Kernel Interpolation of Incident Sound Field in Region Including Scattering Objects

Presentation slides for IEEE WASPAA 2023

1. Kernel Interpolation of Incident Sound Field in Region Including Scattering Objects Shoichi Koyama1, Masaki Nakada, Juliano G. C. Ribeiro, and Hiroshi Saruwatari2 1National Institute of Informatics, Japan 2The University of Tokyo, Japan
2. Sound field estimation Ø Fundamental problem in audio signal processing having various applications Ø Applicable to spatial audio recording, interpolation of acoustic transfer functions, spatial active noise control, and so forth. October 14, 2023 2 Estimating sound field inside target region using multiple mics Microphone Target region
3. Limitation of current sound field estimation methods Ø In some practical applications, incident sound field in region containing scattering objects needs to be estimated Ø E.g., spatial active noise control: October 14, 2023 3 Target region should not contain any scattering objects Estimation accuracy can be degraded owing to scattering effects
4. Goal October 14, 2023 4 Sound field estimation in region containing scattering objects of unknown properties Ø Problem setting: – Estimating pressure distribution of incident sound field in target region – One or more scattering objects exist inside spherical region Ø Goal: – Develop estimation method of in from observation without prior knowledge of scattering properties Incident sound Microphone Scattering objects Incident Total Scattering
5. Method Ø Incident field: Weighted sum of kernel functions to constraint estimates to satisfy homogeneous Helmholtz equation [Ueno+ 2018] Ø Scattering field: Spherical wave function expansion for exterior field up to order October 14, 2023 5 Joint estimation of incident and scattering fields based on representation of spherical wave function expansion Kernel function: Spherical wave function: Weight Expansion coefs ( : # of mics)
6. Method Ø Optimization problem: October 14, 2023 6 Joint estimation of incident and scattering fields based on representation of spherical wave function expansion , , , Weighting matrix
7. Method Ø Optimization problem: Ø Closed-form solution: October 14, 2023 7 Joint estimation of incident and scattering fields based on representation of spherical wave function expansion Weighting matrix
8. Weighting matrix for inducing smoothness Ø is defined so that regularization term for corressponds to Ø Each element of is analytically derived October 14, 2023 8 Weighting matrix inducing smoothness of scattering field in angular directions to alleviate dependence on truncation order Small high-order coefficients compared to low-order coefficients
9. Experiments Ø Settings – Target region: Sphere of radius R = 0.5 m – 25 mics on 2 spherical surfaces of radii 0.5 and 0.55 m (50 mics in total) – Rigid sphere of radius 0.3 m at center of – Point source at (2.0, 2.0, 0.0) m Ø Comparison – Kernel ridge regression (KRR) – Proposed • Weighting matrix: • Truncation order: Ø Evaluation measure October 14, 2023 9 Point source Mic Rigid sphere
10. Experiments Ø NMSE w.r.t. frequency October 14, 2023 10 § Low NMSE was achieved by Proposed below 600 Hz § Dependence on truncation order was alleviated by weighting matrix
11. Experiments Ø Estimated incident pressure and error distributions at 300 Hz October 14, 2023 11 KRR Proposed Pressure Error
12. Conclusion Ø Sound field estimation in region including scattering objects – In some practical applications, incident sound field needs to be estimated without prior knowledge of scattering properties – Representation of incident and scattering fields by weighted sum of kernel functions and spherical wave function expansion, respectively – Closed-form solution for joint estimation of weights of kernels and expansion coefficients – Weighting matrix for expansion coefficients to induce smoothness of scattering field in angular directions – High estimation accuracy was achieved by Proposed below 600 Hz – Dependence on truncation order was alleviated by weighting matrix for expansion coefficients October 14, 2023 12