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Sha et al.

IS&T Members.

In a computer-generated holographic projection system, the image is reconstructed via the diffraction of light from a spatial light modulator. In this process, several factors could contribute to non-linearities between the reconstructed and the target image. This paper evaluates the non-linearity of the overall holographic projection system experimentally, using binary phase holograms computed using the one-step phase retrieval (OSPR) algorithm, and then applies a digital pre-distortion (DPD) method to correct for the non-linearity. Both a notable increase in reconstruction quality and a significant reduction in mean squared error were observed, proving the effectiveness of the proposed DPD-OSPR algorithm.

In a computer-generated holographic projection system, images are generated via the controlled diffraction of coherent light, which is modulated by a spatial light modulator (SLM). Contemporary SLM’s can only modulate either phase or amplitude, hence algorithms are needed to compute amplitude-only or phase-only holograms. Among those, phase-only holograms are generally preferred due to inherently higher energy efficiency as there is no intentional blockage of light during modulation. Classic phase retrieval algorithms include direct binary search [

There are several factors contributing to the non-linearities between the reconstruction of hologram and the target image, including the calculation and quantization of the hologram, modulation of the light and imperfections in the optical setup. This article proposes the digital pre-distorted one-step phase retrieval (DPD-OSPR) algorithm. The digital pre-distortion (DPD) is carried out on the holographic projection system using holograms computed by the OSPR algorithm by measuring the non-linearity experimentally and applying the corresponding pre-distortion curve on target images. DPD can be done via a one-to-one correction curve or a lookup table (LUT) which allows the relationship between the input and output to be adjusted without any heavy computation.

The intuition of the proposed DPD-OPSR algorithm for CGH comes from the gamma correction method for conventional displays, such as cathode-ray tube (CRT) monitor [

The DPD-OSPR method builds on the OSPR algorithm [

The OSPR algorithm is described in Algorithm 1, where the propagation function

The holographic projector used in this experiment is a Fourier projection system developed by Freeman [

Optical setup [

Mechanical components [

The mechanical components are listed in Figure

To determine the DPD curve of the holographic projection system, the non-linearity needs to be measured first. The hologram in Figure

Determining the DPD curve. (a) Input linear grey-scale ramp. (b) Corresponding CGH of (a) with 24-subframe binary phase encoding. (c) Holographic projection replay field of (b). (d) Plot of non-linearity measurement and corresponding pre-distortion curve.

The projection output of the linear grey-scale ramp was then captured and cropped as shown in Fig.

A high degree of non-linearity is exhibited. By taking the mean of the square of the error between the measured output (blue line) and the linear reference (green dashed line), the normalized mean squared error (NMSE) of the measured output was calculated to be 0.0858. To correct for the non-linearity, the DPD curve (red line) was formed by inverting the smoothed non-linearity curve (yellow line) in Fig.

Validation of DPD curve on the grey-scale ramp. (a) Pre-distorted ramp. (b) Corresponding CGH of (a) with 24-subframe binary phase encoding. (c) Holographic projection replay field of (b). (d) Non-linearity measurement after DPD.

Subsequently, the DPD curve (red line in Fig.

Non-linearity results before and after DPD.

NMSE | Percentage | |
---|---|---|

Before DPD | 0.0858 | 100% |

After DPD | 0.0039 | 4.55% |

Thus, as demonstrated in Table

To qualitatively demonstrate the effectiveness of our approach, we project a simple test pattern of a graduated ramp test pattern consisting of 10-step strips in Figure

Application of DPD on the 10-step strips. (a) 10 strips with equal step of pixel value. (b) CGH of (a). (c) Holographic projection replay field of (b). (d) After DPD of (a). (e) CGH of (d). (f) Holographic projection replay field of (e).

Application of DPD on two sample real-word images. (a) Sample image 1: City Scene [

Then the DPD curve was applied to the two sample images as shown in Figure

Projection output of the two sample images before and after DPD. (a) Replay field of Sample image 1 before DPD (NMSE = 0.06139). (b) Replay field of Sample image 2 before DPD (NMSE = 0.04309). (c) Replay field of Sample image 1 after DPD (NMSE = 0.04920). (d) Replay field of Sample image 2 after DPD (NMSE = 0.03635).

The replay fields of the holographic projection of original images are shown in Fig.

As shown in Fig.

In Fig.

DPD results for sample images.

Sample image 1 | NMSE | Percentage |
---|---|---|

Before DPD | 0.06139 | 100% |

After DPD | 0.04920 | 80.15% |

Sample image 2 | MSE | Percentage |

Before DPD | 0.04309 | 100% |

After DPD | 0.03635 | 84.36% |

Hence, as summarised in Table

Finally, as the DPD is a one-to-one mapping, the computation time is negligible. In practice, the computational overhead is too small to be measured against randomness between subsequent runs. DPD can also be further accelerated in hardware using a hardware LUT, so that the DPD can be carried out instantly. This approach is widely adopted in gamma correction for displays.

The non-linearity between target image and reconstructed image was measured for the overall holographic projection system by projecting a linear grey-scale ramp. Then DPD was applied to the grey-scale ramp and successfully reduced the MSE by 95.45%. To examine its effectiveness on real-world images, the DPD method was applied on two sample images, it was observed that more details were shown in the replay field after DPD, and the MSE’s of the two example images were reduced by 19.86% and 15.64%. As the DPD is a one-to-one mapping, the extra computation required is negligible. Thus, we have demonstrated the effectiveness of the proposed DPD-OSPR method to improve reconstruction quality on the existing OSPR algorithm while still keeping its ability for real-time holography.

This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) [EP/S022139/1].