Physique quantique et representation du monde by Erwin Schrödinger
By Erwin Schrödinger
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This certainly allows us to construct a hierarchical representation of the data f (x). However, this transform is not in general invertible. 9 Hybrid Approaches There is some advantage to be gained by working on image data in the Fourier domain and doing the wavelet operations there before transforming back to the real data space. 10 Multifractals It was remarked that the wavelet transform of f (x), F (a, b), is dominated by cusps that represent the scaling behaviour of f (x) in local neighbourhoods.
H3 h 4 0 0 0 0 0 . . 0 h 1 h 2 ⎤ ⎡ g1 g2 g3 g4 0 0 0 . . 0 0 0 ⎢ 0 0 g1 g2 g3 g4 0 . . 0 0 0 ⎥ ⎥ G(2N ×N )) = ⎢ (99) ⎣. . . . . . . . . . . . . . ⎦ . g3 g4 0 0 0 0 0 . . 0 g1 g2 Note that by virtue of (94) the rows of G are orthogonal to one another and by virtue of (93) the rows of H are orthogonal to one another. The rows of G are also orthogonal to the rows of H. We have orthogonality everywhere and such wavelet transforms are referred to a “bi-orthogonal”. We can usefully block the two matrices G and H together to form a version of W whose rows have been rearranged: W(2N ×2N ) = H .
8) 2 (117) . Fig. 9. Top panel: the original data consisting of three wave packets, the central one being a mixture of the other two. Middle panel: Mexican Hat scalogram. Bottom panel: the Gabor scalogram The Sea of Wavelets 39 The analysing wavelets are the Mexican Hat wavelet and the complex Gabor wavelet. The Mexican Hat used here is ψ(x) = x 2π 1 − 2π 1/2 w w 2 exp −π(x/w)2 , (118) where w is a parameter that is varied to generate the scalogram. The data is transformed for a series of values of w.