Motion Clustering Using the Trilinear Constraint over Three Views. in Workshop on Geo- Metrical Modeling and Invariants for Computer Vision. Xidian 7 Concluding Remarks a Appendix: the Trilinear Tensor

b) c) d) Figure 5: Reprojection in diierent methods. a) Reprojection using epipolar line intersection. Fundamental Matrices computed with code distributed by INRIA. b) Reprojection using epipolar line intersection. Fundamental Matrices computed from tensor. c) Reprojection using the tensor equations. d) Original third image. Presented for comparison. with the standard summation convention that an index that appears as a subscript and superscript is summed over (known as a contraction). For details on the derivation of this equation see Appendix A. Hence, we have four trilinear equations (note that l; m = 1; 2). In more explicit form, these functions (referred to as \trilineari-ties") are: Since every corresponding triplet p; p 0 ; p 00 contributes four linearly independent equations, then seven corresponding points across the three views uniquely determine (up to scale) the tensor jk i. More details and applications can be found in 15]. Also worth noting is that these trilinear equations are an extension of the three equations derived by 19] under the context of unifying line and point geometry. The connection between the tensor and homography matrices comes from contraction properties described in Section 4, and from the homography matrices one can obtain the \fundamental" matrix F (the tensor produces 18 linear equations of rank 8 for F, for details see 18]). Fig. 5 shows an example of image reprojection (transfer) using the trilinearities, compared to using the epipolar geometry (recovered using INRIA code or using F recovered from the tensor). One can see that the best results are obtained from the trilinearities directly. a) b) c) d) Figure 3: Second sequence-close objects. a) First original frame. b) Second original frame. The camera was moving and rotating around the objects. c) average of the two original images. d) average of the two images after rotation cancelation. The remaining motion is only due to the original translation. A new robust method to recover the rotation of the camera was described. The main contribution to the robustness is the fact that we do not have to recover the epipoles, and the rotation is computed directly from three homography matrices assuming small rotations. The homography matrices are obtained from three images using the trilinear tensor parameters, and the recovery process does not assume any 3D model. The method can be extended to handle also the case of general rotations by using iterations. In this section we brieey present the trilinear …