Wavelet Thresholding via A Bayesian Approach

Abstract

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coef®cients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior speci®ed, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any speci®c Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coef®cients. However, prior knowledge about a function's regularity properties might be dif®cult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.

Denoising of multicomponent images using wavelet least-squares estimators

“…Standard wavelet thresholding [10] treats the coefficients with magnitudes below a certain threshold as "non significant" and sets these to zero; the remaining, "significant" coefficients are kept unmodified (hard-thresholding) or reduced in magnitude (soft-thresholding). Shrinkage estimators can also result from a Bayesian approach [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], which imposes a prior distribution on noise-free data. Common priors for noise-free data include (generalized) Laplacians [11,18,21], alpha-stable models [20], double stochastic (Gaussian scale mixture) models [24,25] and mixtures of two distributions [13][14][15][16][17] where one distribution models the statistics of "significant" coefficients and the other one models the statistics of "insignificant" data.…”

Analysis of Mass Spectrometry Data Using Bayesian Wavelet-Based Functional Mixed Models

“…This property makes the procedure useful for modeling functions with many local features like peaks. References on wavelet regression can be found in Chapters 6 and 8 of Vidakovic (1999), and in Donoho and Johnstone (1995), Chipman, Kolaczyk, and McCulloch (1997), Vidakovic (1998), Abramovich, Sapatinas, and Silverman (1998), Clyde, Parmigiani, and Vidakovic (1998), and Clyde and George (2000).…”

“…Specifically, the prior for B * ijk , the wavelet coefficient at scale j and location k for fixed effect function i, was a spike-slab prior given by B * ijk = γ ijk Normal(0, τ ij ) + (1 − γ ijk )δ 0 , with γ ijk ∼ Bernoulli(π ij ) and δ 0 being a point mass at zero. This prior is commonly used in Bayesian implementations of wavelet regression, including Clyde, Parmigiani and Vidakovic (1998) and Abramovich, Sapatinas, and Silverman (1998). Use of this mixture prior causes the posterior mean estimates of the B * ijk to be nonlinearly shrunken towards zero, which results in adaptively regularized estimates of the fixed effect functions.…”

Statistical Modeling of Photographic Images

“…For example, when applied to the problem of compression, the entropy of the distributions described above is significantly less than that of a Gaussian with the same variance, and this leads directly to gains in coding efficiency. In denoising, the use of this model as a prior density for images yields to significant improvements over the Gaussian model [e.g., 48,11,2,34,47]. Consider again the problem of removing additive Gaussian white noise from an image.…”