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Smooth SCAD: A Raised Cosine SCAD Type Thresholding Rule for Wavelet Denoising

Radhika Kulkarni, Aluisio Pinheiro, Brani Vidakovic, Abdourrahmane M. Atto

TL;DR

This work addresses wavelet denoising with SCAD-type thresholding by introducing a smooth SCAD rule based on a raised-cosine transition, placing the method inside the continuous SURE framework. The resulting shrinkage function $\rho_{ssc}$ is odd, continuous, and differentiable away from the main threshold, preserving sparsity for small coefficients and near-unbiasedness for large ones. The authors derive an explicit SURE risk, provide a Bayesian interpretation via a smooth nonconvex prior, and develop both global and level-dependent adaptive thresholding strategies with data-driven tuning. Numerical experiments on Donoho–Johnstone benchmarks demonstrate improved performance over classical SCAD and universal thresholding, with practical guidelines for choosing the threshold scale $\lambda$ and the shape parameter $a$.

Abstract

We introduce a smooth variant of the SCAD thresholding rule for wavelet denoising by replacing its piecewise linear transition with a raised cosine. The resulting shrinkage function is odd, continuous on R, and continuously differentiable away from the main threshold, yet retains the hallmark SCAD properties of sparsity for small coefficients and near unbiasedness for large ones. This smoothness places the rule within the continuous thresholding class for which Stein's unbiased risk estimate is valid. As a result, unbiased risk computation, stable data-driven threshold selection, and the asymptotic theory of Kudryavtsev and Shestakov apply. A corresponding nonconvex prior is obtained whose posterior mode coincides with the estimator, yielding a transparent Bayesian interpretation. We give an explicit SURE risk expression, discuss the oracle scale of the optimal threshold, and describe both global and level-dependent adaptive versions. The smooth SCAD rule therefore offers a tractable refinement of SCAD, combining low bias, exact sparsity, and analytical convenience in a single wavelet shrinkage procedure.

Smooth SCAD: A Raised Cosine SCAD Type Thresholding Rule for Wavelet Denoising

TL;DR

This work addresses wavelet denoising with SCAD-type thresholding by introducing a smooth SCAD rule based on a raised-cosine transition, placing the method inside the continuous SURE framework. The resulting shrinkage function is odd, continuous, and differentiable away from the main threshold, preserving sparsity for small coefficients and near-unbiasedness for large ones. The authors derive an explicit SURE risk, provide a Bayesian interpretation via a smooth nonconvex prior, and develop both global and level-dependent adaptive thresholding strategies with data-driven tuning. Numerical experiments on Donoho–Johnstone benchmarks demonstrate improved performance over classical SCAD and universal thresholding, with practical guidelines for choosing the threshold scale and the shape parameter .

Abstract

We introduce a smooth variant of the SCAD thresholding rule for wavelet denoising by replacing its piecewise linear transition with a raised cosine. The resulting shrinkage function is odd, continuous on R, and continuously differentiable away from the main threshold, yet retains the hallmark SCAD properties of sparsity for small coefficients and near unbiasedness for large ones. This smoothness places the rule within the continuous thresholding class for which Stein's unbiased risk estimate is valid. As a result, unbiased risk computation, stable data-driven threshold selection, and the asymptotic theory of Kudryavtsev and Shestakov apply. A corresponding nonconvex prior is obtained whose posterior mode coincides with the estimator, yielding a transparent Bayesian interpretation. We give an explicit SURE risk expression, discuss the oracle scale of the optimal threshold, and describe both global and level-dependent adaptive versions. The smooth SCAD rule therefore offers a tractable refinement of SCAD, combining low bias, exact sparsity, and analytical convenience in a single wavelet shrinkage procedure.
Paper Structure (18 sections, 1 theorem, 67 equations, 2 figures, 1 table)

This paper contains 18 sections, 1 theorem, 67 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and SURE framework
  3. Smooth SCAD via a raised cosine
  4. Verification of structural assumptions and regularity conditions
  5. Structural conditions
  6. Derivative bounds
  7. Risk, SURE, and asymptotic behavior of smooth SCAD
  8. Bayesian interpretation of smooth SCAD
  9. Prior induced by the shrinkage profile
  10. Penalty, prior, and posterior
  11. Threshold where the posterior mode leaves zero
  12. Extension to the full wavelet model
  13. Choice of the threshold for smooth SCAD
  14. Approximate choice of $\lambda$ for smooth SCAD
  15. Choice of the shape parameter $a$
  16. ...and 3 more sections

Key Result

Theorem 1

Assume the underlying function $f$ is supported on a finite interval and uniformly Lipschitz with exponent $\gamma>1/2$. Then its true wavelet coefficients satisfy a sparsity--inducing decay condition of the usual form. For smooth SCAD shrinkage with fixed $a>1$, the oracle optimal risk $R_{J}(\lamb The precise constants and logarithmic factors are inherited directly from the KS analysis and depen

Figures (2)

  • Figure 1: Classical SCAD (a) and smooth SCAD (b) thresholding functions for $\lambda=1$.
  • Figure 2: (a) Doppler test signal of length $N=1024$ with additive Gaussian noise, rescaled to achieve $\mathrm{SNR}=7$ (variance ratio); (b) Smooth SCAD reconstruction using oracle–selected threshold $\lambda^* = \arg \min_\lambda \hbox{MSE}(\lambda) \approx 2.3485,$ showing effective recovery of structure and suppression of high-frequency artifacts. The oracle–selected $\lambda^*$ is smaller than the universal threshold, which in this case is $\lambda_{U} = \sqrt{2 \log(1024)} = 3.724$.

Theorems & Definitions (1)

  • Theorem 1: Asymptotic behavior under Lipschitz regularity