Estimation of large-amplitude motion and disparity fields: Application to intermediate view reconstruction

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1 c 2000 SPIE. Personal use of this material is permitte. However, permission to reprint/republish this material for avertising or promotional purposes or for creating new collective works for resale or reistribution to servers or lists, or to reuse any copyrighte component of this work in other works must be obtaine from the SPIE. IS&T/SPIE Symposium on Electronic Imaging Visual Communications an Image Proc. Jan , 2001, San Jose, CA, USA Estimation of large-amplitue motion an isparity fiels: Application to intermeiate view reconstruction Mustapha Karouchi, Janusz Konra an Carlos Vázquez INRS-Télécommunications, Institut National e la Recherche Scientifique Place Bonaventure, P.O. 644, Montreal QC, H5A 1C6, Canaa karou@inrs-telecom.uquebec.ca Boston University, Dept. of Electrical an Computer Eng., Boston, MA 02215, USA jkonra@bu.eu ABSTRACT This paper escribes a metho for establishing ense corresponence between two images in a vieo sequence (motion) or in a stereo pair (isparity) in case of large isplacements. In orer to eal with large-amplitue motion or isparity fiels, multi-resolution techniques such as blocks matching an optical flow have been use in the past. Although quite successful, these techniques cannot easily cope with motion/isparity iscontinuities as they o not explicitly exploit image structure. Aitionally, their computational complexity is high; block matching requires examination of numerous vector caniates while optical flow-base techniques are iterative. In this paper, we propose a new approach that aresses both issues. The approach combines feature matching with Delaunay triangulation, an thus reliable long-range corresponences result while the computational complexity is not high (sparse representation). In the propose approach, feature points are foun first using a simple intensity corner etector. Then, corresponence pairs between two images are foun by maximizing cross-correlation over a small winow. Finally, the Delaunay triangulation is applie to the resulting points, an a ense vector fiel is compute by planar interpolation over Delaunay triangles. The resulting vector fiel is continuous everywhere, an thus oes not reflect motion or epth iscontinuities at object bounaries. In orer to improve the renition of such iscontinuities, we propose to further ivie Delaunay triangles whenever the isplacement vectors within a triangle o not allow goo intensity match. The approach has been extensively teste on stereoscopic images in the context of intermeiate view reconstruction where the quality of estimate isparity fiels is critical for final image renering. The first results are very encouraging as the reconstructe images are of high quality, especially at object bounaries, an the computational complexity is lower than that of multi-resolution block matching. Keywors: motion estimation, isparity estimation, optical flow, feature corresponence, Delaunay triangulation, stereoscopic image processing, intermeiate view reconstruction 1. INTRODUCTION Estimation of ense corresponences, such as motion fiels in vieo sequences or isparity fiels in stereoscopic image pairs, in the presence of large-amplitue isplacements, is a ifficult problem. Methos base on block matching can hanle large isplacements but they are computationally involve. Moreover, they cannot reliably isambiguate among caniate blocks since simple cumulative error criteria are typically use. Traitional optical flow methos are incremental as they assume local intensity linearity an compute isplacement fiels iteratively in small steps. They are not irectly applicable to large-amplitue isplacements. To eal with this problem, two ifferent approaches have been propose in the literature. One approach uses a multi-resolution representation that enables estimation of large-amplitue vectors at coarse levels of a multi-resolution pyrami an subsequent refinement at finer resolutions. Although successfully use in both block matching an optical flow methos, multi-resolution implementation oes not exploit explicit image structure to help isambiguate among caniate matches. Thus, vector fiel iscontinuities are not well estimate; in block matching, because of vector constancy over a block, an in optical flow since it is very ifficult to ajust smoothness factors at each resolution level an usually oversmoothe fiels result 1,2. In the secon approach, corresponences are explicitly compute base on image structure, e.g., by means of feature point matching, but the resulting vector fiels are sparse since the vectors are efine only at feature points 3,4.

2 In this paper, we propose a new metho that aresses the above issues. Since the intene application is the reconstruction of intermeiate images in a vieo sequence or of intermeiate views in a stereo pair, reliability an computational efficiency of the approach are very important. In orer to aress reliability, we propose to compute feature-base corresponences first. Although the resulting isplacement vectors are mostly reliable, they are also sparse. Thus, to compute ense vector fiels, an o this with efficiency, we propose triangulation-base low-orer spatial interpolation between feature-point vectors. Since some triangles overlap ifferent motion/isparity regions an thus cannot faithfully represent the corresponing iscontinuity, we etect such triangles, further ivie them, an re-interpolate the isplacement vectors for each new triangle. We apply the estimate vector fiel to intermeiate view reconstruction in stereoscopic pairs. Numerous experimental results are shown in Section APPROACH The propose metho consists of three main components: computation of sparse corresponences base on feature points, computation of ense vector fiel base on these sparse corresponences by means of Delaunay triangulation, an recursive refinement of the triangulation. These three components are escribe next Computation of sparse corresponences Extraction of feature points A feature point etector aims at the localization of significant intensity structures in an image. Often, points of abrupt intensity change, as manifeste by large intensity graient, are use as an inication of a significant image structure. These points shoul be etecte consistently an localize precisely because errors in etection or localization are magnifie in the final results. Corner-point etection is one of the more reliable mechanisms allowing to fin feature points. Three ifferent approaches have been propose in the literature to etect corners: contour-base methos 6,7, signal-base methos 8 an template fitting methos 9,10. In this work, we opt for a metho from the secon class, the so-calle Harris etector 5. We apply the Harris etector to each image of the stereo or motion image pair (I 0, I 1 ). The Harris etector s threshol is fixe epening on the average image graient. In our application, this threshol is selecte so that 800-1,200 corner points are recovere from a pixel image. The final results are not particularly sensitive to the number of corner points etecte Matching of feature points Once feature points are extracte in both images, a corresponence between feature points in the two images must be establishe. This may be one by evaluating similarity between small winows aroun two feature points, one from each image. A pair is sai to have been matche if within a search area it results in the highest similarity (lowest issimilarity) measure. There are many possible measures of similarity. Some are taken from the fiel of signal processing 11 while others from statistics 12. In this paper, to measure the similarity we use a statistically-motivate metho base on cross-correlation 13. For each feature point m 0 in image I 0, a local winow centere at m 0 is correlate with a winow centere at each caniate feature point m 1 within a selecte area S of image I 1. The search area S represents an upper boun on the maximum motion/isparity. A constraint on the correlation measure is applie in orer to eliminate unreliable matches; results with correlation uner a certain threshol are rejecte. Then, cross-valiation is applie to eliminate false matches; if m 1 in I 1 is the best match for m 0 in I 0 within the search area S, then also m 0 must be the best match for m 1 within S. Typically, we foun acceptable matches for 800-1,200 feature points etecte in a pixel image. Although the cross-valiation eliminates almost all unreliable/false matches, a small number of them may go unetecte thus affecting the final results. We remove the remaining unreliable matches by postprocessing escribe in Section As a result, we obtain an ensemble of feature point pairs (corresponences), that can be use as control points for ense vector fiel interpolation Computation of ense vector fiel In this section, we present a metho to generate a ense vector fiel from the compute corresponence pairs. First, Delaunay triangulation is performe, then its noes are post-processe to eliminate potential outliers, an finally a ense vector fiel is compute by low-orer spatial interpolation within each Delaunay triangle.

3 p 2 p 1 p 3 p p 4 p 5 Figure 1. Neighborhoo η(p) of corresponence point p Delaunay triangulation of corresponence points The general Delaunay triangulation can be escribe as follows. Let P be the set of all points in a two-imensional plane to be triangulate. The Delaunay triangulation of P, enote DT (P), is the unique triangulation base on the concept of empty overlap with a circles, i.e., { } DT (P) = (p, q, r) P 3, C(p, q, r) P =, where C(p, q, r) is a circle circumscribe on the tree points p, q, an r. In the case consiere here, P is the set of corresponence points in image I 0. Clearly, more feature points exist in active image areas, i.e., at object bounaries or in texture regions. Thus, smaller triangles will result in these active areas, an larger triangles will appear in homogeneous regions Elimination of potential outliers As mentione in Section 2.1.2, unreliable corresponence points may be foun an use for the Delaunay triangulation. Such unreliable points may aversely affect the ense fiel interpolation escribe in the next section. In orer to eliminate them, we use the following proceure: 1. for each corresponence point p in image I 0 ientify its neighborhoo η(p), i.e., all vertices of triangles containing p, except for p itself (Fig. 1), 2. remove unreliable corresponence points; a corresponence point p is consiere unreliable if: p p > ρ, where p is the corresponence vector at p, p is the average corresponence vector over η(p), an ρ is an ajustable threshol. Using this simple proceure, we are able to remove significant number of outlying corresponences (although some correct matches are also remove) at very low computational cost. We teste the propose proceure on several natural stereoscopic images. Results obtaine with the cross-correlation matching an the above post-processing step are presente in Section 4. Typically, at least 16% of the total number of corresponence points are remove by this post-processing step Interpolation of vectors Given the triangulation an the corresponence vectors at its noes, we compute a ense vector fiel uner the assumption that, in spatial coorinates, motion/isparity varies linearly within each triangle. Given three vertices of a triangle, vector = [ 1, 2 ] T at each pixel within the triangle is compute from an affine (linear) moel. This is equivalent to passing one plane in space (x, y, 1 ) an one plane in space (x, y, 2 ), each with respect to triangle vertices. Let p, q, an r be the three noes of a triangle (Fig. 2). If p, q an r are the corresponence vectors associate with p, q an r, respectively, then the vector x for image point x within the triangle forme by the triplet of corners p, q, an r is given by: x = (1 i j) p + i q + j r, (1)

4 θ n ipq p p q jpr q x m r r Figure 2. Interpolation of isplacement vector at x within triangle with noes p, q an r. where i an j relate the coorinates of point x to those of the noe points p, q, an r. Referring to Fig. 2, the equations for the values of i an j are: i = ( px 2 pn 2 ) 1 2 pq cos( π 2 Θ), j = ( px 2 pm 2 ) 1 2 pr cos( π, (2) 2 Θ) where ( pq pr ) Θ = arccos, pm = pq pr pr px, pn = pr pq px. pq with pq enoting a vector joining points p an q, while m an n are orthogonal projections of x onto pr an pq. The ense vector fiel is etermine by first computing the coefficients i an j (2) for each image point within each triangle, an then by applying them to equation (1). Such vector fiel computation presents very interesting properties both from mathematical an practical point of view it efines a vector fiel that is continuous everywhere an oes so with computational efficiency. However, this metho oes not allow to take into account potential vector fiel iscontinuities. We aress this problem in the next section Triangulation refinement It is clear that the performance of the ense fiel computation escribe above epens on the quality of the Delaunay triangulation. One potential problem exists when a triangle overlaps two ifferent objects in the image or an object an backgroun. Then, such a triangle contains an object bounary where the true motion or isparity vector fiel is likely to be iscontinuous. Clearly, a iscontinuity cannot be reprouce by the linear interpolation propose, an a remey must be sought. One option woul be to evelop a more sophisticate interpolation scheme within each triangle, while another to align the triangles better with object bounaries in the image. We opt for the secon metho an we propose to refine (subivie) the triangles that exhibit excessive preiction error between images I 0 an I 1. The mean-square preiction error between the images I 0 an I 1 is efine as follows: E = s Ω(I 1 ( s + s ) I 0 ( s)) 2, where Ω is the (iscrete) image omain, s = (x, y) is a spatial position in Ωm, an s is a isplacement vector at s. Note that Ω is partitione by a triangular mesh. Let Ψ = {ψ i, i = 1, 2,..., N} be the set of N Delaunay triangles; ψ i is the set of image points within triangle number i. Using this triangular mesh, the mean-square preiction error can be written as follows: E ε(ψ i ), ψ i Ψ where ε(ψ i ) is the mean-square preiction error between I 0 an I 1 within the triangle ψ i. This is an approximation only since each triangle sie is counte twice in the computation (it is consiere to belong to two triangles).

5 (a) Figure 3. Refinement of the triangulation: (a) ivision of the triangle of interest into three triangles; (b) four cases of ivision of the neighboring triangles. (b) In orer, to ientify triangles that are likely to overlap motion or isparity iscontinuity, an that nee to be further ivie, we compute the mean-square preiction error for each triangle an we compare this error to threshol σ. If the absolute value of the preiction error for the particular triangle excees σ, an the triangle is sufficiently large, we ivie it into four triangles accoring to Fig. 3.a. The triangle create in the center has noes (vertices) in the mile of the original triangle s sies. In orer to assure triangulation uniformity, neighboring triangles may be split into up to four triangles (Fig. 3.b). For each univie triangle ψ i, we compute the number n(ψ i ) of neighboring ivie triangles. There are four possibilities (Fig. 3.b): if n(ψ i ) = 0, nothing to be one; if n(ψ i ) = 1, the triangle ψ i is ivie into two sub-triangles with the new vertex place on the sie that ajoins the neighboring ivie triangle, if n(ψ i ) = 2, the triangle ψ i is ivie into three sub-triangles; there are two possibilities: we choose the one for which the triangles create are more compact, if n(ψ i ) = 3, the triangle ψ i is ivie into four sub-triangles. Now, that we have new vertices (an triangles), we nee to compute isplacement vectors associate with them. Let i, j an k be the unknown isplacement vectors corresponing to the new noes i, j an k create in a given triangle ψ i with noes p, q an r. Let p, q an r be the corresponence vectors for ol vertices p, q an r (Fig. 4). We assume that the triangle ψ i is sufficiently compact. Uner the assumption that motion/isparity is locally smooth, the isplacements that minimize the mean-square preiction error in the neighborhoo of each vertex i, j, k shoul belong to the set { p, q, r }. Then, the vector l at vertex l is estimate as follows: l = min s { p, q, r} s W l (I 1 (s + s ) I 0 (s)) 2, l {i, j, k}, (3) where W l is a small winow aroun image point l. We chose 3 3 winow for experiments. Once the vectors at new noes have been compute, the ense vector fiel within each new sub-triangle is compute using the metho escribe in Section Having compute this new ense vector fiel, the refinement proceure is repeate until convergence. The convergence is achieve when no triangles unergo ivision. We can summarize the refinement proceure as follows: 1. choose value of the threshols: σ for the mean-square preiction error E, an γ for the triangle size, 2. for each triangle ψ i, compute the mean-square preiction error ε(ψ i ); if ε(ψ i ) > σ an ψ i γ, where ψ i enotes the size of ψ i, ivie the triangle into four sub-triangles (Fig. 3.a), 3. ensure triangulation uniformity; for all univie triangles compute the number of neighboring newly-ivie triangles an split them accoringly (Fig. 3.b),

6 p i p i q k j q k j r r Figure 4. Original triangle with vertices p, q, r, an the newly create vertices i, j, k. 4. compute isplacement vectors for all the new triplets of vertices by performing minimization (3), 5. re-interpolate the ense vector fiel for each new triangle as escribe in Section 2.2.3, 6. go back to 2 unless no more triangles can be ivie. 3. APPLICATION OF THE METHOD TO THE RECONSTRUCTION OF INTERMEDIATE IMAGES IN STEREOSCOPIC PAIRS The application we are intereste in is the reconstruction of intermeiate views in stereoscopic images. Such a reconstruction has a number of applications in the fiel of stereoscopic imaging. Problems relate to parallax ajustment (e.g., too much stereo) 14, for example, are solve through the reconstruction of images coming from virtual cameras with a baseline (istance between optical centers of the cameras) ifferent than that of the true cameras. Continuous look-aroun, i.e., reconstruction of intermeiate images to allow lateral viewer motion in front of stereoscopic isplay, is another possible application 15. In the stereoscopic literature, two approaches have been propose for the reconstruction of intermeiate views 16,17 : 1. methos base on a 3-D moel of the scene: estimation of a 3-D representation of the scene from two or more images (left an right in the stereo case), followe by projection onto a virtual camera place at the esire reconstruction position, 2. methos base on isparity compensation: estimation of a isparity fiel between left an right images an its application to isparity-compensate reconstruction at the esire camera position. For the complex real-worl scenes, the 3-D moeling is too complex an therefore results are usually imprecise. For the intene entertainment applications, that we envisage, the camera convergence angle is very small an the camera baseline is close to that of the interocular istance, limiting the range of isparities an thus favoring the isparity compensation approach. We are intereste in the case of parallel or almost parallel camera geometry, an in virtual cameras having optical centers on the line joining the optical centers of left an right cameras; the optical axes of the virtual cameras are assume parallel to the optical axes of the true cameras. With this geometry, projections of a point onto a virtual camera are on the line passing through the projections of this point on the left an right camera planes. This allows us to reconstruct any intermeiate view by linear isparity-compensate interpolation, assuming that we are able to estimate the isparity between left an right images. Once the isparity is estimate, the reconstruction of an intermeiate image at position γ (0 < γ < 1, γ=0 being the left image, an γ=1 the right one), uner ieal conitions, shoul be as simple as copying either the left or right pixel to the intermeiate image I γ. But ue to noise, illumination effects an aliasing, the matching pixels, even in the case of perfect isparity estimation, are not ientical. In a simple scenario, an image at position γ can be then reconstructe by a linear isparity-compensate interpolation as follows 17 : I γ ( s + (γ α) s ) = (1 γ)i L ( s α s ) + γi R ( s + (1 α) s ).

7 (a) (b) Figure 5. Original pair from the stereoscopic vieo sequence Flowerpot: (a) left image; (b) right image. Note that the isparity fiel { s, s Ω} is efine at position α, i.e., the vectors pass through points of the regular gri Ω at α. Two reconstruction scenarios are possible. In the first scenario, in orer to reconstruct {I γ, s Ω}, isparities are estimate at α = γ; for each γ a separate isparity fiel must be compute thus imposing extreme computational effort. Alternatively, a single isparity fiel is compute at α, but then for γ α samples of I γ are recovere over a set of irregular spatial locations an further reconstruction of regularly-space samples is neee; for a given α, a regularly-space image {I γ ( s), s Ω} must be reconstructe from irregular samples {I γ ( s + (γ α) s ), s Ω}. In particular, for α=0, i.e., when the isparity fiel is estimate at the left image, a regularly-space image {I γ ( s), s Ω} must be reconstructe from irregular samples: I γ ( s + γ s ) = (1 γ)i L ( s) + γi R ( s + s ). (4) Various approaches have been propose to ate to reconstruct an image from irregularly-space samples, varying from simple polynomial interpolation base on triangulation to more complex iterative algorithms. Complex algorithms like those propose in the literature offer better results at the expense of higher complexity an processing time. A low-orer polynomial interpolation is a simpler solution an offers acceptable results for the current stage of our work. To keep the computational complexity of the overall process low, we use a linear interpolation base on Delaunay triangulation of the irregularly space image samples. Let Γ = { s + γ s, s Ω} be the irregular sampling gri. First, we perform the Delaunay triangulation on Γ. Then, we compute pixel values over the regular gri Ω by linear (planar) interpolation over each triangle in the mesh. The overall process can be escribe as follows: 1. isparity estimation using the algorithm propose in Section 2, 2. computation of the irregularly-sample image through isparity compensation (4), 3. reconstruction of the regularly-sample image using linear interpolation base on Delaunay triangulation. 4. EXPERIMENTAL RESULTS We have teste the propose metho on real stereoscopic images. An example stereo pair Flowerpot (see Acknowlegments) is shown in Fig. 5. First, feature points were extracte from the two images using the Harris corner point etector. Then, corresponence pairs were foun by maximizing a cross-correlation measure. Fig. 6.a shows corresponence vectors that join pairs of corresponence points, before post-processing. Fig. 6.b shows the corresponence points after post-processing

8 that removes outliers (Section 2.2.2). It can be seen that most of the false matches have been eliminate, although some correct matches may have been remove as well. The resulting fiel of corresponence vectors is sparse with more vectors locate in texture areas an very few vectors - in uniform areas. The set of resulting corresponence points was then use to perform the Delaunay triangulation an to compute a ense isparity fiel. In the experiments, the preiction error threshol σ was set to 200. Fig. 7.a shows the ense fiel obtaine with linear interpolation before refinement. Overall, the compute isparity fiel closely matches epth in the original scene espite the fact that it is interpolate from a sparse set of corresponence vectors rather than being compute locally for every pixel. Fig. 7.b shows the same fiel after refining the mesh (triangle subivision) as escribe in Section Although the vector fiels before an after refinement are very similar, local ifferences exist an they can be better seen in the enlarge sub-winows in Fig. 9. The triangulation results, superpose on the reference image I 0 (left image from the stereo pair), are epicte in Fig. 8. Fig. 8.a shows the triangulation of corresponence points with no refinement, while Fig. 8.b shows the triangulation after refinement. Note the appearance of many new triangles in texture areas (plants, fruit stan) an aroun object bounaries (aroun umbrellas, flags, men s pants). (a) (b) Figure 6. Corresponence vectors between images I 0 an I 1 : (a) before post-processing; (b) after post-processing to remove outliers (Section 2.2.1). (a) (b) Figure 7. Disparity fiel obtaine from corresponence points by the propose metho: (a) without mesh refinement; an (b) with mesh refinement. For clarity, both vector fiels have been horizontally an vertically subsample by 18 an 6, respectively.

9 (a) (b) Figure 8. Triangulation (a) with no mesh refinement; an (b) after mesh refinement (preiction threshol σ= 200). Finally, we have applie the compute isparity fiels to intermeiate view reconstruction at γ=0.5 as escribe in Section 3. Fig. 9 shows enlarge pixel winows aroun the flower pot that were extracte from the triangulation image (Fig. 8), from ense vector fiels (Fig. 7) an from images reconstructe at γ=0.5, that we are not showing here in full since they look remarkably similar in print. Clearly, the refinement ha a significant impact on the interpolate ense isparity fiel; the iscontinuities in the isparity fiel are more perceptible after refinement (Fig. 9.). This improvement is ue to the fact that several new triangles have been create at the flower pot ege, especially at the winow frame in the backgroun, allowing for a better renition of epth (isparity) iscontinuity. An improvement is also visible in the reconstructe image with refinement (Fig. 9.f), aroun the winow frame above the flower. Note that the isparities between the two images are greater than 8 pixels/frame in magnitue an, in general, not easy to estimate. Clearly, the reconstructe intermeiate images both with an without refinement, are very close in quality to the original (reference) image (Fig. 10), although they are more blurre ue to the inevitable interpolation process involve in combining images I 0 an I 1, an also ue to the applie non-uniform-to-uniform interpolation. A single-view comparison of the reconstruction, however, is not critical enough. A much more emaning scenario is the reconstruction of a series of intermeiate views an there play-out in time, thus simulating a camera translation between the left- an right-camera positions. We have performe such a simulate camera translation by reconstructing 18 intermeiate images between I 0 an I 1. The overall perception of the camera translation was very natural; it truly seeme that the camera was moving horizontally. While in the sequence reconstructe base on the isparity fiel with no refinement errors appeare at some object bounaries, especially aroun the flower pot where the isparity iscontinuity is significant, most of these errors were eliminate when the refine isparity fiel was use. We have also performe some initial comparisons of the propose reconstruction with a reconstruction base on optical flow-type isparity fiel 14. In these early tests the new metho compare favorably at object bounaries, but further etaile comparisons are require. 5. CONCLUSIONS We propose a new metho for the estimation of large-amplitue motion an isparity fiels. This metho is base on feature corresponences an the interpolation of sparse vector ata. The approach has several benefits: it permits to estimate large isplacements an subsequently take into account motion/isparity iscontinuities. Moreover, the metho is very efficient computationally compare to a typical multi-resolution block matching. Initial results are very promising; the metho prouces high-quality intermeiate views for natural (complex) stereoscopic image sequences. However, the results epen on the presence of texture in the images; the metho works well in sequences with strong textures. 6. ACKNOWLEDGMENTS We tank NHK of Japan for proviing us with the stereoscopic test sequence Flowerpot.

10 (a) (b) (c) () (e) (f) Figure 9. Enlarge winow ( ) from the triangulation (Fig. 8), isparity fiel (Fig. 7) an image reconstructe at γ=0.5: (a,c,e) without mesh refinement; (b,,f) with mesh refinement.

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