gasper (version 1.1.6)

LD_SUREthresh: Level Dependent Stein's Unbiased Risk Estimate Thresholding

Description

Adaptive threshold selection using the Level Dependent Stein's Unbiased Risk Estimate.

Usage

LD_SUREthresh(
  J,
  wcn,
  diagWWt,
  beta = 2,
  sigma,
  hatsigma = NA,
  policy = "uniform",
  keepSURE = FALSE
)

Value

A list containing the wavelet coefficient estimates after applying the SURE thresholding.

• wcLDSURE: The wavelet coefficient estimates obtained by minimizing SURE.

• wcLDhatSURE: If hatsigma is provided, this component contains the wavelet coefficient estimates obtained using the hatsigma estimator.

• lev_thresh: If keepSURE is TRUE, this component contains a list of results similar to the output of SUREthresh for each scale.

Arguments

J

Integer. The finest scale, or the highest frequency. This parameter determines the total number of scales that the function will process.

wcn

A numeric vector of noisy spectral graph wavelet coefficients that need to be thresholded.

diagWWt

Numeric vector of weights.

beta

Numeric. The type of thresholding to be used. If beta=1, soft thresholding is applied. If beta=2, James-Stein thresholding is applied (Default is 2).

sigma

Numeric. The standard deviation of the noise present in the wavelet coefficients.

hatsigma

Numeric. An optional estimator of the noise standard deviation. If provided, the function will also compute wavelet coefficient estimates using this estimator.

policy

The policy for threshold setting. It can be either "uniform" (default) or "dependent".

keepSURE

A logical flag. If TRUE, the function will also return a list containing the results of the SURE thresholding for each scale.

Details

This function applies SURE in a level dependent manner to wavelet coefficients, which aims to minimize SURE at each wavelet scale.

In the "uniform" policy, the thresholds are set based on the absolute value of the wavelet coefficients. In the "dependent" policy, the thresholds are set based on the wavelet coefficients normalized by the weights from diagWWt.

References

Donoho, D. L., & Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. Journal of the american statistical association, 90(432), 1200-1224.

de Loynes, B., Navarro, F., Olivier, B. (2021). Data-driven thresholding in denoising with Spectral Graph Wavelet Transform. Journal of Computational and Applied Mathematics, Vol. 389.

Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. The annals of Statistics, 1135-1151.

See Also

SUREthresh for the underlying thresholding method used at each scale.

Examples

if (FALSE) {
# Compute the Laplacian matrix and its eigen-decomposition
L <- laplacian_mat(grid1$sA)
U <- eigensort(L)

# Compute the tight frame coefficients
tf <- tight_frame(U$evalues, U$evectors)

# Generate some noisy observation
n <- nrow(L)
f <- randsignal(0.01, 3, grid1$sA)
sigma <- 0.01
noise <- rnorm(n, sd = sigma)
tilde_f <- f + noise

# Compute the transform coefficients
wcn <- forward_sgwt(f, U$evalues, U$evectors)
wcf <- forward_sgwt(f, U$evalues, U$evectors)

# Compute the weights
diagWWt <- colSums(t(tf)^2)

# Compute to optimal threshold
lmax <- max(U$evalues)
J <- floor(log(lmax)/log(b)) + 2
LD_opt_thresh_u <- LD_SUREthresh(J=J,
                                 wcn=wcn,
                                 diagWWt=diagWWt,
                                 beta=2,
                                 sigma=sigma,
                                 hatsigma=NA,
                                 policy = "uniform",
                                 keepSURE = FALSE)

# Get the graph signal estimator
hatf_LD_SURE_u <- synthesis(LD_opt_thresh_u$wcLDSURE, tf)
}