A Unified View on Deformable Shape Factorizations: Supplemental Material Roland Angst and Marc Pollefeys Computer Vision and Geometry Lab, Department of Computer Science ETH Z¨ urich, Universit¨ atstrasse 6, 8092 Z¨ urich, Switzerland {rangst,marc.pollefeys}@inf.ethz.ch http://www.cvg.ethz.ch/ Abstract. This supplemental material provides additional details and derivations related to the ECCV submission with paper ID 1560 [1]. Sec. 1 presents additional insights related to the basis shape model. These sections are not required for the understanding of the paper. However, additional insights related to the tensor formulation of basis-shape mod- els are given. Sec. 2 describes the steps for the closed-form algorithm in detail. This section does not add any new insight but should facilitate an implementation of this algorithm. The iterative algorithm mentioned in the paper requires the Jacobian of a matrix-valued objective function w.r.t. matrix-valued variables. Sec. 3 provides these Jacobians, render- ing an implementation of the iterative algorithm straight-forward. And lastly, Sec. 4 provides a detailed description of the relation between Za- heer et.al.’s previous work [2] and our method. 1 Basis-Shapes Representation In the upcoming subsections, previously established results are recompiled in tensor algebraic form. These sections will therefore not present new results per se, but rather a new way of deriving and reasoning about them. We think that such a unified presentation of previously dispersed results in a consistent formulation will ultimately lead to a better and clearer understanding. In Sec. 1.1, the non-uniqueness of factorization approaches will be recapitu- lated shortly, while Sec. 1.2 presents a different view on the results of [3]. Finally, Sec. 1.3 gives a generalized view on the widely-used orthogonality constraints. 1.1 Ambiguities In this section, we will look at the ambiguities inherently contained in the low- rank non-rigid trajectories. To start, let us define the two affine transformations for the camera and the structure Q C =[ T C t C 0 1×3 1 ]∈ R 4×4 and Q S =[ T S t S 0 1×d S 1 ]∈ R d S +1×d S +1 . (1)