Single shot large field of view imaging with scattering media by spatial demultiplexing

S i n g l e s h o t l arg e f i e l d o f vi ew im ag in g wi th s c a t t e r i n g m e d i a by s pa ti al d emu lt ip lex in g S U J I T K U M A R S A H O O , 1 , 2 , * D O N G L I A N G T A N G , 1 A N D C U O N G D A N G 1 , * 1 Centr e f or OptoElectr onics and Biopho tonics (C OEB), Sc hool of Electrical and Electr onic Engineering, The Photonic Ins titute (TPI), N anyang T ec hnological U niv ersity Sing apor e, 50 N any ang A v enue, 639798, Sing apor e 2 Department of Statistics and Applied Pr obability , National U niv er sity of Sing apor e, 117546, Sing apor e * sujit@pmail.ntu.edu.sg, hcdang@ntu.edu.sg Abstract: Optically f ocusing and imaging through strongl y scattering media are challenging tasks but hav e widespread applications from scientific research to biomedical applications and daily lif e. Benefiting from the memor y effect (ME) f or speckle intensity cor relations, only one single-shot speckle pattern can be used f or the high quality reco very of the objects and av oiding some complicated procedures to reduce scatter ing effects. In spite of all the spatial inf ormation from a larg e object being embedded in a single speckle image, ME giv es a strict limitation to the field of view (FO V) f or imaging through scatter ing media. Objects be yond the ME region cannot be reco vered and onl y produce unw anted speckle patterns, causing reduction in the speckle contrast and recov ery quality . Here, w e extract the spatial inf or mation by utilizing these una voidable speckle patter ns, and enlarge the FO V of the optical imaging system. Regional point spreading functions (PSFs), which are fix ed and onl y need to be recorded once for all time use, are emplo yed to reco ver cor responding spatial regions of an object by deconv olution algor ithm. Then an automatic weighted av eraging in an iterativ e process is performed to obtain the object with significantly enlarg ed FO V . Our results present an impor tant step tow ard an advanced imaging technique with strongly scatter ing media. OCIS codes: (110.0113) Imaging through turbid media; (110.6150) Speckle imaging; (100.0100) Image processing. References and links 1. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” Jour nal of the Optical Society of America A 66 , 207–211 (1976). 2. R. K. T yson, Principles of adaptive optics (CR C press, 2015). 3. C. Gabr iel, S. Gabr iel, and E. Corthout, “The dielectr ic properties of biological tissues: I. literature surve y , ” Physics in Medicine and Biology 41 , 2231 (1996). 4. V . Ntziachristos, “Going deeper than microscopy : the optical imaging frontier in biology , ” N ature Methods 7 , 603–614 (2010). 5. D. Huang, E. A. Sw anson, C. P . Lin, J. S. Schuman, W . G. S tinson, W . Chang, M. R. Hee, T . Flotte, K. 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Unf or tunatel y , there are still some real-life obstacles such as the atmospher ic turbulence, biological tissues or e v en bead curtains, which degrade the visibility of the interested objects at different lev els. These scattering media contain inhomogeneity of refractive inde x or absor ption coefficient to generate random phase or amplitude distor tions on the ideal light paths from the or iginal objects. This results in complex speckle patter ns on detections and impairs the optical observations. Thus, imaging through those scatter ing media has become a hot topic and an important technical challeng e in the past decades. T o ov ercome these limitations, v ar ious no v el, breakthrough and practical approaches hav e been proposed with the adv ances of the opticall y scatter ing theor y and optoelectronic devices. One of the successful techniques, optical coherence tomography (OCT), can utilize the un-scattered ballistic light and a scanning gate windo w to capture the static images, con versel y the required amount of the ballistic photons will giv e a limitation of the penetration depth through scattering media [5, 6]. As the scatter ing media mainly scramble the phase inf or mation after light propagation in most cases, another straightf or w ard method is tr ying to compensate the phase per turbation. Learning from the adaptive optics in the astronomical observation [2], one can use a spatial light modulator (SLM) to play the role of the def or mable mir rors and reduce the aber rations [7 – 10]. An iterative algorithm with the assistance of a bright guiding star as a par t of the f eedback mechanism effectivel y reduces the aber rations, paving the wa y for f ocusing and imaging through scattering media. By recording the transmitted wa vefront of the scattered light in an optical setup with str ict alignment, one can generate a phase-conjugate wa vefront using SLM [11–15]. Another set of ideas was proposed by exploiting the speckle intensity cor relation for imaging, thank to ME [16 – 27]. The ME states that the random speckle patterns observed after the scattering medium are shifted (i.e. highly correlated) when the illumination angle varies within a cer tain rang e. By estimating this ME region, a very sparse object of size smaller than this can be recov ered nonin vasiv ely with an ultra nar ro w-band illumination. Because of the ME and randomness of the speckle pattern, the autocor relation of the object mostl y preser v es through scattering media. A single-shot speckle pattern is enough to reconstruct the object with the help of a phase retr ie val algorithm [19 – 24]. The ME region on the object plane is in versel y propor tional to the thickness L of the scatter ing medium and propor tional to the distance u from the object plane to the scattering medium, according to the f ormula: u λ / π L [16, 17]. Objects larg er than the ME region produce un wanted speckle patterns, the object autocorrelation can not be preser v ed, and the phase retriev al is not possible anymore. Deconv olution imaging becomes useful in this case, where the scattering medium could be treated as a scatter ing lens [26, 27]. One can make full use of the intrinsic ME region b y suitably positioning the scattering medium and optics components [26]. Ho we ver , the FO V of the optical imaging sys tems is still limited by the ME region of strongl y scattering media, and unw anted speckle patterns only result in the reconstruction ar tif acts. In this work, w e demonstrate a decon v olution imaging technique to produce a larger FO V from a single-shot imag e with scattering media. Regional point spreading functions (PSFs), which are fix ed and only need to be recorded once f or all time use, are employ ed to recov er cor responding spatial regions of an object b y deconv olution algor ithm. Then an automatic weighted av eraging in an iterative process is performed to obtain the object with significantly enlarg ed FO V . W e make use of those uncor related speckle patter ns generated by objects be yond the ME region, which are previousl y considered as un wanted noise. The motiv ation is to maximize the utilization of the information captured by the imager , which is the super position of multiple speckle patter ns coming from the different regions of an e xtra-larg e object. Our results present an impor tant step to ward an advanced imaging technique with strongl y scatter ing media. The detailed pr inciple is described in the f ollowing section. 2. Principle The object is simply a composition of point sources with various positions and intensities. For a linear optical imaging sys tem, an image is expressed as a superposition of the multiple PSFs from different spatial positions with cor responding intensity . The follo wing equation is a mathematical representation of the image formation. I ( x , y ) = Õ ( i , j ) O ( i , j ) P S F i j ( x − i , y − j ) (1) where O is the object, and P S F i j is the optical response function of a point source located at position ( i , j ) . In an ideal scenar io, the image becomes identical to the object I = O when these PSFs are identical impulse functions, which denotes an ideal optical imaging sys tem. In contrast, the scatter ing medium produces not onl y random but also variable PSFs, which are completely uncorrelated for far apar t point sources. Ho w ev er, because of ME, the point sources within a small ME region of the object plane will g enerate similar but shifted speckle patterns (i.e. PSFs) on the image plane. The mathematical representation of the image f or mation can be restated considering the ME as f ollow s: I ( x , y ) = Õ R Õ ( i , j ) ∈ R O ( i , j ) P S F R ( x − i , y − j ) (2) where R is the ME region ha ving nearly identical P S F i j as P S F R . T aking advantag e of the shift-in variant property of P S F i j , the super position can be expressed as a conv olution of different regions of the objects with their corresponding PSFs. I R = O R ∗ P S F R = Õ ( i , j ) ∈ R O ( i , j ) P S F R ( x − i , y − j ) (3) where O R is the por tion of the object within the ME region, i.e. O ( i , j ) f or all ( i , j ) ∈ R . The object O R can be recov ered by deconv olution using P S F R of the optical sys tem, if we kno w its speckle pattern I R . The deconv olution process is ideally expressed as f ollow s. O R = d ec on v ( I R , P S F R ) = F F T − 1  F F T ( I R ) F F T ( P S F R ) c | F F T ( PS F R ) | 2  (4) where ( . ) c is the comple x conjugate, F F T ( . ) and F F T − 1 ( . ) are the Fourier transform and its in verse, respectivel y . The decon volution in equation (4) is possible because the conv olution in spatial domain become the multiplication in Fourier domain. F F T ( O R ∗ P S F E ) = F F T ( O R ) F F T ( P S F R ) (5) This decon volution technique has been demonstrated to recov er high resolution image of object within ME region of scattering media [26, 27]. Here f or object O e xtended ov er the ME region, its speckle patter n I of the object is a composite response of the scatter ing light coming from the various regions of the object. Thus, the captured image can be e xpressed as f ollo ws: I = Õ R I R = Õ R O R ∗ P S F R (6) Because of the random structure of the scatter ing media, spatiall y separated point sources bey ond the ME region produce uncor related speckle patter ns. This can be mathematically e xpressed as f ollow s: P S F R 1 ? P S F R 2 = ( 0 if R 1 , R 2 δ if R 1 = R 2 (7) where ? is the correlation operator, and δ is the spatial impulse function. Because of the relation F F T ( A ? B ) = F F T ( A ) F F T ( B ) c , the follo wing relationship can be deduced: F F T ( I ) F F T ( P S F R ) c = F F T ( I R ) F F T ( P S F R ) c (8) Theref ore, multiple region R of the object can be reconstructed from a single monochromatic image I as below : O R ≈ d ec on v ( I , P S F R ) (9) In essence, each P S F R giv es a limited FO V on the object plane, as seen in Fig. 2, where central FO V of the optical sys tem is determined by central ME region and shown in the red dashed circular line. Actuall y , the scattering media might hav e different effective thickness f or different angle illumination, therefore ME could slightl y vary at different spatial position on the object plane, where spatial points hav e their own range of spatial shifted-inv ar iance PSFs and unique spatial FO V . At the same time, all the spatial information of an object are multiple xed inside a single speckle imag e captured b y the camera. One could use the unique regional PSF to reconstruct the spatial object O R f or each region R , and obtain a full reconstruction of the larg e object by ar ranging the respective regions in space. Projector u v Object Filter Iris SC Camera FOV Fig. 1. Exper imental setup. A common projector displa ys letters ‘NTU EEE OPTIMUS!’ as an object for the imaging system. Light coming from the object passes through a green bandpass filter, an ir is and the scattering medium, then g enerates a scrambled speckle pattern on a camera. Memor y effect of the scattering medium gives a limited FO V on the object plane, as sho wn in the red dashed circle on the object plane. SC: scatter ing medium. 3. Experiment The schematic of our complete optical setup is presented in Fig. 1. A common projector creates a desired object as the input. In our e xper iment, the object is composed of a group of letters ‘NTU EEE OPTIMUS!’ . Light coming from the object passes through a nar ro w bandpass green filter (center wa velength: 550 ± 2 n m , FHWM: 10 ± 2 n m ), an ir is and a thick scatter ing medium then generates a scatter ing speckle image on the camera (Andor Neo 5.5, 2560 × 2160 , pixel size 6 . 5 µ m ). Iris with diameter 2 mm is used to remov e the background light and control the intensity , aper ture, speckle size and signal-to-noise ratio of the speckle patter n. The distance from the object to the scatter ing medium and that from the scatter ing medium to the camera are u ≈ 170 mm and v ≈ 77 . 5 mm , respectiv ely . Theref ore the magnification of the setup is M = v / u ≈ 0 . 456 . The thick scattering medium reduces the ME and theref ore limits central FO V as shown in the dashed red circle in Fig. 1. Thus, only a par tial object located within this FO V could be reconstructed with the central PSF through the decon volution process. As the light from a point source (a single projector’ s pixel) going to the camera sensor is quite small, the e xposure times f or spatial PSF measurements are set as 10s; f or the object with multiple point sources, the e xposure times are about 0 . 03 s ∼ 0 . 4 s . The applied Wiener decon v olution algor ithm [28] is robust to the reconstruction noises and only takes about 0.5 second f or each reconstr uction with MA TLAB on a normal PC (Intel Core i7, 16 GB memor y). For each speckle patterns and PSFs, we divide them b y their lo w frequency env elop to remov e the hallow effect and shar pen the speckles. The resolution of the imaging sys tem is defined by the speckle grain size, which depends on the numer ical aper ture of the sys tem. W einer deconv olution is per f ormed by setting the noise lev el as the mean value of the deconv ol ving PSF . Firs tly , we need quantify and measure the central FO V in this setup. PSFs with respect to various positions of point sources on the object plane are measured as the projector successivel y lights up one pixel from the central position to w ards the cor ner along x direction. The cross-cor relation coefficients between the PSF at the center and PSFs at different positions are calculated to confir m the av ailable central FO V according to the magnification between the object and image plane. Fig. 2a show s these cross-correlation coefficients as a function of the shifted distance on the object plane. W e could see that only the object located in the cyan region could be reconstructed with high fidelity , if we define the FO V is the region with cross-cor relation coefficient greater than 0.5. This ME region is cor responding to a strongly scatter ing medium with the effective thickness of 41.3 um. T o demonstrate this limited FO V , we use the central PSF (Fig. 2b) and the speckle image (Fig. 2c) of the object in Fig. 1 to do deconv olution. The reconstructed result and Fig. 2. Measurements and image reco very for the limited FO V in our optical sys tem. (a) Cross-correlation coefficient between the central PSF and various spatial PSFs to identify the FO V on the object plane. Cyan region indicates a high-quality reconstructed area with cross-correlation coefficients larg er than 0.5. (b, c) Central PSF and a captured speckle image of the object on the camera. (d) Recons truction through a post-process algorithm indicates a successful reco very of the objects located in the dashed red circle, cor responding to the cyan region in (a). (e) Intensity distribution along the central hor izontal direction. Scale bars: 1000 µ m. The ar ro w indicate the central ro w of the image. the intensity across the central horizontal direction are shown in Fig. 2d and Fig. 2e, where w e clearl y obtain the recov ery of the object in the limited 720 µ m radius FO V , same with the cy an region in Fig. 2a. For the reconstr uction process, w e remov e the background with intensity less than 15% of the maximum intensity f or the clear display . The back g round noise comes from reconstruction artifact, which depends on the relationship betw een the actual signal of interest and the noise (in this w ork, noise are mainly from the uncorrelated speckle image generated by nearb y regions of object). The full dimension of the speckle image and the PSF are used f or the reconstruction, because the increase in number of speckles would reduce the reconstruction noise. Ho we ver , the background noise is still una voidable because the non-sparse object makes a poor speckle contrast within the dynamic range of camera. Bey ond the FO V , par tial recov ery of the object could also be identified but the intensity are v ery dim due to the low cross-correlation coefficient (less than 0.5). The abo ve central PSF in Fig. 2 only guarantees successful recov ery image in the optical sys tem with the linear shift-inv ar iant property within a small region, where the deconv olution process reconstructs the object with high quality . Normally , the optical imaging sys tem with scattering media is limited within the central FO V determined by the central PSF . How ev er , the speckle image in Fig. 2c actually is the sum of the various PSFs from ev er y point source on the object plane. In this w ork, we e xplore more inf or mation from the non-paraxial region and find an alter nativ e wa y to e xtend the FO V while imaging with a strongl y scatter ing medium using a single shot. There are multiple spatial FO Vs satisfying the conv olution operation in Equation (6) for different spatial regions on the object plane. Equation (9) show s that the information from various spatial regions could be recov ered from the recorded single speckle image. W e present various spatial point sources on the object plane and measure the cor responding PSFs, then utilize the same speckle image in Fig. 2c to e xecute deconv olution process. Fig. 3 show s f our typical reconstructions with f our different spatial PSFs, the clear reconstruction centers and surrounding dim recov eries hav e same effects as Fig. 2d. At the bottom of each reconstructed image w e ha ve plotted 1D intensity of the pix el line r unning along center . W e need note that the PSFs and av ailable reconstruction regions (ME range) on image plane are unique and different f or different spatial positions due to the random nature of the scatter ing medium. Theref ore one could observe the different reconstr uction FO V , f or ex ample, the FO V in Fig. 3c seems larg er than that in Fig. 3a. (a) (b) (c) (d) 0.3 0.5 0.7 1000 2000 3000 4000 5000 6000 7000 0 x (μm) Intensity (a.u.) 0.3 0.5 0.7 1000 2000 3000 4000 5000 6000 7000 0 x (μm) Intensity (a.u.) 0.3 0.5 0.7 1000 2000 3000 4000 5000 6000 7000 0 x (μm) Intensity (a.u.) 0.3 0.5 0.7 1000 2000 3000 4000 5000 6000 7000 0 x (μm) Intensity (a.u.) Fig. 3. Limited FO Vs with various spatial PSFs. (a-d) A group of successful reconstr uctions of the object and cor responding 1D intensity distributions along central row s. Scale bars: 1000 µ m. The ar ro w indicate the central ro w of the image W e ha ve demonstrated our principle to demultiplexing the spatial inf ormation multiple x ed into one single speckle image through v arious spatial PSFs in Fig. 3. Ho w ev er, f or each individual reconstructed imag e, the inf or mation of interest are shifted to the center , and it is necessar y to arrange them to form a single image with an enlarg e FO V . W e need to shift the reconstructed images according to the region associated with the PSFs, whose positions are predeter mined. In fact, PSFs could record the shift of the speckle patterns and one could confir m their position through calculating the cross-cor relation coefficient (ev en though the value ma ybe v er y low) to ar rang e the regional reconstructed images. If the regional PSF is measured f or the point at position ( u x , u y ) , the reco vered region on image plane should be centered at position ( v x , v y ) , where v / u is the magnification. No w the challenge is to super pose these reconstructed regions to f or m the entire FO V . In reality , ME does not mean identical PSFs in the regional FO V . The cross-cor relation coefficient between two PSFs reduces monotonically with their inter distance. In each FO V region, the cross-cor relation coefficient of central PSF with other PSFs reduces when the point goes from the center to the edge (Fig. 2a); theref ore, the intensity of recov ered imag e decreases from center to the peripher y . As presented abo ve, w e only need a single PSF for each regional FO V to present the function of a scatter ing lens. The position of the point source to take the PSF defines the center of lens and the PSF is the representative f or this region. Any other point source on object plane also creates their own PSF whose cross-cor relation coefficient with representative PSF represents the ‘transmission coefficient’ of light from the point through the regional scatter ing lens in the deconv olution process. These transmission coefficients f or m a matr ix T f or each scattering lens. The scatter ing medium is fixed; the cross-cor relation coefficients f or each region are not much different from the center regions. W e assume that a single matr ix T can be used f or all regions, and it can be written as f ollow s: O R = O [ shift R ] T (10) where O [ shift R ] is the object O being shifted to center of the region R . Unf or tunatel y both O and T are unkno wn and need to be reco v ered. Inspired by embedded pupil function reco very f or Fourier pty chographic microscopy (EPR Y -FPM) [29, 30] we adopt an iterative update strategy to obtain both O and T as shown below . O ( n + 1 ) [ shift R ] = O n [ shift R ] + α T n max ( T n ) ( O R − O n [ shift R ] T n ) , (11) T ( n + 1 ) = T n + β O n ma x ( O n ) ( O R − O n [ shift R ] T n ) (12) In the ne xt experiment, the point sources used f or the PSF measurement are shifted along both x and y direction with a fixed space. Theref ore, we just need to retr ie ve the larg e FO V image from these reconstructed smaller FO V images in accordance with the shift in imaging plane using the af orementioned algorithm. Groups of letters ‘welcome TO our lab’ and ‘NTU EEE OPTIMUS!’ on vertical and hor izontal axis are displa y ed as the object. The spatial PSFs f or our optical sys tem are from the point sources located at the various positions on the object plane, sho wn with red circles in Fig. 4a. A super posed reconstruction in Fig. 4b is 6650 µ m width much larg er than the reconstr ucted region by the central PSF alone ( 2 × 720 µ m ) . W e hav e demonstrated the aim of the proposed technique: enlarging the FO V , which is done b y utilizing the full capacity of the imager . The successful recov er y will depend on the ability to resolv e the contrast of the speckle image within the giv en dynamic rang e of the imager . The speckle contrast is reduced with increase in the illumination bandwidth, and ha ving more br ight areas in the object. Theref ore, the reconstruction quality can be fur ther enhanced with a nar ro w band illumination imaging and relativ ely sparser objects. Superposed reconstruction (a) (b) Object plane Fig. 4. Superposed reconstruction to enlarg e the limited FO V . (a) Spatial distr ibution of the objects on the object plane. Red circles indicate the spatial positions of point sources for measuring the various spatial PSFs. (b) Super posed reconstruction image. Dashed red circle indicates the enlarg ed FO V . Scale bars: 1000 µ m. 4. Conclusion W e demonstrate an enlarg ed FO V for imaging with strongl y scattering media by utilizing multiple ME regions. A single-shot speckle pattern contains essential spatial inf or mation of the large object. All the information is multiplex ed on a single imag e randomly but pre-deter minis tically b y the scattering medium. Any object region of interest can be retr ie v ed b y using the corresponding PSF of that region. The regional PSFs are fixed and only need to be recorded once for all time use. The decon v olution algorithm utilizes the shift inv ar iance (i.e. spatial cor relation) of the PSFs for image reconstruction, while the or thogonality between the regional PSFs (i.e. spatial decorrelation) makes them playing the role of spatial information demultiplex er . An iterative process f or automatic weighted av eraging is used to stitch the multiple regional images and form a large FO V image. The propose technique utilizes the imag er to its full capacity in ter ms of both dimension and dynamic range to enlarg e FO V imaging with strongly scattering media. Funding N an yang T ec hnological U niversity’ s start-up grant; Sing apore Ministry of Education, MOE-A cRF Tier -1 grant (R G70/15); Singapore Ministry of Health’ s National Medical Researc h Council . Ackno wledgments W e would like to thank the financial suppor ts from NTU start-up g rant, Singapore MOE-A cRF Tier -1 grant (R G70/15) and the Singapore Ministry of Health ’ s National Medical Research Council under its .