Threshold selection for de-noising
THR = thselect(X,TPTR)
thselect is a one-dimensional
de-noising oriented function.
THR = thselect(X,TPTR) returns threshold
value using selection rule defined by character vector
Available selection rules are
TPTR = 'rigrsure', adaptive threshold
selection using principle of Stein's Unbiased Risk Estimate.
TPTR = 'heursure', heuristic variant
of the first option.
TPTR = 'sqtwolog', threshold is
TPTR = 'minimaxi', minimax thresholding.
Threshold selection rules are based on the underlying model y = f(t)
+ e where e is a white noise N(0,1).
Dealing with unscaled or nonwhite noise can be handled using rescaling
wden for more information).
Available options are
tptr = 'rigrsure' uses for the
soft threshold estimator, a threshold selection rule based on Stein’s
Unbiased Estimate of Risk (quadratic loss function). One gets an estimate
of the risk for a particular threshold value (t).
Minimizing the risks in (t) gives a selection of
the threshold value.
tptr = 'sqtwolog' uses a fixed-form
threshold yielding minimax performance multiplied by a small factor
tptr = 'heursure' is a mixture
of the two previous options. As a result, if the signal to noise ratio
is very small, the SURE estimate is very noisy. If such a situation
is detected, the fixed form threshold is used.
tptr = 'minimaxi' uses a fixed
threshold chosen to yield minimax performance for mean square error
against an ideal procedure. The minimax principle is used in statistics
in order to design estimators. Since the de-noised signal can be assimilated
to the estimator of the unknown regression function, the minimax estimator
is the one that realizes the minimum of the maximum mean square error
obtained for the worst function in a given set.
% The current extension mode is zero-padding (see
dwtmode). % Generate Gaussian white noise. x = randn(1,1000); % Find threshold for each selection rule. % Adaptive threshold using SURE. thr = thselect(x,'rigrsure') % Fixed form threshold. thr = thselect(x,'sqtwolog') % Heuristic variant of the first option. thr = thselect(x,'heursure') % Minimax threshold. thr = thselect(x,'minimaxi')
Donoho, D.L. (1993), “Progress in wavelet analysis and WVD: a ten minute tour,” in Progress in wavelet analysis and applications, Y. Meyer, S. Roques, pp. 109–128. Frontières Ed.
Donoho, D.L., I.M. Johnstone (1994), “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol 81, pp. 425–455.
Donoho, D.L. (1995), “De-noising by soft-thresholding,” IEEE Trans. on Inf. Theory, 41, 3, pp. 613–627.