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A novel method for the procurement of full-color three-dimensional (3-D) images of real objects has been developed. This method is based on extracting information from 3-D Fourier spectra, which are calculated from several projection images recorded using a white light source. 3-D Fourier spectra for three colors were obtained separately for projection images recorded with a color-CCD camera. Three computer-generated holograms (CGHs) were then synthesized from those Fourier spectra. The resulting numerically and optically reconstructed full-color images are presented.
©2004 Optical Society of America
1. Introduction
Though there are many hologram applications, CGH [1] is one of the most important among them because it can be used to reconstruct virtual objects and modulate wavefronts arbitrarily. Moreover, the development of computers has made the synthesis of CGHs easier. Various kinds of CGHs have been reported such as the Lohmann type [1], kinoform [2] and complex amplitude modulation [3] in order to create virtual three-dimensional objects. There are some methods that make use of projection processes for wide viewing-angles [4, 5]. These methods require calculations of the projections for virtual objects. Abookasis [6] and Sando et. al. [7] proposed new methods for creating real existing objects, as opposed to virtual objects, using projection processes and computational processing. In these methods, CGHs are synthesized from several projection images recorded with white light. The absence of a coherent light source in these methods makes them suitable for CGHs of real existing objects.
The full-color reconstruction of 3-D objects is also a popular research topic. Using the super-position of three rainbow CGHs corresponding to red, green and blue components, full-color 3-D objects can be reconstructed at the cost of vertical parallax [8]. The space division method for full-color CGH was proposed [9].
In this report, we describe a new method for the 3-D reconstruction of full-color real existing objects based on the previous report [7]. In addition, methods for deciding certain parameters such as angular ranges, angular increments and projection numbers are presented for the purpose of making the magnifications constant for each color. Verification of this method by both numerical and optical reconstruction is also presented.
2. The principle of the method
A brief description of the method used is described here. A higher level of detail can be found in reference [7]. The method used in this report requires several projection images recorded with white light. The optical recording system and the virtual system are shown in Fig. 1 and 2, respectively. O 1(x,y,z) indicates the reflection intensity at the objects’ surfaces. The phase distribution of the 3-D objects is treated as spatially unvaried. Our method is based on the relationship between one projection image recorded in Fig. 1 and the Fourier plane defined by Fig. 2. When 3-D objects are regarded as a collection of sectional planes perpendicular to the z-axis, the complex distribution at the Fourier plane is given by
where λ is the wavelength of the reconstructing wave, which can be set to the color of choice.
A projection image is recorded with the system shown in Fig. 1. To be precise, it is not orthogonal projection but rather imaging of the 3-D objects to the CCD plane that is carried out in this system. However, this imaging can be regarded as orthogonal projection as long as the depth of the 3-D objects is much shorter than the distance from the origin to the CCD camera. In addition, since it is assumed that light is isotropically scattered at the object’s surface, the object distribution becomes mere two-dimensional (2-D) curved surfaces. Therefore, under the condition where hidden surfaces do not exist, a projection image recorded at projection angles (θi,ϕj) in this system is represented as follows:
where O′1(xij,yij,zij) is another representation of O 1 in the CCD coordinate system. This method is based on an extraction in the 3-D Fourier space [7]. According to the principle of 3-D computed tomography (CT), calculating the 2-D Fourier transform of a 2-D orthogonal projection image gives the 2-D sectional Fourier plane in the 3-D Fourier space of the objects. To obtain a part of the 3-D Fourier space of the objects, Eq. (2) is 2-D Fourier transformed into
Here, a given component represented by the following equation is extracted from Eq. (3) to become
where λ is the same wavelength as the one for the virtual system in Fig. 2, and α is the magnification of the z-axis to be described in detail in the following section. For simplicity, the value of α is set to 1, and by making use of the conversion of integral variables and small-angle approximations of trigonometric functions, Eq. (5) is transformed into
where x′ and y′ are defined as x′=2fθi and y′=2fϕj, respectively. As can be seen by comparing Eq. (6) and Eq. (1), the right hand sides are completely identical to each other. However, Eq. (6) has only one value dependent on the projection angles (θi,ϕj), while Eq. (1) is a function of two variables. This means that one component of the 2-D distribution in the Fourier plane in Fig. 2 is extracted from one projection image recorded in Fig. 1. So, when the projection angles are changed along both horizontal and vertical directions two-dimensionally, the distribution in the Fourier plane is completed from several projection images because the extracted component is dependent on the projection angles. Therefore, it is possible to calculate the complex amplitude in the Fourier plane using a coherent light from multiple projection images, to synthesize either a Fourier CGH or a Fresnel CGH [7].
3. Some parameters for full-color reconstruction
3.1. Extension to full-color reconstruction
It is easy to extend this method to full-color reconstruction if the projection images are recorded with a color CCD. The Fourier spectra for three colors are obtainable. As can be seen from Eq. (2) and (3), the wavelength λ has no connection to the recording process or the CT principle. Wavelengths are defined freely in the virtual system (Fig. 2). Therefore, if three wavelengths corresponding to R, G and B components are introduced in the synthesizing process, it is easily possible to perform full-color reconstruction.
3.2. The magnification for each direction
The effect of α is discussed here. When the value of α is not 1, Eq. (6) changes to:
Changing the scale of the z-axis by z′=αz gives another representation of Eq. (7):
This equation shows the relationship between the complex amplitude in the Fourier plane in Fig. 2 and the objects magnified α times in the z direction. Consequently, the magnification in the z direction becomes α. It goes without saying the magnifications in the x and y directions are 1.
It is preferable that the magnification in each direction is either 1 or all equivalent, but in this case it is difficult because of the resolution of projection images. Here, we explain how to determine the value of α in consideration of the discretization. It is assumed that the size of a projection image, the pixel number for it and the maximum projection angles are W×W, N×N and ±θ 0,±ϕ 0, respectively. It is sufficient to consider only the x axis in this description. The maximum spatial frequency under this condition is N/2W, which should correspond to the maximum value for ui j. Therefore, the following relation is formed using Eq. (4):
The z-axis magnification depends on the wavelength λ.
3.3. Projection range for each wavelength
It should be noted that the magnification in the z-direction depends on the wavelength. In fullcolor reconstruction, it is very important that the magnifications are constant in spite of wavelength differences. According to Eq. (9) there are a few ways to accomplish this. However W and N cannot be changed because the difference of W gives the affects of the magnification of x and y axes and N is constant. So in our method the projection range is decided and made proportional to the wavelength in order to make the magnification of the z-axis constant. By doing this, it is possible to synthesize CGHs whose pixel number N, pixel size W/N and the positions of reconstructed images do not depend on wavelengths.
4. Experimental results
4.1. Projection images and designing
Several color projection images were generated with a computer for simplification of the recording process. Nine typical examples are shown in Fig. 3. The size and pixel number of each projection image are 1cm×1cm and 256×256, respectively. Three blocks, each with a heart, club, or spade symbol on the front face, were placed at z=-2.7 mm, 0mm and 2.7 mm, respectively. Since He-Ne (632.8 nm), Nd-YAG (532 nm) and Ar+ (457.9 nm) lasers were
used for the red, green and blue components, respectively, the angular ranges were set to ±16°, ±13.5° and ±11.6°, respectively. Given that the angular increment between two successive projections is 1° for all wavelengths, the projection numbers for the three components were 1089, 841 and 625, respectively. All 1089 colored projection images were then generated with a computer. The magnification of the z-axis, α, changes to about 69.0 under these conditions.
4.2. Numerical reconstruction
Based to the conditions described above and reference [7], three Fresnel CGHs were generated with a distance from the origin of the object space to the hologram plane of 61.7cm for all components. This resulted in three reconstructed symbols at z=43.2 cm, 61.7 cm and 80.2 cm from the CGHs in consideration of the magnification of the z-axis. Reference beams with an incident angle of 1° are used on the three Fresnel CGHs. After that, the size of CGHs is adjusted to a desired one (1cm in our case) by photo-reductions. Three reconstructed images from the three CGHs were mutually superimposed. The simulation results for the different positions where objects should have been reconstructed are shown in Fig. 4. Each of the three different objects can be seen clearly at their corresponding positions without blurring in any direction. Moreover, the adjustment of the magnifications for wavelength differences was successful. The 0th order diffraction beam appears on the left side of Fig. 4(a), a characteristic of amplitude-type CGHs used in this method as opposed to phase-type.
4.3. Optical reconstruction
Figure 5 shows the simplest optical system possible for full-color reconstruction. All parameters are the same as in the numerical reconstruction. The distance between each CGH and the color CCD plane is constant. Figure 6 shows the reconstructed images using this optical system. As can be seen from these images, the 3-D full-color reconstruction was also confirmed by optical reconstruction. The white area appearing on the left side of Fig. 6(a) is the effect of the 0th order diffraction.
5. Conclusion
We have proposed a new method for the 3-D full-color display of real objects. When a color CCD is used in the recording of projection images, three CGHs for three colors can easily be constructed. The discussion of the magnifications including discretization effect was also presented. This method resulted in clearly reconstructed images verifiable by both numerical and optical reconstruction. The ease of acquisition of 3-D full-color information potentially has great value to significant applications such as holographic cameras, CG displays and holographic security.
References and links
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