Joint Estimation of Amplitude and Phase from Phase-Diversity Data J.H. Seldin and R.G. Paxman General Dynamics Advanced Information Systems, 1200 Joe Hall Drive, Ypsilanti, MI 48197 rick.paxman@gd-ais.com Abstract: We present the problem of jointly estimating wavefront amplitude and phase from phase-diversity data and provide simulation examples. © 2005 Optical Society of America OCIS codes: (100.3190) inverse problems; (100.5070) phase retrieval 1. Introduction Equations used to perform maximum-likelihood estimation of phase aberrations from Phase-Diversity (PD) data are well understood for the cases of additive Gaussian noise, signal-dependent Poisson noise, and a mixture of both of these noise sources [1]. In this paper we generalize these equations to accommodate joint estimation of phase and amplitude (along with the object) for the specific case of PD data corrupted by additive Gaussian noise. Simulation experiments indicate significant improvement in performance over the conventional estimator when estimating phase in the presence of amplitude aberrations. Simulations also suggest that estimation of the clear aperture may be accomplished with appropriately defined PD data sets, obviating the need for a pupil camera. 2. Imaging Model We begin by introducing terminology for the incoherent-imaging model. The coherent transfer function (CTF) for the kth diversity channel is given by: )] ( ) ; ( [ ) ; ( ) , ; ( u u i k k e u C u H θ α φ β β α + = , (1) where k = index of the PD channel, (2) u = 2-D spatial-frequency coordinate vector, (3) α = phase-aberration parameter vector, (4) β = amplitude-aberration parameter vector. (5) C(u;β ) = unknown CTF amplitude-aberration function, (6) φ(u;α) = unknown CTF phase-aberration function, and (7) θ k (u) = known diversity phase function for channel k. (8) The phase and amplitude aberrations are each expressed as a weighted sum of appropriate basis functions, respectively: ) ( ) ; ( u u j j j ∑ = φ α α φ . (9) 0 ) ; ( ), ( ) ; ( ≥ Ψ = ∑ β β β u C u u C j j j . (10) The incoherent PSF for the kth channel can be discretely modeled as proportional to the squared magnitude of the inverse Discrete Fourier Transform (DFT) of the CTF: { } 2 ) , ; ( ) ( ) , ; ( 1 β α β η β α u H x s k k − = F , (11) where the scale factor η depends on the amplitude aberration. It follows that the Optical Transfer Function (OTF) for the kth channel is just the DFT of the PSF: { } ) , ; ( ) , ; ( β α β α x s u S k k F = . (12) Under a discrete-imaging model in which additive noise dominates, the image data for the kth diversity channel are modeled as n x x s x f x d x k k + − = ∑ ' ) , ; ' ( ) ' ( ) ( β α , (13) where n is a Gaussian random variable, considered uniform across all detector elements. JTuB4