Bayesian Fisher Information Based Photoacoustic Optimal Waveform Design via Fredholm Integral Equation

Photoacoustic imaging or detecting systems rely on accurate characterization of biological and material media through high-precision estimation of key physical parameters, such as absorber location (time delay) and thickness (effective signal duration). This paper develops a Bayesian Fisher information framework to address optimal excitation waveform design for enhanced joint parameter estimation performance in photoacoustic sensing systems contaminated by additive white Gaussian noise. We derive the data-driven Fisher information matrix (FIM) from the photoacoustic transfer function model, and further integrate prior parametric uncertainties by introducing a diagonal prior FIM to form the complete Bayesian total FIM. Under the classic D-optimality criterion, we maximize the determinant of the total FIM to construct an optimal spectral weight function, which adaptively distributes waveform energy toward frequency components carrying the richest parametric information. Constrained to physically realizable time-limited excitation signals, the underlying optimality condition is transformed into a second-kind Fredholm integral equation, whose leading principal eigenfunction directly yields the optimal waveform solution. Unlike conventional SNR-oriented waveform strategies, the proposed method explicitly accounts for inherent parameter coupling and prior uncertainty effects, establishing a systematic information-theoretic design paradigm. An iterative eigenvalue discretization procedure is also outlined to enable efficient numerical implementation and performance validation in practical photoacoustic metrology scenarios.