Computer Vision News - February 2017

Computer Vision News Computer Vision News Research 33 Research Riesz Transform: the Riesz transform is a generalization of the one-dimensional Hilbert transform, in other words this can be thought of as a steerable Hilbert transformer that enables us to compute a quadrature pair of a non-oriented image sub-band that is 90 degrees phase-shifted with respect to the dominant orientation at every point. This generalization allows for phase analysis of 2D non-oriented images. Mathematically, the Riesz transform in 2D is a pair of filters (denoted by R 1 , R 2 ) with functions: (1) To reduce run-time the paper proposes approximating the Riesz transform with the three tap finite difference filters [0.5, 0, −0.5] and [0.5, 0, −0.5]T . These filters have frequency response: (2) where For phase-based magnification, we need to compute the phase at every point and at every scale (sub-band). For breaking the image into sub-bands corresponding to different scales and orientations, an invertible filter bank is used; this forms the real part of the pyramid. To form the imaginary part of the pyramid, the Hilbert transform is used, along with the dominant orientation at every scale. The method approximates the Riesz pyramid only along the main orientation. The input I together with Riesz transform R 1 , R 2 form a triplet that can be converted to spherical coordinates to yield the local amplitude A, local orientation θ, local phase φ, and the quadrative pair Q. The exact derivation with all the details can be found in the paper. The pseudocode implementation will be presented further on. Method: The input to this method is the original video. The output is the motion magnified video. The figure in the following page illustrates the steps of the process. , − − sin ≈ − , − sin ≈ − , ≈ 2 .