On the Relationship Between Dual Photography and Classical Ghost Imaging

Transcription

1 1 On the Relationshi Between Dual Photograhy and Classial Ghost Imaging Pradee Sen University of California, Santa Barbara arxiv: v1 [hysis.otis] 12 Se 2013 Abstrat Classial ghost imaging has reeived onsiderable attention in reent years beause of its remarkable ability to image a sene without diret observation by a light-deteting imaging devie. In this artile, we show that this imaging roess is atually a realization of a aradigm known as dual hotograhy, whih has been shown to rodue full-olor dual (ghost) images of 3D objets with omlex materials without using a traditional imaging devie [1], [2]. Seifially, we demonstrate mathematially that the ross-orrelation based methods used to reover ghost images are equivalent to the light transort measurement roess of dual hotograhy. Beause of this, we are able to rovide a new exlanation for ghost imaging using only lassial otis by leveraging the rinile of reiroity in lassial eletromagnetis. his observation also shows how to leverage revious work on light transort aquisition and dual hotograhy to imrove ghost imaging systems in the future. Index erms dual hotograhy, ghost imaging, single-ixel amera, rojetor-based imaging, orrelated imaging. I. INRODUCION Ghost imaging has been the subjet of onsiderable researh within the alied otis ommunity, given its ability to image a sene (usually a 2D artially-transarent slide) without diret observation by a light-sensing imaging devie. Although initially believed to be only a quantum effet [3], [4], it was later extended to use airs of lassially orrelated hotons [5] [10]. However, debate about whether it was intrinsially a quantum effet or lassial henomenon ontinued (e.g., [11], [12]), and it was only reently that the otis ommunity aeted that ghost imaging ould be erformed with fully lassial, inoherent illumination [13], [14]. he fous on hoton orrelation from early ghost imaging work has influened lassial ghost imaging systems even till today, however. Most systems tyially emloy a rossorrelation tehnique between the illumination and the measured value to erform the ghost imaging [6], although slight variations suh as differential imaging [15] and normalized ghost imaging [16] have been roosed. However, these orrelation-based methods all have the fundamental limitation that they are statistial in nature, and so require a large number of measurements ( 10 6 ) in order to rodue images that aroah aetable quality. Furthermore, they rovide little flexibility in the kind of mathematial tools that an be used to imrove their quality or inrease their SNR. o date, there is no ublished work in the alied otis ommunity that demonstrates full-olor ghost images of a 3D sene with omlex materials (e.g., senes with transluent or seular materials, or highly sattering media) with a hotograhi quality omarable to that of modern digital ameras. sen@ee.usb.edu On the other hand, a highly related line of work has been ursued indeendently by the image-based relighting ommunity in the field of omutational imaging. Seifially, we refer to work on dual hotograhy [1], [2], whih an erform high-quality imaging without using a standard imaging devie by modulating the inident illumination to erform imaging. We note that ghost imaging is effetively equivalent to dual hotograhy, and indeed many of the early results from dual hotograhy resembled the noisy, graysale images rodued by the ghost imaging systems of today. However, dual hotograhy algorithms have imroved, and have been shown to rodue high-quality, full-olor dual (ghost) images of 3D objets with omlex materials from the oint-of-view of strutured light soures, with orret shading effets. he goal of this aer is to resent dual hotograhy within the ontext of ghost imaging so that the relationshi between the two is lear. his will not only allow ghost imaging researhers to benefit from advanes in light transort aquisition that have been develoed by the image-based relighting ommunity, but it would also allow the two ommunities to work more losely together in the future. Although nothing in this aer is atually new (it has all been ublished before [1], [2]), it aomlishes several imortant tasks: We demonstrate mathematially the equivalene between the ross-orrelation method used in ghost imaging and the light transort measurement of dual hotograhy. We use this equivalene to rovide a simle exlanation for ghost imaging using the lassial eletromagneti rinile of reiroity. o our knowledge, ghost imaging had not been reviously exlained in this manner. We show how advanes for dual hotograhy and light transort aquisition ould be used to imrove ghost imaging. For examle, olor imaging, effiient aquisition with few light atterns (e.g., less than 1,000), and imaging of real 3D senes with omlex materials have all been demonstrated for dual hotograhy and are therefore ossible for ghost imaging. Note that we only fous on ghost imaging using lassial, linear otis at marosoi sales, so inoherent illumination without quantum effets is assumed in our disussion. II. BACKGROUND IN DUAL PHOOGRAPHY Work on dual hotograhy stemmed from researh on imagebased relighting [17], [18], where the light transort matrix [1], [19] between a set of light soures and detetors was measured for real senes in order to aly new lighting atterns as a ost-roess. We begin by onsidering the simlest dual-hotograhy system: a single-ixel amera [20] using a

2 2 rojetor rimal onfiguration hotodetetor virtual amera ixel j ixel j t sene t j j dual onfiguration t j t sene =t virtual light soure Fig. 1. Dual hotograhy with single-ixel system. (to) In the rimal domain (i.e., the real world) the rojetor erforms the imaging by rojeting different illumination atterns. In this ase, for examle, a single ixel j is illuminated. he light arriving at the single-ixel detetor is given by t j j, where j is the radiant ower illuminated by the j th ixel and t j is the transort from this ixel to the detetor. (bottom) In the dual onfiguration (done virtually on the omuter), the rojetor is transformed into a amera and the detetor into a virtual light soure (their resetive duals). If the virtual light soure emits radiant ower, then t j will be the ower arriving through the j th ixel of the virtual amera, aounting for all light aths from the virtual light soure to this ixel. he final dual (ghost) image will be from the oint of view of this new amera, while the sene is illuminated by the new light soure with orret shading effets. Images from [1], [24]. o see the lear similarity between dual hotograhy and ghost imaging, omare this image with Fig. S2 in the sulemental materials of [23]. = t rojetor for strutured illumination and a single hotodetetor to measure outgoing light (Fig. 1). his system was first demonstrated by Sen et al. [1], although similar systems were later demonstrated by others [21] [23]. Here, the rojetor s illumination (or sekle field from a satial light modulator) an be reresented by a k 1 vetor, where k is the number of indeendent illumination elements (ixels). If the single-ixel amera measures salar value, the measurement roess (assuming lassial, linear otis) is = t, where vetor t (size k 1) is the light transort between rojetor ixels and the detetor [1]. Seifially, element t j is the fration of radiant ower from rojetor ixel j measured by the hotodetetor integrated over all diret/indiret light aths. he inner-rodut of t and aumulates ontributions from all rojetor ixels and rodues measurement. his framework naturally handles refletion, transareny, and other linear light-transort roesses sine the only assumtion about aths is their linearity. his is therefore a more general view of the roblem than ghost imaging, whih has mostly been limited to diret transmission through 2D slides [13] [16], [25]. Dual hotograhy uses lassial eletromagneti reiroity to virtually exhange the light soures and detetors in a sene [1]. In this ase, transort t j is reversible: if the detetor was a oint light soure, then t j would also be the fration of light arriving at the rojetor through ixel j. In other words, the fration of the light ower arriving at the hotodetetor from ixel j is also equal to the fration of the light ower arriving at ixel j from a virtual light soure ositioned at the dual rimal (a) (b) () (d) Fig. 2. Samle dual (ghost) images, ublished in 2005 [1]. he to row shows the dual images reonstruted by dual hotograhy, as seen by the modulated light soure whih is doing the imaging. On the bottom row are the rimal images, seen from the osition of the detetor with a fully lit rojetor. he first three are natural senes with 3D objets to demonstrate the aability of dual hotograhy and the last is an exeriment to demonstrate that it rodues orret shading effets. (a) Dual hotograhy an handle omlex materials, suh as transluent materials (the Coke bottle) and seular materials (the metal teaot). It is interesting that the austi of the glass bottle in the dual image beomes the atual bottle in the rimal and vie-versa. (b) Sene demonstrating the fairly extreme hange of view oint that an be done with dual hotograhy. he objets are seen from the front in the dual image, while the amera sees them from the side. () Dual hotograhy is able to ature details not visible in the rimal image. Note, for examle, how the rings in the illar are learly visible in the dual image although they annot be seen in the rimal. (d) Finally, the shading effets are orret based on the new ositions of the virtual light soures and imaging devies (see Se. V). Images rerinted from [1]. hotodetetor. Note that in this dual domain, the rojetor is transformed into a amera beause rojetors angularly emit light while ameras angularly measure light. he k-ixel image that would be atured by the rojetor in the dual domain (whih we all the dual or ghost image) is given by = t, where hotodetetor now rovides illumination. Here t is the same as before (beause of reiroity), and is roortional to sine is a simle saling fator. herefore, the roess of dual hotograhy is about measuring the light transort to the hotodetetor(s) and transosing it to rodue an image from the oint-of-view of the light soure. Examles of dual (ghost) images are shown in Fig. 2. It is worth noting that the same rinile of reiroity an exlain lassial ghost imaging as well. he reason ghost imaging is ossible is that there is light transort from the modulated light soure doing the imaging to the simle detetor. his transort is subjet to the rinile of reiroity and an therefore be reversed, roduing a dual (or ghost) image from the oint of view of the light soure that is illuminated by the detetor. As we show in the Aendix of this aer, the orrelation roess at the ore of all ghost imaging algorithms is effetively solving for this light transort, whih is why it an rodue the dual image. III. LIGH RANSPOR MEASUREMEN We now examine how the light transort t between the light soure and the detetor is measured, sine this is essential for erforming dual hotograhy (or ghost imaging). Conetually, we aly a set of n light atterns = t P, where is a vetor of n measurements and P is a k n matrix, and then use our knowledge of and P to solve for t [1], [17].

3 3 amera sene rojetor Fig. 3. Illustration of flying sot amera, ublished in 1928 [26]. In this setu, a sinning erforated disk (a Nikow disk) is illuminated by a strobed ar light to san the light beam aross the sene whih is then measured by four hotodetetors. his signal an then be transmitted over the air to a television whih reonstruts the image by modulating a sanning beam based on the measurements. Note the remarkable similarity between this setu and reent exeriments in ghost imaging, suh as the exerimental setu shown in Fig. 1 of [23]. he key tehnial differene is, of ourse, that the 1928 version simly sans the illumination beam beause random binary atterns were diffiult to ahieve in a mehanial fashion. As we show, the aroah of [23] whih uses binary oded atterns ouled with the orrelation-based reonstrution is equivalent to sanning the beam, sine they both simly measure the light transort between ixels in the light soure and the hotodetetors. However, the orrelation-based aroahes tyially require more measurements, beause the sanning aroah only requires as many measurements as there are ixels in the final image. he image (ourtesy of Wikiedia [27]) has been edited slightly to fit. Of ourse, the inident illumination atterns P must be known, whih an be easily ahieved with digital rojetors or satial light modulators that rodue known sekle atterns. herefore, imaging systems that use ontrollable illumination soures (e.g., [1], [2], [13], [23], [25]) require only a single detetor. On the other hand, early lassial ghost imaging systems (e.g. [5], [6]) did not know the illumination at P a riori and so they required the exliit measurement of the illumination with a seondary beam whih did not interat with the objet. It was the unusual fat that this seond beam ould be simly orrelated with the buket signal to rodue the ghost image that gave ghost imaging a ertain air of mystery in the alied otis ommunity. Assuming a roerly ontrollable light soure, however, the simlest way to measure t is to use an identity matrix for P, effetively sanning the light beam. Indeed, sanning beam systems fitted with simle baksatter detetors rodue images from the oint-of-view of the sanning soure that are illuminated by the detetors, e.g., Baird s flying sot amera for early television [26] (see Fig. 3), or sanning eletron mirosoes [28]. his framework shows that these, too, are examles of ghost imaging systems where the sekle attern is a known delta funtion. When using a digital rojetor, this effet an be ahieved by simly illuminating a single ixel (or a small blok of ixels) at a time and sanning over all ixels in the rojetor to rodue an image of the resolution of the rojetor. However, these aroahes an suffer from low signal-to-noise ratio (SNR) [29] and are relatively slow, given that the number of measurements is equal to the effetive resolution of the illumination devie. One an also measure t by rojeting random atterns and omuting the ross-orrelation between the detetor value and the rojeted ixels [6]. For examle, ixel j in the dual image an be omuted as j = ( )( j j ) where denotes the ensemble average over the n atterns. As we have disussed, this is the aroah tyially used in ghost imaging (a) (b) () (d) Fig. 4. Light transort measurement for omressive dual hotograhy [2]. (a) Exerimental setu with a digital rojetor that rojets strutured illumination atterns onto the sene and a amera (or light detetor) to measure the refleted light. (b) Samle binary light attern shown by the rojetor. In this ase the ixels are either blak or white with 50% robability. () Samle image atured by the digital amera above the sene when the attern in (b) is illuminated. (d) Reonstrution of the dual (ghost) image using omressed sensing [30], [31]. he image is of a quality omarable to that of a digital amera with only 1,000 measurements, while the naïve san would have required 65,536 measurements (image resolution: ). When omared to reent work in ghost imaging (e.g., Sun et al. [23]), we see that the aquisition setu is nearly idential. However, ghost imaging uses orrelationbased methods for reonstrution whih require many more images (10 6 ) to rodue graysale (not olor) results, and these are still visibly noisy and not amera quality. Images rerinted from [2]. [15], [16], [23]. In the Aendix we show that this roess effetively measures transort t j sine this sets j to be equal to 1 4 t j. Hene, ghost imaging also measures light transort u to a sale, whih an be ignored beause of fator in the dual equation. his is the reason why the early lassial ghost imaging systems (e.g. [5], [6]) ould simly orrelate the two beams to omute the dual image: one beam was measuring and the other, and their orrelation was effetively measuring the light transort between the modulated light soure and the detetor. Reiroity was then imliitly leveraged to rodue an image from the oint of view of the light that was illuminated by the detetor. he fat that orrelated ghost imaging simlifies down to lassial light transort measurement also roves it an be a urely lassial effet [1]. Correlation-based aroahes suffer from serious drawbaks, however. For one, they are statistial methods and so they require a large number of measurements ( 10 6 ) to rodue reasonable results, albeit still visibly noisy. he observation that lassial ghost imaging is equivalent to dual hotograhy enables the use of any aroah for measuring light transort (e.g., aroahes that are more robust to noise) for imroving ghost imaging. For examle, a better aroah for measuring the transort is to use an algorithm whih adatively modulates the light atterns in resonse to measurements to omute t [1]. his is better than ross-orrelation in both reonstrution quality and the number of atterns required, although it needs ative roessing during aquisition. An imroved method was demonstrated by Sen and Darabi [2], whih rojets simle random binary atterns and uses omressed sensing to reonstrut the light transort (see Fig. 4). o see how, we rewrite the measurement roess by transosing both sides and reresenting the transort in transform domain Ψ, yielding = P Ψˆt, where ˆt is the light transort in the transform domain and is assumed to be sarse. Comressed sensing is then used to solve for the sarsest ˆt that mathes observations. After ˆt is reonstruted, we an take the inverse transform to get t, whih rodues high quality images with a small number of atterns [2].

4 4 rojetor rimal onfiguration amera virtual amera ixel j ixel j sene dual onfiguration i j i sene ixel i ixel i i j j m ixels m ixels = virtual rojetor = Fig. 5. Dual hotograhy using a amera. (to) If the hotodetetor in Fig. 1 is relaed by an imaging amera, the system now measures the light transort matrix between the ixels in the rojetor and the ixels in the amera. By transosing the transort matrix, we get the dual onfiguration (bottom) where the amera has been relaed by a virtual rojetor. We an now aly arbitrary rojetor atterns at and get the orret dual image, as shown in Fig. 6. Although there is now an imaging devie in the setu, this ature roess is still equivalent to ghost imaging beause the final image is generated from the ersetive of the illumination soure. Images from [1]. We note that a similar aroah using omressed sensing was ursued simultaneously by Peers et al. [32], although their system uses a diffuse light soure (an LCD dislay instead of a rojetor) and hene is only intended to measure light transort and annot erform imaging. Hene, although their method is also related to this disussion, it is not diretly onneted to the aliation of ghost imaging. here was also work in the ghost imaging ommunity that attemted to use omressed sensing [25], but their aroah required muh larger erentage of atterns (around 60%) and was demonstrated using only 2-D senes of grainy, graysale slides. he algorithm of Sen and Darabi [2], for omarison, ould rodue full-olor images of hotograhi quality in some ases with less than 1% of the total number of measurements. Finally, there have been other advaned methods for measuring light transort that have been develoed in the image-based relighting ommunity whih we disuss briefly at the end of the next setion. However, not all of them an be alied to the roblem of dual hotograhy (or ghost imaging). IV. USING CAMERAS FOR SCENE RELIGHING he dual hotograhy setu desribed in Se. II an be extended by relaing the hotodetetor with an imaging amera as shown in Fig. 5. In this new onfiguration, the m- ixel amera takes measurements while the k-ixel rojetor rovides illumination, so the measurement roess beomes =, where is now an m 1 vetor and matrix (size m k) reresents the light transort between rojetor and amera ixels [1]. Here, element ij reresents the fration of radiant ower from rojetor ixel j measured by ixel i of the amera integrated over all diret/indiret light aths. In the dual onfiguration, the rojetor is virtually relaed by a k-ixel amera and the amera by a m-ixel rojetor, with the field of view, fous, and other settings staying exatly fully illuminated dual image dual image relit with arbitrary atterns ground truth rimal our omuted rimal Fig. 6. Image-based relighting results. he first set of images shows how the dual image an be virtually relit by alying arbitrary atterns at the virtual rojetor. he first of these images is with the virtual rojetor fully illuminated while the other two rojet seifi atterns. Note, for examle, how the light transort aurately atures the omlex austis through the glass bottle. he next two images demonstrate the fidelity of the measured light transort matrix using the omressed sensing method of [2]. he first is the rimal image atured by the amera when the rojetor is showing a seifi attern (ground truth). he seond is our alulated image by erforming the matrix-vetor multily between the light transort matrix and the desired light attern ( = ). he two are omarable, indiating that the light transort matrix is aurately reresenting the flow of light in the sene. Images from [1] and [2]. the same for eah. he transort in the dual ase between the i th ixel of the rojetor and the j th ixel of the amera is still ij beause of reiroity 1. herefore, the light transort equation from the virtual rojetor to the virtual amera in the dual domain is given by =. As with the simle hotodetetor onfiguration, this is also an examle of ghost imaging sine the final image is generated from the oint of view of the illumination soure without using a standard imaging devie at that osition. Like before, the roblem redues to measuring the light transort, in this ase matrix, whih an be done with the methods desribed in the last setion. One this light transort matrix has been measured, it an be simly transosed to rodue the dual (ghost) images. Furthermore, we an aly an arbitrary attern at the virtual rojetor to virtually relight the sene as shown in Fig. 6. he ability to relight the final image with arbitrary light atterns as a ost-roess is the reason that the image-based relighting ommunity has foused on measuring the light transort matrix, and not the simle light transort vetor t that ghost imaging systems tyially measure. As mentioned earlier, the relighting ommunity has roosed other advaned methods for measuring light transort more effiiently, suh as Symmetri Photograhy [33], the Kernel Nyström method [34], otial Krylov subsae methods [35], and Primal-Dual oding [36]. However, most of these are not aliable to dual hotograhy (and hene ghost imaging) in their urrent state. For examle, the Kernel Nyström method samles olumns and rows of the light transort matrix and reonstruts it assuming that it is low rank in a nonlinear sae [34]. Measuring olumns of the matrix is straightforward (simly illuminate the aroriate ixels in the rojetor), but samling the rows is muh trikier beause it requires a oaxial rojetor at the amera s osition and a oaxial amera at the rojetor s osition to measure the outut. herefore, these algorithms are only useful for light transort aquisition and not dual (ghost) imaging sine they need an imaging amera at the osition of the light soure, defeating the urose of ghost imaging. However, it would be 1 By reiroity, the light transort between the j th ixel of one devie and the i th ixel of another is the same regardless of whih way the light is atually flowing.

5 5 an interesting subjet of future work to see if any of these tehniques ould be modified to work for dual hotograhy and ghost imaging. V. CORREC SHADING IN HE DUAL IMAGE Sen et al. noted that the shading effets in the dual (ghost) images are orret for the osition of the new virtual light soures [1], whih an be inititally surrising. Assume, for examle, a sene omosed entirely of diffuse (Lambertian) surfaes, where the shading is simly roortional to the dot rodut between the illumination diretion and the surfae normal. If the rojetor is shining orthogonally to a diffuse surfae, these regions will be seen as bright by the detetor (hotodetetor or imaging amera), regardless of its osition. In this ase, suose that the detetor is imaging the surfae from a grazing angle. In the dual domain, the detetor would turn into the light soure and the rojetor into the amera. Sine the virtual light is illuminating the diffuse surfae at a grazing angle, the surfae should be dark. Is it ossible to rodue dark values in this region when the detetor was measuring bright values before? he surrising answer is that this is indeed what haens in dual hotograhy (and ghost imaging). As the roof in Aendix A of [1] shows, the energy transfer between a ixel of the rojetor and a ixel of the amera remains the same in either the rimal or dual onfigurations. However, the shading will hange beause eah amera ixel reeives light transort from a different number of rojetor ixels deending on the onfiguration. For examle, in the rimal domain, a single amera ixel will measure the ontributions from many rojetor ixels beause it is viewing the surfae at a grazing angle. So 1 unit of ower at eah rojetor ixel will be aumulated over many rojetor ixels to rodue one bright amera ixel. his will make the Lambertian surfaes with orthogonal light soures aear bright in the rimal image. On the other hand, when the amera is turned into a virtual rojetor in the dual onfiguration, eah ixel in the virtual rojetor will ma to many dual amera ixels (the same number as before and with the same energy transfer of eah). In this domain, 1 unit of ower from a virtual rojetor ixel will be sread out to many different virtual amera ixels. herefore, the results at eah ixel of the final image will be muh dimmer, as they should be. Hene the shading will be orret with reset to where the virtual light soure is ositioned. his same analysis an be alied to a simle hotodetetor, whih is effetively a 1-ixel amera. As shown in [1], this integration roess over different numbers of ixels an be aounted for by adding another osine term to the light transort equation, whih has also used by others, e.g., the Helmholtz Stereosis work by Zikler et al. [37]. Exerimentally, this has been verified in [1] using a alibrated exeriment, shown here in Fig. 2(d). In this ase, the rimal rojetor is almost orthogonal to the first G in the SIGGRAPH 2005 logo, so this region aears bright in the rimal image. On the other hand, the 2 is at a grazing angle, so it is dark. In the dual image, the light soure and amera are interhanged, so the G is now dark and the 2 is bright. his shows the shading effets are orret based on the osition of the light soure. VI. COMPARISON O RECEN WORK IN GHOS IMAGING Most ghost imaging systems used simle senes omosed of 2D graysale transarent slides [13] [16], [25], so the relationshi with dual hotograhy was diffiult to see. Reently, however, Sun et al. made efforts to extend this to imaging 3D diffuse objets [23], whih made it more lear that this work was a reliation of revious work in dual hotograhy. For examle, Sun et al. s system uses a digital rojetor to rojet binary atterns to erform ghost imaging of a diffuse sene with 4 hotodetetors. his setu is similar to revious dual hotograhy work [1], [2], and the binary atterns and aquisition roess are idential to those of omressive dual hotograhy (see Fig. 4). In their imlementation, Sun et al. use a ross-orrelation method to reonstrut their dual (ghost) images, whih means they need a large number of atterns (10 6 ). his is ineffiient sine the resolution of their dual images (whih are in Fig. 1 of [23]) should require at most 33,180 measurements to rodue an image [1]. With the omressive dual hotograhy method of [2], the number of measurements required would be less than 16,000, at least a 60 imrovement over the results resented in [23]. Furthermore, by following the aroahes resented in earlier work [1], [2], they would be able to rodue results in olor (either laing olor filters in front of the hotodetetors or using multiolor illumination atterns) and with less noise. In the end, Sun et al. s system rodues four dual images from the oint-of-view of the rojetor lit by the different hotodetetors, whih are then fed into a standard shae-fromshading algorithm to outut the final graysale geometry [23]. A drawbak of this simle shae-from-shading algorithm is that it requires the surfaes to be entirely diffuse. For omarison, the image-based rendering ommunity has also used dual hotograhy to reonstrut geometry (e.g., [38]), but these aroahes an reonstrut high-quality, full-olor geometry for omlex senes and an handle glossy materials. VII. CONCLUSION By showing that ghost imaging redues to the roblem of measuring the light transort between ixels in the rojetor (or satial light modulator) and a detetor, we make a onnetion between ghost imaging and revious work in dual hotograhy/light transort aquisition. his observation will rovide researhers in ghost imaging with new tools that ould imrove their results, suh as reduing the number of measurements and roduing high-quality, full-olor results of senes with omlex materials. Although the work to date in dual hotograhy and ghost imaging has been done searately by different researh ommunities, we hoe disussions like this will inrease their ollaboration and lead to imroved imaging tehniques in the future. ACKNOWLEDGMENS We thank our o-authors from the original dual hotograhy aers [1], [2] for valuable disussions. he omressive

6 6 dual hotograhy work [2] was sonsored by NSF CAREER grant #IIS Images from [1] 2005 Assoiation for Comuting Mahinery, In., rerinted by ermission. Images from [2] rerinted by ermission. APPENDIX he following derivation roves that ross-orrelation methods used in ghost imaging, whih omute final ixels in the form j = ( )( j j ) where denotes the ensemble average, are effetively measuring light transort: j = ( )( j j ), = j j j + j, = j j. (1) Assuming random, binary illumination atterns where ixel j has equal robability being 0 or 1, the exeted value j = 1 2. Given that = t, we an omute : = t = t i i = t i i, i i = 1 2 t 1, where 1 is a k 1 vetor of all 1 s. A similar alulation an be done for j : j = i (t i i ) j, = i j (t i i ) j + t j 2 j, = 1 t i t j = 1 4 t t j. i j Putting this all together into the last line of Eq. 1 gives us: j = 1 4 t t j 1 4 t 1 = 1 4 t j. 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