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Furthermore, we let u = [0, 0]T be the center of the encoded image or patch (see Fig. 2). 2: Left: Intrinsic channel coordinate system. The dots indicate channel centers. Right: Similarity parameters governing the location of the patch in the image. now a multi-dimensional array c[˜ x] = 1 I ˜) = wi B(xi − x i 1 I wi B(ui − u ˜ , z(ui ) − ˜ z) . g. homogeneous regions get a low weight since the orientation estimates in these regions are unreliable. When working with derivatives with respect to image transformations, it will be more convenient to use a continuous formulation.

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 . Note that since Φ is independent of our feature values d, it can be computed analytically and inverted once and for all for a specific problem class.

13) then generalize to c{f } = 1 I f (xi − ·) = i K{f }[p, ·] ∗ cmp [p, ·] . 15) p Even though the notation is non-standard and may look confusing, this single equation compactly summarizes more or less everything said so far in this chapter. It is important to fully understand this equation before proceeding into the next section. 26), it is immediately clear that dc{B}[˜ x] = − c{Bu }[˜ x] dbu dc{B}[˜ x] = − c{Bv }[˜ x] . 17) These can be computed by Alg. 1 using K{Bu } and K{Bv }. The derivatives with respect to rotation and scale are more complicated.

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