Fast generation of full analytical polygon-based computer-generated holograms

Ya-Ping Zhang; Fan Wang; Ting-Chung Poon; Shuang Fan; Wei Xu

doi:10.1364/OE.26.019206

1. Introduction

Holography is regarded as an ideal technique for 3D display because it provides all the physiological depth cues without any conflict [1–3]. Computer-generated holography can provide 3-D dynamic [4], full-color holographic [5], large-scale display with high definition [6] as well as 3-D watermarking [7]. However, the computation time due to the massive data involved has become a problem for real-time generation and display. Recently, many computational algorithms directing to improve calculation efficiency have been presented. In general, CGH algorithms can be classified into a point-based method and a polygon-based method. In the former method, a three-dimensional (3-D) object is treated as a collection of mass points and each point is subject to diffraction calculation [8–18]. In general, for the point-based method, the number of sampling points required reach to millions and that slows down the calculation speed tremendously. Ray tracing [11–13], look-up table (LUT) [14–16], use of recurrence formulas [17] and wave-front recording plane [18–20] have been continuously researched for the enhancement of computational efficiency and holographic image quality. The polygon-based method, on the other hand, is calculated by dividing the 3D object into many micro-polygons, for example the use of triangles [21–29].

Undoubtedly, the polygon-based method reduces a vast amount of sampling units compared with the point-based method. The affine transformation based on triangle meshes has been intensively proposed. The frequency spectrum in the hologram plane of every sub-triangle facet can be derived analytically and with a single inverse Fourier transform, we can obtain the light field on the hologram plane, thereby saving plenty of computation time. Early research works with affine transformations involved frequency analysis with rotation of the different tilted planes [30–32], and subsequently angular spectrum theory can be used for the calculation of the object light field to the hologram plane. The algorithm proposed by Kim et al. [22] devised a diffusive surface model for producing texture and shade effects and Pan et al. [26] presented an analytical brightness compensation method to describe the influence of the fabrication of the hologram.

In 2008, Lukas Ahrenberg et al. [23] proposed a novel 2D full-analytical method based on affine transformation between an arbitrary triangle and a primitive triangle, which is able to enhance the computational speed as compared to that of traditional models. It first defines a primitive triangle in the local coordinate system, $f_{Δ} (x, y)$ , and then computes the 2D Fourier transform of $f_{Δ}$ . Next, it transforms an arbitrary triangle into a primitive triangle in the local coordinates. A formula error, however, exists in the final result. The error has been recognized as Zhang et al. derived a correct analytical expression partially [34] and in Section 2, we provide a complete correct expression. In other related research efforts, Pan et al. has developed 3D transformation between an arbitrary triangle and a primitive triangle [27], in which transformation occurs in the frequency domain and adopts interpolated remapping for new frequencies. This 3D affine matrix ( $3 \times 4$ ) contains all shifting, rotational and scaling transformation information. They have further defined a so-called pseudo-inverse $4 \times 4$ matrix that can easily be affine transformed [28]. Unfortunately, the matrix is not accurate enough. In the present paper, we have presented a method that avoids the use of this matrix. The use of conformal geometry based on triangle polygons has also been explored as Zhang et al. obtained successfully the full-analytical optical field distribution for a 3D object and implemented the reconstruction of the achieved computer-generated hologram [35].

In this paper, we derive a complete transformational frequency spectrum of an arbitrary triangle in the three-dimensional coordinate system based on the 2D affine transformation theory proposed by Ahrenberg et al. [23]. The formula error existing in his original theory is first indicated and rectified. The proposed method to acquire computer-generated holograms includes two key steps: rotation in the space domain and affine transformation in two dimensions (2Ds), which are expressed by two matrices. These two key steps are devoted to obtain a 3D transformation matrix from an arbitrary triangle to the primitive triangle with scrupulous derivation as compared to the works of Pan et al. [27, 28]. In Pan et al.’s method, the primitive triangle is defined to locate at $z^{'} = 0$ , i.e., the plane of the local coordinate system, in order to get a solvable transformation matrix, but it is not rigorous enough theoretically and that will be discussed in Section 3. The ultimate frequency spectrum of an arbitrary triangle has been derived using 3D transformation matrices obtained by the proposed method. We have employed some very complex 3D objects with thousands of triangles by utilizing our theory through simulations and optical results to verify the correctness of our proposed method. Besides, in order to obtain reconstructed images with correct shadowing relationship, back-face culling is also considered in the development. Computation time has shown to be decreased owing to the elimination of hidden surfaces through our new algorithm.

The paper is organized as follows. In Section 2, we contrast the already presented triangle-based method to that proposed by our work. In Section 3, the specific theory derivation and implementation approach is described. In Section 4, we show three 3D object models, their computer-generated holograms, simulated reconstruction images and experimental results by a spatial light modulator (SLM). Moreover, recognizing hidden polygons with the method of outer normal will be briefly discussed in this section. We conclude the innovation made in contrast to previous methods and discuss potential ways of speeding up the proposed method in Section 5.

2. Comparison with previous methods

There are many techniques based on the polygon-based method [22–29]. Their common goal is to get the frequency distribution of each triangle at the hologram plane, then sum over all the frequency spectra of all the triangles, and finally take the inverse Fourier transformation to obtain optical field distribution of the whole object.

The traditional method based on triangle meshes has two pivotal procedures before diffraction calculation. Making coordinate transformations through rotation and translation for a spatially tilted triangle until it is parallel to the hologram plane. After that, the spectrum is obtained by Fourier transform and the frequency distribution at the hologram plane is obtained by Fresnel diffraction through the angular spectrum method. It costs much calculation time because we need to take the Fourier transform for each triangle and perform interpolation for new frequencies so as to obtain equispaced sampling when taking fast Fourier transform (FFT).

In the affine transformation method proposed by Ahrenberg et al. [23], an arbitrary triangle is first undergone affine transform to a primitive triangle, that is to say, the coordinates of an arbitrary triangle are described by those of a primitive triangle using affine transformation relation. Because the spectrum of the primitive triangle has been obtained analytically, the spectrum of an arbitrary triangle can now be expressed in terms of the spectrum of a primitive triangle. Next the spectrum of an arbitrary triangle is rotated to make it parallel to the hologram plane. Computational efficiency is enhanced due to the availability of the analytical spectrum of the primitive triangle as the time consumed in FFT and its interpolation has been economized. In that paper, however, a formula error occurs due to the different coordinate assignments when the frequency expression of the arbitrary triangle is presented directly through citing Bracewell’s affine theorem [36]. According to Bracewell’s, the affine transform plane coordinates, $x'$ and $y'$ , of the primitive triangle and the global coordinates, x and y, of the arbitrary triangle are related by

[\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} a & b \\ d & e \end{matrix}] \cdot [\begin{matrix} x \\ y \end{matrix}] + [\begin{matrix} c \\ f \end{matrix}] .

However, in Ahrenberg et al.’s paper, the affine transform plane coordinates, x and y, of the primitive triangle and the arbitrary triangle coordinates, s and t, are given by

[\begin{matrix} s \\ t \end{matrix}] = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] \cdot [\begin{matrix} x \\ y \end{matrix}] + [\begin{matrix} a_{13} \\ a_{23} \end{matrix}] .

In other words, in Eq. (1), the affine coordinates are expressed in terms of the arbitrary triangle coordinates, whereas in Eq. (2), the arbitrary triangle coordinates are expressed in terms of the affine coordinates. Therefore, on the basis of symbols definition in Eq. (2), Eq. (19) in literature [23] should be rectified to be

F_{Γ} (u_{s}, v_{t}) = J \cdot e^{- j \cdot 2 π (a_{13} u_{s} + a_{23} v_{t})} \cdot F_{Δ} (a_{11} u_{s} + a_{21} v_{t}, a_{12} u_{s} + a_{22} v_{t}),

where

u_{s}

and

v_{t}

are spatial frequencies corresponding to s and t, respectively, and

F_{Γ}

is the Fourier transform of the arbitrary triangle

Γ

on the st-plane and

F_{Δ}

is the Fourier transform of the primary triangle

Δ

on the xy-plane, and finally

J

is the Jacobian of the transformation. Equation (3) is derived in Appendix A. Note that all symbols of variables, such as s, t, x, y, a_ij, follow those of the definition of corresponding references [23] and [36].

Moreover, there are other problems in [23]:

$\cdot$ there should be mapping between (1,0) and (s₂, t₂); it incorrectly states (0, 1)
$\cdot$ the constant J used in Eq. (19) is different from J defined in Eq. (16)
$\cdot$ there should be (a₂₂u – a₂₁v)/J in the first argument instead of incorrect (a₂₂u – a₁₂v)/J to $F_{Δ}$ in Eq. (19)
$\cdot$ there should be $f_{Γ} (s, t) = f_{Δ} (x, y)$ in Eq. (18) instead of $f_{Γ} (x, y) = f_{Δ} (s, t)$

The three-dimensional affine theory by Pan et al. [27, 28] sets up a transformation matrix that contains all the information on transformations. The primitive triangle in the local coordinates and the arbitrary triangle in the global coordinates are connected by the following 4 × 4 matrix:

T = [\begin{matrix} a_{11} & a_{12} & a_{13} & t_{1} \\ a_{21} & a_{22} & a_{23} & t_{2} \\ a_{31} & a_{32} & a_{33} & t_{3} \\ 0 & 0 & 0 & 1 \end{matrix}],

where a_ij as well as below symbols are the definition in the literature [27,28]. The coordinates of the arbitrary triangle, (x, y, z), are substituted by the coordinates of the primitive triangle,

(x', y^{'}, z')

,

{\begin{matrix} x = a_{11} x^{'} + a_{12} y^{'} + a_{13} z^{'} + t_{1} \\ y = a_{21} x^{'} + a_{22} y^{'} + a_{23} z^{'} + t_{2} \\ z = a_{31} x^{'} + a_{32} y^{'} + a_{33} z^{'} + t_{3} \end{matrix} .

The matrix, T, could be obtained by

T = B C^{- 1}

, where in the literature [27] C and B are denoted as

[\begin{matrix} x^{'} & y^{'} & z^{'} & 1 \end{matrix}]

and

[\begin{matrix} x & y & z & 1 \end{matrix}]

, which are the vertex vectors of the primitive and the arbitrary triangle, respectively. Obviously, matrix C is not invertible. In order to overcome this difficulty, the authors denote a so-called pseudo-inverse matrix C^†. However, solving the matrix T in MATLAB might not be accurate enough with this newly created pseudo-inverse matrix as it possibly provides computational hidden issues leading to the problem of inaccuracy. Furthermore, the primitive triangle is assumed to lie in the plane

z^{'} = 0

, i.e., the plane of the local coordinate system. Equation, (4-b) therefore can be rewritten as

{\begin{matrix} x = a_{11} x^{'} + a_{12} y^{'} + t_{1} \\ y = a_{21} x^{'} + a_{22} y^{'} + t_{2} \\ z = a_{31} x^{'} + a_{32} y^{'} + t_{3} \end{matrix} .

By doing so, the matrix T could be solved accurately by defining C as

[\begin{matrix} x^{'} & y^{'} & 1 \end{matrix}]

and B as

[\begin{matrix} x & y & z \end{matrix}]

, because C and B are invertible at the moment. Matrix T can be described by P as follows:

P = [\begin{matrix} a_{11} & a_{12} & 0 & t_{1} \\ a_{21} & a_{22} & 0 & t_{2} \\ a_{31} & a_{32} & 0 & t_{3} \\ 0 & 0 & 0 & 1 \end{matrix}] .

However, in this case it does not consider the relative position between the global system and the local system as the value of

z'

of the primitive triangle has been set to be zero. And the correct result could not be obtained, if the distance between the two systems is ignored when calculating hundreds of triangles. In the method we propose in the present paper, we consider the distance of the two systems, and consequently the primitive triangle must be located in a specific position as

z'

is non-zero. Finally, the transformation matrix could be obtained precisely without using a pseudo-inverse matrix, which will be discussed in detail in the next section.

Unlike the three ways discussed above, in the proposed method, we derive a full-analytical expression of the spectrum in the hologram plane for a 3D object in a new way. The method includes three core steps: rotation in the spatial domain that is different from that of Ahrenberg et al.’s [23]; 2D affine transform in the rotated triangle plane with consideration of the distance between the local and global coordinate systems, and finally solving accurately the transformation matrix that differs from that in Pan et al.’s [27, 28]. Some complex objects with thousands of polygons are reconstructed successfully using our proposed method. Simulation results indicate that the computation speed of the proposed method is much faster than all the previous methods.

3. Theory

In the global coordinate system $(x, y, z)$ as shown in Fig. 1, the origin is at $P (0, 0, 0)$ . Three points make up an arbitrary spatial triangle that is slanted in regard to xPy-plane, i.e., the hologram plane. The vertex coordinates of the spatial triangle are $A (x_{1}, y_{1}, z_{1}), B (x_{2}, y_{2}, z_{2}), C (x_{3}, y_{3}, z_{3})$ . Its center of gravity is at $O (x_{0}, y_{0}, z_{0})$ . The novelty of the theory involves three steps. First, we perform rotation transformation for the tilted triangle until the normal of the triangle, $\overset{⇀}{n}$ , points to the negative z-direction while keeping the center of gravity fixed. Second, we perform 2-D affine transform on the rotated triangle to a primitive triangle on the plane of the rotated triangle. Third, we compute the light field distribution on the hologram plane by using the angular spectrum method for diffraction.

Fig. 1 Spatial arbitrary triangle in the global coordinate system (x, y, z) that is tilted from the hologram plane.

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3.1 Step 1: rotation around the center of gravity

Rotation transformation of a tilted plane so that the tilted plane becomes parallel with the xPy-plane has been already proposed by Matsushima et al. [33]. In this paper, rotation is done to a tilted triangle. In particular, the center of gravity of the triangle does not move. In other words, the surface normal, $\overset{⇀}{n}$ , of the triangle will be parallel to the z-axis. The arbitrary triangle is translated so that the center of gravity, O, becomes the origin point, P, of the global system. The surface normal is now denoted by ${\overset{⇀}{n}}_{T}$ , as shown in Fig. 2 on the global xyz system. The translational coordinates of the triangle are given by

{\begin{cases} x_{T} = x - x_{0} \\ y_{T} = y - y_{0} \\ z_{T} = z - z_{0} \end{cases} .

The angle between

{\overset{⇀}{n}}_{T}

and

\overset{⇀}{z}

is denoted as

θ

.

M_{1} (x_{0 T}, y_{0 T}, z_{0 T})

is a point on the normal vector

{\overset{⇀}{n}}_{T}

. The projection of M₁ to the plane of xPy is

M_{1}' (x_{0 T}, y_{0 T}, 0)

. The angle between

\overset{⇀}{P M_{1}'}

and

\overset{⇀}{x}

is denoted as

α

.

Fig. 2 Rotating normal ${\overset{⇀}{n}}_{T}$ by $α$ around the z-axis and then rotating by $θ$ around the y-axis.

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Rotation transformations are applied twice so that the surface normal ${\overset{⇀}{n}}_{T}$ first turns around the z-axis with the angle of $α$ and then it is rotated by the angle of $θ$ around the y-axis. Eventually ${\overset{⇀}{n}}_{T}$ is aligned with the direction of the z-axis, ${\overset{⇀}{n}}_{R}$ . The two rotation transformations are given by

T_{R} = R_{y} \cdot R_{z},

where

R_{y}

and

R_{z}

are

R_{z} = | \begin{matrix} \cos α & \sin α & 0 \\ - \sin α & \cos α & 0 \\ 0 & 0 & 1 \end{matrix} |,

and

R_{y} = | \begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix} | .

The conventions on the rotation matrices are that

α

and

θ

have been rotated anti-clockwise on the xy- and xz-planes, respectively. If rotating clockwise,

\sin α

in the matrix of

R_{y}

is

- \sin α

, and

\sin θ

in

R_{z}

is

- \sin θ

^.

The rotated coordinates of the triangle are given by

{[\begin{matrix} x^{'} & y^{'} & z^{'} \end{matrix}]}^{T} = T_{R} \cdot {[\begin{matrix} x_{T} & y_{T} & z_{T} \end{matrix}]}^{T},

where

(x_{T}, y_{T}, z_{T})

are the coordinates of triangle ABC in the xyz global coordinate system,

(x^{'}, y^{'}, z^{'})

are the rotated coordinates that have been rotated around the origin of the global system, as shown in Fig. 2, and superscript

T

signifies the transpose of a vector. Translating

(x^{'}, y^{'}, z^{'})

back to the original location where the center of gravity is

O (x_{0}, y_{0}, z_{0})

, becomes the rotated triangle that has been rotated around the center of gravity. Its final coordinates is

(x_{R}, y_{R}, z_{R})

given by

{\begin{cases} x_{R} = x^{'} + x_{0} \\ y_{R} = y^{'} + y_{0} \\ z_{R} = z^{'} + z_{0} \end{cases} .

As a result, the relation between the coordinates of the rotated triangle, $(x_{R}, y_{R}, z_{R})$ , that have been rotated around the center of gravity and the coordinates of the arbitrary triangle, $(x, y, z)$ , is obtained by substituting Eq. (6) and Eq. (10) into Eq. (9):

[\begin{matrix} x_{R} \\ y_{R} \\ z_{R} \\ 1 \end{matrix}] = [\begin{matrix} a_{11} & a_{12} & a_{13} & x_{0} \\ a_{21} & a_{22} & a_{23} & y_{0} \\ a_{31} & a_{32} & a_{33} & z_{0} \\ 0 & 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} x - x_{0} \\ y - y_{0} \\ z - z_{0} \\ 1 \end{matrix}],

where

a_{i j}

are the elements of rotational matrix

T_{R}

and note that

a_{i j}

here are different from those in Eq. (2)-(5). Now the final rotated triangle has vertices

A_{R}, B_{R}, C_{R}

, as shown in Fig. 3. The triangle is now parallel to the xy-plane. In addition,

z_{1 R} = z_{2 R} = z_{3 R} = z_{0}

. We now denote the triangle on the sQt-plane for presentation convenience, as shown in Fig. 3.

Fig. 3 Affine transformation principle on the 2D plane. The gray triangle is the rotated triangle, and the blue one is the primitive triangle. (a) The relative position of the local and global coordinate systems. (b) Affine transformation for the arbitrary triangle.

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3.2 Step 2: affine transformation in two-dimensional plane

On the basis of 2D affine transformation [23, 37], the analytical frequency spectrum of an arbitrary triangle is now derived. In step 1, two-dimensional coordinates $(x_{R}, y_{R}, z_{0})$ have been obtained through rotating the tilted triangle around the center of gravity, according to Eq. (11). As illustrating in Fig. 3, a 2D rectangle coordinate system, sQt, is denoted after the rotation of the triangle, where it is now parallel to the hologram plane. The origin Q is on the z-axis, which is $z_{0}$ away from the global system origin, P(0, 0, 0).

The three coordinates of the vertices of the triangle are $A_{R} (x_{1 R}, y_{1 R}, z_{0})$ , $B_{R} (x_{2 R}, y_{2 R}, z_{0})$ and $C_{R} (x_{3 R}, y_{3 R}, z_{0})$ on the sQt-plane. Based on the 2D affine theory, there always is a 2D transformational relation between $Δ A_{R} B_{R} C_{R}$ and the primitive triangle whose vertices are $(0, 0, z_{0})$ , $(1, 0, z_{0})$ and $(1, 1, z_{0})$ on the sQt-plane. The affine transformation relationship is denoted as

[\begin{matrix} x_{R} \\ y_{R} \end{matrix}] = [\begin{matrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix}] \cdot [\begin{matrix} s \\ t \end{matrix}] + [\begin{matrix} b_{13} \\ b_{23} \end{matrix}],

where

b_{i j}

are elements of the matrix for the transformation. A pairwise correspondence of the vertices is set as

A_{R} (x_{1 R}, y_{1 R}) \to (0, 0)

,

B_{R} (x_{2 R}, y_{2 R}) \to (1, 0)

and

C_{R} (x_{3 R}, y_{3 R}) \to (1, 1)

. We can represent the translation and the linear affine mapping using a single matrix multiplication given below:

[\begin{matrix} x_{1 R} \\ \begin{matrix} y_{1 R} \\ 1 \end{matrix} \end{matrix} \begin{matrix} x_{2 R} \\ y_{2 R} \\ 1 \end{matrix} \begin{matrix} x_{3 R} \\ y_{2 R} \\ 1 \end{matrix}] = [\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} s_{1} \\ t_{1} \\ 1 \end{matrix} \begin{matrix} s_{2} \\ t_{2} \\ 1 \end{matrix} \begin{matrix} s_{3} \\ t_{3} \\ 1 \end{matrix}],

The above

b_{i j}

matrix is called an affine transformation matrix,

T_{a}

.

T_{a}

can be represented by

T_{a} = [\begin{matrix} x_{1 R} \\ \begin{matrix} y_{1 R} \\ 1 \end{matrix} \end{matrix} \begin{matrix} x_{2 R} \\ y_{2 R} \\ 1 \end{matrix} \begin{matrix} x_{3 R} \\ y_{2 R} \\ 1 \end{matrix}] \cdot {[\begin{matrix} s_{1} \\ t_{1} \\ 1 \end{matrix} \begin{matrix} s_{2} \\ t_{2} \\ 1 \end{matrix} \begin{matrix} s_{3} \\ t_{3} \\ 1 \end{matrix}]}^{- 1},

which could be solved accurately without the introduction of a pseudo-inverse matrix. For keeping consistency of the matrix dimension with

T_{R}

in Eq. (11),

T_{a}

could be rewritten as

T_{a} = [\begin{matrix} b_{11} & b_{12} & 0 & b_{13} \\ b_{21} & b_{22} & 0 & b_{23} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],

where

b_{i j}

are obtained by Eq. (14). Furthermore, Eq. (13) is equivalent to

[\begin{matrix} x_{R} \\ \begin{matrix} y_{R} \\ \begin{matrix} z_{0} \\ 1 \end{matrix} \end{matrix} \end{matrix}] = [\begin{matrix} b_{11} & b_{12} & 0 & b_{13} \\ b_{21} & b_{22} & 0 & b_{23} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} s \\ t \\ z_{0} \\ 1 \end{matrix}],

where z₀ is the separation between P and Q, shown in Fig. 3.

Combining Eq. (15) with Eq. (11), the transformation relation from the spatial tilted triangle to the primitive triangle can be conclusively expressed as

[\begin{matrix} x \\ \begin{matrix} y \\ \begin{matrix} z \\ 1 \end{matrix} \end{matrix} \end{matrix}] = [\begin{matrix} c_{11} & c_{12} & c_{13} & c_{14} + x_{0} \\ c_{21} & c_{22} & c_{23} & c_{24} + y_{0} \\ c_{31} & c_{32} & c_{33} & c_{34} + z_{0} \\ 0 & 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} s \\ t \\ z_{0} \\ 1 \end{matrix}],

where

c_{i j}

are the elements of the comprehensive transform matrix, T, given by

T = T_{R}^{- 1} \cdot T_{a} .

Note that the third column of T is not equal to zeroes like the matrix in Eq. (5-)b) because we have taken the distance from Q to P into account and that is distinguishable from the 3D affine theory proposed by Pan et al. [27, 28].

3.3 Step 3: diffraction computation

Suppose we have an arbitrary $Δ A B C$ which is given by $f (x, y, z)$ and we denote the primitive triangle as $g (s, t, z_{0})$ on the sQt-plane. Their Fourier transforms in the sQt-plane are defined as $F_{s Q t} (u, v)$ with (x, y; u, v) being the transform variables and $G_{s Q t} (u_{0}, v_{0})$ with (s, t; u₀, v₀) being the transform variables, respectively. Now $F_{s Q t} (u, v)$ is defined as

F_{s Q t} (u, v) = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} f (x, y, z = z_{0}) \cdot e^{- j 2 π (u x + v y)} d x d y .

With the change of variables in the above equation, and with the definition of

g (s, t, z_{0}) = {\begin{matrix} 1 \\ 0 \end{matrix} ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ \begin{matrix}  \end{matrix} ​ \begin{matrix}  \end{matrix} ​ \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix} (s, t) lies inside primitive triangle \\ else \end{matrix},

Equation (18) becomes

F_{s Q t} (u, v) = J_{s t} \cdot \int_{0}^{1} \int_{0}^{s} g (s, t, z_{0}) \cdot e^{- j 2 π [u (c_{11} s + c_{12} t + c_{13} z_{0} + c_{14} + x_{0}) + v (c_{21} s + c_{22} t + c_{23} z_{0} + c_{24} + y_{0}]} d t d s,

where

J_{s t}

is the Jacobian of the transformation.

In the diffraction process from sQt-plane to the hologram plane, the use of angular spectrum is applied. The spectrum of an arbitrary triangle at the hologram plane, $F_{h o l o} (u, v)$ , is obtained by

F_{h o l o} (u, v) = e^{- j \cdot 2 π \cdot z \cdot w} \cdot F_{s Q t} (u, v),

where

w

is

w = \sqrt{λ^{- 2} - u^{2} - v^{2}} .

Solving the definite integral in Eq. (19), $F_{h o l o} (u, v)$ can be analytically given by (See Appendix B for steps leading to this final expression)

F_{h o l o} (u, v) = {\begin{array}{l} \frac{J_{s t}}{2} e^{- j \cdot 2 π (p + q)} ......................................................................................... u^{'} = 0, v^{'} = 0 \\ \frac{J_{s t}}{- j \cdot 2 π u^{'}} e^{- j \cdot 2 π (p + q)} \cdot [e^{- j \cdot 2 π u^{'}} + \frac{1}{j \cdot 2 π u^{'}} \cdot (e^{- j \cdot 2 π u^{'}} - 1)] ............................ u^{'} \neq 0, v^{'} = 0 \\ \frac{J_{s t}}{- j \cdot 2 π v^{'}} e^{- j \cdot 2 π (p + q)} \cdot [\frac{1}{- j \cdot 2 π v^{'}} \cdot (e^{- j \cdot 2 π v^{'}} - 1) - 1] .................................. u^{'} = 0, v^{'} \neq 0 \\ J_{s t} \cdot e^{- j \cdot 2 π (p + q)} \cdot [\frac{1}{- j \cdot 2 π v^{'}} + \frac{1}{4 π^{2} u^{'} \cdot v^{'}} \cdot (e^{- j \cdot 2 π u^{'}} - 1)] ............................... \begin{matrix} u^{'} \neq 0, v^{'} \neq 0, \\ u^{'} + v^{'} = 0 \end{matrix} \\ J_{s t} \cdot e^{- j \cdot 2 π (p + q)} \cdot [\frac{1}{- 4 π^{2} v^{'} \cdot (u^{'} + v^{'})} (e^{- j \cdot 2 π (u^{'} + v^{'})} - 1) + \frac{1}{4 π^{2} u^{'} \cdot v^{'}} \cdot (e^{- j \cdot 2 π u^{'}} - 1)] ... else \end{array},

where J_st, p, q, u’ and v’ are given by

{\begin{matrix} J_{s t} = | c_{11} c_{22} - c_{12} c_{21} | \\ u^{'} = c_{11} u + c_{21} v + c_{31} w \\ v^{'} = c_{12} u + c_{22} v + c_{32} w \\ q = (c_{13} u + c_{23} v + c_{33} w) \cdot z_{0} \\ p = (c_{14} + x_{0}) \cdot u + (c_{24} + y_{0}) \cdot v + (c_{34} + z_{0}) \cdot w \end{matrix} .

From the result of the above derivation, the spectrum of an arbitrary triangle on an object can be obtained quickly owing to the analytical equation given by Eq. (22). Note that in Eq. (23) there is the q factor, which is the result of the distance, z₀, of the two coordinate systems we have considered. q can be considered zero in the 3D affine theory of Pan et al., where they have not considered the relative position of the two coordinate systems.

It is well-known that the rotation of the source plane will introduce a carrier frequency onto the hologram’s spectrum [33]. To eliminate this carrier frequency, we introduce the following frequencies:

Δ u = c_{31} / λ, Δ v = c_{32} / λ .

To offset the carrier spatial frequency effect, and from Eq. (23), we have

(u^{″}, v^{″})

{\begin{matrix} u^{″} = u^{'} - Δ u \\ v^{″} = v^{'} - Δ v \end{matrix} .

This introduction of the frequency offset now makes the two spectra, i.e., ( $u v$ - plane) and ( $u^{″} v^{″}$ - plane), have the same origin. The situation is shown in Fig. 4 for two perspectives: one perspective with rotation of $v^{″}$ and $w^{″}$ as shown in Fig. 4(a), and the other with $u^{″}$ and $w^{″}$ as shown in Fig. 4(b). Figure 4 illustrates a phenomenon that only the frequency range that is within the “red stripe” can be recorded in the hologram and hence the frequencies outside this range (missed frequencies) should be zero padded. This treatment of zero-padding has been previously discussed in literature [26, 32, 33, 37, 38].

Fig. 4 Frequency range (colored red stripe) that is recordable on hologram due to source rotation. (a) $v^{″} w^{″}$ -perspective. (b) $u^{″} w^{″}$ -perspective.

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According to the superposition principle, the whole diffraction field of the 3D object at the hologram plane, i.e., the complex hologram, $U_{o b j e c t}$ , is given by adding the optical fields of all the triangles, which is expressed to be

U_{o b j e c t} (x, y) = \sum_{i}^{t r i_{t o t a l}} F^{- 1} (F_{h o l o, i}) .

where

t r i_{t o t a l}

is the number of triangles making up the 3D object, and

F^{- 1}

is the inverse transform operator.

Up to now, we can compute the optical field distribution of an object on the hologram plane with only once inverse Fourier transform given by Eq. (26). The holographic image is reconstructed through either calculating inverse Fourier transform for light field $U_{o b j e c t}$ in simulations or being modulated by a spatial light modulator (SLM) in optical experiments. In the next section we will give the results of simulations and optical experiments for several 3D objects.

4. Simulations and optical experiments

In this section, we mainly describe the simulation and experimental results of three 3D models that are generated by 3d Max software (Autodesk 3ds Max 2014). First, for proving that the proposed method can realize the reconstruction of a hologram for a 3D polygon, a spatially tilted triangle is made into a CGH and reconstructed in different depths. Next, the parameters of more complicated 3D objects are presented. Third, in order to reconstruct a real 3D image with valid shadowing relationship, the principle of obtaining exterior normal to identify backward surfaces will be introduced, and its merits and demerits are discussed. In the end of this section, simulations with MATLAB and experimental results using SLM with a proof-of-concept system are demonstrated for several 3D models. Computation time with previous triangle-based methods is compared with the proposed method. All dimensions of the 3D models are in millimeters unless otherwise stated.

4.1 Calculation of a single tilted triangle using the proposed method

An arbitrary triangle in the 3D spatial domain is the fundamental calculation unit in the triangle-based method. We primarily compute the computer-generated hologram of a spatial triangle to verify the correctness of our proposed technique. Figure 5(a) shows a tilted triangle relative to the hologram plane. The three vertices have different depth (z) coordinates and the hologram plane is set at z = 0. Vertex A is closest to the hologram plane and vertex C is farthest. According to the proposed method, the three-dimensional transformation matrix is obtained by Eq. (17) that has been derived in step 1 and step 2 of section 3. Equation (22) gives the expression of the triangle’s spectrum distribution at the hologram plane. Ultimately, the calculated computer-generated hologram with 1024 × 1024 pixels is generated in Fig. 5(b).

Fig. 5 Simulation results for an arbitrary triangle using the proposed method. (a) Triangle position in 3D space. (b) Diffracted field of the triangle at the hologram plane z = 0, namely, the obtained computer-generated hologram. (c) Reconstruction of the computer-generated hologram at z = 134mm, where vertex C is located. The yellow squares point to enlarged local views. (d) Reconstruction of the computer-generated hologram at z = 86mm, where vertex A is located.

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Figures 5(c) and 5(d) show the reconstructed images at different depths. Figure 5(c) is reconstructed at the distance of z = 134mm, i.e., at the location of vertex C (−4, −4, 134). It can be seen, from the enlarged portions of the figure that vertex C is sharply focused, whereas vertex A is blurry. Similar observations can be made in Fig. 5(d), where reconstruction is made at z = 86mm, showing vertex A is focused sharply with vertex C blurry. Clearly, the proposed method to compute a 3D triangular mesh is verified convincingly.

4.2 Parameters of 3D objects

Three objects with different numbers of polygons are created in 3ds Max, as shown in Figs. 6(a)-6(c). The ‘sphere’ is made of 162 points with 320 triangles meshes, ‘Baymax’, a carton model, has 1632 points with 3060 triangular surfaces, and 3241 points with 6320 triangles make up a ‘teapot’. The parameters in Fig. 6 indicate that the number of points is around half of that of triangles for the objects.

Fig. 6 3D models applied to calculate CGH using the proposed method. (a) A ‘sphere’ with 320 triangles in which green triangles represent those hidden surfaces; in which red and blue meshes are enlarged and shown in (d). (b) A ‘cartoon’ character, Baymax, with 3060 triangles. (c) A ‘teapot’ with 6320 triangles. (d) The blue triangle, $Δ A B C$ , is used to judge whether it is hidden with the method of outer-normal, and the red ones are its adjacent triangles. ${\overset{⇀}{n}}_{o}$ , ${\overset{⇀}{n}}_{i}$ , and ${\overset{⇀}{e}}_{z}$ are outer-normal, inner-normal, and unit vector of the z-axis, respectively. The hologram plane is on the plane z = 0.

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For matching the size of the SLM used in optical experiments, objects need to be pre-treated and scaled down. All objects are projected to the hologram plane that measures from −7.68 mm to 7.68 mm along x, and from −6.14 mm to 6.14 mm along y. The size of the sphere is 12.86 × 9.61 × 21.98 mm³; its depth along the z-axis is from 493.53 to 515.51. The size of Baymax is 12.86 × 9.78 × 18.72 mm³; its depth is from 489.28 mm to 508.00 mm. The size of the teapot is 12.86 × 9.79 × 25.54 mm³; its depth is from 488.15 mm to 513.69 mm. The propagation distance of all the 3D objects is around 500mm to the hologram plane. The wavelength used in diffraction computation is 532nm, and the hologram plane is sampled by 1024 × 1024 pixels.

4.3 Hidden surfaces removal

In the real world, only the part of the object that is facing toward the viewer can be observed. No rays of light emitted from the opposite side can be viewed. In the numerical simulations, if we do not distinguish the front and the back faces, the observed image will have incorrect occlusion relation. In this paper, we adopt the exterior-normal method to recognize the back face [39]. The principle of this method is explained below briefly.

In back-face culling, exterior-normal is computed as

s = {\overset{⇀}{e}}_{z} \cdot {\overset{⇀}{n}}_{o},

where

{\overset{⇀}{e}}_{z}

is the unit vector of the z-axis, which is also the normal of the hologram plane with the hologram on the z = 0 plane,

{\overset{⇀}{n}}_{o}

is the outer-normal of the triangle mesh that points to outside of the object, which is obtained automatically by 3ds Max software. Both are presented exactly as shown in Fig. 6(d).

s \geq 0

indicates that the triangle either faces away from or is perpendicular to the hologram plane. Hence, these triangular surfaces cannot be viewed at the hologram plane and we can, therefore, remove them from being processed.

It needs to pay attention to that the proposed method to remove the hidden surfaces also exists limitation. Outer-normal-based method such as the one proposed here is more applicable to deal with convex objects. For some concave surfaces in complex models, some of the surfaces are considered hidden but probably still be visible. Removing fully all the true hidden parts for 3D objects is a complicated problem. The z-buffer algorithm might be a better method but with precision issues for depths further away from the observation point, which is the case for objects with large depth.

We first reconstruct the complex hologram of the sphere without removing the back triangles, as illustrated in Fig. 7(a). The non-uniform intensity among triangles indicates overlapped reconstructions in various degrees. The reconstruction result processed by our improved outer-normal-based method is shown in Fig. 7(b) and we observe that the spherical surface intensity distributes uniformly. Figure 7c) shows the optical reconstruction of the phase hologram obtained from the complex hologram (we have used a phase-only SLM for reconstruction) and Fig. 7d) shows computer simulation result of the same phase hologram obtained from the complex hologram.

Fig. 7 Reconstruction images of the sphere using different ways. (a) Numerical reconstruction of the complex hologram for all triangles on sphere that include hidden polygons. (b) Numerical reconstruction of the complex hologram for those triangles facing toward the hologram plane. (c) Optical reconstruction of the phase hologram (extracted from the complex hologram) with the removal of back faces. (d) Numerical reconstruction of the phase hologram with the removal of back faces.

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4.4 Simulations and experimental results

Based on the theory proposed in section 3 and the method for removing back faces proposed in part C, we have calculated CGHs for the three models introduced in part A. The three light fields on the hologram are shown in Fig. 8. With the proposed method, the CGH of the sphere, Baymax and the teapot is generated with 48 seconds, 562 seconds, and 911 seconds, respectively. The hardware includes Intel Core i5-3470 @ 3.20GHz, 4G-byte RAM under the environment of MATLAB 2014a. The computation time has been reduced substantially as compared to that in previous works [28]. A teapot with 6320 triangle meshes is generated to obtain a hologram and the hologram is reconstructed with no more than 911 seconds. And every polygon consumes almost 3.8 seconds using the method by Pan et al. [28], which translates to about 5800 seconds for 1536 triangles. The proposed theory serves as an important role and the contribution from back-face culling cannot be ignored.

Fig. 8 Computer generated holograms based on the proposed method. (a) Sphere. (b) Baymax. (c) Teapot.

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Numerical reconstruction of the complex hologram of the sphere at z = 500mm is shown in Fig. 7(b). The black lines on the sphere surface probably are the result from common sides of adjacent triangles. These lines manifest themselves as black stripes on the optical reconstruction as shown in Fig. 7(c). Figure 9(a) shows numerical reconstruction of Baymax at z = 490mm, which is where the chest is focused. According to the parameters introduced in part B of this section, the Baymax’s head lies in around at z = 500mm, and so the enlarged image in Fig. 9(c) illustrates that the head is not reconstructed clearly. On the other hand, the teapot is reconstructed at z = 500mm, as shown in Fig. 9(b). The reconstruction distance is exactly where the lid of the teapot. Figure 9(d) clearly shows the focused portion of the lid. In addition, it is found that removing the hidden surfaces based on the improved outer-normal principle is not thorough enough for Baymax and the teapot. Because the joints such as between the body and the arms exhibit concavity for Baymax, illustrating non-uniform intensity (or brighter areas) around those areas as shown in Fig. 9(a). So do the joints of the body with the spout and the lid area for the teapot. As we have analyzed in part C, the outer-normal method to recognize hidden faces is limited to process convex objects and do not work well for concave surfaces or those inlaid with each other.

Fig. 9 Numerical reconstruction results of the complex hologram. (a) Reconstruction of Baymax at the chest plane where z = 490mm. (b) Reconstruction of the teapot at the lid plane where z = 500mm. (c) Closeup of the marked area in (a). (d) Closeup of the marked area in (b).

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In the optical experiments, a phase reflective spatial light modulator (Three-Five System MD1280: active area = 15.56mm × 12.48mm, 1024 × 1080 pixels with pixel size of 12μm) is used to reconstruct the holograms. The phase of the hologram is obtained from the light field distribution, $U_{o b j e c t}$ , expressed by Eq. (26). A beam of light with 532nm emitted from a semiconductor laser is collimated by a Fourier lens (focal length of 153mm). The plane wave of light irradiating the SLM is modulated by the phase value of the hologram. The reflected light from the SLM reconstructs the image on a paperboard at around 500 mm. A camera (Canon EOS 700D) records the reconstructed image projected on the paperboard. The reconstructed results of the sphere, Baymax and the teapot are shown in Fig. 7(c), Figs. 10(a) and 10(b), respectively. The reconstructed images verify the proposed method. For comparison with the optical results of Baymax and the teapot in Figs. 10(a) and 10(b), we have also reconstructed numerically the phase holograms of Baymax and the teapot, shown in Figs. 10(c) and 10(d), respectively.

Fig. 10 Experimental and numerical reconstruction results of the phase hologram. (a) Reconstruction of Baymax optically. (b) Reconstruction of the teapot optically. (c) Reconstruction of Baymax numerically. (d) Reconstruction of the teapot numerically.

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5. Conclusion

In conclusion, we have developed a novel 3D affine transformation method to calculate the light field from an arbitrary triangle on the basis of previous 2D affine theory. Performing only once inverse Fourier transformation for all the triangles is advantageous as compared to traditional polygon-based methods. The proposed method needs a transformation matrix that requires a more rigorous derivation. The transformation matrix from a spatial triangle to a primitive triangle is obtained by rotation transformations and the use of 2D affine transformation. The method in this work overcomes the problem of the insolvable matrix presented in the paper by Pan et al.

The proposed method has been verified in simulations and experiments by generating CGH of three 3D models with different numbers of triangles. We have also reconstructed these holograms optically and numerically to illustrate the 3D aspect of the reconstruction. The calculation time of all the three test objects has been presented and shown to be reduced substantially as compared to that in previous methods. In the future, we would like to investigate parallel acceleration of the proposed method combining with GPU and hope to achieve real-time display of a 3D human model with hundreds of thousands of triangles. We believe the proposed theory will help to advance the polygon-based method of computer-generated holography.

Appendix A derivation of Equation (3)

Here, we derive an analytical expression of the spectrum distribution of a 3D object based on symbols and definition used in Ahrenberg et al. An arbitrary triangle is defined as $f_{Γ} (s, t)$ , and a primitive triangle is defined as $f_{Δ} (x, y)$ . Their Fourier transforms are $F_{Γ} (u_{s}, v_{t})$ with (s, t; u_s, v_t) being the transform variables and $F_{Δ} (u, v)$ with (x, y; u, v) being the transform variables, respectively. Equation (2) can be written as

{\begin{matrix} s = a_{11} x + a_{12} y + a_{13} \\ t = a_{21} x + a_{22} y + a_{23} \end{matrix},

where a_ij could be obtained according to mapping relation (0, 0)→(s₁, t₁), (1, 0)→(s₂, t₂), (1, 1)→(s₃, t₃) as

{\begin{matrix} a_{11} = s_{2} - s_{1} & a_{12} = s_{3} - s_{2} & a_{13} = s_{1} \\ a_{21} = t_{2} - t_{1} & a_{22} = t_{3} - t_{2} & a_{23} = t_{1} \end{matrix} .

Hence, the frequency spectrum of an arbitrary triangle,

F_{Γ} (u_{s}, v_{t})

, is obtained by

F_{Γ} (u_{s}, v_{t}) = \int_{- \infty}^{+ \infty} \int_{- \infty}^{\infty} f_{Γ} (s, t) \cdot e^{- j 2 π \cdot (u_{s} \cdot s + v_{t} \cdot t)} d s d t .

With a change of variables governed by (28) and

f_{Γ} (s, t)

is substituted by

f_{Δ} (x, y)

in (30), we have

F_{Γ} (u_{s}, v_{t}) = J \cdot \int_{0}^{1} \int_{0}^{x} f_{Δ} (x, y) \cdot e^{- j 2 π \cdot [u_{s} \cdot (a_{11} x + a_{12} y + a_{13}) + v_{t} \cdot (a_{21} x + a_{22} y + a_{23})]} d y d x,

where

J

is the Jacobian of the transformation, determined by

J = | a_{11} a_{22} - a_{12} a_{21} | .

Because the primitive triangle is defined by

f_{Δ} (x, y) = {\begin{matrix} 1 \\ 0 \end{matrix} ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ \begin{matrix}  \end{matrix} ​ \begin{matrix}  \end{matrix} ​ \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix} (x, y) lies inside primitive triangle \\ else \end{matrix},

Equation (31) becomes

F_{Γ} (u_{s}, v_{t}) = J \cdot e^{- j 2 π \cdot (a_{13} u_{s} + a_{23} v_{t})} \int_{0}^{1} \int_{0}^{x} e^{- j 2 π \cdot [(a_{11} u_{s} + a_{21} v_{t}) \cdot x + (a_{12} u_{s} + a_{22} v_{t}) \cdot y)]} d y d x .

On the basis of Eq. (8) from the literature [23], the integral term of the above equation is given by

\int_{0}^{1} \int_{0}^{x} e^{- j 2 π \cdot [(a_{11} u_{s} + a_{21} v_{t}) \cdot x + (a_{12} u_{s} + a_{22} v_{t}) \cdot y)]} d y d x = F_{Δ} (a_{11} u_{s} + a_{21} v_{t}, a_{12} u_{s} + a_{22} v_{t}) .

So Eq. (34) finally becomes

F_{Γ} (u_{s}, v_{t}) = J \cdot e^{- j \cdot 2 π (a_{13} u_{s} + a_{23} v_{t})} \cdot F_{Δ} (a_{11} u_{s} + a_{21} v_{t}, a_{12} u_{s} + a_{22} v_{t}),

which is presented in Eq. (3).

Appendix B derivation of Eq. (22)

According to Eq. (19) and (20), the frequency spectrum in the hologram plane is obtained by

\begin{array}{l} F_{h o l o} (u, v) = J_{s t} \cdot \int_{0}^{1} \int_{0}^{s} g (s, t, z_{0}) \cdot e^{- j \cdot 2 π \cdot z \cdot w} \cdot \\ \begin{matrix}  \end{matrix} e^{- j 2 π [u (c_{11} s + c_{12} t + c_{13} z_{0} + c_{14} + x_{0}) + v (c_{21} s + c_{22} t + c_{23} z_{0} + c_{24} + y_{0}]} d t d s, \end{array}

where

w

is given by Eq. (21). Because

g (s, t, z_{0})

is equal to 1 inside the primitive triangle, Eq. (37) can be simplified to

F_{h o l o} (u, v) = J_{s t} \cdot \int_{0}^{1} \int_{0}^{s} e^{- j 2 π [u (c_{11} s + c_{12} t + c_{13} z_{0} + c_{14} + x_{0}) + v (c_{21} s + c_{22} t + c_{23} z_{0} + c_{24} + y_{0}) + w (c_{31} s + c_{32} t + c_{33} z_{0} + c_{34} + z_{0})]} d t d s .

Reformatting the above equation, we have

\begin{array}{l} F_{h o l o} (u, v) = J_{s t} \cdot \int_{0}^{1} \int_{0}^{s} e^{- j 2 π [(c_{11} u + c_{21} v + c_{31} w) s + (c_{12} u + c_{22} v + c_{32} w) t + (c_{13} u + c_{23} v + c_{33} w) z_{0}} \cdot \\ \begin{matrix}  \end{matrix} e^{- j 2 π [(c_{14} + x_{0}) \cdot u + (c_{24} + y_{0}) \cdot v + (c_{34} + z_{0}) \cdot w]} d t d s . \end{array}

Substituting the definition of the variables from Eq. (23) into Eq. (39), we have

F_{h o l o} (u, v) = J_{s t} \cdot e^{- j 2 π (p + q)} \int_{0}^{1} \int_{0}^{s} e^{- j 2 π (u^{'} \cdot s + v^{'} \cdot t)} d t d s,

once solving the definite integral in Eq. (40), $F_{h o l o} (u, v)$ is finally given by Eq. (22) in the text.

Funding

Financial support for this study was provided by the National Natural Science Foundation of China (Grant No.: 61565010; 11762009) and National key research and development program (2017YFB0503505).

Acknowledgment

We thank Prof. Sui Wei and her student XiangXiang Wang (Key Laboratory Intelligent Computing & Signal Processing, Ministry of Education, Anhui University, Hefei, China), for their sincere help and support in the aspect of optical experiments.

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Fast generation of full analytical polygon-based computer-generated holograms

Abstract

1. Introduction

2. Comparison with previous methods

3. Theory

3.1 Step 1: rotation around the center of gravity

3.2 Step 2: affine transformation in two-dimensional plane

3.3 Step 3: diffraction computation

4. Simulations and optical experiments

4.1 Calculation of a single tilted triangle using the proposed method

4.2 Parameters of 3D objects

4.3 Hidden surfaces removal

4.4 Simulations and experimental results

5. Conclusion

Appendix A derivation of Equation (3)

Appendix B derivation of Eq. (22)

Funding

Acknowledgment

References and links

Cited By

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Optics Express