Scaling of Three-Dimensional Computer-Generated Holograms with Layer-Based Shifted Fresnel Diffraction  
[PERSON]  
1 State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Tsinghua University, Beijing 100084, China; [EMAIL_ADDRESS]. reconstruct the accurate depth information of the 3D scenes. However, in stereogram-based algorithms, each hogel corresponds to a viewpoint during the rendering procedure, and this means that we need to render multiple times before implementing the CGH algorithm, which would increase the rendering time. Also, the continuity of the motion parallax and occlusion would be affected by the segmentation setup of the CGH. Layer-based techniques are efficient in calculating the CGH, by partitioning the 3D scene in multiple layers, according to the depth information [13; 14; 15; 16]. Fast Fourier transform (FFT)-based diffraction can be used in simulating the wave propagation processes, from the layers to the CGH plane. However, there are sampling restrictions in the current layer-based 3D CGH algorithms. In FFT-based Fresnel diffraction calculations, the sampling interval on the observation plane is proportional to the propagation distance, while the sampling interval would remain unchanged in convolution-based Fresnel algorithm and angular spectrum method [17]. Hence, the flexibility of the holographic 3D display technique, based on the layer-based algorithm, is limited. Although zero-padding and resampling methods can be used to change the sampling rates, this would take a larger amount of memory and a longer computation time, due to more sampling points that need to be processed. A double Fresnel transform algorithm was developed, to calculate the wave propagation, with variable magnification, by splitting the propagation distance into two partial steps, which has been used in generating layer-based 3D CGHs with amplitude modulation [18; 19]. Since there are minimum propagation distances for the two propagation steps, the propagation distance and the adjustable range of the sampling rates are limited. The wavefront recording plane (WRP) method with non-uniform fast Fourier transform (NUFFT) was used to simplify the 3D CGH calculation [20]. Since the WRP needs to be located close to the object points, the overall depth range is limited during reconstruction. Chirp Z-transform with [PERSON]'s algorithm provides a powerful way to perform discrete Fourier transforms with different scaling factors [21; 22], which has been used in the Fourier transform-based Fresnel diffraction calculations. Shifted Fresnel diffraction was introduced to simulate the scalable Fresnel diffraction between shifted parallel planes in computational holography [23]. Later, aliasing-reduced shifted and scaled (ARSS) Fresnel diffraction was introduced in the algorithm, to reduce aliasing by a band-limiting function [24], which was also used in lenseless holographic projections [25]. Similar algorithms was also introduced in digital holography to perform magnified reconstructions of the digital holograms [26]. However, the above applications, based on the scaling diffractions, were mainly focused on 2D reconstructions. Also, the sampling technique in 3D CGH still needs further investigation. In this work, layer-based shifted Fresnel diffraction is implemented to calculate the wave propagation from the partitioned layers to the CGH plane, with adjustable sampling rates. Shifted Fresnel diffraction can simulate the wave propagation between parallel planes with scalable sampling, by using a Bluestein transform. This layer-based processing with scalable diffraction calculation provides an effective way of generating a CGH, with flexible sampling of the 3D scene. By implementing a scalable diffraction calculation into layer-based processing for 3D CGH generation, the reconstructed 3D images are zoomable without an additional optical zoom module. The sampling parameters of the reconstructed 3D images can be adjusted accordingly.