Improved resolution three-dimensional integral imaging using optimized irregularlens-array structureZahra Kavehvash,1,*Khashayar Mehrany,1and Saeed Bagheri21Department of Electrical Engineering,Sharif University of Technology,P.O.Box11365-8639,Tehran,Iran 2Philips Research North America,345Scarborough Road,Briarcliff Manor,New York10510,USA*Corresponding author:zkavehvash@Received29May2012;revised26July2012;accepted26July2012;posted26July2012(Doc.ID169490);published24August2012A rigorous approach is proposed to improve the resolution of integral imaging(InI)by finding the appro-priate form of irregularity in the arrangement of the InI lenslets.The improvement of the resolution isachieved through redistribution of the sampling points in a uniform manner.The optimization process forfinding the optimum pattern of the lens-array irregularity is carried out by minimizing a cost function,whose mathematical closed-form expression is provided.The minimization of the proposed cost functionensures the uniform distribution of sampling points and thus improves the resolution within the desireddepth of field(DOF)and field of view(FOV).A set of standard resolution charts is used to demonstratethe improvement of the quality of the three-dimensional(3D)images obtained by using the optimizedirregular lens array.It is shown that the overall level of the lateral and depth resolutions is improved atthe same time.©2012Optical Society of AmericaOCIS codes:110.6880,110.3000.1.IntroductionThree-dimensional(3D)integral imaging(InI)does not require wearing special eyeglasses to provide autostereoscopic images from a continuous view-point[1–6]and thus counts among the promising 3D imaging methods to be used for commercial3D visualization.Although more than a century-old technique[1],it has recently attracted increasing at-tention thanks to the renewed interest in3D imaging [7].A conventional InI system captures the visual in-formation of a3D object through a regular flat lens array and forms a bunch of elemental images(EIs). Since each EI provides a different perspective of the 3D object,the original3D object is reconstructed once the EIs are displayed on an optical device placed in front of a similar lens array.Despite the many-sided advantages brought by using the InI system,the reconstructed3D images have limited depth of field(DOF)and field of view (FOV)together with a low level of lateral and depth resolution.Many researches have been conducted to improve InI by adding different sorts of irregularity to the conventionally used flat lens array having si-milar lenslets.DOF improvement is already reported when curved lens arrays[8],and flat lens arrays made of lenslets with different focal lengths are em-ployed for InI[9].FOV improvement is also reported when appropriately curved lens arrays are used [10–12].The possibility of achieving a higher level of depth and lateral resolution by finding and adding the proper form of irregularity is,however,not a very well-studied subject.Such a study necessitates a quantitative gauge to measure the level of achiev-able lateral and depth resolution in InI systems. This has not been possible until very recently,when object-independent models based on the concept of1559-128X/12/256031-07$15.00/0©2012Optical Society of America1September2012/Vol.51,No.25/APPLIED OPTICS6031sampling rays were introduced to quantify the lateral and depth resolution[13].A sampling ray connects the center of each pixel on the display de-vice to the center of the corresponding lens in the lens array and thus carries the information necessary for image reconstruction[13–15].The intersection point of each sampling ray and the image plane is referred to as the sampling point[13].Given that two or even more sampling rays can intersect with each other at a specific image plane,the concept of the order of the sampling points was introduced to enumerate the in-tersecting sampling rays.The n th-order sampling point was thus defined to designate the intersection of n 1sampling rays.In the most recent work on the subject,the total number of the first-order sam-pling points is related to the lateral resolution of the InI system while the depth resolvability of different points of the image domain is related to the order of its sampling point[13].In this fashion,the depth and lateral resolution of the InI system is measured and a mathematical cost function is proposed to find the appropriate irregular pattern of the lens array that improves both lateral and depth resolvability in the desired DOF and FOV range.This is,to the best of our knowledge,the first attempt to propose a rigorous approach for finding the optimum irregu-lar pattern achieving higher depth and lateral resolution.The structure of this paper is as follows:we first introduce the overall structure of an irregular lens array,extract the coordinates of the k th-order sam-pling points,and optimize the structure to ensure a more uniform distribution of sampling points—all in Section2.We then present the simulation results in Section3and thus confirm the claimed improvement observed in the quality of the images reconstructed by using the optimized irregular lens array.Finally,we make the conclusions in Section4.2.Optimized Irregular Lens-Array PatternIt is already shown that the distribution of the sampling points is rather nonuniform whenever con-ventional flat lens arrays made of similar lenslets are employed[13].Quite often,the sampling points get accumulated at specific image planes.Since the high-er density of sampling points warrants a higher lat-eral resolution,any such plane is referred to as the lateral resolution plane.Likewise,a lower number of higher-order sampling points occasionally pile up at some other image planes.Since the presence of higher-order sampling points ensures a good depth resolvability,this latter type of planes is referred to as the depth resolution plane.Given that the high-order sampling points are formed by the inter-section of more sampling rays,there is a trade-off be-tween the number and order of sampling points at each lateral image plane and thus the good lateral resolution in the lateral resolution planes is bought at the expense of the depth resolvability[10].It is therefore conceivable that the uniform distribution of sampling points would be very much desired. Therefore,we are trying to rearrange the samplingrays in such a manner that the sampling pointsof different orders become uniformly distributed within the imaging space.This can be achieved byintroduction of a meticulous irregularity in the posi-tion and direction of the lenslets in the lens-array structure.Using the Cartesian coordinate system,it is hereassumed that the3D objects are to be reconstructedwithin a specific region lying between z z1,z z2, x x1,x x2,y y1,and y y2.This specific regionis hereafter referred to as the region of interest(RoI).It is within the RoI that the distribution of thespatial coordinates of the sampling points and their orders is tried to be made uniform.First,a conven-tional regular lens array is made irregular to showthat the distribution of the sampling points of differ-ent orders can be controlled.Then,the coordinates ofthe n th-order sampling points lying within the RoIare extracted in terms of the position and direction of the lenslets in a lens array with arbitrary irregu-larity.The appropriate cost function to be used foroptimizing the irregular pattern of the lenslets is finally introduced.A.Irregular Lens-Array StructureAn irregular lens array with an arbitrary pattern is schematically shown in Fig.1.Three different crosssections of the structure in the xy,yz,and xz planes(front view,top view,and left view)are shown in Figs.1(a),1(b),and1(c),respectively.Assuming thatthe original flat lens array was originally placed inthe xy plane,the p,q th lenslet of the array is shifted along the z-axis by s pq,and rotated byθx;pq andθy;pqaround the x-and y-axes,respectively.Insofar as thesampling ray connects an arbitrary input pixel of a specific EI,viz.P1in Figs.1(b)and1(c),to the centerof its corresponding lens,the irregularity of thelenslets—controlled by s pq,θx;pq,andθy;pq—affects the spatial arrangements of the sampling rays.Thisis clearly demonstrated in Figs.1(b)and1(c),wherethe sampling rays of the input pixel P1in the regular and irregular lens arrays are plotted by dashed andsolid lines,respectively.Once the sampling rays arerearranged by changing s pq,θx;pq,andθy;pq,the popu-lation of the sampling points of different orders is controlled within the RoI.This point is further expounded below,where the coordinates of the n th-order sampling points are extracted.B.Extraction of the n th-Order Sampling PointsThe coordinates of all sampling points can be easilyextracted once the algebraic equation of each sam-pling ray emanating from the i,j th pixel of the p;q th EI is given.Since each sampling ray passes through the center of its corresponding lenslet in the array,the governing equation of a typical sam-pling ray can be written in terms of its pixel co-ordinates x i;y j;−g ,and its lenslet coordinates x p;y q;s pq :6032APPLIED OPTICS/Vol.51,No.25/1September2012z g s pq x −x ix p i y −y j y q j.(1)Here,g is the distance between the display device and the lens-array plane.It should be pointed out that the sampling ray passing through the p;q th lenslet lies within the FOV of the lenslet:αx and αy along the x and y directions,respectively.Therefore,the following two relations are held:tan −1x p −x ig−θx;pq ≤αx tan −1y q −y jg−θy;pq ≤αy .(2)Similarly,the length of the sampling ray is limitedto the DOF of the lenslet lying between z L and z R in the regular lens array.Therefore,the z -coordinate of the sampling ray in the irregular structure is subjected to the following relation:z a − z a −z L cos θx;pq cos θy;pq s pq <z <z az R −z a cos θx;pq cos θy;pq s pq ;(3)where z a is the focused image plane satisfying the lens law .The coordinates of the zeroth-order sampling points at the image plane z i can then be obtained straightforwardly:8>><>>:z z i x z i g x p −x i s pq g x i y z i g y q −y j s pq gyj; 4where z i fulfills the preceding inequality givenby Eq.(3).The coordinates of the higher-order sampling points are not very easy to extract.It requires solving a set of linear equations.The coordinates of the first-order sampling points are extracted by finding the intersection point of two different sampling rays,each one emanated from a different EI.The coordi-nates of the first-order sampling points satisfy the following set of equations:8>><>>:x −x ix p −x i x −x i 0x p 0−x i 0y −y jq −y j y −y j 0q 0−y j 0x p −x i pq g z g x i x p 0−x i 0pq gz g x i 0; 5where p;q ≠p 0;q 0to make sure that the sampling rays are coming from different EIs (sampling rays coming from the same EI does not cross with each other).Once again,the z -coordinate of the intersec-tion point should fulfill the DOF inequality given in Eq.(3).Along the same line,the coordinates of the n th-order sampling points are expected to beextractedFig.1.Regular and irregular lens arrays,their typical sampling rays,and the RoI.Lenslets of the regular and irregular lens arrays and their sampling rays are depicted by dashed and solid lines,respectively in (a)front view ,(b)top view ,and (c)left view .1September 2012/Vol.51,No.25/APPLIED OPTICS6033by finding the intersection point of n 1 samplingrays.It therefore seems that the unknown x-,y-,and z-coordinates of the intersection point should fulfill aset of3n equations.That would leave us with an over-determined set of linear equations.It should,how-ever,be noted that each sampling ray is in real fact a beam of nonzero width and thus the n th-order sampling point is formed not only when n 1 sam-pling rays are crossing each other but also when they are in the vicinity of each other.Therefore,a certainlevel of error can be tolerated in the fulfillment of the 3n equations governed by the intersection of n 1 sampling rays.The three unknown coordinates of the n th-order sampling point are thus extracted by find-ing those specific coordinates whose overall error in satisfaction of the overdetermined set of3n equa-tions is below a certain threshold level.This can be done by following a standard optimization algo-rithm.Once more,the coordinates of the n th-order sampling points are subjected to the DOF inequality expressed in Eq.(3).C.Optimization ProcessNow that the coordinates of the n th-order sampling points are at hand,an appropriate cost function is needed to find the optimum irregular pattern of the lenslets.The appropriate cost function is ex-pected to reach its minimum value when s pq,θx;pq, andθy;pq of each lenslet result in a uniform distribu-tion of all those sampling points that are lying within the RoI.It is therefore necessary to have a mathema-tical expression to quantify the uniformity of the sampling point distribution.The first step toward this end is the measurement of the distance between the neighboring sampling points whose coordinates lie within the RoI cube.Here,the Delaunay triangu-lation algorithm is employed[16].Once the Delau-nay triangles are generated,the length of sides of the Delaunay triangles are recorded in a vector whose average and variance are denoted by E dandσ2d ,respectively.Insofar as higher E d,andσ2dim-plies a lower level of uniformity in the distribution of the sampling points,the proposed cost function is setproportional to E d andσ2d .Given that the minimization of E d andσ2d does notensure that the total volume of the RoI is uniformly covered,the marginal sampling points whose x-,y-, and z-coordinates are close enough to the border of the RoI are further studied.These marginal sam-pling points are selected by pursuing the following algorithm.The first k sampling points whose x-coordinates are closest to x1,the first k sampling points whose y-coordinates are closest to y1,and the first k sampling points whose z-coordinates are closest to z1are all selected.Similarly,The first k sampling points whose x-coordinates are closest to x2,the first k sampling points whose y-coordinates are closest to y2,and the first k sampling points whose z-coordinates are closest to z2are also se-lected.They are hereafter referred to as the marginal sampling points.The average of the distance between different marginal sampling points and the borders of the RoI is calculated together with its variance.They are denoted by E m andσ2m Along the same line,E m andσ2m are expected to be as low as possible in the optimized irregular lens array. Finally,to ensure that not only the sampling points but also their order are uniformly distributed,the frequency of observing the n th-order sampling points should be studied.Assuming that C n denotes the to-tal number of the n th-order sampling points that can be counted within the RoI,it is desired to have al-most equal C n s.Given that equal C n s maximize ΣC2n whenever there are N sampling points,the pro-posed cost function is set to be inversely proportional toΣC2n.It is therefore written as follows:f s pq;θx;pq;θy;pqE2dσ2dE2m σ2mN ΣC2n.(6)Decreasing the first and second terms in the nu-merator of the proposed cost function guarantees the constellation of the sampling points in the vici-nity of each other,and the proximity of them to the borders of the RoI,respectively.Increasing the denominator of the cost function,on the other hand, ensures the uniformity of the distribution of the or-der of the sampling points.Therefore,the optimum s pq,θx;pq,andθy;pq minimize the proposed cost func-tion.It should,however,be kept in mind that there are some constraints imposed on the dynamic range of s pq,θx;pq,andθy;pq.These constraints should be ap-plied when the optimization algorithm is running to minimize the cost function.3.Simulations and AnalysisIn this section,a typical imaging system with a reg-ular lens array is considered.It is composed of21×21similar lenslets.The lens pitch is5mm.Assuming that the distance between the display device and the lens array is g 50mm,and that each EI gets25×25pixels of the display device,there are2060sam-pling points within the RoI,100×100×200mm3. Using the Delaunay triangulation algorithm,it can be shown that Ed 0.77mm andσ2d0.05mm2 for the sampling points.It can be also shown that E m 0.61mm andσ2m 0.1mm2for the marginal sampling points.Furthermore,there are C1 1648,C2 232,C3 60,C4 50,C5 0,C6 40, and C7 30sampling points of the first,second, third,fourth,fifth,sixth,and seventh orders,respec-tively.Counting the total number of first-and seventh-order sampling points at different lateral planes,the resolution and depth planes are found to be at z 400mm and z 416.7mm,respectively. The former has140first-order sampling points and the latter has30seventh-order sampling points. Using the genetic algorithm to minimize the pro-posed cost function,the optimized values of s pq,θx;pq,andθy;pq of each lenslet are found.This time, N 2436sampling points are counted within the6034APPLIED OPTICS/Vol.51,No.25/1September2012RoI and we have E d 0.71mm,σ2d 0.004mm 2,E m 0.65mm,and σ2m 0.01mm 2.Reduction ofσ2d and σ2m stands witness for the fact that the distri-bution of the sampling points is now more uniform.Growth of the total number of the sampling points is also a benefit.Even more,we have C 1 860,C 2 416,C 3 308,C 4 272,C 5 280,C 6 160,and C 7 140and thus C i s ,i 1;…7are made closer to each other.The sampling points of the optimized irregular lens array are distributed in such a uniform fashion that the lateral and depth resolution planes cannot be easily discriminated from each other.Be-fore moving further on examination of this point in detail,it is perhaps necessary to say a few words on explanation of how the EIs are to be recorded and displayed via the designed irregular lens array .Since each lenslet in the irregular lens array is pos-sibly rotated whereas the sensor behind it remains fixed,the lens-array pattern cannot be made irregu-lar by simply rotating the cameras —which would ro-tate lens and sensor both.Still,the EIs to be recorded via the irregular structure can be easily obtained by computational processing of the EIs recorded via the regular structure.This computational process takes advantage from two essential facts.First,rotation of lenslets by θx;pq and θy;pq is equivalent to rotation of the EI plane by -θx;pq and -θy;pq .Second,axial shift of each lenslet by s pq has the same effect as does changing the focal length of each lenslet to f pq :f pq1f 1g −1g s pq−1.(7)Here,the standard computational recording pro-cess [17]is pursued to obtain every p;q th EI.The thus obtained EIs are then postprocessed and the EIs that would be recorded by the irregular struc-ture are obtained.Similarly,the reconstruction stage is performed computationally .It is,however,worth noting that these EIs are also usable in physical re-construction of the 3D scene if one employs spatial-light-modulator to mimic the irregular lens-array pattern [18].Now ,as a demonstrative example,two resolution charts are placed at the lateral and depth resolution planes of the original regular lens array at z 400mm,and z 416.7mm,respectively .The former is hereafter referred to as the plane A ,and the latter as the plane B .The original scene com-posed of these two resolution charts placed at two dif-ferent planes is then reconstructed at the plane A .The reconstructed images of the regular and irregu-lar lens arrays are shown in Figs.2(a)and 2(c),re-spectively.Although the resolution chart placed at the plane B is not expected to be observed when the scene is reconstructed at the plane A ,it imprints a conspicuous shadow because the lateral and depth resolution planes are close to each other.This un-wanted shadow is,however,very strong when the scene is reconstructed by using a regular lens arraysince the good lateral resolution at plane A is actu-ally bought at the expense of the depth resolvability.Figure 2(c)demonstrates that the unwanted effect caused by the shadow of the resolution chart at plane B is partly rectified by introduction of appropriate ir-regularity in the lens array .This means better depth resolvability at the plane A .It should,however,be pointed out that the improved depth resolvability has its own expense as the reconstructed image of the resolution chart at plane A has lost some of its high-frequency contents when it is reconstructed by the irregular lens array .To better demonstrate this fact,the high-frequency contents of the resolu-tion charts;which are marked in Figs.2(a)and 2(c),are enlarged and shown in Figs.2(b)and 2(d).Along the same line,the scene is reconstructed as it is seen at the plane B .Figures 3(a)and 3(c)show the image reconstructed via the regular and the op-timized irregular lens arrays,respectively .This time the resolution chart placed at the plane A is not ex-pected to be observed.Given that the image recon-struction through the regular lens array enjoys the highest level of depth resolvability at the plane B ,the unwanted shadow of the resolution chart placed at the plane A is quite weak.This is not true for the irregular lens array because the sampling points are uniformly distributed and there are neitherdepthFig.2.(Color online)(a)Reconstructed image in the lateral reso-lution plane (plane A )of regular lens array ,(b)its high-frequency contents enlarged to show how good the resolution is,(c)recon-structed image in the lateral resolution plane (plane A )of irregular lens array ,and (d)its high-frequency contents enlarged to show how good the resolution is.1September 2012/Vol.51,No.25/APPLIED OPTICS6035nor resolution planes.Rather,the reconstructed images have almost equal quality when they are re-constructed at different planes.It is worth noting that even though the unwanted shadow is a bit stron-ger for the irregular lens array when the scene is re-constructed at the plane B ,the lateral resolution achieved by the irregular lens array is better than the lateral resolution achieved by the regular lens array.Once again,the high-frequency contents of the resolution charts,which are marked in Figs.2(a)and 2(c),are enlarged and shown in Figs.2(b)and 2(d).Finally,a third resolution chart is added to the original scene.It is placed at z 408.2mm.This plane —which is here after referred to as the plane C —has no non-zeroth-order sampling points and thus has inferior resolution.The scene is now recon-structed at the plane C and the image obtained by the regular and irregular lens array is shown in Figs.4(a)and 4(c),respectively .The poor quality of the reconstructed image is readily noticeable in Fig.4(a),showing the image reconstructed by the regular lens array .The shadows of the resolution charts placed at the planes A and B are conspicuous,and the resolution chart at the plane C has low qual-ity [Fig.4(b)].Thanks to the uniform distribution of the sampling points in the RoI of the optimized irregular lens array ,the poor quality of image recon-struction at the plane C is partly boosted.The unwanted shadows are weaker in Fig.4(c),and thelateral resolution is higher,when the optimized irre-gular lens array is used for image reconstruction.This is clarified in Figs.4(b)and 4(d),where the high-frequency contents of the resolution charts are magnified.All these figures stand witness for the fact that the sampling points in the optimized irregular lens array are rearranged uniformly.There are no lateral reso-lution planes with bad depth resolvability and no depth resolution planes with bad lateral resolution.Rather,a same good level of depth and lateral reso-lution is achieved at all different points.The mini-mum level of the lateral and depth resolution achieved by regular flat lens arrays is therefore in-creased by using the optimized irregular lens array because almost all points enjoy the same level of lat-eral and depth resolution.4.ConclusionThe idea of using irregularity in InI systems,e.g.,using curved or nonuniform lens arrays,is not un-known and has successfully improved the DOF or FOV [8–12].The possibility of using irregularity in InI systems to enhance the lateral and depth resolu-tion was,however,left to be explored.Exploration of the effects of irregularity on the resolution is just made possible thanks to theobject-independentFig.3.(Color online)(a)Reconstructed image in the depth reso-lution plane (plane B )of regular lens array ,(b)its high-frequency contents enlarged to show how good the resolution is,(c)recon-structed image in the depth resolution plane (plane B )of irregular lens array ,and (d)its high-frequency contents enlarged to show how good the resolutionis.Fig.4.(Color online)(a)Reconstructed image in z 408.2mm (plane C )of regular lens array ,(b)its high-frequency contents en-larged to show how good the resolution is,(c)reconstructed image in plane C of irregular lens array ,and (d)its high-frequency contents enlarged to show how good the resolution is.6036APPLIED OPTICS /Vol.51,No.25/1September 2012approach recently proposed to quantify the lateral and depth resolution of InI[13].This approach is in this manuscript followed to first measure the lat-eral and depth resolution of InI and then to find the optimum pattern of irregularity to increase the mini-mum achievable level of resolution.A mathematical cost function is proposed to fulfill this puta-tional InI is carried out and a typical3D scene—formed by resolution charts placed at different planes—is reconstructed.It is shown that the opti-mized irregular lens array found by using the proposed optimization algorithm achieves a higher level of resolution than the typical flat lens array con-ventionally used for InI.References1.M.G.Lippmann,“Epreuves reversibles donnant la sensationdu 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