Transparent Object Reconstruction via Coded Transport of Intensity

Transcription

1 Transparent Objet Reonstrution via Coded Transport of Intensity Chenguang Ma 1 Xing Lin 1 Jinli Suo 1 Qionghai Dai 1 Gordon Wetzstein 2 1 Department of Automation, Tsinghua University 2 MIT Media Lab Abstrat Capturing and understanding visual signals is one of the ore interests of omputer vision. Muh progress has been made w.r.t. many aspets of imaging, but the reonstrution of refrative phenomena, suh as turbulene, gas and heat flows, liquids, or transparent solids, has remained a hallenging problem. In this paper, we derive an intuitive formulation of light transport in refrative media using light fields and the transport of intensity equation. We show how oded illumination in ombination with pairs of reorded images allow for robust omputational reonstrution of dynami two and three-dimensional refrative phenomena. 1. Introdution Imaging and reovering refrative surfaes and volumes has been an ative area of omputer vision researh for deades. Whereas some appliations seek reonstrutions of geometry or refletane of natural phenomena, suh as gases, fluids, solids, or flames [12], others evaluate aerodynami properties of vehiles [26], or synthesize omposites of aptured phenomena and renderings [35]. Reonstruting phenomena with omplex refrative properties, however, is a very hallenging problem. Existing solutions an be roughly lassified as intrusive or non-intrusive methods. For example, intrusive methods over objets of interest with diffuse oatings [7] or immerse them in fluoresent liquids [9] whereas non-intrusive methods measure the distortion of a referene pattern by the refrative phenomenon [22, 34, 24, 3, 1]. In this paper, we propose a non-intrusive tehnique that builds on the transport of intensity equation (TIE) [28, 27]. Derived using wave optis models with oherent illumination, the TIE is known in the optis ommunity and has appliations in phase mirosopy. We present a new derivation of the TIE using inoherent light fields, whih makes it intuitive and pratial for omputer vision appliations. Further, we modify widely-used TIE aquisition setups by introduing oded illumination that failitates improved robustness of tomographi refrative volume reonstrutions. Finally, we evaluate the proposed omputational imaging system in simulation and with a range of physial experiments. 2. Related Work Fluid and Phase Imaging Imaging and reonstruting fluids is a diverse area of researh. An overview of tehniques suh as partile image veloimetry and other flow estimation tehniques an be found in [19]. Phase ontrast mirosopy and differential interferene ontrast are standard tehniques in mirosopy [23]. Shlieren photography is a non-intrusive way of enoding refrative phenomena as hanges in intensity or olor of a aptured image [26]. Traditionally, this is done with intriate optial setups; modern approahes ombine optial flow with oded bakgrounds [6]. More reently, oded 4D light field illumination has been shown to replae both ompliated optial setups in traditional setups and expensive omputation in bakground-oriented Shlieren methods [32, 33]. A popular tehnique for phase imaging with oherent illumination is the transport of intensity equation (TIE) [28, 27]. Using a first or higher-order [31] approximation of the light transport operator, 2D phase information of an objet an be reonstruted from a foal stak ontaining two or more images. An interpretation of these tehniques in phase-spae was presented in [25]. The methods disussed in this paper build on existing oherent TIE formulations, but derive forward and inverse methods using inoherent light fields along with optial oding shemes to make these tehniques pratial for omputer vision appliations. Refrative and Speular Surfae Reonstrution In the omputer vision ommunity, refrative surfae reovery has been an extremely ative area. A reent survey [12] presents a omprehensive overview of the field, whereas [17] lassifies the problem spae based on required aquisition setup and number of refrative events. A body of work has takled the problem using a shape from distortion approah [22, 34, 24, 3, 1]. Ative, oded illumination an be used instead of observing the distortion of stati alibration patterns [21]. A simple approah to aquiring surfaes with omplex refletane properties is to either apply a diffuse oating and san them with standard tehniques [7] or to immerse [9] or dye [11] objets of interest with fluoresent substanes. However, these are highly intrusive approahes. Finally, polarization properties an also be observed to infer information about refration [20, 10]. Our approah is 1

2 related to other shape from distortion tehniques that use ative illumination, but we introdue the TIE to omputer vision, enhane it with oded illumination, and ombine it with tomographi volume reonstrution. Refrative Tomography When imaging threedimensional, inhomogeneous refrative volumes, onventional surfae reonstrution tehniques usually fail. Computed tomography methods [16], however, have been adopted to reover volumes of flames [13], gases [2, 15], and solids [30, 29]. The main differene between these methods is whether the tomographi reonstrution is performed from observed light attenuation [13, 30] or refration [2, 29, 15]. Our method is losely related to [29] by employing the TIE, but we follow [15] by applying a regularizer based on Fermant s priniple to the tomographi reonstrution. To the best of our knowledge, this is the first time that the TIE is derived in the geometrial optis domain using a light field notation. We also augment the TIE with oded illumination that allows us to estimate ray-ray orrespondenes for more robust tomographi reonstrutions. 3. Image Formation and Reonstrution 3.1. Light Propagation in Refrative Media Using a geometrial optis model, the propagation of light in materials with a heterogeneous refrative index field n is governed by the Eikonal equation or ray equation of geometrial optis [4]: s ( n x ) = n. (1) s As illustrated in Figure 1, x is the lateral position of a light ray on some plane that is defined to be zero along the optial axis and s is its trajetory through the medium. We an write this as two ordinary differential equations [14]: n x s = ν ν = n. (2) s The loal diretion of propagation is ν. Equation 2 states that the hange in diretion a light ray undergoes in the medium is equal to the lateral gradient of the refrative index field. Using the notion of light fields [18], we an desribe the effets of a refrative medium to an arbitrary distribution of light rays in an intuitive manner. A light field l(x, ν) fully defines a radiane distribution using rays as its primitives, that are parameterized by a position x and the propagation diretion ν = tan(θ), where θ is the angle of propagation. We use an intuitive 2D light field notation throughout the paper; full 4D formulations and extended illustration an be found in the supplement. The refration of a light field l ( ) (x, ν) inident on a thin ollimated light refrated light refrative objet sreens Figure 1. Ray shemati. Collimated illumination is refrated by a thin surfae. The resulting intensity is measured at two distanes, z and z+ z. The transport of intensity equation relates the refrative surfae gradients with the differene of those measurements, allowing the surfae to be reovered. refrative objet is then l(x, ν) = l ( ) (x, ν n(x)), (3) where l(x, ν) is the light field on the same loation but after refration. As illustrated in Figure 1, the refrative event basially shifts the angles of the light field. For volumetri index fields, one must trae a light ray through the volume along its urved trajetory. From the Eikonal equations, we know that the outgoing angle of a ray is ν out = ν in + n(x, z)ds, (4) for any ray with inident angle ν in. The volumetri refration a light field undergoes is expressed as ( ) l 0 (x, ν) = l ( ) 0 g(x, ν), ν n(x, z)ds, (5) where g(x, ν) is a general funtion desribing the lateral displaement of a ray by the refrative volume. We note that this formulation neglets effets suh as refletion, absorption, sattering, diffration, and dispersion. The refrative volume is assumed to be differentiable. Abrupt hanges in a refrative index field, suh as at the intersetion of air and water, will be smoothed out in this formulation unless an additional sparse gradient prior is employed, as suggested in Setion Transport of Intensity The transport of intensity equation (TIE) [28, 27] is often used in the optis ommunity to model light propagation through refrative index fields using a first-order approximation of the transport operator. Whereas formulations exist using wave models of light, this setion derives what we believe to be the first formulation of the TIE using light fields as a geometrial optis model. Let us onsider transport of a light field in free spae from z by a distane z. This is a shear along the spatial oordinates: l z+ z (x, ν) = l z (x zν, ν). (6)

3 Following Semihaevsky and Testorf [25], the propagation operator an be approximated to first order using a Taylor expansion: l z (x zν, ν) l z (x, ν) zν x l z (x, ν). (7) We note that this is a good approximation under paraxial assumptions, i.e. when the light field is propagating with small deviations along the optial axis z. Higher-order approximations an be employed if this is not the ase [31]. The transport of intensity equation requires two intensity measurements to be taken at a slight axial distane z between eah other. These an be reorded with a bare amera sensor or by photographing a white, diffuse sreen with a onventional amera. The intensity on some axial position z is defined as the projetion of a light field over angular domain : I z (x) = l z (x, ν) dν. (8) Combining Equations 6, 7 and 8 yields an expression for the image intensity measured at z + z: I z+ z (x) l z (x, ν) dν z ν x l z (x, ν) dν. (9) A forward finite differenes approximation of intensity along the optial axis z is then I z (x) I z+ z (x) I z (x) (10) z z Hene, ombining Equations 8, 9, and 10 yields: I z (x) ν z x l z (x, ν) dν, (11) whih is the foundation of the transport of intensity equation. Semihaevsky and Testorf [25] refer to the quantity νl z (x, ν) dν as the first-order angular moment of the light field TIE-based Refrative Objet Reonstrution Using Equation 11, we an derive the transport of intensity equation for both thin and volumetri refrative phenomena. However, ollimated inident illumination is required, i.e. l ( ) 0 (x) = δ (ν) I (x). Following Equation 3, the light field after refration by a thin objet is given as l 0 (x, ν) = δ (ν n (x)) I (x). (12) The right-hand side of Equation 11 now beomes ν x l 0 (x, ν) dν ( I (x) = x ) νδ (ν n (x)) dν = (I (x) n (x)), (13) x reonstrution volume ameras refrated light path oded ollimated light soure Figure 2. Tomographi volume reonstrution. The volumetri refrative index field is observed from multiple amera perspetives. A transport of intensity equation is solved for eah independently. The resulting projetions of the volume are then proessed using omputed tomography to reover the volume. resulting in the transport of intensity equation for thin refrative phenomena: I (x) = (I (x) n (x)). (14) z x This formulation is losely related to the wave models derived in [28, 27] and to interpretations in phase-spae [25]. Figure 1 illustrates the desribed setup using a ray diagram. Imaging and reonstruting volumetri refrative index fields is also possible. Again, for ollimated inident light l ( ) 0 (x, ν) = δ(ν)i(x), this results in a variant of the TIE: I (x) = ( ) I(x) n(x, z)ds. (15) z x 3.4. Tomographi Volume Reonstrution For the speifi ase of imaging volumetri refrative index fields, the TIE (Eq. 15) an only be solved for the aumulated amount of refration that eah ray underwent in the volume. This is a projetion of all refrative index gradients along the optial path n(x, z)ds. As illustrated in Figure 2 and proposed in previous work (e.g., [2, 15, 29]), omputed tomography methods [16] (e.g., the simultaneous algebrai reonstrution tehnique (SART)) an be used to solve for the volumetri index field from projetions aptured from different amera perspetives. For this purpose, the refrative index field is represented as a linear ombination of i = 1... k basis funtions φ i with assoiated weights α i, leading to the following expression: ( ) n ds = α i φ i ds = ( ) α i φ i ds. i i (16) Common hoies for φ i are ubi voxels or radially symmetri basis funtions [2]; in either ase, the number of basis funtions is equal to the number of unknown voxels.

4 b = Φα + Γ (α). Illustration of Setup Coded Collimated Illumination Using this formulation in hand, Equation 15 an be solved independently for multiple different amera perspetives to give one projetion of the 3D index field eah. Staking all resulting projetions in a vetor b Rl, l being the total number of all amera pixels, allows us to formulate a tomographi reonstrution problem that solves for the unknown oeffiients α Rk (17) l k Here, olumn i of matrix Φ R ontains the preomr puted light path integrals over the basis funtions φi ds for some measurement. An additional visual hull restrition and total variane regularizer Γ ( ) an be employed if the equation system is ill-posed. The visual hull an be used to determine the basis funtions with possibly nonzero oeffiients [2] and the regularizer is a diret 3D extension of the ommonly-used 2D TV regularizer. Detailed explanation an be found in the supplement. Equation 17 an be omputed with any standard linear solver Inorporating Light Path Approximation Unfortunately, the atual ray paths through the volume are usually unknown. Common assumptions inlude straight ray approximation [29] or approximating the path by a single refrative event lose to the enter of the volume [2]. Ji et al. [15] reently proposed a method that optially aquires the orrespondenes between light rays inident on the volume and the same, but refrated rays emerging on the other side using light field probes [32]. These ray-ray orrespondenes have been demonstrated to ahieve higher-quality reonstrutions by imposing Fermat s priniple as a regularizer to the tomography problem. We believe this to be an effetive way of onstraining the possible set of solutions for volumetri refrative index fields. Establishing ray-ray orrespondenes in the proposed framework is straightforward: instead of illuminating the sample objet with uniform ollimated light, i.e. I(x) = onst, we an ode the intensity distribution I(x) suh that ray-ray orrespondenes an be estimated using optial flow. We employ a simple olor gradient, although more sophistiated illumination patterns, suh as gray odes, are possible. We note that ray-ray orrespondenes require both position and diretion of an inident ray to be mathed with one of the refrated rays. Whereas [15] ahieve this using 4D light field probes, our method relies on ollimated illumination thereby unambiguously identifying inident rays using purely spatial illumination odes. 4. Analysis and Evaluation In this setion, we evaluate the proposed methods. We proeed by simulating and evaluating system parameters and noise resiliene for oded aquisition and reonstrution of a thin, two-dimensional refrative objet. We also Target Objet Reonstrutions Figure 3. Reonstrution of thin refrative objet. Two images, separated by distane z, are aptured (top) and proessed to reover the refrative index (bottom, right). show tomographi reonstrutions of three-dimensional volumetri senes, both for homogeneous and inhomogeneous refrative index fields. Finally, we ompare our method with previous work. Thin Refrative Objet Figure 3 illustrates the setup of a syntheti experiment. A thin refrative objet distorts the inident ollimated illumination, whih is then measured at two different distanes from the sample objet. By solving the transport of intensity equation (Eq. 14), we an reover the refrative index field. Figure 3 shows two reonstrutions with different levels of iid Gaussian sensor noise. For this experiment, the objet is simulated to be 4 4 m, the inident illumination is oded with a simple olor gradient, and the refrative index varies smoothly between 1 and Equation 14 is solved by onverting the measured olor images to graysale. Reonstrution performane w.r.t to sensor noise and a varying distane z is further evaluated in Table 1. Whereas the peak signal-to-noise ratio (PSNR), measured between 3D ground truth index field and reonstrution, improves with a dereasing distane in the absene of noise, the subtle hanges in images reorded at suh small distanes are easily perturbed by noise. An optimal tradeoff has to be found depending on the employed amera s noise harateristis. Homogeneous Volumetri Refrative Index Field To reover a full, three-dimensional refrative index field, multiple projetions of the volume need to be aptured from different perspetives. We simulate a voxelized bunny with a refrative index of 1.4 surrounded by air in Figure 4. Image pairs are reorded from eight different amera perspetives and individually proessed with Equation 14 to yield

5 Table 1. Evaluation of resiliene to sensor noise and variations in distane between the two aptured images. Reonstrution errors are given in peak signal-to-noise ratio (PSNR). Based on these simulations, we see that a smaller distane is favorable for lower sensor noise, but larger distanes are more resilient to noise. Ground Truth Sensor Noise Standard Deviation σ Fuel Dataset Sreen Image Reonstrution e-4 Error z in m Diff Optial Flow -1e-4 Figure 5. Volumetri reonstrution. Left: fuel dataset (ground truth), one of the two images aptured at slightly different axial positions, their differene, and estimated optial flow from oded illumination. Right: slies of original volume, reonstruted volume, and error. Figure 4. Reonstrution of a homogeneous refrative volume. From left: target objet, projetions reonstruted via Eq. 14 from eight different perspetives, tomographially-reonstruted 3D volume. projetions of the refrative index volume. Using the tomographi reonstrution method outlined in Equation 17, we an reover the refrative volume. Inhomogeneous Volumetri Refrative Index Field Using the fuel injetion dataset DFG SFB 382 ( we evaluate the proposed method for an inhomogeneous refrative index field. Figure 5 shows a 3D rendering of the volume (left) as well as individual slies (right). We simulate aquisition by seven equally-spaed perspetives in a half ring setup. From eah perspetive, two images are reorded at slightly different axial positions (one shown in Fig. 5, left); their differene is used to solve for the 2D projetion of the refrative index field via Equation 14. The oded illumination allows for ray-ray orrespondenes to be estimated using optial flow (Fig. 5, left), whih makes a tomographi reonstrution of the 3D volume more robust. For this experiment, we employ a 3D total variation (TV) regularizer for the reonstrution (Eq. 17) and assume that the surrounding medium has the refrative index of air. The reonstrution quality of the proposed 3D tomographi method w.r.t. the number of amera perspetives is evaluated in Table 2. As expeted, an inreasing number of perspetives makes the volumetri reonstrution more robust and leads to lower root-mean-square errors (RMS) and peak signal-to-noise ratios (PSNR). The proposed reonstrution is similar to SART [16]. Whereas diret inversion methods, suh as filtered bakprojetion using the Radon transform, ould be signifiantly faster, these are only appliable when a dense set of projetions of a fullring setup are available. We also implemented the methods Simulation Errors w.r.t. Number of Cameras # of ameras Proposed RMS Method PSNR Atheson RMS et al PSNR Ji RMS et al PSNR Table 2. Evaluation of tomographi reonstrution using a varying number of ameras in a half-ring setup. Compared to Atheson et al. [2] and Ji et. at [15], the proposed method always performs better under the simulated onditions. proposed by Atheson et al. [2] and Ji et. at [15]. The results show that our method has a higher RMS/PSNR than theirs for all simulated onfigurations. This is intuitive, beause we follow [15] by using a light path approximation prior but our setups allows us to apture higher-resolution data. Additional results and omparisons an be found in the supplement. 5. Implementation Hardware Figure 6 shows our prototype apture devie. We use a Texas Instruments LightCrafter as a programmable light soure, whih provides a resolution of pixels and three primary olor hannels. A lens is mounted at its foal distane to the projetor and ollimates the projeted optial ode a simple olor gradient whih then rear-illuminates the refrative sample. A beam splitter separates the optial path on the other side of the sample suh that two intensity projetions at distanes spaed by 1m an be imaged by two mahine vision ameras (PointGrey Flea2-08S2, px) simultaneously. The refrative sample is mounted on a rotation stage; for volumetri reonstrutions, multiple images are aptured showing the sample from different perspetives.

6 Figure 6. Prototype apture devie. A ollimated projetor provides oded illumination that is refrated by the sample and imaged on the other side of the setup. Calibration Both TIE and tomographi volume reonstrution require preise image registration. For this purpose, we use standard amera alibration tehniques [5] to orret for intrinsi distortions. Further, we register the two aptured images using features observed in a alibration image displayed by the projetor. Figure 7. Reonstrutions of refrative index fields. These haraters are omposed of lear optial adhesive with a refrative index that is different from the underlying glass plate. By apturing two differently-foused images from the same perspetive, we an reover 2D projetions of the refrative index fields, as illustrated in the enter and bottom rows. Software After this one-time alibration proedure, the system simultaneously reords alibrated image pairs with a resolution of pixels. For thin refrative objet, we further down-sample them to pixels and use a Fast Poisson Solver ( harper/ poisson.htm) to solve Equation 14. This solver is implemented in Matlab and takes approx. 76 seonds on an Intel i7 PC with 8 GM RAM. For volumetri refrative index fields, we use the CVX toolkit [8] to solve Equation 17. Reovered volumes are disretized to voxels. A reonstrution from eight different perspetives takes approx. 3 minutes. Both the Poisson solver and the tomographi reonstrution ould be signifiantly aelerated using more advaned algorithms. of Figure 7. The two foal planes are separated by a distane z = 1 m. Other system parameters are desribed in Setion 5. 2D Reonstrution of Dynami Index Fields Using the dual-amera setup desribed in Setion 5, we an apture dynami senes from a single perspetive but foused at two different axial planes. As shown in Figure 8, the hot air plume of a andle an be aptured and reovered using the proposed method. Similar to the reonstrutions of Figure 7, the transport of intensity equation atually omputes a two-dimensional projetion of the three-dimensional refrative index field. System parameters for this experiment are idential to the previous one. 3D Tomographi Reonstrution of Stati Index Fields Finally, we also show a tomographi reonstrution of a solid three-dimensional glass objet in Figure 9. This is aptured from multiple perspetives by mounting it on a rotation stage and reording photographs at seven settings that are equally spaed apart by 30. The two foal planes for eah perspetive are separated by a distane of z = 0.5 m. For eah of the perspetives, we reover the 2D projetion of the volumetri refrative index field via Equation 15 and then solve the tomographi problem, i.e. Equation 17. The latter inorporates light path approximation onstraints using the ray-ray orrespondenes that are estimated from the illumination oding sheme desribed in Setion Experimental Results In this setion, we desribe a number of experiments we have performed with the prototype setup desribed in the previous setion. 2D Reonstrution of Stati Refrative Index Fields Figure 7 shows results of stati objets reovered with the proposed tehnique. For this experiment, we drew haraters on a thin glass1 plate using lear optial adhesive with a refrative index of Photographs of the target objets are shown in the top row, whereas the enter row shows reonstrutions of the refrative index field with a resolution of Please note that the objets are not atually thin, so the reonstrutions show integrals over the refrative index field as desribed by Equation 15. Height field renderings of the reonstrutions are shown in the bottom 1 The glass plate has a refrative index of Disussion In summary, we propose a tehnique to reonstrut two and three-dimensional refrative index fields using a new, geometrial optis formulation of the transport of intensity

7 Figure 8. Reonstrutions of a dynami refrative index field. Even the small differenes in refration aused by the heat above a andle an be faithfully reovered via oded transport of intensity. In this experiment, we apture time-synhronized videos with two ameras foused at different foal planes (shown for a single frame of the animation, top) and reonstrut eah frame separately (bottom). Figure 9. Three-dimensional tomographi reonstrution of a volumetri objet (left). We apture pairs of photographs from seven perspetives (not shown) and reover the volumetri refrative index field as desribed in the text. Volume and surfae renderings of the reonstrutions are shown from perspetives that are different from those aptured. equation. In addition to deriving it using light field notation, we enhane the models used in the optis ommunity by oded bakground illumination that allows us to inorporate more sophistiated regularizers, as reently proposed in the omputer vision ommunity [15]. We evaluate the proposed tehnique in simulation and demonstrate that it ahieves higher-quality three-dimensional, tomographi reonstrutions than alternative tehniques. We also build a prototype devie and show reonstrutions of stati and dynami refrative index fields. optial flow may fail to reover preise ray-ray orrespondenes. Our prototype system has limitations as well. The finite projetor aperture provides rear-illumination that is not perfetly ollimated and the optial odes are limited by the minimal observable hanges in pixel intensity and hue by the amera. Dynami senes aptured with a finite amera exposure time may ontain motion blur Future Work 7.1. Limitations The proposed method makes a number of assumptions that may not be met for arbitrary refrative objets. These inlude the absene of refletion, sattering, diffration, and dispersion. Further, the projetion of the refrative index field is assumed to be differential, hene smooth. While the transport of intensity equation inherently separates absorption from refration, the proposed optial oding strategy requires that the medium is either non-absorptive or that absorption is onstant over the imaged area. Otherwise, the In the future, we would like to experiment with an array of ameras that aptures dynami, three-dimensional index fields from multiple perspetives simultaneously. We ould redue the number of required devies by using a single light field amera for eah perspetive and synthesize multiple intensity projetions at different foal distanes. This approah, however, would ome at the ost of redued image resolution. Finally, apturing more than two intensity projetions from eah perspetives an ahieve better reonstrutions [31].

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