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Factorization - The rigid case |
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The factorization method decomposes a generic matrix W in the product of two
lower rank matrices M (Motion matrix) and S (Structure matrix). The decomposition is performed by Singular
Value Decomposition (SVD) maintaining the first k most significant singular values where k = rank(W). |
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W = MS |
| Tomasi&Kanade exploited the rank property of a measurement matrix W containing image point tracks of an image sequence to infer the geometric structure of a rigid object under orthographic viewing condition. Under these assumptions the overall rank of W is close to 3 (see the related papers for details). The motion matrix M, after a proper affine correction, is composed by projection matrices for each frame while the structure matrix S represents the points 3-D coordinates. |
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The algorithm can infer the 3-D location of a small set of image points tracked for few frames.
A video (avi, 1.42MB) shows the
reconstruction preserving the geometric properties of the rigid object.
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Factorization - The non-rigid case |
| The rigidity constraint allows to describe a shape by a set of 3-D points and by a related projection matrix for each frame. A shape that can deforme/evolve in time requires a set of basis-shapes that can describe properly the shape modes of deformation. The BHB method (Bregler, Hertzmann, Biermann - see details in the references on factorization) infers the basis-shapes considering a rank k higher than 3. The algorithm performs a linear combination of the basis-shapes to obtain the reconstructed shape for each frame. |
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The algorithm can infer the 3-D shape of the subject face from a set of reliable tracks (avi, 1.06MB) using markers.
A video (avi, 1.93MB) shows the reconstruction
of the non-rigid object.
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References on factorization |
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C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: a factorization method. International
Journal of Computer Vision, vol. 9, no. 2, pages 137-154, November 1992. C. Bregler, A. Hertzmann, H.Biermann. Recovering Non-Rigid 3D Shape from Image Streams. IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), June 2000. M. E. Brand. Morphable 3D Models from Video. IEEE Computer Vision and Pattern Recognition (CVPR), December 2001. |
| Updated Aug 2006 by Alessio Del Bue |