Ideal Denoising in an orthonormal basis chosen from a library of bases

Abstract

Suppose we have observations y i = s i +z i , i = 1; :::; n, where (s i ) is signal and (z i ) is i.i.d. Gaussian white noise. Suppose we have available a library L of orthogonal bases, such as the Wavelet Packet bases or the Cosine Packet bases of Coifman and Meyer. We wish to select, adaptively based on the noisy data (y i ), a basis in which best to recover the signal ("de-noising"). Let M n be the total number of distinct vectors occcuring among all bases in the library and let t n = p 2 log(M n ). (For wavelet packets, M n = n log 2 (n).) Let y[B] denote the original data y transformed into the Basis B. Choose ? 8 and set n = ( \\Delta (1 + t n )) 2 . Define the entropy functional E (y; B) = X i min(y 2 i [B]; 2 n ): Let B be the best orthogonal basis according to this entropy: B = arg min B2L E (y; B): Define the hard-threshold nonlinearity j t (y) = y1 fjyj?tg . In the empirical best basis, apply hard-thresholding with threshold t = p n : s i [ B] = j ...