A method is presented that makes it possible to reconstruct an arbitrary sound field based on measurements with a spherical microphone array. The proposed method (spherical equivalent source method) makes use of a point source expansion to describe the sound field on the rigid spherical array, from which it is possible to reconstruct the sound field over a three-dimensional domain, inferring all acoustic quantities: sound pressure, particle velocity, and sound intensity. The problem is formulated using a Neumann Green's function that accounts for the presence of the rigid sphere in the medium. One can reconstruct the total sound field, or only the incident part, i.e., the scattering introduced by the sphere can be removed, making the array virtually transparent. The method makes it possible to use sequential measurements: different measurement positions can be combined, providing an extended measurement area consisting of an array of spheres, and the sound field at any point of the source-free domain can be estimated, not being restricted to spherical surfaces. Because it is formulated as an elementary wave model, it allows for diverse solution strategies (least squares, ' 1 -norm minimization, etc.), revealing an interesting perspective for further work.

This study examines a near-field acoustic holography method consisting of a sparse formulation of the equivalent source method, based on the compressive sensing (CS) framework. The method, denoted Compressive-Equivalent Source Method (C-ESM), encourages spatially sparse solutions (based on the superposition of few waves) that are accurate when the acoustic sources are spatially localized. The importance of obtaining a non-redundant representation, i.e., a sensing matrix with low column coherence, and the inherent ill-conditioning of near-field reconstruction problems is addressed. Numerical and experimental results on a classical guitar and on a highly reactive dipole-like source are presented. C-ESM is valid beyond the conventional sampling limits, making wide-band reconstruction possible. Spatially extended sources can also be addressed with C-ESM, although in this case the obtained solution does not recover the spatial extent of the source.

Capturing the impulse or frequency response functions within extended regions of a room requires an unfeasible number of measurements. In this study, a method to reconstruct the response at arbitrary points based on compressive sensing (CS) is examined. The sound field is expanded into plane waves and their amplitudes are estimated via CS, obtaining a spatially sparse representation of the sound field. The validity of the CS assumptions are discussed, namely, the assumption of the wave field spatial sparsity (which depends strongly on the properties of the specific room), and the coherence of the sensing matrix due to different spatial sampling schemes. An experimental study is presented in order to analyze the accuracy of the reconstruction. Measurements with a scanning robotic arm make it possible to circumvent uncertainty due to positioning and transducer mismatch, and examine the accuracy of the reconstruction over extended regions of space. The results indicate that near perfect reconstructions are possible at low frequencies, even from a limited set of measurements. In addition, the study shows that it is possible to reconstruct damped room responses with reasonable accuracy well into the mid-frequency range.