Compressive Object Tracking using Entangled Photons
Omar S. Magaña-Loaiza, Gregory A. Howland, Mehul Malik, John C. Howell, Robert W. Boyd
CCompressive Object Tracking using Entangled Photons
Omar S. Maga˜na-Loaiza, a) Gregory A. Howland, Mehul Malik, John C. Howell, andRobert W. Boyd b)1) The Institute of Optics, University of Rochester, Rochester, NY 14627,USA Department of Physics and Astronomy, University of Rochester, Rochester,NY 14627, USA (Dated: 31 October 2018)
We present a compressive sensing protocol that tracks a moving object by removingstatic components from a scene. The implementation is carried out on a ghost imagingscheme to minimize both the number of photons and the number of measurementsrequired to form a quantum image of the tracked object. This procedure tracks anobject at low light levels with fewer than 3% of the measurements required for araster scan, permitting us to more effectively use the information content in eachphoton. a) [email protected] b) Also at Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada. a r X i v : . [ qu a n t - ph ] J un ompressive sensing (CS) has recently been of great utility in quantum optical and low-light level applications, for instance, single-photon level imaging, entanglement characteri-zation and ghost imaging . CS provides a resource-efficient alternative to single-photonarrayed detectors, permitting us to reduce operational problems involved in systems em-ploying raster scanning .CS applies optimization to recover a signal from incomplete or noisy observations of theoriginal signal through random projections . These ideas applied to the field of imaging allowone to retrieve high resolution images from a small number of measurements . Recently,the quantum optics community has employed CS for quantum state tomography, todemonstrate nonclassical correlations and to form compressed ghost images .Ghost imaging is a technique which employs the correlations between two light fields toreproduce an image. For example, entangled photons exhibit strong correlations in manyproperties such as time-energy and position-momentum . One photon of an entangled pairilluminates an object and is collected by a bucket detector, which does not provide spatialinformation. Its entangled partner photon is then incident on a spatially resolving detectorgated by the first photon’s bucket detector. Remarkably, an image of the object appears onthe spatially resolving detector, even though its photon never directly interacted with theobject .Compressive ghost imaging allows one to replace the spatially resolving detector witha bucket detector. This procedure reduces both acquisition times for systems based onraster scanning and the required number of measurements for retrieving images . Theseimprovements have motivated an ongoing effort to implement technologies based on ghostimaging such as image encryption , quantum sensors , object identification and mostrecently ghost imaging ladar .In spite of the advantages that technologies based on ghost imaging offer, they can behard to implement in practice. Most current quantum optical technologies work at the singlephoton level, and are unfortunately vulnerable to noise and are inefficient, requiring manyphotons and many measurements . To reduce these limitations, we apply an efficient formof compressive sensing. This allows us to overcome the main problems which underminethe practical application of many attractive correlated optical technologies. To demonstratethese improvements, we implement a ghost object tracking scheme that significantly outper-forms traditional techniques. This opens the possibility of using correlated light in realistic2pplications for sparsity-based remote-sensing.We present a proof-of-principle experiment based on a quantum ghost imaging schemethat allows us to identify changes in a scene using a small number of photons and many fewerrealizations than those established by the Nyquist-Shannon criterion. Object tracking andretrieval is performed significantly faster in comparison to previous protocols . Thisscheme uses compressive sampling to exploit the sparsity of the relative changes of a scenewith a moving object. With this approach we can identify the moving object and reveal itstrajectory. Our strategy involves removing static components of a scene and reduces theenvironmental noise present during the measurement process. This leads to the reductionof the number of measurements that we take and the number of photons required to forman image, both important issues in proposals for object tracking and identification . Thereduction of noise and removal of static components of a scene is carried out by subtractingtwo observation vectors, corresponding to two realizations of a scene. We call this techniqueghost background subtraction. Our results demonstrate that this technique is adequate forobject tracking at low light levels.Consider the ghost imaging scheme depicted in Fig. 1. A laser pumps a nonlinear crystaloriented for type-I spontaneous parametric down-conversion (SPDC). The approximatedoutput state is given by first order perturbation theory, which leads us to the followingtwo-photon entangled state: | Ψ (cid:105) = (cid:90) d(cid:126)k g d(cid:126)k o f ( (cid:126)k g + (cid:126)k o )ˆ a † g ( (cid:126)k g )ˆ a † o ( (cid:126)k o ) | (cid:105) . (1)We refer to the down-converted photons as the ghost and object photons denoted by thesubindices g and o , respectively. The two-photon probability amplitude, which is responsiblefor the transverse momentum correlations existing between the ghost and object photons, isrepresented by the non-factorizable function f ( (cid:126)k g + (cid:126)k o ), where k is the transverse wavevectorof the ghost or object photon. The form of this function depends on the phase-matchingconditions, but it is often approximated by a double gaussian function . This two-photonentangled state is strongly anti-correlated in transverse momentum, such that if the trans-verse momentum of the object photon is measured, the transverse momentum of the ghostphoton is found to have the same magnitude and opposite direction. These momentumanti-correlations allow us to perform ghost imaging.In our experiment, we use digital micromirror devices (DMDs) to impress spatial infor-3 IG. 1. Entangled photons at 650 nm are generated in a Bismuth Barium Borate (BiBO) crystalthrough type-I degenerate spontaneous parametric downconversion (SPDC). The far field of theBiBO crystal is imaged onto two digital micromirror devices (DMDs) with a lens and a beamsplitter (BS). One DMD is used to display the object we want to track, while the other is usedto display random binary patterns. Single-photon counting modules (SPCMs) are used for jointdetection of the ghost and object photons. mation onto the entangled photon pair. The DMDs work by controlling the retro-reflectionof each individual pixel on the display. After each photon is reflected by a DMD, a single-photon counting module (SPCM) counts the number of photons in it. The correlationsbetween the two down-converted photons allows one to correlate the images displayed in theDMDs.We jointly detect photons pairs reflected off a changing scene O and a series of randommatrices A m . The subindex m indicates the m - th realization. The coincidence countsbetween the two detectors are given by J m ∝ (cid:90) d(cid:126)ρ DMD (cid:12)(cid:12)(cid:12)(cid:12) A m (cid:18) (cid:126)ρ DMD m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O (cid:32) (cid:126) − ρ DMD m o (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2)where A m and O are the reflectivity functions displayed on the DMD g located in theghost arm and on DMD o in the object arm, respectively. Meanwhile m r and m o are theircorresponding magnification factors. These are determined by the ratio of the distancebetween the nonlinear crystal to the lens and the distance from the lens to DMD g or DMD o .4n our experiment m r and m o , are equal. (cid:126)ρ DMD represents the transverse coordinates of oneof the DMDs.Eq. 2 critically shows that the joint-detection rate is proportional to the spatial overlapbetween the images displayed on DMD o and DMD g . This behavior can be interpretedas a nonlocal projection, which demonstrates the suitability for implementing compressivesensing techniques nonlocally with ghost imaging .Compressive sensing uses optimization to recover a sparse n -dimensional signal from aseries of m incoherent projective measurements, where the compression comes from the factthat m < n . Image reconstruction via compressive sensing consists of a series of linearprojections . Each projection is the product of the image O consisting of n pixels, witha pseudorandom binary pattern A m . Each pattern produces a single measurement, whichconstitutes an element of the observation vector J . After a series of m measurements, asparse approximation ˆ O of the original image O can be retrieved by solving the optimizationproblem, known as total variation minimization , given by Eq. 3. min ˆ O ∈ C n (cid:88) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D i ˆ O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ˆ O − J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3) D i ˆ O is a discrete gradient of ˆ O at pixel i , µ is a weighting factor between the two terms,and A is the total sensing matrix containing all the pseudorandom matrices A m . Eachmatrix A m is represented into a 1D vector and constitutes a row of the total sensing matrix A . The algorithm known as Total Variation Minimization by Augmented Lagrangian andAlternating Direction (TVAL3) allows us to solve the aforementioned problem. The solutionof the optimization problem allows us to recover the image ˆ O , which is the compressed versionof the original image O , with a resolution given by the dimensions of the matrix A m . Theoriginal image O is characterized by a sparsity number k , which means that the image canbe represented in a certain sparse basis where k of its coefficients are nonzero. The numberof performed measurements m is greater than the sparsity number k , but far fewer than thetotal number of pixels n contained in the original image. The constraints imposed in therecovery algorithm minimize the noise introduced during the measurement process.We are able to compressively track and identify a moving object in a scene by discardingstatic pixels. A scene with a moving object possesses static elements that do not provide5nformation about the object’s motion or trajectory. These redundancies can be discrim-inated from the moving object as follows. Let us consider the projection of two differentframes onto the same pseudorandom pattern. Each projective measurement picks up littleinformation about the components of a frame. If the two projective measurements producethe same correlation value, it would imply that the two frames are identical and we areretrieving meaningless information which can be ignored. The opposite case would revealinformation about the changes in a scene.This protocol is formalized as follows. Two different correlation vectors, J j and J j − ,corresponding to two consecutive frames are subtracted, giving ∆ J . This introduces thefollowing important modification to Eq. 3. min ˆ O ∈ C n (cid:88) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D i ∆ ˆ O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ∆ ˆ O − ∆ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)The subtracted vector ∆ J is sparser than both J j or J j − , thus requiring fewer measurementsfor its reconstruction. This corresponds to fewer realizations of A m , and hence smallersensing matrix A . Furthermore, subtracting the background in this manner mitigates theenvironmental noise present during the tracking process. The retrieved image ∆ ˆ O willprovide information about the relative changes in the scene.Our experimental setup is sketched in Fig. 1. A 325 nm, continuous-wave HeCd laserpumps a type-I phase matched BiBO crystal to produce degenerate entangled photon pairsat 650 nm. Two interference filters are placed after the nonlinear crystal. The first is a lowpass filter that removes the pump and the second is a 650/12 nm narrowband filter thattransmits the down-converted photons. A beam splitter probabilistically separates the twophotons into ghost and object modes. An 88 mm focal length lens puts the far field of thecrystal at the location of DMD. Two free space detectors receive the light reflected from theDMDs by means of two collection lenses with a 25 mm focal length. One DMD is used todisplay a scene with a moving object while the other is used to impress a series of randombinary patterns. Coincidence counts are obtained within a 3 ns time window.We apply this method to a scene with a flying object. The static components of the sceneare a house, the moon and a tree. The object moves a certain distance in each iteration of thescene (insets of Fig. 2). We first reconstruct a compressed ghost image of the static frameof the scene, which represents the background. In order to do this, we put 2000 differentrandom patterns on DMD g , with DMD o displaying the background scene. These realizations6epresent 49% of a raster scan. For each random pattern, we count coincidence detectionsfor 8 s. Typical single count rates were 13 . counts/s for the ghost and object armswith the coincidence counts approximately 2% of the single counts. Fig. 2(a) shows theretrieved background scene ˆ O . After this, subsequent frames of the scene with the object indifferent positions are displayed on DMD o . After applying the optimization algorithm, themoving object was clearly identified as shown in Figs. 2(b)-(f). The reconstructions weredone using 400 patterns, which represents 9.7% the measurements of a raster scan. Thenegative values in the retrieved images are due to background subtraction and fluctuationsin the measurements process.A straightforward examination of the limits of our protocol is carried out by reducingthe number of measurements used to track an object. The images shown in Fig. 3 werereconstructed with only 200 and 100 measurements, corresponding to 4.88% and 2.44% ofthe measurements of a raster scan. The metric employed to characterize the fidelity of thesereconstructions is the mean-squared error defined as M SE = (1 /n ) (cid:107) O − ˆ O (cid:107) . The M SE is seen to increase as the number of measurements is decreased. Although, it is still possibleto detect the object trajectory with just 100 measurements.The photon efficiency is studied by estimating the dependence of the
M SE on the num- !" ./0 .90 "*$ !"
FIG. 2. Compressed ghost image of (a) the background of the scene and (b-f) the tracked objectin different positions. These reconstructions were obtained by defining different ∆ J vectors with400 elements, corresponding to the number of measurements. The insets show the original framesof the scene displayed on the DMD. " !" '!!$./0123/./451$ %!"+$%!",$%!"-$!$!"-$ @ABC!"!- !"-$ !" -!!$./0123/./451$ !" FIG. 3. Reconstructed ghost image of (a-e) tracked object with 200 measurements. (f-j) sameobject with 100 measurements. ber of photons per measurement, for a fixed number of measurements. A simulation ofthe protocol was carried out by using the data employed in the experiment. In order toachieve realistic experimental conditions, dark and shot noise were introduced by means ofpoissonian distributions. The amount of dark noise was modeled based on the frequencydistribution of counts obtained when both of the DMDs were turned off. We have consid-ered reconstructions employing 100 and 400 measurements. Fig. 4 shows the dependence ofimage quality on the number of detected photons per measurement. The minimum numberof photons per measurement needed to distinguish the silhouette of the object by eye are 500photons/measurement and 200 photons/measurement for 100 and 400 measurements respec-tively. The estimated thresholds correspond to a
M SE oscillating around 0.04. For the sit-uation where an object was tracked with 100 measurements and 500 photons/measurement,we estimate that we can impress approximately 0.082 bits/photon. This is considering thatfor a binary image the number of pixels corresponds to the number of bits .The maximum object velocity that we can track is limited by the number of photons thatwe are able to detect. In our setup, each scene reconstruction took 13.3 minutes (for the caseof 100 measurements) due to the low photon flux. If we were to use a high brightness sourceof entangled photons, we could shorten the acquisition time needed to retrieve a compressedghost image with an M SE below the threshold shown in Fig. 4. As such, there is no hardtheoretical limit on the maximum object velocity that can be tracked using this method.In conclusion, we have proposed and demonstrated a proof-of-principle object-tracking8 """ !" '!""" . / - FIG. 4. (Color online) Calculated mean-squared error of the compressed tracked object at theposition shown in Fig. 2(b). Green (Red) line indicates the MSE using 400 (100) measurements.The thresholds indicate that a low quality image is retrieved and is not possible to track the object. protocol in a ghost imaging scheme. This protocol uses compressive sensing to exploit thesparsity existing between two realizations of a scene with a moving object. It also reducesthe environmental noise introduced during the measurement process. Further, it allows us toperform image retrieval significantly faster by employing single pixel detectors. Our methodis photon-measurement efficient, allowing us to track an object with only 2.44 % of thenumber of measurements established by the Nyquist criterion, even at low light levels. Thiseconomic procedure shows potential for real-life applications.
ACKNOWLEDGMENTS
The authors would like to thank M. Mirhosseini and A.C. Liapis, P. Ybarra-Reyes andJ.J. Sanchez-Mondragon for helpful discussions. This work was supported by the DARPAAFOSR GRANT FA9550-13-1-0019 and the CONACYT.
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