Improved Compressed Sensing Based 3D Soft Tissue Surface ...scbms.whu.edu.cn/pdf/Improved Compressed Sensing... · motion capture system (PPT2); (3) experimental material: Memory

Improved Compressed Sensing Based 3D Soft Tissue

Surface Reconstruction

Sijiao Yu, Zhiyong Yuan*, Qianqian Tong, Xiangyun Liao, and Yaoyi Bai

School of Computer, Wuhan University, Wuhan, China [email protected]

Abstract. This paper presents a 3D soft tissue surface reconstruction method based on improved compressed sensing and radial basis function interpolation for a small amount of uniform sampling data points on 3D surface. We adopt radial basis function interpolation to obtain the same amount of data points as to be reconstructed and propose an improved compressed sensing method to re-construct 3D surface: we design a deterministic measurement matrix to signal observation, and then adopt the discrete cosine transform to the 3D coordinate sparse representation and use weak choose regularized orthogonal matching pursuit algorithm to reconstruct. Experimental results show that the proposed algorithm improves the resolution of the surface as well as the accuracy. The average maximum error is less than 0.9012 mm, which is smooth enough to provide accurate surface data model for virtual reality based surgery system.

Keywords: Compressed Sensing; Deterministic Measurement Matrix; Radial Basis Function Interpolation; Sparse 3D Discrete Point.

1 Introduction

3D reconstruction is an important research direction in the field of computer vision and it has been widely used in Virtual Reality (VR), object recognition and visualiza-tion. In recent years, 3D reconstruction technology has become a new hotspot in med-ical research, and VR based surgery is one of its applications. It provides a powerful tool for both surgical training and its evaluation, in which the accurate acquisition of 3D soft tissue structural information is primary premise.

A lot of research has been conducted on 3D surface reconstruction. Different kind of data can deal with different kind of methods and the results produced by the algo-rithm is highly depends on the types of data [1]. In typical 3D surface reconstruction methods, data sets are usually obtained from different sources such as medical image-ry, laser range scanner and mathematical models [2].

For approaches based on images, Mohammed E. et.al [3] presented an image-based modeling technique based on polynomial texture mapping. Ningqing Qian [4] pre-sented an efficient poisson-based surface reconstruction of 3D model from a non-homogenous sparse point cloud. However, their limitation is that the generated mod- * Corresponding author

els are only planar nature. For approaches based on point cloud through laser range scanner, lots of techniques usually used dense data set to calculate the connectivity to avoid holes appear which can cause the shape incomplete and not perfect [5], but there exists some data redundancy problems. Nina Amenta et.al [2] is the first that offers provable guarantee to the surface reconstruction problem from unorganized sample points and they proposed a Voronoi-based surface reconstruction algorithm. Later, Patrick et.al [6] proposed a modular framework for robust 3D reconstruction from unorganized, unoriented, noisy, and outlier ridden geometric data. This approach is scalable while robust to noise, outliers, and holes. But, it is for closed surface and relies on large scale of points. For approaches based on mathematical models, such as implicit surfaces, this approach uses different standard to fit the implicit surfaces to-ward inputs point by minimizing the energy that represent different distance functions [7]. To summarize, using existing methods to reconstruct accurate surface data must take advantage of large amounts of data, which will bring data redundancy and affect the speed of reconstruction.

To reduce the amount of data collected and reconstruct the 3D surface with high accuracy, we proposed a method using the recently proposed sampling method, com-pressed sensing or compressive sampling (CS) theory [7-9] to reconstruct for a small count of uniform point. CS theory has the ability to compress a signal during the pro-cess of sampling, which means that CS can collect compressed data at a sampling rate much lower than that needed in Shannon's sampling theorem. Experimental results show that the proposed algorithm improved the resolution of the surface as well as the accuracy.

2 Surface Reconstruction

The goal of this paper is to reconstruct 3D surface with high accuracy and strong ro-bustness. We now describe the model and implementation of our method. Firstly, we collect a small amount of random 3D tissue surface data sets 1S , as described in litera-

ture [10]. Secondly, we obtain the same amount of data points 2S as to be reconstruct-

ed by using Radial Basis Function (RBF) interpolation. Then, we adopt the Discrete Cosine Transform (DCT) to the 3D coordinate sparse representation respectively, and design a deterministic measurement matrix to signal observation. Finally, we choose Weak choose Regularized Orthogonal Matching Pursuit (WROMP) algorithm as re-construction algorithm. The reconstruction process is described in Fig.1.

1S

2S Φ 3S1S

Fig. 1. Reconstruction process

2.1 RBF Interpolation

Radial basis functions are commonly used for all kinds of scattered data interpolation problems. RBF interpolation is to find a spline ( )S x that passes as close as possible to

the data points and is as smooth as possible. In this paper, we use Thin Plate Spline (TPS) [11] as basis function. For a set of discrete data sets

{ , } , 1,2,...,di ix f R R i n and basis function

2 -

2

( ) ln ,:

( )

k d

k d

x x x d evenR R

x x d odd

， (1)

Let,

( ) , (1,..., ,...)Tj d j m

E x X x

, the solution can be expressed as

( ) ( , )0 0

T T

T

A E fS x X

E

(2)

We first collect only a small amount of data points 1S . In order to ensure the accu-

racy of the final reconstruction, we combine fitting with RBF interpolation. We get the connectivity information by fitting. When interpolation, we first interpolate in a rough layer to get 2S so that the data points distribute uniformly, then interpolate

again. The steps are summarized in Table 1.

Table 1. RBF interpolation algorithm

Input: Data set 1 ( , , )S x y z

Output: Data set 2 2 2 2( , , )S x y z

1: Fitting

2: For each column ( , )x z

3: Construct fitting curves of ( , )x z to get R

4: Compute the inserted x coordinates 1x

5: Compute the inserted z coordinates 1z accord-

ing to R 6: end 7: RBF Interpolate

8: Train RBF neural network N based on 1S

9: Let 1 1( , )x z as input of N , obtaining the inserted y

coordinates 1y

10:Combine 1 ( , , )S x y z and 1 1 1( , , )x y z to get interpolated data

set 2 2 2 2( , , )S x y z

2.2 CS reconstruction algorithm

Mathematical Model. After preprocessing, we focus on CS reconstruction. CS is a nonlinear theory of

sparse signal reconstruction. Suppose nx R is a digital signal, if it is a K sparse or

compressible signal, then we can estimate it with few coefficients by linear transfor-

mation. We can represent this process mathematically as , my x y R Φ .

( ( ))m n m nΦ is a measurement matrix and represents a dimensionality reduction.

We can rebuild x from y by solving the optimal problem below:

0

ˆ arg min , .x x s t y x s s Φ ΦΨ Θ (3)

When log( )m K n and Φ has restricted isometry property (RIP) [12]. Our main

work is to design a measurement matrix Φ that fulfills the above properties and can recover the original signal x from measured value y .

Deterministic Measurement Matrix Construction. In this paper, we propose an improved CS algorithm. Our main contribution is to propose a practical deterministic measurement matrix construction algorithm. We note that random matrices, such as Gaussian and Bernoulli matrix, generally satisfy RIP, but sometimes they are impractical to be built in hardware. Thus, we construct deterministic measurement matrix using training method.

The criterion of constructing deterministic measurement matrix [13-14] is as fol-

lows: Let Gram matrix TG = Θ Θ . If G is a symmetric matrix and nonnegative defi-

nite matrix, and the eigenvalues of G are in[1- ,1+ ]k k , then the Θ will meet the

conditions of RIP, where k is constrained equidistant constant.

In work [15], -averagedt mutual-coherence is defined as the average of all absolute

and normalized inner products between different columns in Θ (denoted asij

g ) that

are above t . Put formally,

1 , ,

1 , ,

( )( )=

( )

i j N i j ij ij

t

i j N i j ij

g t g

g t

Θ (4)

Our goal is to minimize ( )t Θ : the main idea is to construct an adaptive measure-

ment matrix set BΦ firstly, and then select the matrix with smallest column coherence

as the measurement matrix Φ .Based on Gaussian random matrix, we construct adap-

tive measurement matrix iΦ according to literature [16]. Then according to the sparsi-

ty of signal, we adopt Elad algorithm [15] to modify certain columns of iΦ . While

modifying, we have to make sure that the modified matrix satisfies RIP and can make

a small coefficient closer to zero, so that the signal is sparse enough to be better re-constructed. We update Gram matrix G by formula (5)

= ( )

ijij

ij ij ij

ijij

g tg

g t sign g t g t

g t g

(5)

Where the value of ,t is very crucial, which will affect the satisfiability of RIP.

The detailed algorithm is explained in Table 2.

Table 2. Deterministic measurement matrix construction algorithm

Input: sparse signal s ;sparse matrix Ψ ;iterations iter ; output: measurement matrix Φ

1. For 1:i iter

2. Generate Gaussian matrix 0 ( )m nΦ ;

3. Let 1 2( , ,... )Tms s s s ;Find the 0k big components

1 2 0, ,...,

kir ir irs s s and their positions 1 2 0{ , , ..., }kA r r r ;

4. Divide Ψ and normalize 0 0, ,

ˆ ˆ( , )Tm k n k nΨ Ψ Ψ ;

5. Calculate threshold matrix 11 1 Δ Ψ Ψ , where

0

1 & & ( , {1,2,..., })

10 & &q

i j

i j j A i j n

i j j A

Δ ;

6. Calculate 0

=iΦ Φ Δ and normalize each column;

7. Calculate ( )t i

Θ by formula (5);

8. if ( ) && ( ) ( )t i t i t

t Θ Θ Θ

9. Let iΦ = Φ ;

10. else

11. Update iΦ by Elad algorithm;

12. end 13.end

WROMP algorithm.

We found that the reconstruction result of WROMP algorithm [17] is accurate and stable. The weak selection criterion of each element is as formula (6):

{ | max }, , (0,1]i j

j

i gg gg i J (6)

Where ig represents the correlation of iteration residua r and measurement ma-

trix Φ , J represents candidate set and represents weak choosing factor. In the case of different r and different atomic correlations of Φ , the weak atomic selection crite-ria can better select the representation of atomic groups of the original signal from candidate set. So WROMP algorithm can reconstruct signal better, enhancing stability.

3 Experiment results

For comparison purposes, we perform simulation experiments on four algorithms: linear interpolation (LI), RBF interpolation (RBFI), CS and RBF_CS (proposed algo-rithm), using memory pillow, and then analyze the results. Our experimental platform includes: (1) hardware: Intel Xeon CPU, 2.40GHz, Operating System windows7-32bit; (2) software: Matlab8.1 (R2013a), Precision Position Tracker with 2 Cameras motion capture system (PPT2); (3) experimental material: Memory pillow ( 450 250 100 mm3), and the deformation range inY is100 mm .

3.1 Data Sets

We sample the memory pillow surface evenly using the method in literature [10] to obtain its 3D surface data 1S with 299(23 13) points. We collect by the step of 30mm

in X direction and by the step of 20mm in Z direction. Three typical surfaces data points are shown in Fig.2 (a), (b), and (c). We suppose the data sets we collect through this method is the actual coordinates of 3D surface. For comparison, we select points set 11(5 13)S from 1S as reference. We select 5

rows from the third line with the step of 4 as reference to ensure the reliability and the remained data 12 (18 13)S is selected as the basis of interpolation. On the basis of

12 (18 13)S , we carry out RBFI according to the algorithm in Table 1. After the first

interpolation, we get 299 points and then we conduct the second interpolation. During

the second interpolation, we first interpolate by row and get1443(111 13) points; then

we interpolate by column and get data sets 2 (111 52)S .

(a) (b) (c)

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240

(d) (e) (f)

Fig. 2. (a), (b) and (c) are original data point coordinates we collect on conditions of no defor-mation, rather small deformation and pretty large deformation; and (d), (e) and (f) are the corre-

sponding data point coordinates after RBF interpolation

3.2 Comparison of different measurement matrix

To verify the deterministic measurement matrix we constructed possesses better per-formance, we performed experiments on three matrices (Gaussian Random matrix, Adaptive matrix and proposed matrix). The experiments are under three kinds of de-formation of coordinate Y : no deformation (No), rather small deformation (Small) and pretty large deformation (Large). We mainly performed tests on mean square error (eMSE) and maximum error (eMAX). The reconstruction result along with defor-mation is shown in Table 3.

Table 3. Reconstruction results of different measurement matrices mm

Deformation No Small Large

eMSE eMAX eMSE eMAX eMSE eMAX

Gaussian 0.1593 1.0391 0.2033 1.2372 0.403 2.6019

Adaptive 0.1498 1.0219 0.2444 1.6239 0.3939 1.3948

Proposed 0.1071 0.6908 0.1135 0.7854 0.1220 0.8967

In the test, the sampling rate is 0.5. As can be seen, the reconstruction error is small when using CS. However, the error of Gaussian and Adaptive matrix increases apparently when there is larger deformation. While for the proposed matrix, the max-imum reconstruction error is within 1mm and has perfect stability, which fully shows the proposed matrix is better.

3.3 Comparison of LI and RBFI

Table 4 shows the reconstruction results of two kinds of interpolation algorithm, LI and RBFI. We can learn that the reconstruction results are of no difference in small deformation. But as the deformation increases, the error of LI enlarges significantly, while the value of eMSE and eMAX of RBFI is relatively stable. As can be seen from Fig.2 (d), (e), and (f), after RBF interpolation, the resolution of surface is increased by a magnitude. The number of points is increased from 234(18 13) points to

5772(111 52) points.

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3.4

During the CS reconstruction, we

spectively. We apply DCT to the sparse re(FFT) as sparse base and use the proposed constructing algorithm construct measurment matrixcombine RBF and CS

interpolation. I

t

error of memory pillow under

represents CS algorithm and the green can be seen, the but the value maximum error is large; and for the proposed algorithm, both the eMAX

Deformation

3.4



interpolation. I

0.2, 0.9t


In Figrepresents CS algorithm and the green can be seen, the but the value maximum error is large; and for the proposed algorithm, both the

MAX are the smallest.

Deformation

LIRBF

Comparison with other methods



interpolation. I

0.2, 0.9


In Figrepresents CS algorithm and the green can be seen, the but the value maximum error is large; and for the proposed algorithm, both the

are the smallest.

0.000.050.100.150.200.250.30

0.000.200.400.600.801.001.201.401.60

Deformation

LI RBF



spectively. We apply DCT to the sparse re(FFT) as sparse base and use the proposed constructing algorithm construct measurment matrix. combine RBF and CS

interpolation. I

0.2, 0.9


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are the smallest.

0.000.050.100.150.200.250.30

0.000.200.400.600.801.001.201.401.60

Deformation



spectively. We apply DCT to the sparse re(FFT) as sparse base and use the proposed constructing algorithm construct measur

. Finally we adopt WROMP algorithm to reconstruct. In this paper, we combine RBF and CS

interpolation. In the experiment

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are the smallest.

data0

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0.000.200.400.600.801.001.201.401.60




Finally we adopt WROMP algorithm to reconstruct. In this paper, we combine RBF and CS

n the experiment

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of eMSE

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data0

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data0

eMAX

Table

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0.1590.1244




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n the experiment

when constructing deterministic matrix. We can s


Fig.

Fig.

3 and Fig. represents CS algorithm and the green

maximum

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are the smallest.

data0

/mm

data0

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Table 4

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n the experiment



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data1

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1.0391 1.





n the experiment


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The comparison of

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and Fig

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data6

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column s ,

is small while and


value of casesseen that the rwhich fully shows our algous ashown in Fi

Deformation

Proposed

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Propos

Fig.

4

In this paper, we present a creatively 3D surface reconstruction algorithm based on improved CS with matrix construcillustrate that the proposed algorithm precision requirement of tissue parameter measurement in terms of objective evalution (rather accurate data model for how to accelerate reconstruction.

Acknowledgment.Foundation of China

value of cases, and the average maximum error is less than 0.9012mm.seen that the rwhich fully shows our algous ashown in Fi

Deformation

RBF

CS

Proposed

Deformation

RBF

CS

Propos

Fig. 5.

In this paper, we present a creatively 3D surface reconstruction algorithm based on improved CS with matrix construcillustrate that the proposed algorithm precision requirement of tissue parameter measurement in terms of objective evalution (erather accurate data model for how to accelerate reconstruction.


value of , and the average maximum error is less than 0.9012mm.

seen that the rwhich fully shows our algous advantages. shown in Fi

Deformation

RBF

Proposed

Deformation

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Proposed

.

Conclusions

In this paper, we present a creatively 3D surface reconstruction algorithm based on improved CS with matrix construcillustrate that the proposed algorithm precision requirement of tissue parameter measurement in terms of objective evalu

eMSE

rather accurate data model for how to accelerate reconstruction.


value of eMAX

, and the average maximum error is less than 0.9012mm.seen that the rwhich fully shows our alg

vantages. shown in Fig.

Deformation

Deformation

(a),

Conclusions

In this paper, we present a creatively 3D surface reconstruction algorithm based on improved CS with matrix construcillustrate that the proposed algorithm precision requirement of tissue parameter measurement in terms of objective evalu

SE, erather accurate data model for how to accelerate reconstruction.


MAX can be within 1.1mm under any deformati, and the average maximum error is less than 0.9012mm.

seen that the recowhich fully shows our alg

vantages. g. 5.

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Conclusions

In this paper, we present a creatively 3D surface reconstruction algorithm based on improved CS with matrix construction algorithm which can be applied in practical.illustrate that the proposed algorithm precision requirement of tissue parameter measurement in terms of objective evalu

MAX)rather accurate data model for how to accelerate reconstruction.


can be within 1.1mm under any deformati, and the average maximum error is less than 0.9012mm.

conwhich fully shows our alg

vantages. T5.

Data0

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0.1071

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0.6908

(a)

(b) and rather small deformation and pretty large deformation

Conclusions

In this paper, we present a creatively 3D surface reconstruction algorithm based on improved CS with

tion algorithm which can be applied in practical.illustrate that the proposed algorithm precision requirement of tissue parameter measurement in terms of objective evalu

MAX), and rather accurate data model for how to accelerate reconstruction.



nstruwhich fully shows our alg

The reco

Table

Data0

0.1244 0.1688

0.1071

Data0

1.0981 1.0575

0.6908

(a)

and (c)rather small deformation and pretty large deformation

Conclusions

In this paper, we present a creatively 3D surface reconstruction algorithm based on improved CS with RBF interpolation.


and rather accurate data model for how to accelerate reconstruction.

Acknowledgment. The research was supported Foundation of China (Grant No.


struction results by our algwhich fully shows our alg

he reco

Table

Data1

0.1635 0.2649

0.1135

Table

Data1

1.3786 1.2538

0.7854

(c) are final reconstruction surface after on conditions of no drather small deformation and pretty large deformation

Conclusions

In this paper, we present a creatively 3D surface reconstruction algorithm based on BF interpolation.


and is more rather accurate data model for how to accelerate reconstruction.

The research was supported Grant No.


tion results by our algwhich fully shows our algorithm is of high acc

he reconstructed su

Table 5.

Data1

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Table 6

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are final reconstruction surface after on conditions of no drather small deformation and pretty large deformation



is more rather accurate data model for how to accelerate reconstruction.



tion results by our algrithm is of high accstructed su

. The

Data1 Data2

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0.1135

6. The

Data1 Data2

1.3786 1.2538

0.7854




is more realistic from visual effects. Thereforerather accurate data model for VR based how to accelerate reconstruction.




The comparison of

Data2

0.15570.2186

0.1355

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Data2

1.41411.3106

1.0383




realistic from visual effects. ThereforeVR based

how to accelerate reconstruction.

The research was supported 61372107)



comparison of

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0.1355

comparison of

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1.0383







tion results by our algrithm is of high accstructed surfaces under

comparison of

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comparison of

Data3

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1.0383

(b) (c)


In this paper, we present a creatively 3D surface reconstruction algorithm based on BF interpolation. We p

tion algorithm which can be applied in practical.illustrate that the proposed algorithm based onprecision requirement of tissue parameter measurement in terms of objective evalu




tion results by our algrithm is of high acc

faces under

comparison of

Data3

0.17440.1785

0.0925

comparison of

Data3

1.12460.7849

0.7488

(b) (c)


In this paper, we present a creatively 3D surface reconstruction algorithm based on We put forward

tion algorithm which can be applied in practical.based on

precision requirement of tissue parameter measurement in terms of objective evalurealistic from visual effects. ThereforeVR based surgery

The research was supported 61372107), the


tion results by our algorithm have a smaller degree of error, rithm is of high accuracy, strong rob

faces under

comparison of eMSE

Data3

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the National Nature ScienceNational Basic R

on, less than 1mm in most Therefore, it can be

rithm have a smaller degree of error, ustness and has obv

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on, less than 1mm in most herefore, it can be


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are final reconstruction surface after on conditions of no d


Experimental results sparse data points satisf

precision requirement of tissue parameter measurement in terms of objective evalurealistic from visual effects. Therefore, it can provide


ional Nature Sciencesearch Program of



deformations

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are final reconstruction surface after on conditions of no deformation,


Experimental results sparse data points satisfies

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rithm have a smaller degree of error, ustness and has obvi-

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mm

mm

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a-it can provide


ional Nature Science search Program of

China (Grant No. 2011CB707904), and the Open Funding Project of State Key La-boratory of Virtual Technology and Systems, Beihang University (Grant No. BUAA-VR-13KF-15).

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