Low Photon Count Phase Retrieval Using Deep Learning

Abstract

Imaging systems' performance at low light intensity is affected by shot noise, which becomes increasingly strong as the power of the light source decreases. In this paper we experimentally demonstrate the use of deep neural networks to recover objects illuminated with weak light and demonstrate better performance than with the classical Gerchberg-Saxton phase retrieval algorithm for equivalent signal over noise ratio. Prior knowledge about the object is implicitly contained in the training data set and feature detection is possible for a signal over noise ratio close to one. We apply this principle to a phase retrieval problem and show successful recovery of the object's most salient features with as little as one photon per detector pixel on average in the illumination beam. We also show that the phase reconstruction is significantly improved by training the neural network with an initial estimate of the object, as opposed as training it with the raw intensity measurement.Many imaging systems only yield partial or distorted information about the object being imaged. Typical causes include loss of spatial frequencies, lack of phase information, unknown scatterers in the optical train, aberrations, and noise in the illumination or detection. In these situations, the mathematical operator describing the imaging system becomes ill-posed and usually requires regularization. A regularizer is an operator designed to favor solutions that match our prior knowledge about the object, if any. The choice of the regularizer itself is often arbitrary and based on practical experience. Recently, Deep Neural Networks (DNNs) have attracted much attention in the field of computational imaging, for they provide a way to regularize a problem adaptively. As of today, DNNs have been proven efficient solvers in many imaging applications such as deblurring [1], undersampled imaging [2], ghost imaging [3], phase retrieval [4][5][6][7][8][9], adaptive illumination microscopy [10], adaptive optics [11], and optical tomography [12,13]. In particular, in the context of phase retrieval, it has been demonstrated numerically that machine learning can improve the imaging condition [7].In this paper, we demonstrate experimentally for the first time, to our knowledge, that DNNs can solve a coherent phase retrieval problem affected by strong shot noise at various levels. We also provide corresponding numerical simulations. In situations where the light source is weak, the detection signal to noise ratio (SNR) is ultimately limited by the quantized nature of light. Because of its fundamental nature, shot noise cannot be avoided and regularization schemes must be devised to handle it. As the noise becomes more significant, reconstruction algorithms' performance in general deteriorates; this is the regime where we expect the biggest payoff from the DNN, assuming that it has been successfully trained to recover the object features that best explain the observed signal distribution. Best results are obtained for objects within restricted classes, i.e. s...

Learning to synthesize: robust phase retrieval at low photon counts

“…The prior may be defined explicitly, e.g. as a minimum energy [61,62] or sparsity [20][21][22][23][24] criterion; or learned from examples as a dictionary [25][26][27][28] or through a deep learning scheme [29,[31][32][33][34][35][36][37][38][39][40][41][42][43].…”

“…Here, as in earlier works on direct phase retrieval [37][38][39][40][41][42][43], and due to the nonlinearity of the forward model, we adopt the End-to-End and Approximant methods. These we denote as End-to-End:f = DNN(g); and (8)…”

“…In addition to requiring a single raw image to retrieve the phase through a learned recombination of the spectral channels, the LS method presented here has the desirable property of resilience to noise, especially in the case of weak photon flux down to a single photon per pixel. We achieved that by using an Approximant filter [39] to pre-process the raw image before submitting it to the two spectral channels. The Approximant produces an inverse estimate that expressly uses the physical model (a single iteration of the GSF algorithm in [39] and here).…”

“…We achieved that by using an Approximant filter [39] to pre-process the raw image before submitting it to the two spectral channels. The Approximant produces an inverse estimate that expressly uses the physical model (a single iteration of the GSF algorithm in [39] and here). For very noisy inputs, the Approximant is of very poor quality; yet, if the subsequent learning architecture is trained with this low-quality estimate as input, the final reconstruction results are significantly improved.…”

“…For very noisy inputs, the Approximant is of very poor quality; yet, if the subsequent learning architecture is trained with this low-quality estimate as input, the final reconstruction results are significantly improved. The LS method with Approximant, as presented here, drastically improves over [39], especially in the reconstruction of fine detail, as [39] did not use separate spectral channels to rebalance frequency content.…”