Parabolic Curve fitting as an 3D Trajectory Estimation of the Soccer ...

Figure 2: Extracting ball candidate blobs. n throw corresponds to n th camera. Left, middle andright columns show original, foreground and ballcandidates images respectively. Ball positions aremanually marked for readers better understanding.3 Ball Trajectory EstimationBall tracking is also a process of filtering one mostlikely sequence of 3D positions out of a sequence ofnoisy multi-view 2D observation sets. Before goingfurther let us define some sets.Q ∗ 1:T = {q ∗ t } T t=1(1)S = {Q t1 :t 2|1 ≤ t 1 ≤ t 2 ≤ T } (2){ {q }i It TU = t i=1}(3)t=1{ { } NP = {p n t (h)} H nh=1(4)n=1} Tt=1where p and q denote points in 2D and 3D respectively,N is the number of cameras and H is thenumber of observations. Q ∗ 1:T is the optimal trajectory,S is a set of 3D trajectory segments, U isa set of 3D ball candidates and P is a sequence ofnoisy multi-view 2D observation sets. As in reverseorder, U is built from P , then S is from U as wellas Q ∗ 1:T is extracted from S.To build U from P , for each time t, 3D ball candidatesare generated from all the possible pairs ( )C2Nof synchronized N views. Given the camera parameters,a point, for example the one q g on the pitchground, on a ray from the camera center, q c areprojected on a point on the image. Ideally if thereexists a 3D object(as a point) and a pair of cameraproject it, the rays from each camera center tothe projected point meet at the 3D point. However,due to some noise, the rays may not meet each otherposing some distance. If the distance is tolerable,the mid-point between the rays is taken as a 3Dball candidate. The mid-point q mid between tworays passing through two points q1 c and q g 1 , and qc 2and q g 2 respectively is computed as following.where⎡A =⎢⎣q1,2 mid = qclosest1 + q2closest]2(5)= A −1 B (6)[ qclosest1q closest2q1 c (z) 0 µ g,c1 (z) 0 0 00 q1 c (z) µ g,c1 (z) 0 0 00 0 0 q2 c (z) 0 µ g,c2 (x)0 0 0 0 q2 c (z) µ g,c2 (y)µ c,g1 (x) µ c,g1 (y) µ c,g1 (z) µ g,c1 (x) µ g,c1 (y) µ g,c1 (z)µ c,g2 (x) µ c,g2 (y) µ c,g2 (z) µ g,c2 (x) µ g,c2 (y) µ g,c2 (z)(7)B = (q g 1 (x) qc 1 (z) , q g 1 (y) qc 1 (z) , q g 2 (x) qc 2 (z) , q g 2 (y) qc 2 (z) , 0, 0) T(8)q1 closest and q2 closest are the points on the two raysclosest to each other and µ a,bi = qi a − qb i . Then U isthe set of mid-points :{ { ()} } N TU = qm midN (n),m N (n+1) p j m N (n) (t) , pk m N (n+1) (t) n=1t=1(9)where 1 ≤ j ≤ Hm t N (n) , 1 ≤ k ≤ Ht m N (n+1) andm N (n) = MAX (mod (n, N + 1) , 1).From U, S is built by extending all the possibletriplets, three consecutive 3D ball candidates,as long as possible to give trajectory segment candidates.A sequence of three ball candidates in U isqualified to be a triplet if their acceleration and velocitiesshow that of ballistic motion under gravity:S =whereβ t =δ t =ϕ t ={}{q t } t 2t=t 1|β t δ t δ t+1 ϕ t > 0, ∀t : t 1 ≤ t ≤ t 2 − 2(10){ 1 Tlβ< q t+2 (z) − 2q t+1 (z) + q t (z) < Tβ(11)u0 otherwise{1 Tlδ< ‖q t+1 − q t ‖ 2< Tδu (12)0 otherwise{ (1 cos −1 (qt+1−q t)·(q t+2−q t+1)‖q t+1−q t‖ 2 ‖q t+2−q t+1‖< T ϕ2)2 (13)0 otherwiseTo get Q ∗ 1:T from S, the longest one among the segmentcandidates is chosen and fitted to a parabolic⎤⎥⎦