Inverse legendre polynomial approach in acoustic travel-time TOMography of the 2D and 3D indoor temperature distribution

• Reconstruction of continuous indoor temperature profile using Acoustic travel-time TOMography. • An inverse Legendre polynomial algorithm was developed for reconstructing the indoor temperature field. • Reconstructing continuous fields without dividing the tomography area into voxels as in the case of algebraic-based techniques. Acoustic Travel-Time TOMography (ATOM) is a system that can reconstruct the unknown indoor temperature that affects the propagation of acoustic waves in a tomography area of interest. The forward problem of indoor ATOM is based on collecting the time-of-flight (ToF) data from early reflections of the measured room impulse responses (RIRs), while the backward problem's goal is to estimate the temperature distributions given the ToF input data using an inverse algorithm. In the present work, we propose a new inverse algorithm based on the Legendre polynomials (LP) method for reconstructing a temperature distribution of an indoor tomography area (1.33 × 1.0 × 1.27) m. By approximating the polynomial functions, we naturally get 2D or 3D continuous profiles without dividing the tomography area into voxels as in the case of algebraic-based techniques. We demonstrate how exploiting the polynomial approximations leads to improving the ATOM performance and promising predicting continuous fields even in the empty areas of ray paths. Meanwhile, we show that the proposed LP algorithm successfully recovers a highly accurate indoor temperature distribution with a max error estimated at approximately 0.14°C.