Neural network and wavelet transform for scale-invariant data classification.

Abstract

Given an astrophysical observation with an arbitrary carrier frequency and an unknown scale under an additive white noise, s'(t)\ensuremath{\equiv}s(\ensuremath{\alpha}t)+n(t), its wavelet transform is W'(a,b)\ensuremath{\equiv}(s'(t),${\mathit{h}}_{\mathit{a}\mathit{b}}$(t)), as computed by the inner product with a daughter wavelet ${\mathit{h}}_{\mathit{a}\mathit{b}}$(t)\ensuremath{\equiv}h((t-b)/a)/a. W'(a,b) equals the original transform W(a,b)\ensuremath{\equiv}(s(t),${\mathit{h}}_{\mathit{a}\mathit{b}}$(t)) displaced along the radial direction W'(a,b)=W(\ensuremath{\alpha}a,\ensuremath{\alpha}b) plus noise in the time-scale joint-representation plane. A bank of wedge-shaped detectors collects those displaced transforms W'(a,b) to create a set of invariant features. These features are fed into a two-layer feed-forward artificial neural network, to interpolate discrete sampling, as demonstrated successfully for real-time-signal automatic classification. Useful wavelet applications in turbulence onset, spectrum analyses, fractal aggregates, and bubble-chamber particle-track pattern-recognition problems are indicated but are modeled, in the interest of simplicity, in a one-dimensional example.