Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2021; 5(4): 409-420
Published online August 25, 2021 https://doi.org/10.3807/COPP.2021.5.4.409
Copyright © Optical Society of Korea.
Dabin Min, Kyosik Min, Jae-Hyeung Park
Corresponding author: *jh.park@inha.ac.kr, ORCID 0000-0002-5881-7369
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a method to synthesize a color non-hogel-based computer-generated-hologram (CGH) from light field data of a three-dimensional scene with a hologram pixel pitch shared for all color channels. The non-hogel-based CGH technique generates a continuous wavefront with arbitrary carrier wave from given light field data by interpreting the ray angle in the light field to the spatial frequency of the plane wavefront. The relation between ray angle and spatial frequency is, however, dependent on the wavelength, which leads to different spatial frequency sampling grid in the light field data, resulting in color aberrations in the hologram reconstruction. The proposed method sets a hologram pixel pitch common to all color channels such that the smallest blue diffraction angle covers the field of view of the light field. Then a spatial frequency sampling grid common to all color channels is established by interpolating the light field with the spatial frequency range of the blue wavelength and the sampling interval of the red wavelength. The common hologram pixel pitch and light field spatial frequency sampling grid ensure the synthesis of a color hologram without any color aberrations in the hologram reconstructions, or any loss of information contained in the light field. The proposed method is successfully verified using color light field data of various test or natural 3D scenes.
Keywords: Computer-generated hologram, Full-color, Light field
OCIS codes: (090.1000) Aberration compensation; (090.1705) Color holography; (090.1760) Computer holography; (090.1995) Digital holography
We propose a novel method that synthesizes a color computer-generated-hologram (CGH) from light field data. CGH plays an important role in three-dimensional (3D) holographic displays [1, 2]. For the hologram synthesis, 3D objects are represented in various forms, and one of them is the light field. Light field data is expressed as a set of spatial and angular rays of light coming from 3D objects [3_{–}7]. The representation of these rays is equivalent to an array of views that look at the 3D object from various directions. Creating holograms from light field data has the advantage that the light field acquisition of real objects is easily achieved with a light field camera [4]. Another advantage of the light field CGH is that scene details such as occlusion and material reflection properties are already included in the light field data, and they can be reflected in the hologram with proper processing.
Synthesizing holograms from light field data has been researched for decades [5_{–}8]. Most of the conventional methods divide the holographic plane into small areas called hogels [9, 10]. Each hogel is processed with a corresponding view and the processed hogels are assembled together, completing the hologram. These hogel based methods, however, generally have a limitation on the number of hogels because of the tradeoff relationship between the number of hogels and the number of pixels in each hogel at a given hologram resolution. This hogel number limitation reduces the maximum spatial resolution of the reconstructed 3D images. Another limitation of the hogel based methods is the phase mismatch. Since reconstruction from each hogel has phase mismatch with the one from the neighboring hogel, continuous wavefront over the hologram plane cannot be reproduced.
Recently, a non-hogel-based CGH method has been introduced [11]. The non-hogel-based CGH method processes all the views in the light field data globally, solving the shortcomings of the hogel-based method. Applicability of arbitrary carrier wave or phase distribution on the 3D object surface is another advantage, enabling optimization of the hologram for each specific application. Although the requirement of a densely sampled large amounts of light field data was problematic in the initial proposal of the non-hogel-based CGH [11], a more efficient calculation scheme has been developed later, reducing the computation time significantly [12].
Despite the advantages, the non-hogel-based CGH has only been demonstrated for a single color. The non-hogel-based CGH method involves sampling and manipulation of the light field data in a spatial frequency domain. Different wavelength results in different spatial frequency for the same ray direction. As red, green, and blue color channels of each view in the light field data share the same field of view (FoV) and observing direction, each color channel has a different spatial frequency sampling grid whose efficient processing is not trivial and has not been developed yet.
In this paper, we propose a method for calculating color non-hogel based CGH. The proposed method interpolates the two-dimensional (2D) spatial frequency grid of green and blue color channels using the sampling interval of a red color channel. Meanwhile, the spatial frequency range is set by the blue channel and the red and green channels are zero-padded to match the range. The proposed spatial frequency grid manipulation using the red sampling interval and blue grid range ensures efficient processing without loss of information. The proposed method is verified by comparing the reconstructions of color CGHs synthesized with and without the proposed method.
The non-hogel-based CGH technique synthesizes complex fields from light field data of the 3D scenes. In a plane, the light field can be denoted by
where λ is the wavelength. The light field is now represented by
Hologram synthesis from the light field
where (
where
Equation (4) indicates that each (
Implementation of the non-hogel-based CGH explained in the previous section involves discrete signals with adequate sampling. The wavelength dependency of the sampling requirement makes the color hologram synthesis non-trivial.
Suppose that a light field data
where Eq. (5) is deduced from the maximum diffraction angle of the hologram with a pixel pitch ∆
Another sampling consideration is for the
where
Suppose a color light field data is given with the same FoV for all color channels. The red, green, and blue channels of the light field data
where λ_{R}, λ_{G}, and λ_{B} are the wavelengths of the red, green, and blue color channel. The sampling interval of the spatial frequency can also be obtained by dividing the range in Eq. (8) by the number of samples
Equations (8) and (9) indicate that the spatial frequency range and the sampling interval in
To synthesize a color hologram without information loss of the light field, the proposed method sets a single common hologram pixel pitch and performs 2D interpolation along the
To satisfy three conditions of Eq. (10) together, the hologram pixel pitch is set to be
following the blue condition of the shortened wavelength.
Once the pixel pitch condition is determined, the FoV of each color channel is extended with zero-padded in the proposed method. As mentioned earlier, it is advantageous that the sampling interval ∆
as illustrated in Fig. 1(b). Note that the manipulated FoVs, i.e.,
Finally, each color channel of the light field data
where Eqs. (8), (9), and (12) are used. In Fig. 2 (a), (b), illustrate the extended FoV with zero-padding and the new interpolation of the proposed method, respectively. Note that by the proposed method all three color channels have the same sampling points
The proposed method is verified by synthesizing color holograms from light field data of various 3D scenes. In all calculations, the red, green, and blue wavelengths are set to be λ_{R} = 633 nm, λ_{G} = 532 nm, and λ_{B} = 488 nm. Figure 3 shows light field data of cross target objects. The scene consists of 4 cross targets which are in the same
Figure 6 shows the light field data of a 3D scene with multiple depths. The scene consists of 4 × 4 resolution targets located at different depths from
Finally, the proposed method is tested with a natural scene with continuous depth 3D objects. A 3D scene model is loaded to a software Blender and orthographic images are rendered as shown in Fig. 9. Each orthographic view has 1563 × 1563 pixels with ∆
In this paper, we propose a method to calculate a full-color non-hogel-based CGH from light field data of 3D scenes. All color channels of the light field data share the same FoV. But the wavelength dependency of the angle equivalent spatial frequency makes their sampling condition different, making a synthesis of color hologram with a shared pixel pitch complicated. The proposed method resamples the light field data along the spatial frequency axes such that all red, green, and blue color channels have the same sampling grid. The sampling range and interval along the spatial frequency axes are set by the ones of blue and red color channels, respectively, which ensure no loss of information. The proposed method is verified using various 3D scenes including single depth plane test target, multiple plane test targets at different depths, and natural continuous 3D objects distributed in space. The reconstructions of the color holograms synthesized by the proposed method show clear focus of the 3D objects at their depths without any color aberrations, successfully confirming the feasibility of the proposed method.
This work was supported by INHA UNIVERSITY Research Grant.
Curr. Opt. Photon. 2021; 5(4): 409-420
Published online August 25, 2021 https://doi.org/10.3807/COPP.2021.5.4.409
Copyright © Optical Society of Korea.
Dabin Min, Kyosik Min, Jae-Hyeung Park
Department of Electrical and Computer Engineering, Inha University, Incheon 22212, Korea
Correspondence to:*jh.park@inha.ac.kr, ORCID 0000-0002-5881-7369
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a method to synthesize a color non-hogel-based computer-generated-hologram (CGH) from light field data of a three-dimensional scene with a hologram pixel pitch shared for all color channels. The non-hogel-based CGH technique generates a continuous wavefront with arbitrary carrier wave from given light field data by interpreting the ray angle in the light field to the spatial frequency of the plane wavefront. The relation between ray angle and spatial frequency is, however, dependent on the wavelength, which leads to different spatial frequency sampling grid in the light field data, resulting in color aberrations in the hologram reconstruction. The proposed method sets a hologram pixel pitch common to all color channels such that the smallest blue diffraction angle covers the field of view of the light field. Then a spatial frequency sampling grid common to all color channels is established by interpolating the light field with the spatial frequency range of the blue wavelength and the sampling interval of the red wavelength. The common hologram pixel pitch and light field spatial frequency sampling grid ensure the synthesis of a color hologram without any color aberrations in the hologram reconstructions, or any loss of information contained in the light field. The proposed method is successfully verified using color light field data of various test or natural 3D scenes.
Keywords: Computer-generated hologram, Full-color, Light field
We propose a novel method that synthesizes a color computer-generated-hologram (CGH) from light field data. CGH plays an important role in three-dimensional (3D) holographic displays [1, 2]. For the hologram synthesis, 3D objects are represented in various forms, and one of them is the light field. Light field data is expressed as a set of spatial and angular rays of light coming from 3D objects [3_{–}7]. The representation of these rays is equivalent to an array of views that look at the 3D object from various directions. Creating holograms from light field data has the advantage that the light field acquisition of real objects is easily achieved with a light field camera [4]. Another advantage of the light field CGH is that scene details such as occlusion and material reflection properties are already included in the light field data, and they can be reflected in the hologram with proper processing.
Synthesizing holograms from light field data has been researched for decades [5_{–}8]. Most of the conventional methods divide the holographic plane into small areas called hogels [9, 10]. Each hogel is processed with a corresponding view and the processed hogels are assembled together, completing the hologram. These hogel based methods, however, generally have a limitation on the number of hogels because of the tradeoff relationship between the number of hogels and the number of pixels in each hogel at a given hologram resolution. This hogel number limitation reduces the maximum spatial resolution of the reconstructed 3D images. Another limitation of the hogel based methods is the phase mismatch. Since reconstruction from each hogel has phase mismatch with the one from the neighboring hogel, continuous wavefront over the hologram plane cannot be reproduced.
Recently, a non-hogel-based CGH method has been introduced [11]. The non-hogel-based CGH method processes all the views in the light field data globally, solving the shortcomings of the hogel-based method. Applicability of arbitrary carrier wave or phase distribution on the 3D object surface is another advantage, enabling optimization of the hologram for each specific application. Although the requirement of a densely sampled large amounts of light field data was problematic in the initial proposal of the non-hogel-based CGH [11], a more efficient calculation scheme has been developed later, reducing the computation time significantly [12].
Despite the advantages, the non-hogel-based CGH has only been demonstrated for a single color. The non-hogel-based CGH method involves sampling and manipulation of the light field data in a spatial frequency domain. Different wavelength results in different spatial frequency for the same ray direction. As red, green, and blue color channels of each view in the light field data share the same field of view (FoV) and observing direction, each color channel has a different spatial frequency sampling grid whose efficient processing is not trivial and has not been developed yet.
In this paper, we propose a method for calculating color non-hogel based CGH. The proposed method interpolates the two-dimensional (2D) spatial frequency grid of green and blue color channels using the sampling interval of a red color channel. Meanwhile, the spatial frequency range is set by the blue channel and the red and green channels are zero-padded to match the range. The proposed spatial frequency grid manipulation using the red sampling interval and blue grid range ensures efficient processing without loss of information. The proposed method is verified by comparing the reconstructions of color CGHs synthesized with and without the proposed method.
The non-hogel-based CGH technique synthesizes complex fields from light field data of the 3D scenes. In a plane, the light field can be denoted by
where λ is the wavelength. The light field is now represented by
Hologram synthesis from the light field
where (
where
Equation (4) indicates that each (
Implementation of the non-hogel-based CGH explained in the previous section involves discrete signals with adequate sampling. The wavelength dependency of the sampling requirement makes the color hologram synthesis non-trivial.
Suppose that a light field data
where Eq. (5) is deduced from the maximum diffraction angle of the hologram with a pixel pitch ∆
Another sampling consideration is for the
where
Suppose a color light field data is given with the same FoV for all color channels. The red, green, and blue channels of the light field data
where λ_{R}, λ_{G}, and λ_{B} are the wavelengths of the red, green, and blue color channel. The sampling interval of the spatial frequency can also be obtained by dividing the range in Eq. (8) by the number of samples
Equations (8) and (9) indicate that the spatial frequency range and the sampling interval in
To synthesize a color hologram without information loss of the light field, the proposed method sets a single common hologram pixel pitch and performs 2D interpolation along the
To satisfy three conditions of Eq. (10) together, the hologram pixel pitch is set to be
following the blue condition of the shortened wavelength.
Once the pixel pitch condition is determined, the FoV of each color channel is extended with zero-padded in the proposed method. As mentioned earlier, it is advantageous that the sampling interval ∆
as illustrated in Fig. 1(b). Note that the manipulated FoVs, i.e.,
Finally, each color channel of the light field data
where Eqs. (8), (9), and (12) are used. In Fig. 2 (a), (b), illustrate the extended FoV with zero-padding and the new interpolation of the proposed method, respectively. Note that by the proposed method all three color channels have the same sampling points
The proposed method is verified by synthesizing color holograms from light field data of various 3D scenes. In all calculations, the red, green, and blue wavelengths are set to be λ_{R} = 633 nm, λ_{G} = 532 nm, and λ_{B} = 488 nm. Figure 3 shows light field data of cross target objects. The scene consists of 4 cross targets which are in the same
Figure 6 shows the light field data of a 3D scene with multiple depths. The scene consists of 4 × 4 resolution targets located at different depths from
Finally, the proposed method is tested with a natural scene with continuous depth 3D objects. A 3D scene model is loaded to a software Blender and orthographic images are rendered as shown in Fig. 9. Each orthographic view has 1563 × 1563 pixels with ∆
In this paper, we propose a method to calculate a full-color non-hogel-based CGH from light field data of 3D scenes. All color channels of the light field data share the same FoV. But the wavelength dependency of the angle equivalent spatial frequency makes their sampling condition different, making a synthesis of color hologram with a shared pixel pitch complicated. The proposed method resamples the light field data along the spatial frequency axes such that all red, green, and blue color channels have the same sampling grid. The sampling range and interval along the spatial frequency axes are set by the ones of blue and red color channels, respectively, which ensure no loss of information. The proposed method is verified using various 3D scenes including single depth plane test target, multiple plane test targets at different depths, and natural continuous 3D objects distributed in space. The reconstructions of the color holograms synthesized by the proposed method show clear focus of the 3D objects at their depths without any color aberrations, successfully confirming the feasibility of the proposed method.
This work was supported by INHA UNIVERSITY Research Grant.