Using Pedestrians Walking on Uneven Terrains for Camera Calibration

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1 Machine Vision and Applications manuscript No. (will be inserted by the editor) Using Pedestrians Walking on Uneven Terrains for Camera Calibration Imran N. Junejo Department of Computer Science, University of Sharjah, P.O.Box 27272, Sharjah, U.A.E Received: date / Revised version: date Abstract A calibrated camera is essential for computer vision systems. The prime reason being that such a camera acts as an angle measuring device. Once the camera is calibrated, applications like 3D reconstruction or metrology or other applications requiring real world information from the video sequences can be envisioned. Motivated by this, we address the problem of calibrating multiple cameras, with an overlapping field of view (FoV) observing pedestrians in a scene walking on an uneven terrain. This problem of calibration on an uneven terrain has so far not been addressed in the vision community. We automatically estimate vertical and horizontal vanishing points by observing pedestrians in each camera and use the corresponding vanishing points to estimate the infinite homography existing between the different cameras. This homography provides constraints on intrinsic (or interior) camera parameters while also enabling us to estimate the extrinsic (or exterior) camera parameters. We test the proposed method on real as well as synthetic data, in addition to motion capture dataset and compare our results with the state of the art. 1 Introduction Due to an exponential increase in the computational power of the present day computers, real-time applications of computer vision algorithms has not only been possible, but has acquired a great deal of interest from governments, commercial companies, security agencies and even general public. The area that has attracted the most attention is camera surveillance. Most of the video surveillance systems involves monitoring people (or pedestrians) in a scene. The system can be monitoring, for instance, a building entrance, an airport lobby, stairways, or mall escalators) using stationary or rotating cameras. The goal for such a system can be to model the behavior of objects (e.g. cars or pedestrians, depending on the situation), event reconstruction, or action recognition etc. In this paper, we present a novel method to auto-calibrate each camera in a system of multiple cameras monitoring a particular area of interest by observing only pedestrians in the scene.

2 2 Imran N. Junejo Camera calibration has now become an essential step for any meaningful computer vision based system. For a surveillance system, it is known that due to the effects of perspective projection, the measurements made from the images do not represent metric data. This is evident from a simple observation: the objects grow larger and move faster as they approach the camera center, or two objects moving in parallel direction seem to converge at a point in the image. The projective camera thus makes it difficult to characterize objects - in terms of their sizes, motion characteristics, length ratios and so on - unless more information is available about the camera being Reference object-based calibration: A traditional approach to camera calibration, is performed by observing a calibration object, typically a calibration rig, whose 3D geometry in space is known in advance with a very good precision. The basic idea is to establish correspondences between known points in the world coordinate frame and their respective projection onto the image plane, so that both the internal and external geometry of the imaging system can be determined (cf. Tsai [24]). Recently some research has focused on using a planar pattern for an accurate calibration as well [22,25,26]. The technique requires elaborate setup and expensive calibration apparatus. Auto-calibration: Self-Calibration or Auto Calibration refers to camera calibration from uncalibrated images. Techniques in this category do not require any calibration object and require only the image point correspondence. Just by moving a camera in the static scene, the rigidity of the scene is used to provide constraints on the internal parameters. Since the seminal work of Faugeras et al. [8], auto-calibration has appealed to the vision community due to its simplicity and ease of use [9, 23]. used. Another application for such a surveillance system is making measurements in the image plane i.e. metrology [7]. Measurements like a person s height or true walking speed can be estimated easily once the camera is calibrated [3]. In terms of camera auto-calibration (or self-calibration), these techniques can be classified into two categories: Reviews and surveys of these methods can be found in [11]. More classically, auto-calibration may be performed from sets of the vanishing points corresponding to the directions which are orthogonal in the scene. Caprile and Torre [5] describe a method that requires three vanishing points corresponding to orthogonal direction for camera calibration with known aspect ratio and skew. Objects in the real world typically contain orthogonal directions, for example buildings, giving this method an edge over the object-based calibration approach. Other works on auto-calibration under various configuration include [6], [17], and [4]. The proposed method belongs to this category of solutions. In this paper, we propose a novel solution to the problem of camera calibration when the pedestrians are walking on an uneven terrain. The setup includes multiple cameras looking at the area of interest. We do not restrict pedestrians to walk in a certain manner or with a constant velocity. See Fig. 1 for

3 Using Pedestrians Walking on Uneven Terrains for Camera Calibration 3 an example of the scenario. The detected top (head) and bottom (feet) locations on a person, over at least two instances, are used to estimate the vanishing points in each view. These vanishing points are then used to estimate infinite homography matrix between the cameras. We estimate three camera parameters i.e. the focal length (f), and the principal point (u o, v o ). The noise in data points is minimized by using all the vanishing points obtained from all detected people and all the frames. We demonstrate results on synthetic as well as on the real data. The rest of the paper is organized as follow: Section 2 lays down the founding principles of the proposed method. The method for estimating the vanishing points by observing pedestrians and estimating camera parameters is detailed in Section 3. We rigourously test the proposed method on synthetic data, motion capture dataset and on two real sequences, as described in Section 4 before conclusion. We discuss the related work on camera auto-calibration by observing pedestrians. The first work to deal with camera calibration from pedestrians was that of Lv et al. [18]. They recover vertical vanishing point by observing the scene as well as walking people. The horizon line is estimated by observing the human motion in different directions. However, their formulation does not handle robustness issues. Similarly, Krahnstoever and Mendonça [15,16] proposed a Bayesian approach for auto-calibration by observing pedestrians. Footto-head homology is decomposed to extract the vanishing point and the horizon line for calibration. They also incorporate measurement uncertainties and outlier models. However, their method requires prior knowledge about some unknown calibration parameters and prior knowledge about the location of people; and their algorithm is also non-linear. Recently, Junejo and Foroosh [13, 14] presented two methods for camera calibration from pedestrians. They decompose the 1.1 Related Work Often in real world scenarios, people do not walk on flat surfaces but rather on uneven terrains. For example, people fundamental matrix induced by different instances of a walking pedestrians to impose linear constraints on camera intrinsics [13]. However, all of these methods only apply to people walking on a smooth horizontal surface. on escalators at the airports or at the shopping malls, people climbing stairs, or walking from road to a sidewalk or on the grass etc. Recently, some researchers have addressed 2 Background the problem of calibration from observing pedestrians walking on a flat surface (i.e. ground plane), but they fail to handle this general behavior of the objects. However, we do not restrict pedestrian movements to any particular direction or even require a constant velocity. [ ] T The projection of a 3D scene point X X Y Z 1 onto [ ] T a point in the image plane x x y 1, for a perspective camera can be modeled by the central projection equation:

4 4 Imran N. Junejo Writing P [ p 1 p 2 p 3 p 4 ], the vanishing points are given as: v p 1 and v p 3. Moreover, two families of parallel lines in mutually orthogonal direction intersect on a line H ij p v i,t v p i,t Fig. 1 Setup: Top-Bottom locations of two different instances of a walking pedestrians provides a vanishing point in each observing camera. There also exists a homography H i,j that maps vanishing points between the different cameras. ` called the horizon line. The horizon lines is given by l = p 1 p 2. The aim of camera calibration is to determine the calibration matrix K. Instead of directly determining K, it is common practice [11] to compute the symmetric matrix ω = K T K 1 referred to as Image of the Absolute Conic(IAC). IAC is then decomposed uniquely using the Cholesky Decomposition [21] to obtain K. λf γ u o [ ] x K R RC X, K = 0 f v o }{{} P (1) 3 Method The scenario that we address in this work is when multiple cameras, having an overlapping FoV, observe pedestrians where indicates equality up to a non-zero scale factor [ ] T and C = C x C y C z represents camera center. Here R = ] R x R y R z = [r 1 r 2 r 3 is the rotation matrix and RC is the relative translation between the world origin and the camera center. The upper triangular 3 3 matrix K encodes the five intrinsic camera parameters: focal length f, aspect ratio λ, skew γ and the principal point at (u o, v o ). As argued by [2, 20], it is safe to assume λ = 1 and γ = 0. Image of a family of parallel lines pass through a common point in the image. This point is referred to as the vanishing point. Since the proposed method uses only two vanishing points, without loss of generality, we refer to them as v for the vertical direction direction and v for the x-direction. walking in an area of interest, indoors or outdoors. Although the method applies to n number of cameras, for sake of simplicity, we consider the case of two cameras i and j from here on. The various steps involved in the proposed method are: (1) foreground object extraction, (2) estimation of the top and the bottom locations, (3) estimation of vanishing points obtained from observing a pedestrian, (4) estimating the infinite homography matrix between different cameras, and (5) camera calibration, i.e. estimating the intrinsic and extrinsic camera parameters. A pedestrian needs to be detected and tracked in the video sequence. Note that we are not solving the background sub-

5 Using Pedestrians Walking on Uneven Terrains for Camera Calibration 5 traction and tracking problem, and therefore use one of the well know methods for performing this task [12]. time, i denotes camera i, and p = 1, 2,..., k denotes label of a person in the scene. Similarly, at any two time instances during the walk of a person, the line joining the head to the 3.1 Estimating Top and Bottom Locations We do require that top and bottom locations be correctly detected from the tracked pedestrians. In this regard, we adapt the approach proposed by [13] and [15]. Whereas Lv et al. [18] performs eigendecomposition of the detected blob to extract top and bottom locations, we calculate the center of mass and the second order moment of the lower and the upper portion of the bounding box of the foreground region to detect these points for each instance of a walking person seen from each camera (cf. Fig. 1). For a camera i at time instance t, we denote the detected top point for a pedestrian p as T p i,t and similarly the bottom point as B p i,t. 3.2 Estimating The Infinite Homography Under projective transformation, parallel lines in the world intersect at a point on the image plane, called the vanishing point. While observing a pedestrian of height h at different time instances in a camera, and assuming that h does not change between different instances: if we draw a line between the head (or the top) position of a pedestrian at these two time instances then this line is parallel to the line obtained by joining the feet (or bottom) positions at these same corresponding time instances. This is shown in Fig. 1. Let the intersection of these parallel lines be projected in camera i as: v p i,t = T p i,t T p i,t+1 Bp i,t Bp i,t+1, where t = 1, 2,..., denotes feet at time instance t should be parallel to the line joining the head to feet in time instance t + 1. These lines intersect at another vanishing point lying on the plane at infinity, as: v p i,t = T p i,t Bp i,t T p i,t+1 Bp i,t+1. All these vanishing points lie on the plane at infinity. In the situation where people are walking only on a flat surface, as in [13, 15] or [18], the obtained vanishing points v and v are orthogonal to each other and satisfy the polepolar relationship with respect to ω. However, this is not the case when the pedestrians are walking on an uneven terrain and hence we cannot use these methods to obtain any constraints on the unknown parameters. We overcome this problem by estimating the infinite homography between different cameras viewing the scene. Consider two cameras i and j observing walking pedestrians. The mapping of points from camera i to camera j over the plane at infinity π is given by H i,j = K j R i,j K 1 i, (2) where R i,j is the relative rotation between the cameras and H i,j, the infinite homography, maps the points lying on the plane at infinity in camera i to their corresponding points lying on the plane at infinity in camera j. In practice, H i,j is often estimated between two images by matching corresponding points. Since H i,j has eight degrees of freedom, a minimum of four point correspondences are necessary to com-

6 6 Imran N. Junejo pute it [11]. However, for a robust solution to H i,j, we use all the estimated vanishing points extracted from the observed pedestrians. Once these vanishing points are estimated, they 18, 20], it is safe to assume that the aspect ratio λ = 1 and also the skew γ = 0. This gives us two linear constraints on the unknown parameters: satisfy: [ ] [ ] v p j,t v p j,t... = H i,j v p i,t v p i,t... (3) c 1,2 = 0 (7) c 1,1 = c 2,2 (8) which relates the obtained vanishing points in camera i to their corresponding vanishing points in camera j. Once sufficient number of such vanishing points are obtained, H i,j is estimated by the DLT (Direct Linear Transform) algorithm [11]. 3.3 Camera Calibration Using the property R i,j = R T i,j, we transform (2) to: ω j = H T i,j ω ih i,j 1 where ω j is the IAC for camera j having the form: 1 τ 0 uo τ ω j λ2 τ λ2 v o τ 1 where represent duplicate symmetric values, τ = ( λ 2 f λ 2 v o 2 + u o 2 ), and f 2 is the focal length of camera 2. by: (4) (5) Due to the symmetry of ω j, (4) can be simply represented ω 1,1 ω 1,2 ω 1,3 ω 2,2 ω 2,3 1 j c 1,1 c 1,2 c 1,3 = c 2,2 c 2,3 1 where * again indicates symmetric values, and c a,b contain the unknown parameters of the RHS. As argued by [2,13,15, i (6) corresponding to ω 1,2 = 0 (from γ = 0) and ω 1,1 = ω 2,2 (from λ = 1), respectively. Using (7) and (8) we express ω 1,3 and ω 2,3 in terms of ω 1,1. Here we introduce a cost function on the algebraic distance of the principal point from the center of the image (I x, I y ), which gives an extra weak constraint on ω: [u o v o ] = arg min ( ω 1,3 ω 1,1 + I x ) 2 + ( ω 2,3 ω 2,2 + I y ) 2 (9) We solve this constraint by an application of Levenberg- Marquardt algorithm [21]. By substituting ω 1,3 and ω 2,3 obtained from solving (7) and (8) into (9) and minimizing it, ω 1,1 can be estimated, which in turn determines ω 1,3 and ω 2,3 (note that ω 1,1 = ω 2,2 ). The relative rotation between the two cameras is obtained by R i,j = K 1 j H i,j K i. Given two vanishing points v and v from each view of a single camera, the absolute rotation of camera i (i.e. its extrinsic parameters) with respect to a common world coordinate system can be computed as: r 3 = ± K 1 i v K 1 i 1 i v K 1 v, r 1 = ± K i v, r 2 = r 3 r 1 r 3 r 1, (10) where r 1, r 2 and r 3 represent three columns of the rotation matrix. The sign ambiguity can be resolved by the cheiral-

7 Using Pedestrians Walking on Uneven Terrains for Camera Calibration 7 ity constraint [11] or by known world information, like the maximum rotation possible for the camera. 4 Results Relative Error (%) Relative Error in f Noise Level (pixels) Relative Error (%) Error in f Noise Level (pixels) In this section, we show an extensive set of evaluations on both synthetic and real data to test the proposed solutions and compare with the state of the art. To simulate multiple Relative Error (%) (a) Error in u o Error in v o Absolute Error in Angle R x R y R z (b) and controlled view settings, in addition to the real dataset, we have used 3D motion capture data from CMU dataset (mocap.cs.cmu.edu). 4.1 Synthetic data Noise Level (pixels) (c) Frobenius Norm ( R true R est ) Noise Level (Pixels) Frobenius Norm (d) We performed detailed experimentation on the effect of noise on camera parameter estimation. In order to preform this simulation, we randomly generated 50 vertical synthetic objects in the 3D synthetic world. These vertical objects were generated on an uneven terrain i.e. care was taken so that z-axis values are not same but rather are all randomly generated. Once these objects are constructed, we project the 3D data on two views and compute the unknown camera parameters. For the first camera, we set the parameters: principal point at Noise Level (pixels) (e) Fig. 2 Performance of auto-calibration method VS. Noise level in pixels. (a) and (b) show the relative error for the estimated focal length in camera 1 and camera 2, respectively. (c) depicts the error rate in computing the principal point (u o, v o). The extrinsic parameters are also computed and (d) depicts the averaged absolute error obtained per noise level. (e) shows the Frobenius norm obtained between the estimated relative orientations and the ground-truth relative orientations. (u o, v o ) = (320, 240) for an image size of , focal length f 1 = 1000, the skew γ = 0, and the aspect ratio as λ = 1. We subjectively rotate this camera around the vertical object at a different location with relative rotation angles (R x, R y, R z ) = (20, 32, 82 ). Also, the focal length for the second camera was set to be f 2 = In order to simulate practical situations where image points are generally affected by the noise in camera, tracking errors, or the image sampling, we gradually add a Gaussian noise with µ = 0 and σ 1.5 pixels to the data-points making up the vertical lines and 1000 trials per noise level were then

8 8 Imran N. Junejo performed. Taking two vertical lines at a time, the vertical vanishing point v is obtained by intersecting the two lines the Frobenius norm, shown in Fig. 2(e). As can be seen in the figure, the error rate increases linearly as well. that are obtained by joining the head and bottom points of the observations, as described in Section 3. Similarly, H i,j is recovered by intersecting the two lines obtained by joining the heads locations and the bottom locations of the two observations, respectively. Once enough vanishing points are obtained (at least 4), the infinite homography is estimated [11] by direct linear transformation algorithm. The image points are normalized first so that their centroid is at the origin and their mean distance from the origin in 2 [10]. Error in the estimated calibration parameters is shown in Fig. 2. The y-axis for Fig. 2(a)-(c) depicts the relative error (in percentage) with respect to the focal length, as argued by [23,25]. For a maximum noise level of 1.5 pixels [25], the error rate for f 1 and f 2 is less than 1%. Similarly the error rate for the estimated principal point is found to be less than 0.4%. The curves show that the relative error increases linearly as a function of the increasing noise. Note that the downward trend in the error curves is due to minimizing (9) in the vicinity of the image center. In addition to the camera intrinsic parameters, the relative orientations angles are also estimated, as shown in Fig. 2(d). The red curve shows the absolute error in the estimated relative rotation angle around the x-axis, green curve as the relative rotation error around the y-axis, and blue as the relative rotation around the z-axis, respectively. In order to measure the accuracy of the computed relative rotation matrix between the different viewpoints of the cameras, we compute 4.2 Motion Capture (mocap) data To simulate multiple and controlled view settings we have used 3D motion capture data from CMU dataset. Trajectories of 3 points on the body are projected on two different views with pre-defined orientation w.r.t. to the human body. The camera intrinsic parameters for the synthetic camera are set to f = 1000, λ = 1, γ = 0, and (u o, v o ) = (320, 240). The setup for this process is shown in Fig. 3(a). One point is the head, and the other two points are the left and the right foot. From these settings, we obtain the top and bottom points as required for our method as follows: we make the observation head-lfoot when the left foot is in line with the head of the person, and another observation as head-rfoot when the right foot is in line with the head location [19]. All these corresponding points are collectively used in estimation of the camera parameters. Some of the sample images from the dataset are shown in the Fig. 3(b)-(e) and their synthetic reconstruction is shown in Fig. 3(f)-(i). In the sequence, a person is walking on an uneven surfaces made from cardboard stacked on the ground. Using the proposed method, the estimated parameters are: f = , and (u o, v o ) = (336.06, ). Even though we do get the exact location of the head and the feet locations (as this is a mocap data), the error arises due to the movement of the head and also at some instances the non-verticalness of the body, as can be seen in Fig. 3(d)(e). However, even in presence of such noise, the

9 Using Pedestrians Walking on Uneven Terrains for Camera Calibration 9 cam6 cam4 cam3 cam1 cam2 cam5 cam1 (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. 3 Experiments with motion capture data: (a) shows our setup for the experiment, two synthetic cameras looking at the person performing the action of walking on a uneven surface. (b)-(e) shows different instances from the sequence. (f)-(i) shows the synthetic reconstruction of the sequence. error is within acceptable limits and very close to the groundtruth values mentioned above. 4.3 Real data We test on two sequences of real data. The first data was captured from cameras having an image resolution of We employed two such cameras looking at a scene with a number of people walking on a uneven/slanted ground plane. Fig. 4 shows two synchronized instances from the dataset consisting of more than 6000 frames. Each column of the figure represents a camera. Walking people are extracted, by applying the background subtraction method and the head/feet locations are determined. The top and the bottom points are marked with a red circle in the figure. The homography was extracted after estimating the vanishing points from the moving pedestrians, as described in Section 3. The extracted intrinsic parameter for camera one are: K 1 =

10 10 Imran N. Junejo by applying the classic camera calibration method of [24]. In comparison to their computed parameters, f = and (u o, v o ) = (376.05, ), we obtained f = and (a) (b) (u o, v o ) = (388, 284). The results obtained from our method are very closed to the given solution. 5 Conclusion (c) (d) Fig. 4 Instances from real data sequence: Each column represents a unique camera when a number of people walk on an uneven surface. The head/bottom points are estimated from the background subtraction method. See text for more details. Real world scenarios involve people walking on uneven terrains. We propose a novel method to obtain camera intrinsic and extrinsic parameters for such scenarios when multiple cameras are looking at the area of interest. To the best of our knowledge, no work exists that deals with camera calibration for this specific scenario. Thus, this method can be very where as the focal length for the second camera is found to be f 2 = Note that the estimated principal point is useful for many of the existing multi-camera video surveillance systems that observe people indoors or outdoors. In the very close to the center of the image. Also note that f 1 and f 2 are very close to each other - indicating, qualitatively, the correctness of the estimated parameters. Another test sequence was obtained from the ETISEO dataset [1]. We test on the sequence ETI-VS2-BE-19. The dataset contains multiple cameras installed at an entrance of a building. Some samples from this dataset are shown in Fig. 5. Each row in the figure represents different views of the scene. In the sequence shown, a person comes out of the building holding a package in his hand and descends the stairs and then continues walking on the road. In addition, a car arrives and parks in the parking lot and the driver walks out. In summary, this is a very interesting and relevant sequence. The dataset also contains the camera parameters estimated proposed method, tracking each walking pedestrian between the frames enables us to obtain horizontal and vertical vanishing points in each camera view. These corresponding vanishing points are then used to estimate the infinite homography that exists between any two cameras viewing the scene. We present novel constraints to solve for three of the camera intrinsic as well as the extrinsic parameters. We put the proposed method to rigorous tests on synthetic data in the presence of considerable noise. We show results and experimentation on motion capture dataset. We also test the method on two real data sequences of multiple cameras observing pedestrians and compare our results with the state-of-the-art calibration method. The encouraging results demonstrate the practicality and the utility of the proposed method.

11 Using Pedestrians Walking on Uneven Terrains for Camera Calibration 11 Camera #2 Camera #1 Fig. 5 ETISEO dataset: A person walks out of the building and descends down the stairs and continues walking on the road. The sequences is view from multiple views, and each of the two rows shows different vies of the scene. A tracked person is show by a blue surrounding rectangle. References 1. Video understanding evaluation project: L. D. Agapito, E. Hayman, and I. Reid. Self-calibration of rotating and zooming cameras. Int. J. Comput. Vision, 45(2): , X. Cao and H. Foroosh. Metrology from vertical objects. Proc. of BMVC, X. Cao and H. Foroosh. Simple calibration without metric information using an isosceles trapezoid. Proc. ICPR, pages , B. Caprile and V. Torre. Using vanishing points for camera calibration. Int. J. Comput. Vision, 4(2): , R. Cipolla, T. Drummond, and D. Robertson. Camera calibration from vanishing points in images of architectural scenes. Proc. of BMVC, pages , A. Criminisi, I. Reid, and A. Zisserman. Single view metrology. Int. J. Comput. Vision, 40(2): , O. Faugeras, T. Luong, and S. Maybank. Camera selfcalibration: theory and experiments. In Proc. of ECCV, pages , R. I. Hartley. Self-calibration from multiple views with a rotating camera. In Proc. ECCV, pages , R. I. Hartley. In defense of the eight-point algorithm. IEEE Trans. Pattern Anal. Mach. Intell., 19(6): , R. I. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, ISBN: , second edition, O. Javed and M. Shah. Tracking and object classification for automated surveillance. In the seventh European Conference on Computer Vision (ECCV), I. Junejo and H. Foroosh. Trajectory rectification and path modeling for video surveillance. In Eleventh IEEE International Conference on Computer Vision (ICCV), I. Junejo and H. Foroosh. Euclidean path modeling for video surveillance. Elsevier Journal of Image and Vision Computing (IVC), 26(4): , N. Krahnstoever and P. R. S. Mendonca. Bayesian autocalibration for surveillance. Tenth IEEE International Conference on Computer Vision, 2005.

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