Holographic Three-Dimensional Imaging of Terra-Cotta Warrior Model Using Fractional Fourier Transform

Journal of Imaging

Article Holographic Three-Dimensional Imaging of Terra-Cotta Warrior Model Using Fractional Fourier Transform

Zhi-Fang Gao, Hua-Dong Zheng * and Ying-Jie Yu

Department of Precision Mechanical Engineering, Shanghai University, Shanghai 200072, China * Correspondence: [email protected]

Received: 14 May 2019; Accepted: 24 July 2019; Published: 26 July 2019

Abstract: Holographic three-dimensional (3D) imaging of Terra-Cotta Warrior model using Fractional Fourier Transform is introduced in this paper. Phase holograms of Terra-Cotta Warrior model are calculated from 60 horizontal viewing-angles by the use of fractional Fourier transform (FRT). Multiple phase holograms are calculated for each angle by adding proper pseudorandom phase to reduce the speckle noise of a reconstructed image. Experimental system based on high-resolution phase-only spatial light modulator (SLM) is built for 3D image reconstruction from the calculated phase holograms. The texture of the Terra-Cotta Warrior model is rough. The calculation of rough texture is optimized in order to show better model details. The eﬀects of computing distance and layer thickness on imaging quality are analyzed ﬁnally.

Keywords: holographic imaging; fractional Fourier transform; spatial light modulator

1. Introduction Holographic display is a three-dimensional (3D) display method for showing 3D information of real objects. Holography can record the light wave information of objects in space and then perform optical redisplay [1]. Holographic 3D display can record the depth information of objects, so that it can improve the reality sense of 3D image. Valuable cultural relics are recorded and redisplayed by the holographic method in the protection of cultural relics. It can not only redisplay the material information of cultural relics, but also protect the original cultural relics from repeated exposure to the air. Holographic imaging can show three-dimensional information of cultural relics, which can show us more useful information, when compared with the traditional two-dimensional (2D) plane photo imaging method. Holograms for holographic display can be generated by optical holographic system or computational methods [2–4]. Computer generated hologram (CGH) has an important advantage [5]. Computing virtual object hologram avoids the complicated and strict operation of optical hologram acquisition. Recently, some methods for generating CGH from three-dimensional objects have been proposed. [6–10]. There are three types: pure amplitude hologram, pure phase hologram, and complex hologram (including amplitude and phase information), according to the wavefront information in the hologram plane. Kinoform is a phase hologram with high diﬀraction eﬃciency [11]. It can reduce the noise of the reconstructed image, but the reconstruction error of the amplitude distribution in the image plane is introduced, because the amplitude of the wavefront in the hologram plane is neglected. The distribution of this kind of noise is usually irregular. We also call it speckle noise or random noise. This kind of noise will destroy the texture details of the image, reduce the quality of the image, and reduce the comfort degree of human observation. [12]. The algorithm of reducing speckle noise arises in order to improve the image quality. These methods mainly include two types: one is for the processing of reconstructed images and the other is focused on heterogeneous optimization to improve

J. Imaging 2019, 5, 67; doi:10.3390/jimaging5080067 www.mdpi.com/journal/jimaging J. Imaging 2019, 5, 67 2 of 13 the quality of reconstructed images. In the first processing method, random diffusers (such as ground glass plates) are usually used to reduce the speckle noise in holographic reconstruction settings [13,14]. Moving aperture can reduce that speckle noise that is caused by time averaging effect [15]. In [16], Zhang et al. proposed the method of displaying full color 3D objects using fractional Fourier transform (FRT). They redisplayed a toy model and analyze the noise by image superposing method. However, the effect of computing distance on image quality was not analyzed, and interval of slices may affect image quality of objects with rough surface. We hope that we can redisplay more details when we redisplay the Terra-Cotta Warrior model, and analyze the effect of computing distance on image quality. This paper presents a method to realize holographic 3D imaging of Terra-Cotta Warrior model using FRT. By superimposing the image reconstructed by multiple phase holograms, the noise in the holographic reconstruction image can be significantly reduced. Section2 will describe methodological theory. Section3 is the digital simulation and photoelectric reconstruction experiments, and the results are presented and analyzed. Additionally, in Section3, the Terra-Cotta Warrior model surface texture is more complex and contains more information, in order to better observe the model when reproducing, we studied the calculation optimization of the Terra-Cotta Warrior model texture. The discussion and conclusion will be given finally.

2. Principle of The Proposed Method

2.1. Spatial Coordinate Transformation for Calculating Holograms of Terra-Cotta Warrior Model Generally, holograms for 360-degree viewing angle of the model are needed if we want to see 360-degree information of Terra-Cotta Warrior model. A cylindrical coordinate system is established to calculate 360-degree holograms of the Terra-Cotta Warrior model. As shown in Figure1, the Terra-Cotta Warrior model is in the center of the cylinder. The radius and height of the cylinder are set to r = 30 mm and h = 200 mm, respectively. The resolution is set as 195.3 µm in the horizontal, as well as vertical, direction of object plane. The hologram plane is set at the distance of d0 in the coordinate system (x0 y0z0). We calculate the holograms of the model along a circle route line, with 60 viewpoints at intervals of 6 degrees [17]. ! ! ! x cosθ sinθ x 0 = − (1) z0 sinθ cosθ z

y0 = y (2) where (x0Oy0) plane is in parallel with (u, v) hologram plane, θ is the rotation angle along the y(y0) axis, and the distance between y(y0) axis and the hologram plane is d0. J. Imaging 2019, 5, 67 3 of 13 J. Imaging 2019, 5, x FOR PEER REVIEW 3 of 13

FigureFigure 1. 1.CoordinatesCoordinates for for one-dimensional one-dimensional rotation rotation transformation. transformation.

2.2.2.2. FRT FRT and and Hologram Hologram Generation Generation TheThe optical optical FRT FRT of of a atwo-dimensional two-dimensional object object function function can can be be expressed expressed as as [16]: [16]:

" 22222+++2 2 2 + # p xx + yy u+ u v+ v xu yvxu + yv Euv(),,,exp2==×p F{} f () xy f() xy jππ − j dd xy， (3) E(u, v) = F f (x, y) = xf (x, y) exp jπλλfptanπ/2() j fp2π sinπ/2() dxdy, (3) × λeefe tan(pπ/2) − λ fe sin(pπ/2) { } ( ) ( ) wherewhere 𝐹Fp · denotesdenotes thethe FRT FRT with with an an order order p (0 p<| p(0<|p|<2).|<2). f (x, y𝑓) and𝑥, 𝑦 E (andu, v )𝐸are𝑢, the 𝑣 complexare the amplitudecomplex {·} amplitudedistributions distributions in the object in planethe object and fractional plane and Fourier fractional plane. Fourierλ is the plane. wavelength λ is the ofilluminating wavelengthlight. of illuminating light. 𝑓 is the standard focal length. fe is the standard focal length. TwoTwo Formulas Formulas (4) (4) and and (5) (5) are are added added in in order order to to mo morere clearly clearly explain. explain. These These two two formulas formulas can can be be usedused to to better better explain explain the the fraction fractionalal Fourier Fourier calculation calculation principle. principle. 𝑝𝜋 𝑝𝜋 𝑄=sinpπ,𝑅 =tanpπ (4) Q = sin 2 , R = tan 4 (4) 2 4 𝑓 𝑓 = ,𝑑fe =𝑅𝑓 (5) f =𝑄 , d = R f (5) Q e Figure 2 is an optical principle model of Lohmann type-I FRT, when light emitted or reflected Figure2 is an optical principle model of Lohmann type-I FRT, when light emitted or reﬂected from from an object propagates freely a distance 𝑑=𝑅𝑓 from space. Light travels through a lens with a an object propagates freely a distance d = R fe from space. Light travels through a lens with a focal focal length of f 𝑓 = , and then propagates in another space at the same distance, which is length of f = e , and then propagates in another space at the same distance, which is symmetrical with symmetrical withQ the space through which the light first passes. When the object light reaches the the space through which the light ﬁrst passes. When the object light reaches the focal distance of the focal distance of the lens, its Fourier transform can be obtained. The Fourier hologram can record not lens, its Fourier transform can be obtained. The Fourier hologram can record not only the information only the information of the object, but also the information of the system parameters, because of the of the object, but also the information of the system parameters, because of the characteristics of the characteristics of the Fourier transform. Fourier transform.

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Figure 2. Lohmann type-I optical setup for performing a FRT. Figure 2. Lohmann type-I optical setupsetup for performing a FRT.

According toto thethe optical optical model model of Lohmannof Lohmann type-I, type we-I, havewe have proposed proposed an improved an improved optical optical model According to the optical model of Lohmann type-I, we have proposed an improved optical tomodel analyze to analyze a three-dimensional a three-dimensional object [object16]. We [16]. assume We assume that the that depth the ofdepth a three-dimensional of a three-dimensional object model to analyze a three-dimensional object [16]. We assume that the depth of a three-dimensional consistsobject consists of many of layersmany inlayers diﬀerent in different depths, depths, as shown as shown in Figure in 3Figure. 3. object consists of many layers in different depths, as shown in Figure 3.

Figure 3. Optical configuration for generating FRT holograms of 3D objects. Figure 3. Optical configuration for generating FRT holograms of 3D objects. Figure 3. Optical configuration for generating FRT holograms of 3D objects. Suppose that we divide the object into L layers, that is, L layers of object planes. The distance Suppose that we divide the object into L layers, that is, L layers of object planes. The distance fromSuppose each layer that to thewe lensdivide is dithefferent. object In into Figure L layers,3, we assumethat is, thatL layers the distanceof object from planes. the firstThe layerdistance to from each layer to the lens is different. In Figure 3, we assume that the distance from the first layer to thefrom lens each is dlayer1, and to thethe distancelens is different. from the In second Figure layer 3, we to assume the lens that is dthe2. Thedistance distance from between the first thelayer two to the lens is d1, and the distance from the second layer to the lens is d2. The distance between the two planesthe lens can is d be1, expressedand the distance as from the second layer to the lens is d2. The distance between the two planes can be expressed as planes can be expressed as ∆d = 2(d d ) (6) 2 − 1 ∆𝑑 =( 2 𝑑 −𝑑) (6) According to Formulas (4) and (5), when∆𝑑 the = plane( 2 𝑑 −𝑑 extends) to the first plane, the distance from(6) the planeAccording to the lens to is Formulasd , and the (4) standard and (5), focal when length the planfe aree extends given, the to fractionalthe first plane, order thep of distance the ith object from According to Formulasi (4) and (5), when the plane extends to the first plane, thei distance from planethe plane can to be the expressed lens is d asi, and the standard focal length 𝑓 are given, the fractional order pi of the ith the plane to the lens is di, and the standard focal length1 𝑓 are given, the fractional order pi of the ith object plane can be expressed as p = 4 tan− (d / fe)/π (7) object plane can be expressed as i i Due to the scaling factor λ fe = 1, for𝑝 a=4tan give wavelength(𝑑/𝑓)/𝜋 of 632.8 nm, fe = 1580 mm. Subsequently,(7) 𝑝 =4tan (𝑑/𝑓)/𝜋 (7) the FRT of the ith object plane (0 < i < L) for a certain viewpoint of θ can be described as Due to the scaling factor 𝜆𝑓 =1, for a give wavelength of 632.8 nm, 𝑓 = 1580 mm. Subsequently, Due to the scaling factor 𝜆𝑓 =1, for a give wavelength of 632.8 nm, 𝑓 = 1580 mm. Subsequently, 2 2 2 2 the FRT of the ith objectp n planeo (0 < i < L) for a certain viewpoint ofx +θy can+u +bev describedxu +asyv the FRTEθ ,oft,i( uthe, v) ith = objecti fθ, iplane(x, y) (0= < i f<θ ,iL()x ,fory) expa certain[jϕt(x, viewpointy)] exp[jπ of θ can be describedj2π as ]dxdy, (8) λ fe tan(piπ/2) λ fe sin(piπ/2) F s ×2222 − x2222+++ y u v xu + yv E (,)uv== pi {} f (, xy ) f (, xy )exp[j(,ϕπ xy )]exp[ × jx+++ y u v − j 2 π xu + yv ]dd, xy θθθ,,tipi , i , i t (8) Eθθθ(,)uv== {} f (, xy ) f (, xy )exp[j(,ϕπ xy )]exp[ × jλπ − j 2 π λπ ]dd, xy where,,,tifθ,i(x, y) is the , i complex , i amplitude tof the ith objectλπfp planeeitan( in / the 2) viewpoint λπ fp ei sin( of θ /and 2) Eθ,t,i(u, v(8)) is fpeitan( / 2) fp ei sin( / 2) the contributed complex amplitude of the ith object plane for the tth hologram in the viewpoint of θ. where, 𝑓,(𝑥,) 𝑦 is the complex amplitude of the ith object plane in the viewpoint of θ and 𝐸,,(𝑢, 𝑣) twhere,= 1, 2, 3𝑓,...(𝑥,T () 𝑦T is the complex total calculation amplitude times of ofthe holograms). ith object planeϕt(x in, y )theis theviewpoint dynamic-changed of θ and 𝐸,, random(𝑢, 𝑣) is the contributed complex amplitude of the ith object plane for the tth hologram in the viewpoint of phaseis the contributed factor for the complextth hologram amplitude calculation of the process,ith object which plane is for used the totth smoothen hologram the in spatialthe viewpoint spectrum of θ. t = 1, 2, 3…T (T is the total calculation times of holograms). 𝜑(𝑥,) 𝑦 is the dynamic-changed distributionθ. t = 1, 2, 3… of theT (T object is the information. total calculation By adding times thisof holograms). random phase 𝜑(𝑥, factor) 𝑦 is [0, the 2π dynamic-changed] to object planes, random phase factor for the tth hologram calculation process, which is used to smoothen the spatial therandom contribution phase factor of all for the the surface tth hologram layers to thecalculation hologram process, in the viewpointwhich is used of θ canto smoothen be expressed the spatial by the spectrum distribution of the object information. By adding this random phase factor [0, 2π] to object followingspectrum formula.distribution of the object information. By adding this random phase factor [0, 2π] to object planes, the contribution of all the surface layers to the hologram in the viewpoint of θ can be planes, the contribution of all the surface layers to the hologram in the viewpoint of θ can be expressed by the following formula. expressed by the following formula.

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𝐸 (𝑢, 𝑣) =𝐸XL (𝑢, 𝑣) (9) , ( ) = ,, ( ) Eθ,t u, v Eθ,t,i u, v (9) i=1 Thus, the tth FRT phase-only hologram of a 3D object is expressed as Thus, the tth FRT phase-only hologram of a 3D object is expressed as 𝐻 (𝑢, 𝑣) =arg{𝐸 (𝑢, 𝑣)} , n , o (10) H (u, v) = arg E (u, v) (10) where, arg{·} denotes the argument of theθ,t complex amplitude.θ,t where, arg{ } denotes the argument of the complex amplitude. 3. Digital Reconstruction· and Electro-Optical Reconstruction Experiment 3. Digital Reconstruction and Electro-Optical Reconstruction Experiment 3.1. Digital Reconstruction 3.1. Digital Reconstruction According to the previous theory, all reconstructed holographic images are superimposed on the finalAccording reconstructed to the plane previous in order theory, to allreduce reconstructed the speckle holographic noise caused images by random are superimposed phase. According on the tofinal the reconstructedexperimental planeresults, in the order more to reduceimages the superimposed speckle noise at caused the same by randomangle, the phase. better According the visual to effectthe experimental of the final synthesized results, the moreimage, images the less superimposed noise contained at the in samethe image, angle, and the the better better the visualthe texture effect detailsof the finalon the synthesized model can image,be seen. the However, less noise the contained more images in the are image, superimposed, and the better the themore texture computing details timeon the and model tasks can arebe needed seen. However,in the computing the more process, images areand superimposed, more storage space the more is needed. computing In practical time and application,tasks are needed the number in the computing of overlapped process, images and morecan be storage adjusted space according is needed. to In the practical actual application,situation, becausethe number the human of overlapped eye's observation images can beability adjusted is limited. according For tothis the experiment, actual situation, T is becauseset as 20. the Table human 1 showseye’s observationtwo redisplaying ability schemes. is limited. The For first this experiment,one is that, forT is seteach as viewing 20. Table 1angle, shows only two one redisplaying image is redisplayedschemes. The from first single one hologram. is that, for The each second viewing one angle, is a hologram only one sequence image is for redisplayed each viewing from angle. single Thathologram. is to say, The T second images one for is the a hologram same viewing sequence angle for with each different viewing angle.random That noises, is to say,andT thenimages these for randomthe same noises viewing are supressed angle with by di ffaveragingerent random effect. noises, The original and then images these randomfrom different noises perspectives are supressed are by shownaveraging in Figure effect. 4, The the original images imagesredisplayd from by di thefferent first perspectives scheme are areshown shown in Figure in Figure 5, and4, the Figure images 6 showsredisplayd the second by the scheme first scheme of reproducing are shown the in images. Figure 5It, is and obvious Figure that6 shows the noise the secondin Figure scheme 5 is more of obviousreproducing than that the images.in Figure It 6. is The obvious reconstructed that the noise image in Figurein Figure5 is 6 more is visually obvious clearer. than that in Figure6. The reconstructed image in Figure6 is visually clearer. Table 1. Kinoform sequence schemes for dynamic three-dimensional (3D) holographic display. Table 1. Kinoform sequence schemes for dynamic three-dimensional (3D) holographic display. Viewing Angle 0° 6° … 354°

SchemeViewing I(𝐻 Angle,) 𝐻, 0◦ 𝐻6,◦ …... 𝐻,354◦ SchemeScheme II(𝐻 I(H,θ,t)) 𝐻,,𝐻,,𝐻H,0,1 ,…,𝐻, 𝐻,,𝐻,,𝐻H6,1,,…,𝐻, …... 𝐻,,𝐻,,𝐻H354,1,,…,𝐻, Scheme II(Hθ,t) H0,1, H0,2, H0,3, ... , H0,T H6,1, H6,2, H 6,3, ... , H6,T ... H354,1, H354,2, H354,3, ... , H354,T

(a) θ = 0 (b) θ = 60° (c) θ = 120°

Figure 4. Cont.

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(d) θ = 180° (e) θ = 240° (f) θ = 300° (d) θ = 180° (e) θ = 240° (f) θ = 300° (d) θ = 180° Figure 4. Original 3D (e)images θ = 240° view ed at different angles. (f) θ = 300° FigureFigure 4. 4.Original Original 3D images images view vieweded at atdifferent diﬀerent angles. angles. Figure 4. Original 3D images viewed at different angles.

(a) θ = 0 (b) θ = 60° (c) θ = 120° (a) (a)θ = θ 0 = 0 (b) θθ = = 60° 60° (c) θ (c) = 120°θ = 120°

(d) θ = 180° (e) θ = 240° (f) θ = 300° (d) θ = 180° (e) θ = 240° (f) θ = 300° Figure 5. Digital reconstructed images at different viewing angles with kinoform sequence scheme I. FigureFigure(d) 5. Digital θ5. Digital= 180° reconstructed reconstructed images images atat didifferentﬀ (e)erent θ =viewin viewing 240°g angles angles with with kinoform kinoform sequence sequence(f) θ scheme= 300° scheme I. I.

Figure 5. Digital reconstructed images at different viewing angles with kinoform sequence scheme I.

(a) θ = 0 (b) θ = 60° (c) θ = 120° (a) θ = 0 (b) θ = 60° (c) θ = 120°

Figure 6. Cont.

(a) θ = 0 (b) θ = 60° (c) θ = 120°

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(d) (d)θ = θ 180° = 180° (e) (e) θ θ = = 240° 240° (f) θ (f)= 300° θ = 300° (d) θ = 180° (e) θ = 240° (f) θ = 300° FigureFigure 6. DigitalDigital 6. Digital reconstructed reconstructed reconstructed images images at different different viewin viewingviewing anglesg angles with withwith kinoform kinoformkinoform sequence sequencesequence scheme schemescheme II. II.II. Figure 6. Digital reconstructed images at different viewing angles with kinoform sequence scheme II. WhenWhen we we calculate calculate the the holograms holograms by slicing the the Terra-Cotta Terra-Cotta Warrior Warrior model modelmodel with withwith a thickness aa thicknessthickness of of 5 mm.5When mm. Figure Figure we 77 showscalculateshows 7 shows thethe thethe reconstructedreconstructed reconstructed holograms image.by image. slicing The TheThe the im impactimpact Terra-Cottapact of ofof these these noises Warrior noises on our on model ourTerra-Cotta Terra-Cotta with a Warrior thickness WarriorWarrior of model is very significant, as we can clearly see the striped noise on the model. The thickness of layers 5model mm. Figure is very 7 significant,significant, shows the asreconstructedas wewe cancan clearlyclearly image. seesee thethThee stripedstriped impact noisenoise of these onon thenoisesthe model.model. on our The Terra-Cotta thickness ofWarrior layers is too large. When we slice the model with a thickness of 1 mm, the image obtained is shown in Figure modelisis too large. large.is very When Whensignificant, we we slice slice as the we the model can model clearly with with a see thickn a th thicknesse stripedess of 1 mm, ofnoise 1 mm,the on imagethe the model. image obtained The obtained thicknessis shown is shown inof Figurelayers in 8. We can see that the fringe effect is weakened. It means that the layer thickness is sensitive to the is8. tooWe large. can see When that we the slice fringe the effect model is with weakened. a thickn Itess means of 1 mm, that the the image layer obtainedthickness is is shown sensitive in Figure to the Figureimaging8. We quality. can see that the fringe e ffect is weakened. It means that the layer thickness is sensitive to 8.theimaging We imaging can quality. see quality. that the fringe effect is weakened. It means that the layer thickness is sensitive to the imaging quality.

Figure 7. Layering Terra-Cotta Warrior model, thickness of layering set at 5 mm, and the inappropriate fringes are redisplayed on the model. Figure 7. Layering Terra-Cotta Warrior model, thickness of layering set at 5 mm, and the Figure 7. Layering Terra-Cotta Warrior model, thickness of layering set at 5 mm, and the inappropriate Figureinappropriate 7. Layering fringes Terra-Cottaare redisplayed Warrior on the model, model. thickness of layering set at 5 mm, and the fringes are redisplayed on the model. inappropriate fringes are redisplayed on the model.

Figure 8. Layering Terra-Cotta Warrior model, thickness of layering set at 1 mm.

Figure 8. LayeringLayering Terra-Cotta Warrior model,model, thicknessthickness ofof layeringlayering setset atat 11 mm.mm. Figure 8. Layering Terra-Cotta Warrior model, thickness of layering set at 1 mm.

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3.2. Electro-Optical Reconstruction Experiment 3.2. Electro-OpticalFigure 9 shows Reconstruction the experimental Experiment setup of dynamic 3D electro-optical reconstruction. He-Ne laser with power of 15 mW and wavelength 632.8 nm is used in the setup; the pixel number of the SLM is Figure9 shows the experimental setup of dynamic 3D electro-optical reconstruction. He-Ne laser 1920 × 1080, and the refresh rate is 60 fps. with power of 15 mW and wavelength 632.8 nm is used in the setup; the pixel number of the SLM is According to the display mode of our hologram, we also need to add a polarizer and an analyzer, 1920 1080, and the refresh rate is 60 fps. which× can adjust the working mode of SLM to phase mode.

FigureFigure 9. 9.Experimental Experimental setupsetup for dynamic dynamic 3D 3D electro-optical electro-optical reconstruction. reconstruction.

3.3.According Image Quality to the Evaluation display mode of our hologram, we also need to add a polarizer and an analyzer, which can adjust the working mode of SLM to phase mode. We use a numerical speckle index (SI) [18] to evaluate the severity of image noise to evaluate the severity of image speckle noise. The formula for calculating SI is 3.3. Image Quality Evaluation 1 𝜎(𝑚, 𝑛) We use a numerical speckle indexSI (SI) = [18] to evaluate the severity of image noise to evaluate(11) the 𝑀𝑁 𝜇(𝑚, 𝑛) severity of image speckle noise. The formula for calculating SI is In this formula, M × N is the size of image. 𝜎(𝑚,𝑛) and 𝜇(𝑚, 𝑛) are used to calculate the M N standard deviation and average value of the1 3 X× 3 neighborhoodX σ(m, n) around the image point P(m, n), SI = (11) respectively. Figure 10a illustrates the SIs ofMN the reconstructedµ(m, nimages) from 20 holograms, which are holograms that are computed from the same imagem=1 ofn= the1 model, but with different random phases. FigureIn this 10b formula, shows Mthe trendN is theof SI size of different of image. nuσmber(m, n of) and superposedµ(m, n) imageare used reconstructed to calculate from the standardthe holograms. According× to the formula, the smaller the SI index, the higher the quality of the deviation and average value of the 3 3 neighborhood around the image point P(m, n), respectively. representative image, and the smaller× the noise. According to the changing trend of SI index in the Figuregraph, 10a it illustrates can be seen thethat, SIs with of the the increase reconstructed of superimposed images images, from 20 the holograms, SI index becomes which smaller are holograms and thatsmaller, are computed and with from the increase the same of image image superimpos of the model,ition, butthe SI with index di decreasesﬀerent random increasingly phases. slowly. Figure 10b shows the trend of SI of diﬀerent number of superposed image reconstructed from the holograms. According to the formula, the smaller the SI index, the higher the quality of the representative image, and the smaller the noise. According to the changing trend of SI index in the graph, it can be seen that, with the increase of superimposed images, the SI index becomes smaller and smaller, and with the increase of image superimposition, the SI index decreases increasingly slowly.

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Speckle index Speckle index Speckle index Speckle index

Sequence of reconstructed image Number of superposed images Sequence of reconstructed image Number of superposed images

(a) (b) (a) (b) Figure 10. (a) Speckle index (SI) of reconstructed image from single hologram calculated by different FigureFigurerandom 10.10. ((a phases.)) SpeckleSpeckle (b) indexSI of superposed (SI) of reconstructed reconstructed imageimage images fromfr fromom single different hologram number calculatedof holograms. by didifferentfferent randomrandom phases.phases. (b)) SISI ofof superposedsuperposed reconstructedreconstructed imagesimages fromfrom didifferentfferent numbernumber ofof holograms.holograms. 3.4. Electro-Optical Reconstruction of Kinoforms 3.4.3.4. Electro-Optical Reconstruction ofof KinoformsKinoforms We use this experimental device to redisplay the photoelectric image. Figure 11 is the observed 3DWeWe image useuse this this when experimental experimental we use scheme device device I in Table to to redisplay redisplay 1. The hologram the the photoelectric photoelectric is loaded image.on image.SLM Figureand Figure then 11 refreshed is11 the is the observed at observed 60 3D image3D imagefps. when The when 3D we image usewe use scheme of thescheme model I in I Table inis displayed.Table1. The 1. hologramThe The hologram reconstruction is loaded is loaded result on SLM son that SLM and are thenandshown then refreshed in Fig.refreshed 11 atare 60 at fps. 60 Thefps. 3DThefrom image 3D only image ofone the hologramof model the model is displayed.at each is displayed. angle, The but reconstruction The the reconstructionreconstruction results results result that are sin that shownFigure are inshown12Figure are fromin 11 Fig. are20 11 from are holograms at each angle. Obviously, in accordance with our simulation results, Scheme II can onlyfromone only hologram one hologram at each angle,at each but angle, the reconstruction but the reconstruction results in Figureresults 12 in are Figure from 2012 hologramsare from 20 at significantly suppress the speckle noise when compared with Scheme I. eachholograms angle. Obviously,at each angle. in accordance Obviously, with in ouraccordance simulation with results, our Schemesimulation II can results, significantly Scheme suppress II can For example, the SI of reconstructed image in viewpoint of 0° with scheme I is 0.2603; however, the speckle noise when compared with Scheme I. significantlythe SI of reconstructed suppress the image speckle in viewpoin noise whent of 0° compared with scheme with II is Scheme 0.1692. I. For example, the SI of reconstructed image in viewpoint of 0° with scheme I is 0.2603; however, the SI of reconstructed image in viewpoint of 0° with scheme II is 0.1692.

(a) θ = 0 (b) θ = 60° (c) θ = 120°

(d) θ = 180° (e) θ = 240° (f) θ = 300°

Figure 11. Electro-optical reconstructed images at diﬀerent viewing angles with kinoform sequence

scheme I.

(d) θ = 180° (e) θ = 240° (f) θ = 300°

J. Imaging 2019, 5, x FOR PEER REVIEW 10 of 13

J. Imaging 2019, 5, 67 10 of 13 Figure 11. Electro-optical reconstructed images at different viewing angles with kinoform sequence scheme I.

(a) θ = 0 (b) θ = 60° (c) θ = 120°

(d) θ = 180° (e) θ = 240° (f) θ = 300°

FigureFigure 12. 12.Electro-optical Electro-optical reconstructed reconstructed images at at differ diﬀerentent viewing viewing angles angles with with kinoform kinoform sequence sequence schemescheme II. II.

3.5. Relationship Between Computational Distance and Imaging Quality For example, the SI of reconstructed image in viewpoint of 0◦ with scheme I is 0.2603; however, the SI ofAccording reconstructed to the image previous in viewpoint content, we of success 0◦ withfully scheme obtain II the is 0.1692. reconstructed image from the holograms of the model. In the experiment above, the distance that is used for calculating the image 3.5.is Relationship 800 mm. When Between we calculate Computational and simulate Distance the andreconstructed Imaging Quality image, we evaluate the image quality of the same image at different imaging distances. In article [16], the relationship between speckle According to the previous content, we successfully obtain the reconstructed image from the noise and the superimposed image is analyzed under two different imaging distances, but the trend hologramsof noise ofvariation the model. is not In analyzed the experiment in detail. above,In this paper, the distance we calculate that ismany used noise for calculating evaluation indices the image is 800 mm.at different When distances we calculate and analyze and simulate their changing the reconstructed trends. image, we evaluate the image quality of the sameFigure image 13 at shows different the relationship imaging distances. between imaging In article distance [16], theand relationshipimage quality. between Here, we specklestill use noise andSI the to superimposed evaluate the image image quality. is analyzed As can be under seen twofrom di thefferent figure, imaging the overall distances, trend of butimage the quality trend ofis noise variationincreasing, is not because analyzed SI inis detail.decreasing. In this Some paper, fluctuations we calculate can manybe seen noise in the evaluation statistical indiceschart, which at diff erent distancesmeans andthat analyzethe curve their is not changing smooth. The trends. reason for these fluctuations is the random phase simulated byFigure the image 13 shows in the theprocess relationship of calculation. between The nois imaginge trend distance that is generated and image by quality.these random Here, phases we still use SI tomay evaluate be high the or imagelow, which quality. leads As to canhigh beor seenlow image from quality. the figure, the overall trend of image quality is Subsequently, we calculated the 10 phase holograms of single viewing angle at each computing increasing, because SI is decreasing. Some fluctuations can be seen in the statistical chart, which means distance. These holograms contain different phase distributions, which result in different random thatnoises. the curve We isalso not calculated smooth. the The SI reason of each for recons thesetructed fluctuations image, and is the the random SIs of the phase 10 reconstructed simulated by the imageimages in the at processthe same of distance calculation. are averaged. The noise trend that is generated by these random phases may be high or low,Figure which 14 shows leads tothe high relationship or low imagebetween quality. the average SI of 10 images and the calculated distance.Subsequently, We can wesee calculatedthat, the farther the 10the phase calculatio hologramsn distance of is, single the lower viewing the average angle at SI each is, and computing the distance.trend of These this curve holograms is smoother contain than diFigurefferent 13. phase distributions, which result in different random noises. WeNext, also we calculatedcalculate the the SI of SI the of superimposed each reconstructed images from image, 10 holograms and the SIs for ofone the viewing 10 reconstructed angle. imagesIn Figure at the 15, same we distancecalculate the are SI averaged. of many different images at different computational distances. We Figure 14 shows the relationship between the average SI of 10 images and the calculated distance. We can see that, the farther the calculation distance is, the lower the average SI is, and the trend of this curve is smoother than Figure 13. J.J. ImagingImaging 20192019,, 55,, xx FORFOR PEERPEER REVIEWREVIEW 1111 ofof 1313

cancan seesee fromfrom thethe graphgraph thatthat thethe qualityquality ofof thethe imagimagee willwill bebe significantlysignificantly improved,improved, andand thethe longerlonger thethe distancedistance is,is, thethe lowerlower thethe SISI is.is. TheThe improvemenimprovementt ofof imageimage qualityquality maymay bebe thethe influenceinfluence ofof depthdepth of field is longer significant in the far computing distance than in the near computing distance. J. Imagingof field2019, 5is, 67longer significant in the far computing distance than in the near computing distance.11 of 13 Accordingly,Accordingly, thethe imageimage willwill bebe clearer.clearer.

J. Imaging 2019, 5, x FOR PEER REVIEW 11 of 13

can see from the graph that the quality of the image will be significantly improved, and the longer the distance is, the lower the SI is. The improvement of image quality may be the influence of depth of field is longer significant in the far computing distance than in the near computing distance. Accordingly, the image will be clearer. Speckle index Speckle index Speckle index Speckle index

ComputationalComputational DistanceDistance (dm)(dm)

FigureFigureFigure 13. 13.The13. TheThe Relationship RelationshipRelationship between betweenbetween Computational ComputatioComputationalnal DistanceDistance Distance andand and ImagingImaging Imaging Quality.Quality. Quality. Speckle index Speckle index

Computational Distance (dm)

Figure 13. The Relationship between Computational Distance and Imaging Quality. Average Speckle index of 10 images index of 10 images Average Speckle Average Speckle index of 10 images index of 10 images Average Speckle

ComputationalComputational DistanceDistance (dm)(dm)

FigureFigureFigure 14. 14.The14. TheThe Relationship RelationshipRelationship between betweenbetween Computational ComputatioComputationalnal DistanceDistance Distance andand and ImagingImaging Imaging Quality.Quality. Quality.

Next, we calculate the SI of the superimposed images from 10 holograms for one viewing angle. In Figure 15, we calculate the SI of many different images at different computational distances. We can see from the graph that the quality of the image will be significantly improved, and the longer the distance is, the lower the SI is. index of 10 images Average Speckle The improvement of image quality may be the influence of depth of field is longer significant in the far computingComputational distance than Distance in the (dm) near computing distance. Accordingly, the image will beFigure clearer. 14. The Relationship between Computational Distance and Imaging Quality. iae iae Speckle index of 10 superimposed of Speckle index Speckle index of 10 superimposed of Speckle index

ComputationalComputational DistanceDistance (dm)(dm) iae

Speckle index of 10 superimposed of Speckle index

Computational Distance (dm)

Figure 15. TheRelationshipbetweenComputationalDistanceandImageQualityof10SuperimposedImages.

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4. Discussion If we superimpose enough reconstructed images from the same angle, we can eﬀectively reduce the speckle noise of image. However, the maximum refresh rate of SLM is 60 fps. That is to say, the longer the reconstructed image stays at the same angle, the slower the speed of three-dimensional display will be. Suppose that ﬁve holograms are used at one angle and a total of 60 viewpoints, it will take 5 seconds to display the whole holograms for the model, such as if 20 reconstructed images are superimposed from each angle, as in our Scheme II, 60-angle images need to be displayed, then it will take 20 seconds to see the whole reconstructed images of the model. In addition, as the number of superposed images increases, the computer needs increased time to calculate the hologram. We can reduce the number of superposed images to improve the display speed, but this will inevitably reduce the visual quality of the image. If we want to ensure the quality of the image and increase the display speed, we need to improve the refresh rate of the SLM. In addition, we discuss the relationship between imaging quality and calculating imaging distance as well as layer thickness. Ideally, the farther the distance is, the better the image quality is.

5. Conclusions In this paper, we introduce an advancement in FRT by calculating many noise evaluation indices at different distances and then analyze their changing trends while simultaneously responding in technical challenges of rough surfaces. When calculating the holograms of Terra-Cotta Warrior model by the slice-based method, the smaller the interval between layers, the smaller the speckle index and the more effective details can be shown. Otherwise, some bad stripes will be produced. At the same time, with the increase of computing distance, the image quality has been significantly improved, but the improvement of image quality is not obvious after a certain distance. An optoelectronic 3D image reconstruction system that is based on SLM is established. The phase holograms of a Terra-Cotta Warrior model with 60 viewpoints are calculated with an interval of 6◦. In the process of image reconstruction, adding random phase on the object plane can significantly reduce the speckle noise by overlapping the reconstructed image.

Author Contributions: Conceptualization, H.-D.Z.; Data curation, Z.-F.G.; Formal analysis, Z.-F.G.; Investigation, Z.-F.G. and H.-D.Z.; Methodology, H.-D.Z.; Project administration, H.-D.Z. and Y.-J.Y.; Resources, Y.-J.Y.; Software, Z.-F.G.; Supervision, Y.-J.Y.; Validation, H.-D.Z.; Visualization, Z.-F.G.; Writing—original draft, Z.-F.G.; Writing—review & editing, H.-D.Z. Funding: This research was funded by the National Natural Science Foundation of China (61875115). Acknowledgments: We would like to extend my sincere thanks to all those who have helped us to write this article. We are deeply grateful to Vivi Tornari, who gave us instructive suggestions. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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