Multiple View Image Reconstruction: A Harmonic Approach


 Jeffry Hancock
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1 Multple Vew Image Reconstructon: A Harmonc Approach Justn Domke and Yanns Alomonos Center for Automaton Research, Department of Computer Scence Unversty of Maryland, College Park, MD, 2742, USA Abstract Ths paper presents a new constrant connectng the sgnals n multple vews of a surface. The constrant arses from a harmonc analyss of the geometry of the magng process and t gves rse to a new technque for multple vew mage reconstructon. Gven several vews of a surface from dfferent postons, fundamentally dfferent nformaton s present n each mage, owng to the fact that cameras measure the ncomng lght only after the applcaton of a lowpass flter. Our analyss shows how the geometry of the magng s connected to ths flterng. Ths leads to a technque for constructng a sngle output mage contanng all the nformaton present n the nput mages.. Introducton The lght feld enterng a camera s, n general, hghly dscontnuous. Ths sgnal can be sad to contan an nfnte amount of nformaton. All cameras, however, can only measure a fnte amount of nformaton. To cope wth ths, good cameras lowpass flter, or blur the ncomng lght before measurng t. Ths s sometmes called antalasng. Note that ths flterng s a contnuous process that takes place n the optcs of the camera. Now, suppose we have two mages of a surface, taken from dfferent postons. If the surface s Lambertan, the dscontnuous lght felds enterng the camera at each poston contan the same sgnals, merely warped dependng on the geometry. However, the real, measured sgnals (after lowpass flterng) are qute dfferent, even accountng for the warpng. To warp and then flter s not the same as to flter and then warp. Ths effect can be understood n a dfferent way. Take two mages of a surface. The ncomng lght (the deal mages ) contans the same underlyng sgnal. Hence the real, If the lowpass flter removes everythng above the Nyqust frequency, the contnuous sgnal can be reconstructed from dscrete samples. Ths paper wll use contnuous mathematcs for standard, dscretely sampled mages, ths can be thought of as referrng to the reconstructed sgnal. observed mages are warped versons of the same (deal) mage, fltered wth an dentcal lowpass flter. We wll show n Secton 2 that they can be thought of nstead as the same (deal) mage, convolved wth dfferent, warped versons of the lowpass flter. Ths nsght makes an analyss of these effects much easer. In Secton 3 we wll show how ths can be understood n the frequency doman. Dependng on the effectve flterng n each mage, dfferent frequences are attenuated to dfferent degrees. Ths leads to a smple dea for multple vew mage reconstructon to create an output mage, just select each frequency from the nput mage n whch t was least fltered. Ths leads to several techncal problems what range of frequences to take from each mage, and how to pull out a specfc range of frequences from a gven mage. These problems are addressed n Sectons 4 and 5. Fnally, n Secton 6 we summarze the algorthm for reconstructon n the case of affne transformatons. Ths s then generalzed to arbtrary, contnuous transformatons n Secton 6. Ultmately, we wll fnd that one can reconstruct an output mage smply as a sum of the nput mages (warped to a common frame) convolved wth dfferent flters. Fg. summarzes the approach... Prevous Work The most closely related prevous work s by Wang et al. [6]. There, the authors model the process of creatng one mage from another through the steps of reconstructng the contnuous sgnal, warpng, preflterng, and resamplng. The mage flterng process s modeled as an deal lowpass flter. If the columns of mages are stacked together as vectors, ths ultmately leads to a numercal lnear constrant on the desred mage wth respect to each observed mage. The desred mage s then obtaned usng least squares to fnd a soluton approxmately satsfyng each of these constrants. Another related area s superresoluton [5]. In superresoluton, the nput mages are generally taken from nearby postons, and the goal s to take advantage of alasng effects to construct a hgherresoluton mage. Unlke ths related work, our method makes no attempt to nvert the
2 3 Add to produce reconstructed mage Fgure. A graphcal summary of the method for the affne case. Top row: Input mages. Second row: Images warped to a common coordnate system. Thrd row: Flters computed for each mage. Fourth row: Result of convoluton of flters wth mages from the second row. Bottom: Reconstructed mage. For nonaffne moton, the only change s that the flters become spatally varant.
3 orgnal blurrng kernel. Instead, here we essentally just combne the nformaton n the dfferent nput mages nto one reconstructed mage. It turns out that ths can stll be done, wth out modelng the blurrng process. Thus, ths method would offer lttle mprovement when the nput mages are nearby. The pontspread functon n real cameras s dffcult to estmate, and does not appear to exactly obey dealzatons lke Gaussan or deal lowpass [4]. If the blurrng kernel s modeled, better results (perhaps greatly better) could be obtaned by fusng a deconvoluton strategy wth the constrant we use here. Other work based on smlar prncples s n the graphcs lterature for texture mappng. Greene and Heckbert proposed the ellptcal weghted average (EWA) flter [3]. Essentally, a crcular flter n the mage space s projected to the texture space, resultng n an ellptcal flter. Smlar prncples are used here. 2. Theory 2.. Notaton Gven some functon f, andamatrxa, we wll use the notaton f A to represent f, warped under the moton A. That s, f A s the functon such that, x,f A (x) =f(ax). () We wll use lowercase letters (, orj) to represent functons n the spatal doman, and uppercase letters (I, orj) for the frequency doman. Boldface letters (x) represent vectors Dervaton Suppose that, f x n the frst mage corresponds to x 2 n the second mage, then x 2 = Ax. So f s a functon representng the frst deal mage, and 2 s a functon representng the second, Hence, for all x, x 2 = Ax = 2 (x 2 )= (x ). (2) 2 (x) = A (x). (3) and 2 should be thought of as essentally arbtrary real functons. Now, the fundamental assumpton here s that we do not observe these deal mages. One can only measure the deal mage after convoluton wth some lowpass flter, e. Letj and j 2 denote the real, measured mages. j (x) =[ e](x) (4) j 2 (x) =[ 2 e](x) (5) Now, use Eqn. 3 to change the expresson for j 2,tomake ts relatonshp to j more explct. j 2 (x) =[ A e](x) (6) Now, we wll use the followng Theorem. (see Appendx) [f A g](x) = [f ga ](Ax) (7) A Applyng ths, we have, j 2 (x) = A [ e A ](A x). (8) So, fnally, j A 2 (x) = A [ e A ](x) (9) Contrast ths wth the expresson for j n Eqn. 4. It s as f, nstead of beng fltered wth e, t was fltered wth A e A. Ths smple result has mportant mplcatons f we have multple vews of some surface, even after we warp the mages nto a common coordnate system, dfferent sgnals are present. However, these sgnals can be understood smply as the result of the same deal mage, convolved wth warped versons of the same flter. Ths s most ntutve n the frequency doman. Let E denote the Fourer transform of e, J that of j, and so on. Then, by the convoluton theorem, J (u) =[I E](u) () The analogous equatons s also true for J 2, but we wll contnue our analyss from Eqn. 9. We wll use the followng theorem, whch s essentally a specal case of the Affne Fourer Theorem [] [2]. If F{f(x)} = F (u),thenf{f A (x)} = A F A T (u) () Now apply the theorem to e A. F{e A (x)} = A EA T (u) (2) So we can clearly see the contrast between the Fourer transform of the frst mage, and the Fourer transform of the warped verson of the second mage F{j A 2 (x)} =[I E A T ](u) (3) In the frequency doman, t s as f, rather than beng fltered wth E, t was fltered wth E A T.
4 E A T ( u) (a) E A T 2 ( u) (b) max(e A T ( u), E A T 2 ( u)) Fgure 2. Dfferent flters n the frequency doman. 3. Intuton Suppose we have many vews of a surface, warped under dfferent motons A. Suppose also that E s a pure lowpass flter. (Ths s only for the purpose of explanaton. We assume only that E s a symmetrc, decreasng functon.) Ths stuaton s pctured n Fg. 2 for the two motons, A = [.75 ], A 2 = [.75 It s natural to thnk that f we have these two mages, we could reconstruct an mage whch contans the best frequency nformaton from each. Ths s shown n part (c). Now, for mages wth affne moton, the followng smple algorthm wll perform multple vew reconstructon. (We emphasze that ths s only for the sake of explanatonthe results n ths paper do not use ths algorthm.) (c) ].. Input a set of mages, j, j 2,... j n, and a correspondng set of motons, A, A 2,... A n 2. Warp each mage to the central coordnate system, to obtan j A (x). 3. For each mage, j compute the Fourer Transform of the warped mage, F{j A }(u) =[I E A T ](u). 4. Create the reconstructed mage n the Fourer doman. For all u, set K(u) = F{j A l l }(u), where l =argmn A T u. 5. Output the mage n the spatal doman. k(x) = F {K}. Steps 3 smply create the frequency doman representatons of the nput mages n a common coordnate system. To understand step 4, notce that for each frequency, we would lke to take t from each mage n whch t has been fltered least. Now, snce we assume that E(u) s a monotoncally decreasng functon of u only, we would lke to u s least. However, ths smple algorthm could only work when the global moton s affne. We wll present a dfferent algorthm that operates completely n the spatal doman. We select the moton for whch A T wll later show that ths allows us to do reconstructon for transformatons that are only locally affne for example full projectve transformatons. Nevertheless, the algorthm operates on the same prncples as the smple one above. Before we can present the algorthm, we need to develop two tools. 4. The Frequency Slce Flter Here, the goal s to create a flter that wll pass a certan range of frequences. More specfcally, gven θ and θ 2, we would lke a flter c such that, n the frequency doman (Fg. 3(a)), (where u denotes the angle of u: f u = r[cos θ, sn θ] T,then u = θ) { θ u θ 2 C θ,θ 2 (u) = (4) else Frst, suppose we would lke to create a flter that passes exactly those frequences n the frst and thrd quadrants. (That s, θ =, θ 2 = π/2.) Navely, we would just plug these values nto the above equaton. However, f we do ths, the nverse Fourer Transform wll not converge. A convenent fact s useful here the mages have already been lowpass fltered. Hence, t does not matter what the flter does to very hgh frequences. So, nstead, we wll defne the followng flter, cuttng off hgher frequences. (Fg. 3(b)) C,π/2,r (u) = { u π/2, and u r else (5) The flters correspondng to dfferent r wll always be dfferent. Nevertheless, once r s suffcently hgh (above the hghest frequency present n an mage), the result of applyng the dfferent flters to an mage wll be the same. Before extendng ths to the case of other θ, θ 2, we wll fnd the nverse Fourer transform. = u r c,π/2,r (x) =F {C,π/2,r (u)} (6) = u C,π/2,r (u)exp(2πu T x)du (7) [exp(2πu T x)+exp( 2πu T x)]du (8) = u r 2cos(2πu T x)du (9) c,π/2,r (x) = [cos(2πrx)+cos(2πry) cos(2πr(x + y)) ] 2π 2 xy (2)
5 C π/,4π/ C,π/2,.5 C π/,4π/,.5 (a) (b) (c) Fgure 3. The frequency slce flter n the frequency doman. c c,π/2,.5 π/,4π/, (a) (b) (c) Fgure 5. Frequency Segmentaton. [, ] T. Hence only those frequences n the correct range of angles wll be passed. Fg. 3 (c) shows an example wth θ = π/, θ 2 =4π/ Fgure 4. The frequency slce flter n the spatal doman. Examples are shown n Fg. 4. Notce that ths functon has lnear decay n the spatal doman. Ths s a potental problem, snce the flter wll need to be large to capture the entre profle. In practce, however, we have found that ths was not a problem. We used matrces of sze 5 by 5 to represent the flters, and found that usng larger szes resulted n lttle change to the output. Ths wll be mportant later, because n projectve reconstructon, the transformaton must be close to affne n a wndow the sze of ths flter. Regardless, t would certanly be better to use a flter wth faster decay. Ths s left as future work. Now, to defne the flter for arbtrary angles, we wll agan use the specal case of the Affne Fourer Theorem, as n Eqn.. Gven θ,andθ 2, defne the followng matrx: [ ] cos θ cos θ V = 2 (2) sn θ sn θ 2 Now, we can defne the frequency slce flter for arbtrary angles. c θ,θ 2,r(x) = V c V T,π/2,r (x) (22) To see ths, apply Eqn. to the rght hand sde F{ V c V T,π/2,r (x)} = V V T C (V T ) T (u) (23),π/2,r F{ V c V T,π/2,r (x)} = CV,π/2,r (u) (24) To understand the presence of V, notce that t wll send [cos θ, sn θ ] T to [, ] T, and [cos θ 2, sn θ 2 ] T to 5. Parttonng Frequency Space Gven the frequency slce flters, we are nearly prepared to do multple vew reconstructon for affne transformatons. The man remanng problem s the followng: Suppose we are gven a set of mages warped to a common frame, {j A (x)}, as well as the set of motons producng those mages, {A }. For a gven frequency u havng angle θ, n whch mage s that frequency least fltered? (Notce that ths wll depend only on the angle of u, and not on ts magntude.) Recall from above that F{j A (x)} =[I E A T ](u). So, for each angle θ, we would lke to choose a moton A, such that E(A T [cos θ, sn θ] T ) s maxmum. Ths wll be the case when A T [cos θ, sn θ] T s mnmum. We can pcture the stuaton by drawng a curve, where for each angle θ, we use the length A T [cos θ, sn θ] T. (Fg 5 (a).) To carve up the space of frequences we need two steps.. For each par of motons A,andA j, fnd ponts for whch the curves meet. That s, fnd u such that A T u T (A A T u = A T j u. A j A T j )u = Notce that ths does not depend on the magntude of u. If we assume that ether the frst or second component of u s nonzero, ths can be solved by settng u =[u x, ] T,oru =[,u y ] T, and solvng a quadratc equaton. Complex values as a soluton ndcate that the two curves do not ntersect. 2. Fnd the angles of all the ponts u found n the prevous step and sort them. Now, form pars from all adjacent angles. Ths results n a sequence of pars of angles, <θ,θ 2 >, < θ 2,θ 3 >... < θ m,θ m >. It now remans to fnd the moton that has the smallest value
6 n each regon. The smplest procedure s smply to test all motons and fnd, [ ] arg mn A T cos(.5(θj + θ j+ ). (25) sn(.5(θ j + θ j+ )) Ths works, but has a worstcase tme complexty of O(n 3 ),wheren s the number of nput mages. However, an optmzaton can reduce ths to O(n 2 log n): for each angle θ j keep track of the motons whose ntersecton produced that angle. Then, f some moton A s mnmum n the regon <θ j,θ j >, A wll also be the mnmum n the regon <θ j,θ j+ >, unless θ j was produced by the ntersecton of A wth some other moton A k,anda k s smaller than A n the sense of Eqn. 25. Ths means that we only need to test at most two motons n each regon. The frequency segmentaton process s llustrated n Fg. 5. (a) shows the magntude of A T [cos θ, sn θ] T, for each moton A, and each angle θ. In (b), the angles are found where the motons ntersect. In (c), the shape s shown, where for each angle, the magntude s taken from the best moton. 6. Affne Reconstructon Algorthm summarzes the method for reconstructon of mages warped under affne transformatons. The method s farly smple frst the frequency space s parttoned, then the fnal mage s reconstructed as the sum of the nput mages convolved wth dfferent flters. Notce that ths algorthm takes place entrely n the spatal doman. In our mplementaton, the convoluton n step 4 takes place dscretely, rather than as an ntegraton. For the results shown n ths paper, we use r =.3. Ths s hgh enough to avod flterng out frequences that are present n the nput mages, but low enough to allow for the flters to be reasonably approxmated wth dscrete samples. 7. General Reconstructon The theory developed thus far has all been for the case of affne moton. We can observe, however, that t s essentally a local process the flters have a small area of support. It turns out that we can extend the method to essentally arbtrary dfferentable transformatons. Ths s because any dfferentable transformaton can be locally approxmated as affne. We wll gve examples here for projectve transformatons, but t s smplest to frst show how to approxmate a general transformaton. Suppose that some functon t(x) gves the transformaton, so f x 2 = t(x ),then 2 (x 2 )= (x ). (26) It follows that Algorthm Affne Reconstructon. Input a set of mages, j, j 2,...j n, and a correspondng set of motons, A, A 2,...A n 2. Warp each mage to the central coordnate system, to obtan j A (x). 3. Use the method descrbed n Secton 5 to partton the frequency space. Obtan a set of pars of angles, along wth the best moton n that regon, <θ,θ 2,A >. 4. Output k = [ja c θ,θ 2,r] x, 2 (x) = (t (x)). (27) Now, wrte j 2 n the usual way. j 2 (x) =[ 2 e](x) (28) j 2 (x) = (t (x x ))e(x )dx x (29) Notce here that e(x ) wll be zero unless x s small. So, we wll use a local approxmaton for the transformaton. t(x x ) t(x) J x x (3) Where J x denotes the Jacoban of t, evaluated at the pont x. Now, take the nverse. t (x x ) t (x) Jx x (3) Substtute ths n the above expresson for j 2. j 2 (x) = (t (x) Jx x )e(x )dx (32) x Now, change varables. Set y = Jx x. j 2 (x) = y (t (x) y)e(j x x ) J x dy (33) j 2 (x) = J x [ e Jx ](t (x)) (34) So fnally, we have a smple local approxmaton. j 2 (t(x)) = J x [ e Jx ](x) (35) The method for general reconstructon s gven as Algorthm 2. Conceptually, the only dfference wth affne reconstructon s that the fnal mage k s the sum of spatally varyng flters convolved wth the nput mages.
7 Algorthm 2 General Reconstructon. Input a set of mages, j, j 2,... j n, and a correspondng set of transformatons, t, t 2,... t n 2. Warp each mage to the central coordnate system, to obtan j t (x). 3. For each pont x, (a) For each transformaton t, compute the Jacoban at x, J x,. (b) Use the method descrbed n Secton 5 to partton the frequency space. Obtan a set of pars of angles, along wth the best moton n that regon, <θ,θ 2,J x, >. (c) Set k(x) = [jjx, c θ,θ 2,r](x) 4. Output k. 7.. Projectve Transformatons We wll gve specfc results for the case of projectve transformatons. The only ssue s how to compute the local Jacoban of the transformaton. In nonprojectve coordnates, let x = [x,y ] T, x 2 = [x 2,y 2 ] T. Then we can wrte the transformaton as x 2 =(h x + h 2 y + h 3 )/(h 7 x + h 8 y + h 9 ), (36) y 2 =(h 4 x + h 5 y + h 6 )/(h 7 x + h 8 y + h 9 ). (37) The Jacoban s smply a matrx contanng four partal dervatves. These are easly evaluated. (See addtonal materal.) ] 8. Dscusson J x = [ x2 x 2 x y y 2 y 2 x y Fgure 6 shows an example reconstructon for affne moton. In ths experment, mages are of the same surface under four dfferent transformatons. Notce that when warped to a common coordnate system, each mage s vsbly blurred. Close observaton reveals that ths blurrng s dfferent n each of the warped mages. Stll, the reconstructed mage s qute sharp. The meanng of ths s perhaps surprsng all the hghfrequency content n the reconstructed mage s present n the nput, just dstrbuted among the dfferent mages. Fgure 7 shows a smlar example for reconstructon under projectve moton. In ths case all four nput mages are shown. (Please see the extra materal for more experments.) Ths paper has presented an analyss of the flterng done by the magng process n multple vews taken of a surface. We have seen that f we warp mages to a common frame, they can be understood as the same deal mage, convolved wth dstorted versons of the same lowpass flter. Here ths has led to an mage reconstructon technque, but the same deas could be useful n dfferent areas. One mportant applcaton s the problem of mage matchng. To correspond surface regons, t s necessary to understand the relatonshp between the underlyng sgnals. Ths analyss could be useful n creatng nvarant regon descrptors for matchng when there s large moton between the mages. References [] R. Bracewell, K.Y. Chang, A. Jha, and Y.H. Wang. Affne theorem for two dmensonal fourer transform. Electroncs Letters, 29(3):34, 993. [2] R. N. Bracewell. Twodmensonal magng. PrentceHall, Inc., Upper Saddle Rver, NJ, USA, 995. [3] N. Greene and P. S. Heckbert. Creatng raster omnmax mages from multple perspectve vews usng the ellptcal weghted average flter. IEEE Comput. Graph. Appl., 6(6):2 27, 986. [4] D. Hong and V. Kenneth. Effects of pontspread functon on calbraton and radometrc accuracy of ccd camera. Appled optcs, 43(3):665 67, 24. [5] S. Park, M. K. Park, and M. G. Kang. Superresoluton mage reconstructon: a techncal overvew. IEEE Sgnal Processng Magazne, 2(3):2 36, 23. [6] L. Wang, S. B. Kang, R. Szelsk, and H.Y. Shum. Optmal texture map reconstructon from multple vews. In CVPR (), pages , 2. A. Appendx Theorem [f A g](x) = [f ga ](Ax) A (38) Proof [f A g](x) = f(ax Ax )g(x )dx x (39) We would lke to change varables. Defne y = Ax. Then, we can rewrte the ntegral. [f A g](x) = f(ax y)g(a y)dy (4) A y [f A g](x) = [f ga](ax) (4) A
8 Input Images Sngle Coordnate System Input Images Sngle Coordnate System Reconstructon: Reconstructon: Fgure 6. An example affne reconstructon. Left column: Input mages. Rght column: Input mages warped to a common coordnate system. Bottom: Reconstructed output mage. Fgure 7. An example projectve reconstructon. Left column: Input mages. Rght column: Input mages warped to a common coordnate system. Bottom: Reconstructed output mage.
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