Active Learning in an Information-rich Age Early 20 th century, how do you study physics? Early 21 st century, how do you study wavelets?

Use Tools Like Caveman Did Publish or Perish (a good tool for scientific research)

Small-World Phenomenon Donoho Johnstone Coifman Vertteli DeVore Daubechies Sweldens Mallat Mayer Morlet, Grossmann Vaidyanathan Schröder

Signal Denoising Study by Toy Examples taken from Donoho’s paper “Translation-invariant Denoising”

Four Classes of Signals

Noisy Observations Y=X+W W~N(0,1)

An Easy Attack Use a Low-Pass filter such as moving averaging Why low-pass? What if noise is periodic? How far are we from the optimal performance?

A Different Class of Signal

A Different Class of Noise ECG signal

Signal in Nature Self-similar property (Global) –1/f noise is a signal or process with a frequency spectrum such that the power spectral density is proportional to the reciprocal of the frequency Transient/nonstationary property (Local) –Changes in space and time –Motivate the study of wavelet transforms

Noise in Nature Always Gaussian? Stationary or transient? Random or structural (perceptually meaningful)?