Multiple-view 3-D Reconstruction Using a Mirror

Transcription

1 Multiple-view 3-D Reconstruction Using a Mirror Bo Hu Christopher Brown Randal Nelson The University of Rochester Computer Science Department Rochester, New York Technical Report 863 May 2005 Abstract We propose a 3-D object reconstruction method using a stationary camera and a planar mirror. No calibration is required. The mirror provides the extra views needed for a multiple-view reconstruction. We examine the imaging geometry of the camera-mirror setup and prove a theorem that gives us the point correspondences to compute the orientation of the mirror. The correspondences are derived from the convex hull of the silhouettes of the images of the object and its mirror reflection. The distance between the mirror and the camera can be then obtained by a single object point and a pair of points on the mirror surface. After the pose of the mirror is determined, we have multiple calibrated views of the object. We show two reconstruction methods that utilize the special imaging geometry. The system setup is simple. The algorithm is fast and easy to implement. Keywords: Mirror, 3-D reconstruction, silhouette, visual hull, pose, camera calibration, catadioptric sensor This work is supported by the NSF Research Infrastructure grant No. EY EIA and the NIH grant No.

2 1 Introduction Computer vision technologies have advanced significantly since its coming into being. But few of the technologies, even those well understood in the research community, have entered the consumer market. The reason used to be that average consumers have only limited computational power and limited access to imaging devices. Now desktop PCs and digital cameras are as good as those in research labs. Yet we still face some obstacles to bringing the technologies into homes. One such technology is 3-D object reconstruction. Given a set of images or a clip of video of an object, 3-D reconstruction methods generate a 3-D model of the object. In general, except for very simple objects, the images are taken from different view points. Calibration is then required to establish the relationships between the view points. Usually employing a calibration rig, multipleview calibration is a non-trivial feat. The process can be hardly automated. Furthermore, unless multiple cameras are connected rigidly, the calibration can be easily broken and needs to be re-done. We here propose a 3-D reconstruction method that uses a stationary camera and a planar mirror. The camera is only minimally calibrated. Namely, one only needs to know the focal length of the lens. There are no multiple coordinate systems in our method. Therefore no external calibration is necessary. Knowing the focal length is in fact the minimum requirement for any Euclidean reconstruction. In practice, one can find out the focal length from the reading on the lens. The setup is simple. We place the object before a stationary camera and take a picture of the object. We then move a planar mirror into the scene and take pictures of the object and its mirror image (Fig. 1). We show a method that computes the orientations and the distances of the mirror automatically. What is new in the pose determination is that no feature correspondence is needed when computing the orientation. Once we know where the mirror is placed, we will have multiple calibrated views of the object and we can apply any 3-D reconstruction method. We show two silhouette-based methods that take advantage of the special imaging geometry. The first one is direct volume intersection. The second method builds a depth map for each object point seen by the stationary camera. This representation is very similar to that of image-based visual hull [16]. But in our case the intersection of epipolar lines takes a very simple form. Silhouette methods generate the visual hull [10] of the object, but it is usually a good approximation of the original object. The system described in this paper can be considered as a type of so-called catadioptric sensor, which combine mirrors and lenses. The majority of catadioptric sensors use curved, e.g., quadric, mirrors to increase the angle of view. Finding the pose of these sensors can be quite complex. E.g., Paulino and Araujo [12] prove that four corners of a rectangle determine the pose of a central catadioptric system. As a corollary, a rectangle (in fact, any four coplanar points [5]) determines the pose of a planar mirror too. Decomposing the pose to orientation and distance, our method only requires one point on the object or two points on the mirror surface. Doubeck and Svoboda [3] used a hyperbolic mirror to triangulate scene points. Their method recovers 3-D information of discrete points, while ours creates the whole shape. Gluckman and Nayar [8] investigated how to construct a system of planar mirrors and a camera so that the stereo images captured by the camera are rectified. Like most catadioptric sensors, the relative position of the camera and the mirror in their system is fixed and pre-calibrated. In our method, the mirror can freely change position, which makes the imaging system much more flexible. Mitsumoto et al.[11] used a planar mirror to reconstruct simple polyhedral objects. They introduced the VP (vanishing point) constraint between an object point and its mirror image to find point correspondences. It is the same constraint that we illustrate in Section 2. Zhang et al.[17] used a planar mirror to simulate mirror-symmetric objects, which allow simple 3-D reconstruction algorithms [7]. The work relies on hand-selected point correspondences. What is new in our method is that we establish the correspondences without identifying feature points. The shape from silhouette problem has a history as long as the discipline of computer vision. Silhouette-based algorithms are robust and able to create complete surfaces. One representative 1

3 Figure 1: Using a mirror to generate a new view. piece of work done by Szeleski [14] uses octrees to recover 3-D shape from a turn-table sequence. Similar algorithms are widely used; however, calibrating a turn-table sequence is not much easier than general multiple-camera setups [6]. Apple s QuickTime VR [1], which enables users to create and view an object from multiple view points, is perhaps the singular example of mature computer vision techniques entering the consumer market. It is however purely image-based and no 3-D models are created. The remaining of this paper is organized as following. In Section 2, we examine the imaging geometry of the mirror-camera setup. We prove the theorem that gives us the orientation of the mirror in Section 3. After the orientation of the mirror is known, there are different methods to determine the distance of the mirror from the camera. We will discuss two of them in Section 4. We show two 3-D reconstruction methods in Section 5. Lastly, we present some experimental results and point out opportunities for future work. Notation We use capital letters, e.g., A for 3-D points and primed letters for their mirror image, e.g., A is the mirror image of A. We use lower case letters for 2-D image points (a is the image of A and a is the image of A.) Vectors are in Bold-face. 2 The geometry of mirror imaging The imaging geometry of a camera and a mirror is depicted in Fig. 2. The stationary camera establishes a coordinate system, with its optical axis being the z-axis. The camera is at the origin O and the imaging plane is at z = f, where f is the focal length. The mirror surface can be expressed by a plane equation in this coordinate system: n x + d = 0, where n is the normal vector of the mirror plane. The normal vector always points toward the camera, which makes d positive and thus the distance from the camera to the plane. The point a is the image of an object point A. The point a is the image of A, the mirror image of point A. It is apparent that AA n. In fact, the line connecting any object point and its mirror image is parallel to the normal of the mirror. If we have another pair B and B and their images b and b respectively, we can see that aa and bb must intersect at the single point e, which is the vanishing point of the normal direction of the mirror. 2

4 B Mirror O A B A z Retina plane e a a b O b focal length Figure 2: The imaging geometry of a camera and a mirror. A and B are two object points. A and B are their mirror image. The camera is at the origin O. The image of A and A is a and a, respectively. The mirror image of the camera is a virtual camera at O. The point e is the epipole of the virtual camera. It is also the vanishing point of the normal direction of the mirror plane. It is interesting to point out that an object and its cast shadow share similar geometry as in the camera-mirror case. In fact, the lines aa and bb are what Shafer called illumination vectors [13]. Shafer observed that vectors from a shadow point to its corresponding occluder point intersect at a single point, which is the vanishing point of a directional light source, or the image of a point light source. The theorems we will prove in the next section therefore apply to the cast shadow cases too. We can also imagine there is a virtual camera, the mirror image of the real camera, looking at the object (its location is shown as O in Fig. 2). The vanishing point of n turns out to be the epipole of the virtual camera. This thinking leads us to the more familiar multiple cameras setting. Notice that the handedness of the virtual cameras is opposite to the real camera. If the normal direction of the mirror is parallel to the imaging surface of the camera, the camera cannot see the reflected object. Evidently, the vanishing point is not defined in this case. In general, consider the view sphere centered at the object. Let the tails of the normal vectors lie on the sphere and the heads point into the inside of the sphere. The sphere is cut in half by a plane parallel to the imaging plane. The admissible mirror normal directions are on the semi-sphere that is on the opposite side of the plane to the camera. 3 Finding the orientation of the mirror In Section 2, we see that given two pairs of corresponding points (aa and bb ) we can find out the the orientation of the mirror, whose vanishing point is the intersection of the two lines. The problem is how we can find the correspondences. The difficulties arise for mainly two reasons. One is that the two views (the views from the real camera and from the virtual camera) are in general far apart. 3

5 3-D Mirror hull Mirror image Object Mirror A mirror line Figure 3: The concept of a mirror hull. Figure 4: A case that the visibility assumption is not satisfied. Finding correspondences from long baseline images is a known difficult task. The other reason is that for smooth objects with little texture, e.g, the ceramic tea cup in Fig 1, it is nearly impossible to find the correspondences even by hand, let alone automatically by computers. Here we prove that under a very simple assumption, we can find a pair of correspondences from just the silhouettes of the images of the object and its mirror reflection. Let s call the convex hull of the object and its mirror image the 3-D mirror hull. The 3-D mirror hull consists of the convex hulls of the object and its mirror image and a cylindrical surface of mirror lines connecting an object point and its corresponding mirror image (Fig. 3). Our assumption is Assumption (Visibility). The whole 3-D mirror hull is always visible by the stationary camera. The assumption is a rephrasing of the requirement that both the object and its image must be seen in the camera. This is easily satisfied and in fact a prerequisite for a full reconstruction. A case that violates the assumption is shown in Fig 4, where part of the reflected object is not visible from the camera. If we imagine that the mirror-reflected object emits light, some light rays are blocked by the mirror in this case. Go back to the 2-D image (e.g., Fig. 5). We call the region composed by the image of the object points the object region and that of its mirrored points the mirror region. The convex hull of both 4

6 Supporting line a a b Object region Mirror region b Figure 5: Finding the supporting lines from the object region and the mirror region. The regions are segmented from the right most image in Fig. 7(a). regions is the projection of the 3-D mirror hull. The visibility assumption ensures us that we can see the entirety of the convex hull. If the two regions do not overlap, the convex hull includes two straight lines that connect the two regions (Fig. 5). We call the two straight lines supporting line. A more rigorous definition of supporting lines is in [9]. They can be understood in an intuitive way. Given the object region and the mirror region, let a straight line approach the regions from infinity until the line touches both regions. This line is a supporting line. The following theorem tells us that a supporting line provides a correspondence between an object point and its mirrored point. Theorem 1. A supporting line provides a correspondence between an object point and its mirror image. Proof. The visibility assumption guarantees that the convex hull of the object region and the mirror region is the projection of the 3-D mirror hull. So the supporting line is the projection of a 3-D mirror line, which means the supporting line provides the correspondence. The theorem appears to be trivial. But it imposes a surprising constraint on the object region and its mirror region. In Fig. 5, the handle of the tea cup is not seen by the camera but is visible in the mirror image (or by the virtual camera). The image of the handle must be bounded by the supporting lines aa because of the theorem. So the handle cannot be arbitrarily large even if we do not see it in the camera view. Algorithm 1 is used to find the supporting lines. Step 1 of the algorithm is for speeding up the second step and it is not necessary for the correctness of the algorithm. The intersection of two supporting lines gives us the vanishing point of the parallel mirror lines, or the vanishing point of the normal vector of the mirror plane. The natural question to ask is that how many supporting lines we can find from one pair of object and mirror regions. The following theorem tells us that the best we can do is two supporting lines. Theorem 2. Each object-mirror pair provides at most two supporting lines. Proof. By contradiction. Assume there are three or more supporting lines, which are edges of a convex polygon. These supporting lines must intersect at a single point. However, we know that three or more edges of a convex polygon do not intersect at a single point. 5

7 Input: Labeled object region and mirror region, with the number of points in the two regions being N. Output: Supporting lines of the object region and the mirror region. 1. Find boundary points by inspecting the 4-neighbors of each point. If all its 4 neighbors are region points, detect as an inside point and remove it. Otherwise, it s a boundary point. This is an O(N) operation. The resulting number of boundary point is O( N). 2. Use Graham s algorithm [2] to find the convex hull. Graham s algorithm works by maintaining a stack of vertices of the convex hull. The time complexity is O( N log N). 3. For every pair of adjacent points in the stack, if the labels (object or mirror point) are different, the line defined by the two points is a supporting line. This is an O(h) operation, where h is number of vertices of the shadow hull. Algorithm 1: Finding supporting lines. 4 Determining the distance of the mirror We have shown how to compute the orientation of the mirror, which is given by the vanishing point of its normal vector. If we adopt the virtual camera intuition of the mirror, we know the epipole of the virtual camera. In general two-camera stereo, knowing the epipole alone is not enough to determine fully the relationship between the two cameras. But here the virtual camera is the mirror image of the real camera. By reflecting the axes of the real camera according to the mirror, we obtain the pose of the virtual camera with regard to the real camera, except the length of the baseline, which is the line connecting the two cameras. For two-camera stereo, we can set the baseline to unit length and achieve a Euclidean reconstruction up to a scale factor. For multiple-view stereo, however, we have to establish a common scale factor among all the views. In other words, we need to find out the distance between the mirror and the camera (up to a common scale factor). There are two ways to determine this distance. The first is to locate an object point in the camera view and all mirror views. In Fig. 2, let the coordinates of the point A be (x 0, y 0, z 0 ) T. The coordinates of its mirror image A are thus x = x 0 2dn x y = y 0 2dn y, z = z 0 2dn z where d is the distance from the mirror to the camera and n = (n x, n y, n z ) T is the normal vector of the mirror plane. The image of A is a = (u, v) T : u = x 0 z 0 v = y 0, z 0 and the image of A is a = (u, v ) T : u = x 0 2dn x z 0 2dn z v = y 0 2dn. y z 0 2dn z The reason that there is no f in the above equations is because we use the normalized image coordinates [4]. Solving the equations for d, we have d = uz 0 2(u n z n x ), (1) 6

8 O a C1 e C2 A c 1 a Epipolar line C1 c 2 Mirror region a Object region b e c 1 a O (a) (b) Figure 6: Computing the depth map from the mirror view. The left pane is a 2-D image of the object and the mirror reflection. The epipole e is the intersection of two supporting lines. Given a point a in the object region, the epipolar line ea intersects the mirror region at c 1 and c 2. The right pane shows the epipolar plane defined by ea. The 3-D point A whose image is a lies on the half-line Oa and is in between C 1 and C 2, which are computed from c 1 and c 2. where u = u u. The only unknown quantity is z 0. The equation says if we can locate a point A in all the views, we can determine the distances of the mirror by letting z 0 be the unit length. The other way is to mark the mirror. If we place two marker points F 1 and F 2 on the mirror surface, we have F 1 = t 1 f 1 and F 2 = t 2 f 2, where t 1,2 are the distances of the two markers from the camera center and f 1,2 are the unit vectors defined by the image points of the two markers. Because the two markers are on the mirror, we have { t 1 n f 1 + d = 0. t 2 n f 2 + d = 0 Or t 1 = d n f 1 t 2 = d n f 2 Letting the distance between the two markers be L = F 1 F 2, we have L = d f 2 n f 2 f 1 n f 1 = d f 2 n f 2 f 1 n f 1. (2) We can drop the absolute sign because d 0. So if we track the two markers in all the views, we can obtain the distances of the mirror in terms of L. Note that we do not need to know the exact positions of the markers. Nor do we need to know the actually value, say in inches, of L. 7

9 5 Multiple-view 3-D reconstruction Having determined the mirror positions, or equivalently, the poses of all the virtual cameras, we are left with a conventional multiple-view 3-D reconstruction problem. The countless 3-D reconstruction methods can be roughly divided into point-based and silhouette-based. The former compute the 3-D position of individual points by triangulation. The latter are usually some variant of volume intersection. That is, the silhouettes and their associated camera centers extend to form view cones, whose intersection gives the 3-D shape. In theory, we can plug in any reconstruction method. In the following, we show two methods that have been customized to use the fact that each virtual camera is symmetric to the real camera. 5.1 Volume intersection Volume intersection is the most straightforward method if one has a robust CSG (constructive sold geometry) package. We can easily construct a cone whose apex is at the camera center and whose bottom face is defined by the silhouette. We then perform a set intersection operation on all the cones using the CSG package. The result is usually a polygonal mesh, which can be projected back to each view to be textured. Since in our case the virtual cameras are mirror images of the real camera, the cone can be actually constructed in the real camera s view and then be reflected using the plane equation of the mirror. Given a point P = (x, y, z) T and the mirror equation n x + d = 0, its mirror image P is P = P 2(P n + d)n. (3) So after constructing the cone in the real camera s view, we use Eq. 3 to transform the cone into the virtual camera s view and then perform the intersection. 5.2 Depth intersection Implementing a robust CSG package can be very tricky and demanding. The quality of the polygonal mesh degrades rapidly as new views being added. We therefore propose another method that finds the depth of each point of the object region. Let s look at the object region and one mirror region (Fig. 6(a)). Given an object point a, its corresponding mirror point a is on the epipolar line ea and inside the mirror region. But it is generally difficult to identify which point is the real correspondence. Yet we know the it is in between the intersections of the epipolar line and the mirror region. Each such intersection gives us a depth of a along Oa (Fig. 6(b)). The mirror plane is n x + d = 0 and the distance of OO is 2d, O being where the virtual camera is. A is a 3-D point and a is its image in the real camera s view. The points c 1 and c 2 are the intersections of the epipolar line ea with the mirror region (Fig. 6(a)). The corresponding point a of a is in between c 1 and c 2. Or the 3-D point A that we are looking for is in between C 1 and C 2. In OO C 1, let β = C 1 OO and α = C 1 O O, we have cos β = n a cos α = n c 1, where a is the unit vector from O to a and c 1 from O to c 1. The second equation holds because of symmetry ( C 1 O O = C 1 OO ). Now the depth z 1 = OC 1 can be computed by the law of sines in OO C 1 z 1 sin α = 2d sin(π α β) The other depth z 2 = OC 2 is computed similarly. Each pair of the intersections give us a depth range [z 1, z 2 ] of a. If more views are added, the depth ranges computed from individual views are (4) 8

10 intersected. The result is a volumetric representation in the real camera s view. That is, for each point in the object region, there is a depth range. This volumetric representation can be easily converted to a surface representation by the marching-cubes algorithm. 6 Summary of the algorithm Putting all the pieces together, we have the following algorithms. 1. Place the mirror in a position that the visibility assumption is met. Take an image. 2. Segment the object region and mirror region in the image and find the silhouettes. 3. Compute the supporting lines using Algorithm 1. Compute the normal vector n of the mirror from the intersection of the supporting lines. 4. Extract the marker positions. From f 1,f 2 and n, compute the distance d between the mirror and the camera. 5. Extract the contours of the object region and the mirror region. Construct the view cones. Transform the cone of the mirror region using Eq Do the above for all the views. Note that the view cone extended by the object region is constructed only once. 7. Intersect all the cones and we obtain the 3-D shape. 8. Back-project the shape to each view and texture the shape. Algorithm 2: Multiple-view 3-D reconstruction using mirrors (1). If we use the depth map method, the step 5 to step 7 become 5. For each point in the object region, compute the intersections of the epipolar line of the mirror region. Compute the depth range using Eq Do the above for all the views and intersect the depth ranges. 7. Using the marching cubes algorithm to extract the surface determined by the depth map. Algorithm 3: Multiple-view 3-D reconstruction using mirrors (2). 7 Experiments The setup consists of a cardboard box that is covered with a piece of black fabric. Because of the uniform background, a simple background subtraction gives us very good segmentation results. Two markers are sticked on the mirror. PostIt notes make good material for the markers. The markers can be put anywhere on the mirror, as long as they can be seen by the camera. We used circular markers and their centroids as the fiducial points. Although centroids are not projectively invariant, in practice we found no need to correct the error caused by projection. Note that a closed-form correction is possible because the orientation of the mirror is computed without using the markers. 9

11 (a) (b) (c) Figure 7: Reconstruction using volume intersection. (a) The five images used in the reconstruction. (b) The view cones and their intersection. (c) The reconstructed cup. We track the markers and the mirror image of the object when the mirror is moved. For the volume intersection approach, the regions are saved and the view cones are formed and intersected offline. If the depth intersection algorithm is used, the reconstruction is done in near real time on a 2GHz Xeon workstation. The result is a depth map and the marching-cubes algorithm is performed to obtain a surface representation. Fig. 7 shows the reconstruction of a tea cup using volume intersection. Five images, or equivalently six views, are used. Six view cones are generated (Fig. 7(b)). The intersection is shown in (Fig. 7(c)). The reconstructed surface is not smooth because of the small number of views. Fig. 8 shows the result of using the depth map method. Nine images, five of which are shown in Fig. 8(a), are used. The ten contours used to compute the depth map are drawn together in Fig. 8(b). The center contour is from the object directly and the surrounding contours are from its mirror image. Two views of the recovered toy dog are shown in Fig 8(c). 8 Discussion and Future work We have demonstrated a method of 3-D reconstruction using a stationary camera and a planar mirror. No other devices and no calibration are needed. It thus has the potential to reach average consumers. There are two issues with the proposed method. The first is that the orientation of the 10

12 (a) (b) (c) Figure 8: Reconstruction using depth intersection. (a) Five of the nine images used in the reconstruction. (b) The contours drawn in one picture. (c) Two views of the reconstructed toy dog. 11

13 mirror is determined by the intersection of two lines and the computation of the distance relies on the orientation. It requires the estimation of the two supporting lines to be very accurate, which in turn calls for accurate segmentation. Because our setup is a highly controlled environment, we can achieve near perfect segmentation. How to make the orientation estimation more robust is our main future research topic. The second issue is that we need to accommodate the object and its mirror image in one image frame, which lowers the effective resolution of one image. But since 5-mega-pixel cameras are not uncommon these days, it is a lesser issue. It is also worth pointing out that Theorem 1 not only applies to planar mirrors, but also curved mirrors. Of course a curved mirror has more external parameters than a normal vector and a distance. For example, using the theorem, we can recover the pose of a quadric mirror with two cameras. Details can be found in [9]. We recognize that determining the pose of the mirror can be done in other ways. A minimum three points are sufficient to compute the pose of the mirror, the so-called P3P (perspective 3 points [5]) problem, though the solution is not unique and therefore not suitable in an automatic system like ours. If there are four points (P4P) on the mirror, the solution is unique. Besides involving solving non-linear equations, both P3P and P4P methods need to know the exact positions of the markers. Not a tremendous effort even for laymen, but nonetheless a hassle. Moreover, they are not inherently more accurate than the supporting line method. If more markers are available, we can compute the pose more robustly, e.g., using the venerable Tsai s method [15]. But we then have to worry about the occlusion between the markers and the mirror image. The collection of the images of the object and its mirror reflections are themselves an interesting 3-D representation, which we call the mirror-based hologram. New views can be generated directly from the images without explicit 3-D reconstruction. How to store the images efficiently and how to generate new views fast are promising research directions. 12

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