Model Implementation and Analysis of a True Three-dimensional Display System

: To model a true three-dimensional (3D) display system, we introduced the method of voxel molding to obtain the stereoscopic imaging space of the system. For the distribution of each voxel, we proposed a four-dimensional (4D) Givone – Roessor (GR) model for state-space representation — that is, we established a local state-space model with the 3D position and one-dimensional time coordinates to describe the system. First, we extended the original elementary operation approach to a 4D condition and proposed the implementation steps of the realization matrix of the 4D GR model. Then, we described the working process of a true 3D display system, analyzed its real-time performance, introduced the ﬁ xed-point quantization model to simplify the system matrix, and derived the conditions for the global asymptotic stability of the system after quantization. Finally, we provided an example to prove the true 3D display system ’ s feasibility by simulation. The GR-model-representation method and its implementation steps proposed in this paper simpli ﬁ ed the system ’ s mathematical expression and facilitated the microcontroller software implementation. Real-time and stability analyses can be used widely to analyze and design true 3D display systems.


Introduction
In recent years, with advances in optical and computer technology, stereoscopic display technology has undergone accelerated development. As a result, people are pursuing a more realistic display effect from the traditional two-dimensional (2D) display to three-dimensional (3D) display [1]. Among these developments, the most representative display is the true 3D display system. For a true 3D display system, each voxel's brightness and color should be controllable, and the relative spatial positional relationship between voxels should also be truly embodied. These goals require finding a suitable model to build a state space for the true 3D display system. Furthermore, the implementation steps and the results of performance analyses must be detailed.
In multidimensional system theory, the Givone-Roessor (GR) [2] and Fornasini-Marchesini (FM) models [3] are two widely used local state-space models. Both have been employed in the state-space modeling of multidimensional systems, including wireless sensor networks [4][5][6][7][8]. Compared with the FM model, the GR model can quantitatively express each dimension's variables and then be used to analyze each dimension's influence on the entire system. Therefore, we established a 4D GR model by introducing 3D position coordinates and one-dimensional (1D) time coordinates to model the system's state space.
Galkowski [9] used the forward transfer operator to represent the transfer function and obtain the system implementation matrix through the matrix transformation for implementing the state-space model. The concept is simple and easily calculable. It can also be used to evaluate the influence of coefficient values on the realization matrix in implementing a multidimensional system. This flexibility, however, also can result in an infinite number of possible intermediate operations in constructing the feature matrix, making it difficult to obtain a general algorithm. Moreover, this method is not easily implemented by a computer program. Xu et al. [10] used the unit retardation factor to represent the transfer function and introduced the elementary operation approach (EOA) to give the implementation method of the system implementation matrix. In this method, however, only one order can be reduced in each supplementary operation. Although it is possible to increase the supplementary operation's efficiency by some decomposition, only a few transfer functions can be applied to this decomposition method. The matrix method is also used to obtain a GR model with a lower order of the matrix [11], but it cannot be used to analyze the coefficient value's influence on the implementation matrix. Xiong [12] proposed an improved 3D EOA algorithm. Through matrix operation, the state-space model of the GR model can be obtained from the transfer function, and the order of the model is significantly reduced. In this paper, the EOA algorithm is extended to four dimensions and applied to the modeling and implementation steps of a true 3D display system. When a true 3D display system works, one needs each voxel to transfer the information to the microcontroller as soon as possible for data analysis; thus, it has high real-time and stability requirements. Therefore, one must analyze the real-time functionality and stability of the GR model. Kokil [13] and Xin et al. [14] introduced 2D discrete system stability. However, if it is directly extended to multidimensional systems, many limitations still exist. Agathoklis et al. [15] introduced the boundedinput, bounded-output stability of the traditional multidimensional system, but the practical application still has many limitations. In a true 3D display system with a finite size, the voxel calculation results often must be quantified to achieve the perfect display effect. Therefore, in the present study, we introduced a quantitative model to obtain the necessary and sufficient conditions for the true 3D display system's stability after quantization, and then provided an example to verify the model.

Method of Voxel Molding
When a true 3D display system is working, a full-body cylinder-voxel space composed of several spatially discrete voxels will exist. It can be imagined that this space is a flask mold, and each voxel in the interior is sand. The sand in a sandbox has a unique 3D space coordinate and a 1D time coordinate. The 3D mathematical model of the displayed object is considered to be a mold. We assume that upon putting the mold into the sandbox, the mold will replace the sand's space in the original sandbox and form a mold cavity. Based on this assumption, we proposed a 3D model voxel generation method-that is, the method of voxel molding (MVM).
We converted the original space's image data to voxels according to the display's requirements, which conform to the display unit's geometric characteristics. Then we inputted the voxels to the display unit for calibration and calculation.
In the cylinder space generated by an LED screen's rotation, if each voxel's coordinates generated at a certain angle (such as 3 ) are ðx; y; zÞ, then the time corresponding to each voxel is t 0 ; t 1 ; t 2 ; t 3 ; …; t n .
Assuming that the 3D model from the acquisition module is stored in the form of a point cloud, the MVM steps are as follows: 1. We put the 3D model into the voxel space, and the edges of the model coincided with or approximated some voxels in the space, as shown in Fig. 1. 2. The voxel ðx i ; y j ; z k Þ with the same time t 2 t 0 ; t 1 ; t 2 ; t 3 ; …; t n ½ f g was set as A ¼ fðx i ; y j ; z k ; t n Þg, that is, 3. We reflected the voxels' data to the LED screen to obtain a set of 3D coordinates.
Considering the voxel space of size N 1 Â N 2 Â N 3 , the display of each voxel is a linear process. Because the 3D mathematical model must be observable and realizable, the entire voxel space display is a process of linear causality in the first octant. The state-space model can represent the causality in the first octant.

Implementation Steps of GR Model with EOA Transformation
We performed EOA transformation to simplify the implementation of the GR model to the supplementary and transformation operations of the multidimensional characteristic polynomial matrix. This method featured easy calculation and could be used analyze the coefficient correlations on the system implementation matrix. We proposed a 4D EOA transformation and obtained the state-space model of the GR model through matrix operation.

Elementary Transformation of Matrices
Numerous elementary transformations of matrices are needed for obtaining the state-space matrix. Several of these transformations are defined below.
Let M denote a matrix and the following four kinds of row (column) transformations for a matrix denote the elementary row (column) transformations of a matrix.

4D EOA Transformation
Next, we implemented the matrix transformation for the 4D GR model in the SISO case (p ¼ q ¼ 1) as follows: ; (2) As shown in Y. Xiong [12], the 4D GR model's implementation problem is to convert M 0 into matrix M by supplementing operations and elementary transformation without changing the determinant value.
The matrix M requires the following properties: 1. The first element on the diagonal can only be x. 2. Other elements on the diagonal can only be 1D linear polynomials about variables z k ; k 2 f1; 2; 3; 4g, and the constant term can only be 1. 3. In addition to the first line, the off-diagonal element can only be a linear monomial about variables z k ; k 2 f1; 2; 3; 4g. 4. In addition to x, the elements of the first row are constant terms. 5. The elements of the same row can only contain the same elements z k ; k 2 f1; 2; 3; 4g, and from the second row all the rows are arranged in the order z 1 ; z 2 ; z 3 ; z 4 . 6. The first element x on the diagonal is just a symbol, not a variable. During transformation, the position and expression of x cannot be changed.
The following operations are then performed: Then, the new rows generated by each operation in p 3 , q 3 , and M 1 are sequentially operated so that terms with z 1 in M 1 become linear monomials concerning z 1 .
Each row in M 1 performs the same operation on the variables z 1 , z 2 , z 3 , and z 4 in turn, so that the diagonal elements other than x in M 1 are all 4D linear polynomials with the constant term of 1.
Step 2: Assume that the matrix M 1 obtained through Step 1 is where * and # are both linear polynomials; a i , b i , c i , and d i are all coefficients, and i = {1,2,3,4}.
Convert matrix M 1 obtained in the first step to M 2 so that the diagonal elements in M 2 except x are 1D linear polynomials with a constant term of 1, and the non-diagonal elements except the first row are all linear monomials about the variables z k ; k 2 f1; 2; 3; 4g.
Perform the following operation on M 1 in Eq. (8): In the same row, perform similar operations on variables z 3 and z 4 . Then, matrix M 2 is finally obtained and the first three elements of the second row in M 2 are a 1 z 1 , 1 þ b 1 z 1 , and c 1 z 1 , respectively; the remaining elements are −1: Step 3: Through appropriate row and column transformations, each row in M 2 is arranged in the order z 1 , z 2 , z 3 , and z 4 , and all of the 1D linear polynomial elements are moved to the diagonal position. Then, the term −1 is eliminated through column transformation. Finally, matrix M 3 is obtained: