Reducing Motion Blur in Ghost Imaging Via the Hessian Matrix

applied sciences

Article Reducing Motion Blur in Ghost Imaging Via the Hessian Matrix

Chen Chang 1,†, Dongyue Yang 1,†, Guohua Wu 1, Bin Luo 2,* and Longfei Yin 1

1 School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China; [email protected] (C.C.); [email protected] (D.Y.); [email protected] (G.W.); [email protected] (L.Y.) 2 State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China * Correspondence: [email protected] † These authors contribute equally to this work.

Abstract: Different from conventional imaging, ghost imaging (GI) is an indirect modality of imaging that needs multiple measurements of the second-order correlation of data collected from two detectors. In some particular cases, the exposure time of two detectors or the rotation speed of the ground glass may not meet the need of experimental condition, resulting in motion blur that reduces the quality of the reconstructed image. In this paper, we propose a method to solve this problem. By convolving the data from the reference arm with the Hessian matrix, the intensity of the light in the data is replaced by the gradient of intensity and the inﬂuence of the motion blur in the reconstructed image can be reduced.

Keywords: motion blur; Hessian matrix; ghost imaging; Gaussian function

1. Introduction

As an indirect modality of imaging, ghost imaging (GI) is a new method of imaging that needs multiple measurements for one imaging process. In the experiment system, the

Citation: Chang, C.; Yang, D.; Wu, G.; pseudo-thermal light produced by a laser passing through a rotating ground glass (RGG) is Luo, B.; Yin, L. Reducing Motion Blur split into two arms by a beam splitter (BS); the arm with the illuminated object is called the in Ghost Imaging Via the Hessian signal arm and the data without spatial resolution from this arm is recorded by the bucket Matrix. Appl. Sci. 2021, 11, 323. detector, which means that the data from the detector is the sum of all of the light intensities https://doi.org/10.3390/app11010323 within the captured range of the detector. The other arm is called the reference arm in which a series of data is recorded by the Complementary Metal Oxide Semiconductor Received: 25 November 2020 (CMOS) camera [1,2]. After the collection of the data, the image can be reconstructed by Accepted: 28 December 2020 some algorithms with data from both arms. When the RGG rotates through the size of Published: 31 December 2020 the laser spot, the distribution of the laser speckle is gradually changed. Until it rotates out of the entire range of the laser spot, which corresponds to the “coherence time” of the Publisher’s Note: MDPI stays neu- pseudo-thermal light [3], the distribution of the laser speckle could be totally different from tral with regard to jurisdictional clai- the original one. In order to observe the second-order intensity correlation, the coherence ms in published maps and institutio- time should be close to or shorter than the integration time (exposure time) of the detector nal afﬁliations. in the reference arm, which requires a shorter exposure time or a lower rotation speed of the RGG [4]. Compared with conventional imaging, GI hosts a number of advantages such as

Copyright: © 2020 by the authors. Li- high resolution and high sensitivity and could be widely applied in many ﬁelds ranging censee MDPI, Basel, Switzerland. from remote imaging to lidar detection and microscopy [5,6]. The ﬁrst visualization of This article is an open access article GI was realized using entangled photon pairs in 1995 [3] and then it was proved that the distributed under the terms and con- experiment could also be realized with pseudo-thermal light sources and even true-thermal ditions of the Creative Commons At- light [7–10]. Recently, modulated light and other types of light sources were applied in an tribution (CC BY) license (https:// experiment [11–14]. Due to its convenience in application, the pseudo-thermal source is creativecommons.org/licenses/by/ still commonly used in GI experiments. 4.0/).

Appl. Sci. 2021, 11, 323. https://doi.org/10.3390/app11010323 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 9

Appl. Sci. 2021, 11, 323 2 of 9 experiment [11–14]. Due to its convenience in application, the pseudo-thermal source is still commonly used in GI experiments. In recent years, with the development of ghost imaging, the object is no longer lim- In recent years, with the development of ghost imaging, the object is no longer limited ited to static targets [15–17] and different kinds of algorithms and methods have been to static targets [15–17] and different kinds of algorithms and methods have been investi- investigated to deal with the problems in the imaging process with moving objects [18,19]. gated to deal with the problems in the imaging process with moving objects [18,19]. One of One of these problems is motion blur, which causes the visual quality decline in the re- these problems is motion blur, which causes the visual quality decline in the reconstructed constructed image. Due to that, ghost imaging requires multiple samplings in one imag- image. Due to that, ghost imaging requires multiple samplings in one imaging. The position ing. The position of the object and speckle should remain relatively static in each acquisi- of the object and speckle should remain relatively static in each acquisition process when tion process when it is a moving object, which means a short exposure time and coherence it is a moving object, which means a short exposure time and coherence time are needed, time are needed, leading to the requirement of a fast speed of the RGG and a fast speed of leading to the requirement of a fast speed of the RGG and a fast speed of the RGG leads to the RGG leads to motion blur. motion blur. To overcome this defect, in this paper we introduce the Hessian matrix to compensate To overcome this defect, in this paper we introduce the Hessian matrix to compensate forfor the the inﬂuenceinfluence ofof motion blur blur caused caused by by th thee fast fast speed speed of of the the RGG. RGG. Based Based on ona table a table top topexperimental experimental setup, setup, we wemeasured measured the theimage image quality quality of a of reconstructed a reconstructed image image under under dif- differentferent numbers numbers of ofcoincident coincident measurements measurements ranging ranging from from under-sampling under-sampling to over-sam- to over- sampling.pling. The The experimental experimental results results demonstrated demonstrated that, that, with with the the post-processing post-processing method method us- usinging the the Hessian Hessian matrix, matrix, GI GIcan can effectively effectively reco reconstructnstruct an image an image with with a higher a higher image image qual- qualityity and and less less influence inﬂuence of motion of motion blur. blur.

2.2. MaterialsMaterials and and Methods Methods TheThe experimental experimental setup setup of of GI GI with with the the Hessian Hessian matrix matrix experiment experiment is shownis shown in Figurein Figure1 . A1. 532 A 532 nm lasernm laser passing passing through through a RGG a (EdmundRGG (Edmund 100 mm 100 diameter mm diameter 220 grit) 220 made grit) a pseudo- made a thermalpseudo-thermal light source. light The source. pseudo-thermal The pseudo-therm light wasal splitlight into was two split arms into by two a BS. arms The by signal a BS. armThe penetratedsignal arm a penetrated transitive object a transitive (0.3 mm object square (0.3 “GI” mm pattern) square and“GI” was pattern) summed and to was be registeredsummed to by be a bucketregistered detector by a bucket (Thorlabs detector PDA100/A) (Thorlabs while PDA100/A) the reference while arm, the reference with no objectarm, with on the no path, object was on recordedthe path, bywas a CMOSrecorded camera by a CMOS (xiQ MQ003MG-CM). camera (xiQ MQ003MG-CM).

FigureFigure 1. 1. ExperimentExperiment system: RGG: RGG: rotating rotating ground ground gl glass;ass; BS: BS: 50:50 50:50 beam beam splitters; splitters; Object: Object: “GI” “GI” (ghost(ghost imaging) imaging) pattern. pattern. In the experiment, we set up the experiment system according to Figure1 and set the In the experiment, we set up the experiment system according to Figure 1 and set the exposure time of the CMOS as 15 ms. Five sets of data were collected including data from theexposure reference time arm of the and CMOS the signal as 15 arm ms. underFive sets different of data rotation were collected speeds including of the RGG, data which from the reference5 arm and the25 signal arm under different rotation speeds of the RGG, which varied from 36 rad/s to 36 rad/s. The data from the reference arm occupied a field of view varied from rad/s to rad/s. The data from the reference arm occupied a field of view of 300 × 300 pixels and the number of samplings in each set of data was 10,000. of 300The × 300 reconstructed pixels and imagethe number of the of object samplings in ghost in imaging each set canof data be written was 10,000. as [15 ]: The reconstructed image of the object in ghost imaging can be written as [15]: N 1 O(x, y) = ∑1 (Bi − hBi)(Ii(x, y)) (1) 𝑂(𝑥,) 𝑦 =N i=1(𝐵 − ⟨𝐵⟩)(𝐼 (𝑥,) 𝑦 ) (1) 𝑁 where Bi was the response of the bucket detector in the ith sampling, hBi was the ensemble averagewhere 𝐵 of Bwasi, Ii (thex, y )responsewas the intensityof the bucket distribution detector of in the the light 𝑖th field sampling, recorded ⟨𝐵 by⟩ was the CMOSthe en- camerasemble andaverageN was of the𝐵, number𝐼(𝑥,) 𝑦 was of samplings. the intensity Usually, distributionBi and ofIi (thex, y light) could field be recorded writtenas by thethe integral CMOS ofcamera the detectors’ and N was response the number during of the samplings. exposure timeUsually,τe. This 𝐵 assumesand 𝐼(𝑥, that) 𝑦 couldτe is setbe towritten be an appropriateas the integral value of andthe thedetect reconstructedors’ response image during has the no motionexposure blur. time The 𝜏 speed. This ofassumes the RGG that was 𝜏v 0 isand set theto be speed an appropriate of the RGG value was then and set the to reconstructedv0 + ∆v for moving image objecthas no imaging.motion blur. The speckleThe speed field of was the generated RGG was by 𝑣 directing and the thespeed laser of through the RGG the was RGG then and set the to speckle𝑣 +∆𝑣 in for the moving current object moment imaging. was produced The speckle by displacement field was generated of it in the by previous directing moment; the laser thus, the faster speed led to a longer distance, which is also called motion blur. Appl. Sci. 2021, 11, 323 3 of 9

The coherence time could be calculated according Equation (2):

τ = l/(ω × r) (2) where τ was the coherence time, l was the diameter of the laser spot, ω was the angular velocity (RGG) and r was the radius at the laser spot; thus, the coherence time of five sets 5 of data were between 28.64 ms and 5.72 ms (l = 0.5 mm, r = 0.04 m, ω varied from 36 rad/s 25 to 36 rad/s). In order to mimic the case of motion blur caused by fast speed of the RGG, the camera’s exposure time was set to be 15 ms and the rotation speed of the RGG was set to different values. This made the coherence time vary from less than the exposure time to over the exposure time, which led to a motion blur in both the reference samples and the reconstructed images. We then introduced the Hessian matrix and applied it in the reference samples as a filtering process to get rid of the GI image degradation. As this is different from the conventional image calculation procedure of GI, before we did the correlation, we first calculated the Hessian matrix of the data in the reference arm. The Hessian matrix can be described as the second-order partial derivative of a matrix [20] and it represents the gradient of the intensity the image. According to the concepts of linear scale-space theory, taking the second-order partial derivative of the image can be written as the convolution of the image and the derivatives of Gaussian functions [21]; thus, Equation (1) can be written as: N 2 0 1 ∂ G(x, y; σ) O (x, y) = ∑(Bi − hBi) Ii(x, y) ⊗ (3) N i=1 ∂x∂y where ⊗ is the convolution operator. In our method, we first set the value of standard deviation σ of the Gaussian function empirically and obtained the convolution window w. We then calculated the second-order partial derivative of the Gaussian function, which we think of as the Hessian matrix, then we used the matrix to process the data collected in the reference arm. The data processed by the Hessian matrix were then transferred to compute the correlation with the data from the signal arm instead of the data collected by the CMOS. To state how the Hessian matrix works, we demonstrated the change of the point spread function (PSF) between it being processed by Hessian matrix and not being processed by the Hessian matrix. The PSF of a reconstructed image with motion blur in GI and its projection in the x-z plane and y-z plane is shown in Figure2. Due to motion blur, the shape of the projection in the x-y plane changed from a circle to an oval. Usually, the projection in the x-z plane and the y-z plane are called a sombrero function. As shown in Figure3, when we put the sombrero functions into the same coordinates, the difference between the functions was clear. The full width at half maximum (FWHM) of the sombrero function in the x-z plane was larger than that in the y-z plane, which meant that the gradient of the intensity of the sombrero function in the y-z plane dropped faster, leading to a bigger absolute value of the gradient. When the Hessian matrix was introduced to process the data recorded in the reference arm, we actually used the gradient of the intensity instead of the intensity in the data to reconstruct the image. After being filtered by the Hessian matrix, the value of the sombrero function in the y-z plane was increased compared with the function in the z-x plane. This equivalently broadened the FWHM, making the projection in the x-y plane close to a circle; thus, the influence of motion blur was reduced. Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 9 Appl. Sci. 2021, 11, 323 4 of 9

(a) (b) (c)

Figure 2.2.( (aa) ) Point Point spread spread function function (PSF) (PSF) with with motion motion blur; (blur;b) projection (b) projection in the x-z in plane;the x-z (c )plane; projec- (c) pro- jectiontion in thein the y-z plane.y-z plane.

Figure 3. Change of PSF after being processed by the Hessian matrix.

3. Results Figure 3. Change of PSF after being processed by the Hessian matrix. A brief, related work was published earlier [22]. The results proved that the method 3.worked Results well when dealing with motion blur. Here, we performed further research. We ver- ified the effect of the method under different rotating speeds of the RGG, compared the methodA brief, with anotherrelated work deblur was algorithm published and studiedearlier [22]. the influence The results of parameters proved that in the our method workedmethod. Thus,well when we may dealing provide with some motion reference blur. for othersHere, towe use performed this method. further research. We verifiedFive the groups effect of of data the were method collected under in diff theerent experiments. rotating The speeds rotation of the speed RGG, of the compared 5 10 15 20 25 theRGG method was set with to 36 anotherrad/s, 36deblurrad/s, algorithm36 rad/s, and36 rad/sstudied and the36 influencerad/s and of each parameters group in our method.was repeated Thus, several we may times provide in different some rotationreference speeds for others of the to RGG. use this We method. used second- orderFive correlation groups (SC) of data to reconstruct were collected the images in the and experiments. compared the The results rotation with thespeed images of the RGG reconstructed by SC with the Hessian matrix and the image filtered by the Lucy–Richardson was set to rad/s, rad/s, rad/s, rad/s and rad/s and each group was repeated filter [23]. The PSF Lucy–Richardson filter could be calculated during reconstructing the severalimages (astimes shown in different in Figure2 rotationa) and the speeds number of of the iterations RGG. inWe the used Lucy–Richardson second-order was correlation (SC)set to to be reconstruct 1000. The comparison the images was and implemented compared on the the results platform with of athe MATLAB images 2019b reconstructed with by SCan Intelwith Core the Hessian i7-5500U matrix CPU and and 12 the GB image RAM. Thefiltered computation by the Lucy–Richardson time of the SC fluctuated filter [23]. The PSFbetween Lucy–Richardson 96 s and 112 s, while filter the could time of be the calcul SC withated the during Hessian matrixreconstructing increased tothe about images (as 153 s to 177 s and the time of the SC filtered by the Lucy–Richardson fluctuated between shown in Figure 2a) and the number of iterations in the Lucy–Richardson was set to be 122 s to 164 s according to the iteration time. 1000.Figure The 4comparison shows the reconstructed was implemented image ofon the the Hessian platform enhanced of a MATLAB GI experiment 2019b at with an Inteldifferent Core rotation i7-5500U speeds CPU of and the RGG 12 GB under RAM. 10,000 The samplings. computation As we time can of see the from SC thefluctuated betweenfigure, compared 96 s and with 112 the s, Lucy–Richardson while the time filter,of the the SC boundary with the of Hessian the object matrix in the image increased to aboutreconstructed 153 s to by 177 the SCs and with the the Hessiantime of matrixthe SC (II) filtered was much by more the Lucy–Richardson continuous and clearer, fluctuated betweenwhile the 122 noise s to of 164 the reconstructeds according to images the iteration filtered time. by the Lucy–Richardson filter was muchFigure lower. In4 shows the condition the reconstructed proposed in the image experiment, of the Hessian motion blur enhanced was not GI caused experiment by at differentthe moving rotation blur but speeds the high of rotational the RGG speed under of the10,000 RGG, samplings. making the As SC we with can the see Hessian from the fig- matrix better than the Lucy–Richardson algorithm if we needed a clearer object. When the ure, compared with the Lucy–Richardson25 filter, the boundary of the object in the image rotational speed was higher than 36 rad/s, it was almost impossible to reconstruct the reconstructed by the SC with the Hessian ma5 trix (II) was much more continuous and image. When the rotational speed was lower than 36 rad/s, motion blur was no longer an clearer,important while factor the in noise reducing of the the reconstructed image quality. images filtered by the Lucy–Richardson filter was much lower. In the condition proposed in the experiment, motion blur was not caused by the moving blur but the high rotational speed of the RGG, making the SC with the Hessian matrix better than the Lucy–Richardson algorithm if we needed a clearer object.

Appl. Sci. 2021, 11, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 9 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 9

When the rotational speed was higher than rad/s, it was almost impossible to recon- When the rotational speed was higher than rad/s, it was almost impossible to recon- Appl. Sci. 11 2021, , 323 struct the image. When the rotational speed was lower than rad/s, motion blur was5 of no 9 struct the image. When the rotational speed was lower than rad/s, motion blur was no longer an important factor in reducing the image quality. longer an important factor in reducing the image quality.

(a) (b) (c) (d) (e) (a) (b) (c) (d) (e) Figure 4. Reconstructed images of different methods at different rotations of the RGG under 10,000 samplings: I. Second- Figureorder 4.correlation Reconstructed (SC), II.images SC with of different the Hessian methods matrix, at III.different SC filtered rotations by the of Lucy–Richardson. the RGG under 10,000 (a) samplings: rad/s, (b) I. Second- rad/s, (c) order correlation (SC), II. SC with the Hessian matrix, III. SC filtered by the Lucy–Richardson. (a) 5rad/s, (b) 10rad/s, (c) order rad/s, correlation (d) rad/s, (SC), II.(e) SC with rad/s. the Hessian matrix, III. SC ﬁltered by the Lucy–Richardson. (a) 36 rad/s, (b) 36 rad/s, (c) rad/s,15 rad/s, (d) (d) rad/s,20 rad/s, (e) (e )rad/s.25 rad/s. 36 36 36 TakingTaking the the third third line line of dataof data shown shown in Figure in Fi4gure as an 4 exampleas an example where thewhere rotation the rotation speed Taking the third15 line of data shown in Figure 4 as an example where the rotation ofspeed the RGG of the was RGG36 wasrad/s, rad/s, we introduced we introduced the signal-to-noise the signal-to-noise ratio (SNR)ratio (SNR) to evaluate to evaluate the speed of the RGG was rad/s, we introduced the signal-to-noise ratio (SNR) to evaluate performancethe performance of the of Hessian the Hessian ﬁlter. filter. The SNR The canSNR be can written be written as as the performance of the Hessian filter. The SNR can be written as 2 ! ∑ ∑∑∑T(x𝑇, y(𝑥,) ) 𝑦 SNR = 10log ( x y ) SNR = 10 log10 ∑∑ 𝑇(𝑥,) 𝑦 (4) (4) ∑∑ (𝑂(𝑥,) 𝑦 − 𝑇(𝑥,2 𝑦)) SNR = 10log∑( x ∑y(O(x, y) − T(x, y)) ) (4) ∑∑ (𝑂(𝑥,) 𝑦 − 𝑇(𝑥, 𝑦)) wherewhereT (𝑇(𝑥,x, y) represented 𝑦) represented the the value value of the of object the object expected expected as shown as shown in Figure in5 Figureand O (5x ,andy) wherewas𝑂(𝑥, the 𝑇(𝑥, reconstructed𝑦) was the 𝑦) represented reconstructed image. the image.value of the object expected as shown in Figure 5 and 𝑂(𝑥, 𝑦) was the reconstructed image.

Figure 5. The expected object: ‘GI’ pattern. Figure 5. The expected object: ‘GI’ pattern. Figure 5. The expected object: ‘GI’ pattern. AsAs is is shown shown in in Figure Figure6, 6, the the SNR SNR increased increased as as the the measurement measurement number number increased. increased. WhenWhenAs the theis measurementshown measurement in Figure number number 6, the was SNRwas less lessincrea than thansed 1000, 1000, as the the the measurement information information in innumber the the data data increased. collected collected Whenwas not the enough measurement to fully reconstructnumber was the less image. than In1000, contrast, the information when the measurement in the data collected number was over 4000, motion blur became an important factor that reduced the image quality. As shown in Figure4, the Lucy–Richardson ﬁlter reduced the noise in the reconstructed Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 9

Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 9

Appl. Sci. 2021, 11, 323 was not enough to fully reconstruct the image. In contrast, when the measurement6 of 9 number was over 4000, motion blur became an important factor that reduced the image qual- wasity. not As enough shown to in fully Figure reconstruct 4, the Lucy–Richa the image. rdsonIn contrast, filter whenreduced the the measurement noise in the num- recon- berimagesstructed was over effectively, images 4000, motioneffectiv making ely, theblur SNRmaking became of images the an SNRim reconstructedportant of images factor by reconstructed SCthat with reduced the Hessian by the SC image matrixwith thequal- Hes- ity.andsian As those matrixshown ﬁltered andin Figure those by Lucy–Richardson filtered4, the Lucy–Richa by Lucy–Richardson ﬁlter nearlyrdson equal. filter filter reduced nearly the equal. noise in the reconstructed images effectively, making the SNR of images reconstructed by SC with the Hes- sian matrix and those filtered by Lucy–Richardson filter nearly equal.

Figure 6. 6.The The signal-to-noise signal-to-noise ratio ratio (SNR) (SNR) of an of image an image reconstructed reconstructed by the by SC, the SC filteredSC, SC byfiltered Lucy– by Lucy– Richardson and and SC SC with with a Hessian a Hessian filter. filter. Figure 6. The signal-to-noise ratio (SNR) of an image reconstructed by the SC, SC filtered by Lucy– RichardsonInIn addition, addition,and SC with it isit worthisa Hessianworth noting notingfilter. that therethat ther weree twowere important two important parameters: parameters: the standard the stand- deviation σ and the convolution window w when calculating the Gaussian second-order ard deviation 𝜎 and the convolution window 𝑤 when calculating the Gaussian second- partial derivative, which made a big difference to the reconstructed images. The equation In addition, it is worth noting that there were two important parameters: the stand- betweenorder partialσ and derivative,w can be written which as [made24] a big difference to the reconstructed images. The ardequation deviation between 𝜎 and theσ and convolution w can be window written as𝑤 [24] when calculating the Gaussian second- order partial derivative, which made σ −a big0.8 difference to the reconstructed images. The w = 𝜎−0.8+ 1 × 2 + 1 (5) equation between σ and w can be writtenw=0.3 as [24]+1×2+1 (5) 0.3 where w determined the range of the pixels𝜎−0.8 where the gradient of intensity was to be where w determined the rangew= of the pixels+1×2+1 where the gradient of intensity was (5)to be calculated; thus, it influenced the performance0.3 of the Hessian matrix. In the experiment, thecalculated; standard thus, deviation it influencedσ varied from the 1performanc to 5, makinge theof convolutionthe Hessian window matrix.w Invary the fromexperiment, where4.3the (consideringstandardw determined deviation that the numberrangeσ varied of of the pixelsfrom pixels 1 is to integer wher 5, makie and thengw gradient thehere convolution was of set intensity to be window 5) towas 31. to𝑤 bevary calculated;Thefrom reconstructed 4.3 thus, (considering it influenced images that of the thethe SC numberperformanc with the of Hessian pixelse of the matrixis integerHessian are shownand matrix. 𝑤 in here FiguresIn thewas 7experiment, andset 8to. be 5) to the31. standard TheAs isreconstructed shown deviation in Figures σ images varied7 and 8 offrom, the the different 1SC to with 5, valuemaki the ng ofHessian theσ led convolution tomatrix different are reconstructed windowshown in 𝑤 Figures vary 7 fromimages.and 4.3 8. (considering To provide a that reference the number for others of to pixels use this is integer method, and we studied𝑤 here the was relationship set to be 5) to 31. betweenThe reconstructed the optimal valueimages of σofand the the SC data with collected. the Hessian As the matrix method are changed shown the in shape Figures 7 of the PSF from an oval to a circle, we speculated that the optimal value of σ related to the and 8. ratio of the major axis to the minor axis of the projection in the x-y plane.

(a) (b) (c) (d) (e)

σ = 1 σ = 2 Figure 7. Reconstructed(a) images of( bthe) SC with the Hessian(c) matrix in different(d )standard deviations.(e )( a) ; (b) ; (c) σ = 3; (d) σ = 4; (e) σ = 5. FigureFigure 7. Reconstructed 7. Reconstructed images images of ofthe the SC SC with with the the Hessian Hessian matrixmatrix in in different different standard standard deviations. deviations. (a) σ (a=) 1σ;( =b) 1σ; (=b)2 σ; = 2; (c) σ =(c) 3σ; (=d)3; σ ( =d) 4σ; =(e)4; σ (e =) σ 5.= 5. Appl. Sci. 2021, 11, 323 7 of 9 Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 9

Figure 8. 8. TheThe SNR SNR of ofa reconstructed a reconstructed image image of SC of with SC the with Hessian the Hessian filter in ﬁlterdifferent indifferent values of values of standard deviation. deviation.

AsAs is listed shown in in the Figures Table 71 and, the 8, ratio the different of the five value sets of ofσ dataled to were different 1.5, reconstructed 1.8, 2.1, 2.4 and images.2.9 and To the provide optimal a reference value of σfor, asothers shown to use in Figurethis method,9, was we 2, studied 2, 2, 2 andthe relationship 1, respectively. Accordingbetween the to optimal the speculation, value of 𝜎 the and optimal the data value collected. of σ should As the havemethod been changed 2, 2, 2, the 2 and shape 3. Here weof the calculated PSF from the an SNR oval ofto fivea circle, sets ofwe data speculated under differentthat the optimal rotational value speeds of 𝜎 of related the RGG to and the ratio results of arethe shownmajor axis in Figure to the9 minor. With axis the exceptionof the projection of the lastin the data x-y group, plane. the optimal of σ was aboutAs listed equal in the to theTable ratio; 1, the thus, ratio our of speculation the five sets should of data have were been 1.5, 1.8, correct. 2.1, 2.4 Unfortunately, and 2.9 andif the the rotational optimal speedvalue of the𝜎, as RGG shown keeps in Figure growing, 9, was it is 2, difficult 2, 2 ,2 and to reconstruct 1, respectively. the image Ac- no cordingmatterwhat to the algorithm speculation, is used,the optimal making value it difficult of 𝜎 should to explore have thebeen relationship 2, 2, 2, 2 and between 3. Here the weoptimal calculated of σ and the theSNR rotational of five sets speed of data of the under RGG different further. rotational speeds of the RGG and the results are shown in Figure 9. With the exception of the last data group, the opti- Tablemal of 1. 𝜎The was ratio about and equal the optimal to the value ratio; of thus,σ under our different speculation rotational should speeds have of been the RGG. correct. Unfortunately, if the rotational speed of the RGG keeps growing, it is difficult to recon- structRotational the image Speedno matter of the what RGG algorithm is used, Ratio making it difficult Optimal to explore Value the of rela-σ tionship between5/36 the rad/s optimal of 𝜎 and the rotational 1.5 speed of the RGG further. 2 10/36 rad/s 1.8 2 Table 1. The ratio15/36 and rad/sthe optimal value of 𝜎 under different 2.1 rotational speeds of the 2RGG. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 9 20/36 rad/s 2.4 2 Rotational Speed of the 25/36 rad/sRatio 2.9Optimal Value 1 of 𝝈 RGG 5/36 rad/s 1.5 2 10/36 rad/s 1.8 2 15/36 rad/s 2.1 2 20/36 rad/s 2.4 2 25/36 rad/s 2.9 1

FigureFigure 9. 9. TheThe SNR SNR of of the the reconstructed reconstructed image image of ofthe the SC SC with with the the Hessian Hessian matrix matrix in different in different values values of standard deviation at different rotation speeds of the RGG. of standard deviation at different rotation speeds of the RGG. 4. Discussion In this paper, we did several experiments to simulate motion blur caused by the improper setting between the exposure time of the detectors and the coherence time. From the experiments, we saw motion blur in the images reconstructed by the SC and SC filtered by Lucy–Richardson, which reduced the quality of reconstructed image. We therefore introduced the Hessian matrix into ghost imaging to reduce the influence caused by motion blur. The results showed that when data were filtered by the Hessian matrix, the reconstructed images had smoother edges, thus improving the quality of the reconstructed images and reducing the influence from motion blur. However, as the cost of the method, when we used the Hessian matrix to deal with motion blur, the resolution was reduced, which may cause some problems if the object has a complex structure. In the further experiments, we hope to find another algorithm to solve the problem that the resolution is reduced when applying the Hessian matrix to the ghost imaging, making the method a promising approach to deal with motion blur in ghost imaging.

Author Contributions: C.C. carried out the experiments, validation and did the writing and original draft preparation. G.W. was responsible for the supervision, D.Y. was responsible for the methodology and formal analysis, B.L. was responsible for the writing, review and editing, L.Y. was responsible for the supervision and writing, review and editing. All authors have read and agreed to the published version of the manuscript. Funding: National Natural Science Foundation of China (61801042, 61631014, 61401036, 61471051 and 61531003), the Youth Research and Innovation Program of BUPT (2015RC12) and the BUPT Excellent Ph.D. Students Foundation (CX2019224). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available because the need of large storage space made it not easy to transfer. Conflicts 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.

References 1. Wu, G.; Li, T.; Li, J.; Luo, B.; Guo, H. Ghost Imaging under Low-Rank Constraint. Opt. Lett. 2019, 44, 4311–4314. Appl. Sci. 2021, 11, 323 8 of 9

4. Discussion In this paper, we did several experiments to simulate motion blur caused by the improper setting between the exposure time of the detectors and the coherence time. From the experiments, we saw motion blur in the images reconstructed by the SC and SC filtered by Lucy–Richardson, which reduced the quality of reconstructed image. We therefore introduced the Hessian matrix into ghost imaging to reduce the influence caused by motion blur. The results showed that when data were filtered by the Hessian matrix, the reconstructed images had smoother edges, thus improving the quality of the reconstructed images and reducing the influence from motion blur. However, as the cost of the method, when we used the Hessian matrix to deal with motion blur, the resolution was reduced, which may cause some problems if the object has a complex structure. In the further experiments, we hope to find another algorithm to solve the problem that the resolution is reduced when applying the Hessian matrix to the ghost imaging, making the method a promising approach to deal with motion blur in ghost imaging.

Author Contributions: C.C. carried out the experiments, validation and did the writing and original draft preparation. G.W. was responsible for the supervision, D.Y. was responsible for the methodology and formal analysis, B.L. was responsible for the writing, review and editing, L.Y. was responsible for the supervision and writing, review and editing. All authors have read and agreed to the published version of the manuscript. Funding: National Natural Science Foundation of China (61801042, 61631014, 61401036, 61471051 and 61531003), the Youth Research and Innovation Program of BUPT (2015RC12) and the BUPT Excellent Ph.D. Students Foundation (CX2019224). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available because the need of large storage space made it not easy to transfer. Conﬂicts 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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