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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This work presents a method to efficiently train object detectors in the presence of geometric transformations that can be represented as vector-matrix multiplications. This has recently been developed for the case of translation transformations ([1,8,14]) but has not been too obvious for other transformations, such as rotations, let alone non-rigid deformations. The authors propose to adapt the Fourier-based method that originally allowed the development of efficient algorithms for the case of translations so that we can now deal with rotations and other `cyclic' signal transformations (e.g. the walking pattern of a pedestrian). The condition for this to hold is that the transformation is norm-preserving, can be represented as a matrix multiplication, x_transformed = Q x, has an inverse Q^{-1} = Q^T and for some s Q^s = I. The starting point for the previous works was the fact that the'data matrix' obtained by'stacking' together all translated versions of a signal is a circulant NxN matrix, where N is the length of the signal, and as such can be diagonalized using the discrete harmonic basis (or, Discrete Fourier Transform-DFT matrix), Eq. 3, and reference [5].