Dual-Port Conditional Invertible Neural Network for Sound Intensity Compensation in Sound Source Localization

Determining the precise direction of arrival (DOA) presents a substantial inverse problem in the field of sound source localization (SSL). Intensimetry, a method known for SSL, provides accurate sound direction estimation at low Helmholtz numbers (kd), which facilitates miniaturization and scalability of microphone array modules. However, its performance in accurately estimating DOA at high kd values is limited, which restricts its practical application. This limitation arises from an inverse problem in which the measured intensity becomes biased due to uneven directivity in sound intensity measurement, depending on the direction of the sound source. The issue of sound intensity bias errors becomes more pronounced in systems with a limited number of microphones and is exacerbated at higher Helmholtz numbers. This study introduces the dual-port conditional invertible neural network (Dual-port CINN), a deep learning (DL) approach designed to address intensity bias errors inherent in SSL. Integral to our model is the normalizing flow (NF), a foundational component of the invertible neural network (INN) framework, which enables the Dual-port CINN to effectively estimate complex distributions for accurate DOA compensation. The model employs an INN architecture, equipped with two specialized conditional ports: a global condition port for referencing the Helmholtz number and an observance condition port for refining biased DOA corrections. This design makes the model exceptionally effective for conditional regression tasks that require the estimation of complex distributions. Consequently, it provides a solution to the challenging inverse problem of DOA estimation based on intensity, which is complicated by spatially nonuniform directivity. The proposed model was trained and validated across a kd range of 0– $\boldsymbol {\pi }$ in a simulation environment using a tetrahedral array composed of four microphones. Furthermore, an experiment was conducted in an anechoic chamber with a cutoff frequency of 80 Hz using various microphone array sizes, including arrays with microphone distances of $d=0.14$ m and a smaller array with $d=0.01$ m. This research addresses a critical barrier in intensity-based SSL and presents a robust solution for practical SSL applications with a compact microphone array system.