimated direction of the sound image \((\hat{\theta}_{\Omega},\hat{\phi}_{\Omega})\). As shown in Figure 10, the position was estimated by drawing a straight line from each recording position to the estimated direction. Then, the center position of the surrounding area by the five lines was treated as the estimated position of the sound image \((\hat{r},\hat{\theta},\hat{\phi})\). From the four procedures, the position of the 3-D sound image was estimated and the accuracy of the sound image localization for the proposed method was evaluated by conducting two trials of Experiment 1. Moreover, the error between the positions of the reconstructed sound image and that of the target sound was evaluated. The error \(D_{\text{Err}}\) is defined as:  
Figure 9: Weighting function \(\beta(\omega)\) and \(1-\beta(\omega)\). \[D_{\text{Err}} =\sqrt{(\Delta X)^{2}+(\Delta Y)^{2}+(\Delta Z)^{2}}, \tag{27}\] \[\Delta X =\hat{r}\sin\hat{\theta}\cos\hat{\phi}-r^{\text{Tar}}\sin\theta^{ \text{Tar}}\cos\phi^{\text{Tar}},\] (28) \[\Delta Y =\hat{r}\sin\hat{\theta}\sin\hat{\phi}-r^{\text{Tar}}\sin\theta^{ \text{Tar}}\sin\phi^{\text{Tar}},\] (29) \[\Delta Z =\hat{r}\cos\hat{\theta}-r^{\text{Tar}}\cos\theta^{\text{Tar}}. \tag{30}\]  
By Equation (27), the error in terms of the Euclidean distance can be evaluated. The error \(D_{\text{Err}}\) was calculated for each trial. Then, the average of the errors was evaluated. Figure 11 shows the estimated position of the reconstructed sound image for each condition (a)-(h) on the first trial and Figure 12 shows the error \(D_{\text{Err}}\) for each position. In Figure 11, the closer the red and black markers, the higher the accuracy of the sound localization. From Figures 11 and 12, the proposed method can reconstruct the sound image close to the target position. The distance error \(D_{\text{Err}}\) of the proposed method is 0.17 m smaller than that of the conventional method on average. These results show the effectiveness of the proposed method, which includes not only the elevation angle \(\theta\) and the azimuth angle \(\phi\) but also the distance \(r\). Here, the error \(D_{\text{Err}}\) of the proposed method is about 0.87 m at positions (a), (c), (e), and (g) at which the distance of the target sound image is 1.0 m, although the error \(D_{\text{Err}}\) of the conventional method is about 1.02 m. However, the error \(D_{\text{Err}}\) of the proposed method is about 1.30 m at positions (b), (d), (f), and (h) at which the distance of the target sound image is 1.5 m, although the error \(D_{\text{Err}}\) of the conventional method is about 1.49 m. These results indicate that the sound pressure and phase largely affect the sound localization in the case that the recording position, that is, the listening position, is close to the position of the sound image.