Single Shot Digital Holography Using Iterative Reconstruction with Alternating Updates of Amplitude and Phase

We present an image recovery approach to improve amplitude and phase reconstruction from single shot digital holograms, using iterative reconstruction with alternating updates. This approach allows the flexibility to apply different priors to amplitude and phase, improves phase reconstruction in image areas with low amplitudes, and does not require phase unwrapping for regularization. Phantom simulations and experimental measurements of a grating sample both demonstrate that the proposed method helps to reduce noise and resolve finer features. The improved image reconstruction from this technique will benefit the many applications of digital holography. Introduction Digital holography has many versatile applications, including microscopy [1], phase contrast [2], 3D displays [3], tomography [4], and terahertz imaging [5]. This technique enables the measurement of both amplitude and phase, useful for quantifying path lengths, measuring index contrast, or viewing biological samples [6, 7]. Various experimental methods can extract amplitude and phase from a measured interference pattern: for example, phase-shifting interferometry involves incrementally stepping the phase of the reference beam, from which the object phase can be deduced [8]. However, this method requires multiple images to be recorded on a vibration-free optical table, typically with expensive devices such as high frame rate cameras. Off-axis interferometry, based on the interference of an object and reference beam slightly offset in angle, can compute amplitude and phase with a single measurement. Conventional processing spatially filters the Fourier transform of the hologram [9]. While simple, Fourier filtering suffers from some drawbacks: The filter window size is subjective, and good reconstruction requires the zero order and cross terms to be well-separated. A large separation between object and reference beams results in a high carrier frequency hologram, in which the zero-order and sidebands are well-separated in the spectrum. Although a high carrier frequency is ideal for Fourier filtering, other factors may prevent perfect reconstruction: The experimental configuration may not allow a large angular separation of the object and reference beams, the sampling requirements of the camera pixels may limit the angular separation, or the object may contain high spatial frequencies. To overcome these drawbacks, other approaches include subtracting the zero order term from the hologram [10], iteratively solving for the field in the spatial domain [11] and the frequency domain [12], and applying a nonlinear filter [13]. Optimizationbased approaches include formulating holography as a nonlinear least squares problem [14], as constrained optimization [15], as penalized likelihood with simulated data [16], or as a nonlinear inverse problem [17]. In this work, we propose a new image recovery approach that calculates an object field from a single hologram, using iterative reconstruction with alternating updates of amplitude and phase. To our knowledge, this work is the first application of the alternating update strategy to single shot digital holography. This approach offers multiple advantages: 1. It allows prior knowledge, such as object smoothness, to be applied separately to amplitude and phase. For example, although nearly transparent biological samples like cells are smooth in amplitude, they exhibit sharp edges in phase. 2. Regularizing phase separately, rather than regularizing the field Aeiφ , mitigates the effects of poor signal areas caused by low amplitudes, aiding phase reconstruction. 3. Our algorithm regularizes phase without requiring phase unwrapping, as discussed in the Theory section. This work is organized as follows. In the Theory section, we present continuous and discrete mathematical models of holographic measurements, followed by an iterative reconstruction algorithm that recovers amplitude and phase via alternating updates. In the Experiment section, we discuss results from a phantom simulation and from experimentally measured holograms of a grating sample. Finally we provide some concluding remarks. Theory Continuous Model In digital holography, an object field, o(x) = A(x)eiφ(x), and a reference field, r(x) = Ã(x)eiφ̃(x), combine to form an interference pattern measured on a camera, Iideal(x) = |o(x)+ r(x)|2 = ∣∣∣A(x)eiφ(x)+ Ã(x)eiφ̃(x)∣∣∣2 = A2(x)+ Ã2(x)+2A(x)Ã(x)cos [ φ(x)− φ̃(x) ] , (1) where x ∈ R2 is the spatial coordinate in the camera plane, and φ̃(x) typically represents linear phase. The measured hologram is generally noisy, which we can model as Gaussian noise that corrupts the interference pattern Iideal(x). Let I(x) denote the measured hologram. It is also possible to model the measured hologram using a Poisson distribution [18, 19]. We formulate the problem as the minimization of a cost function with the general ©2016 Society for Imaging Science and Technology DOI: 10.2352/ISSN.2470-1173.2016.19.COIMG-158 IS&T International Symposium on Electronic Imaging 2016 Computational Imaging XIV COIMG-158.1 form c(A,φ) = L(A,φ)+βAR(A)+βφ R(φ), where L(A,φ) is the negative log-likelihood function corresponding to our model of the noisy hologram as a Gaussian distribution, βA and βφ are scalar regularization parameters, and R(A) and R(φ) are roughness penalty functions for amplitude and phase, respectively. In the continuous formulation, the likelihood function has the form