Multi-depth hologram generation using stochastic gradient descent algorithm with complex loss function

Abstract

The stochastic gradient descent (SGD) method is useful in the phase-only hologram optimization process and can achieve a high-quality holographic display. However, for the current SGD solution in multi-depth hologram generation, the optimization time increases dramatically as the number of depth layers of object increases, leading to the SGD method nearly impractical in hologram generation of the complicated three-dimensional object. In this paper, the proposed method uses a complex loss function instead of an amplitude-only loss function in the SGD optimization process. This substitution ensures that the total loss function can be obtained through only one calculation, and the optimization time can be reduced hugely. Moreover, since both the amplitude and phase parts of the object are optimized, the proposed method can obtain a relatively accurate complex amplitude distribution. The defocus blur effect is therefore matched with the result from the complex amplitude reconstruction. Numerical simulations and optical experiments have validated the effectiveness of the proposed method.

High-contrast, speckle-free, true 3D holography via binary CGH optimization

“…First, since we take TM to reduce the speckle noise, a proportionally increased computational burden is an remaining issue for real-time implementation. We believe it will be resolved in the near future with the rapidly developing computational holographic works such as complex amplitude optimization 34 , and the deep-learning hologram generation 5 , 6 , 35 – 37 . Note that the computing cost for optimizing a single B-SGD hologram has similar cost for a conventional single 8-bit phase-only SGD hologram.…”

Automotive Holographic Head‐Up Displays

“…[ 112 ] This algorithm‐only approach was utilized to display replay field results with multiple depths by using a stochastic gradient decent (SGD) algorithm with complex loss function. [ 129 ] Previous algorithms calculated the loss function generally by comparing the obtained amplitude of the reconstructed image with the target images at different depth planes. [ 130 ] Hence, the optimization time is a crucial challenge.…”

“…Hence, the optimization time of the algorithm is close to the single‐depth optimization time. [ 129 ] Fourier and Fresnel methods have been used to generate CGHs. [ 131 ] The Fresnel method generates arbitrary large images with object depths.…”

Compact Computational Holographic Display

“…From Eqs 8, 9, it is clear that we need to optimize real-valued loss functions with complex variables, that is, f(z): C → R. However, a nonconstant real-valued function of a complex variable is not (complex) analytic and therefore is not differentiable. Generally, the same real-valued function viewed as a function of the real-valued real and imaginary components of the complex variable can have a (real) gradient when partial derivatives are taken with respect to those two (real) components, that is, f(z) f(x, y): R 2 → R. However, taking the real or imaginary part of a complex number (Peng et al, 2020;Chen et al, 2021), do not satisfy the Cauchy-Riemann equations and cannot be addressed via a complex differentiation. In this work, we use the Wirtinger derivative (Remmert, 1991;Kreutz-Delgado, 2009), which can rewrite a real differentiable function f(z) as two-variable holomorphic function f(z, z*), where z = x + jy and z* = x − jy.…”