Beyond sparse denoising in frames: minimax estimation with a scattering transform
This addresses the limitation of sparse estimators in harmonic analysis for adapting to complex signal regularities, offering a new approach for noise suppression and geometric regularity specification in image processing, with potential implications for bridging harmonic analysis and deep learning.
The paper tackles the problem of denoising cartoon images with unknown Lipschitz exponents by introducing a scattering transform-based estimator that minimizes and maximizes ℓ¹ norms of subsets of scattering coefficients. The result shows that this estimator reaches the minimax asymptotic bound for all α ≤ 2, as supported by numerical experiments and stated as a mathematical conjecture.
A considerable amount of research in harmonic analysis has been devoted to non-linear estimators of signals contaminated by additive Gaussian noise. They are implemented by thresholding coefficients in a frame, which provide a sparse signal representation, or by minimising their $\ell^1$ norm. However, sparse estimators in frames are not sufficiently rich to adapt to complex signal regularities. For cartoon images whose edges are piecewise $\bf C^α$ curves, wavelet, curvelet and Xlet frames are suboptimal if the Lipschitz exponent $α\leq 2$ is an unknown parameter. Deep convolutional neural networks have recently obtained much better numerical results, which reach the minimax asymptotic bounds for all $α$. Wavelet scattering coefficients have been introduced as simplified convolutional neural network models. They are computed by transforming the modulus of wavelet coefficients with a second wavelet transform. We introduce a denoising estimator by jointly minimising and maximising the $\ell^1$ norms of different subsets of scattering coefficients. We prove that these $\ell^1$ norms capture different types of geometric image regularity. Numerical experiments show that this denoising estimator reaches the minimax asymptotic bound for cartoon images for all Lipschitz exponents $α\leq 2$. We state this numerical result as a mathematical conjecture. It provides a different harmonic analysis approach to suppress noise from signals, and to specify the geometric regularity of functions. It also opens a mathematical bridge between harmonic analysis and denoising estimators with deep convolutional network.