Limited-memory BFGS Optimisation of Phase-Only Computer-Generated Hologram for Fraunhofer Diffraction

Abstract

We implement a novel limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimisation algorithm with cross entropy (CE) loss function, to produce phase-only computer-generated hologram (CGH) for holographic displays, with validation on a binary-phase modulation holographic projector.

Multi-depth phase-only hologram optimization using the L-BFGS algorithm with sequential slicing

“…The most common multi-depth CGH optimization methods either evaluate their loss against the entire multi-depth 3D target, which is time-consuming, or evaluate the hologram for each plane and then sum the holograms, which introduces quality degradation. This paper therefore extends on the previous research, which proved the ability of L-BFGS algorithm to generate phase-only hologram for a 2D image [13], and proposes the sequential slicing (SS) technique for the optimization of CGH for multi-depth 3D target, which evaluates the loss for a single slice at each iteration, aiming for quicker hologram generation with proper overall quality and low quality imbalance across the multiple depths enabled by the second-order nature of the L-BFGS optimization algorithm. The following sections start from the background knowledge of numerical optimization including L-BFGS algorithm, then introduces and carries out the optimization of multi-depth CGH with sequential slicing (SS) technique, with results analysed.…”

“…It is a measure of how much a probability distribution 𝑃 is different from another probability distribution 𝑄. Both CE and RE are usually computed between the true probabilistic distribution and the predicted probabilistic distribution, while the images are not probability distributions, the pixel values are normalized to decimal numbers in the range of 0 to 1 so that CE and RE can be applied.The effectiveness of L-BFGS optimization of phase-only CGH for a single slice target image (𝑛 = 1) has been demonstrated in the previous research[13]. However, for a 3D target consisted of multiple slices at different depths, the optimization of CGH becomes challenging.The typical technique is to sum the losses computed for each slice for each iteration during optimization, which is called the Sum-of-Loss (SoL) method in this paper.…”

Multi-Frame Binary-Phase Holograms Batched Optimization

“…A few time-multiplexed multi-frame holograms generation methods have been explored in the literature, including the One-Step Phase-Retrieval (OSPR) algorithm 12 and the Adaptive One-Step Phase-Retrieval (AD-OSPR) algorithm 13 ; however, both OSPR and AD-OSPR are still subject to defects in reconstruction quality. This paper therefore extends on the previous research using the L-BFGS optimization algorithm for single-frame phase-only hologram generation 9,10 , and proposes a novel time-multiplexed multi-frame holograms generation method using L-BFGS optimization, called Multi-Frame Holograms Batched Optimization (MFHBO), to produce better reconstruction quality than the existing OSPR and AD-OSPR methods.…”

“…The classic phase-retrieval algorithms include direct binary search 1 , simulated annealing 2 and Gerchberg-Saxton 3 . With the developments in modern numerical optimization methods and increase in computational power, phase retrieval with new numerical optimization methods has also been found in the literature such as: gradient descent 4,5 , its stochastic variations [6][7][8] , and its derivative L-BFGS 9,10 . However, all of these are single-frame hologram generation methods.…”

Information capacity of phase-only computer-generated holograms for holographic displays

“…Fig.6plots the NMSE between the reconstruction of the hologram and the target image against the hologram bit depth for every 5 images (as there are a total of 800 images, only every 5 images are shown in the plot to avoid overcrowding, where the full data can be accessed from the published research data18 ). It can be observed that, for each target image, the NMSE between the resulting reconstructions and their according target images decreases as the hologram bit depth increases.…”