By Erik Jonsson
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Additional info for Channel-coded feature maps for computer vision and machine learning
3, this formulation will be used for tracking, and in Chapter 11, this will be one component of a view-based pose estimation method. In order to minimize E with a local optimization, we need the derivatives of f with respect to the similarity parameters. 3). We focus on a ˜ . 2 Derivatives of Channel Coded Feature Maps 31 compact, we define ˜ ) = B(u − u h(x) = B(x − x ˜ , z(u) − ˜ z) . 4) Furthermore, we ignore the weight function w(u) for a moment. 3) as c= h(u, z(u)) du . 5) Since the expressions get rather lengthy anyway, this will be more convenient to work with.
FN +1 }, which can easiest be seen by decomposing p∗ into p∗ = 22 Channel-Coded Scalars 2 q1 + q2 , where q1 ∈ Q1 and q2 ∈ Q⊥ = q1 2 + q2 2 . 1 . Since q1 ⊥ q2 , we have p∗ But q2 ⊥ fn for all feature functions fn , so q2 does not affect the constraints and must be zero in order to minimize p∗ 2 . Hence p∗ = q1 ∈ Q1 , which implies that p∗ can be written as p∗ (x) = αn fn (x) . 41) k giving the αn ’s as a solution of a linear system of equations. 42) where α = [α1 , . . , αN +1 ]T and Φ is the Gram matrix Φ = [ fi , fj ]ij .
For each spatial channel, there is a soft histogram of chromacity and orientation, giving in total a 4D histogram. orientation histograms and downsampled color images are constructed efficiently by box filters using integral images . This is possible since rotation of the tracked object is not considered. g. in ), the descriptors are computed at fixed positions without attempting to fine-tune the similarity parameters. The channel-coded feature maps generalize all these approaches, allow for arbitrary basis functions, and support derivatives with respect to rotation, translation and scale changes.
Channel-coded feature maps for computer vision and machine learning by Erik Jonsson