Composite Wavelet Matrix-Based Transforms and Applications

TL;DR

This paper obtains new unitary transforms that generally fall outside the strict wavelet filterbank class, yet remain fully invertible and numerically stable by combining orthogonal wavelet matrices through products, Kronecker products, and block-diagonal constructions.

Abstract

Orthogonal wavelet transforms are a cornerstone of modern signal and image denoising because they combine multiscale representation, energy preservation, and perfect reconstruction. In this paper, we show that these advantages can be retained and substantially enhanced by moving beyond classical single-basis wavelet filterbanks to a broader class of composite wavelet-like matrices. By combining orthogonal wavelet matrices through products, Kronecker products, and block-diagonal constructions, we obtain new unitary transforms that generally fall outside the strict wavelet filterbank class, yet remain fully invertible and numerically stable. The central finding is that such composite transforms induce stronger concentration of signal energy into fewer coefficients than conventional wavelets. This increased sparsity, quantified using Lorenz curve diagnostics, directly translates into improved denoising under identical thresholding rules. Extensive simulations on Donoho-Johnstone benchmark signals, complex-valued unitary examples, and adaptive block constructions demonstrate consistent reductions in mean-squared error relative to single-basis transforms. Applications to atmospheric turbulence measurements and image denoising of the Barbara benchmark further confirm that composite transforms better preserve salient structures while suppressing noise.

Figures (9)

• Figure 1: Lorenz Curves for four Donoho--Johnstone test signals. Product matrix $\bm W_1 \bm W_2$ yields a substantially more disbalanced energy distribution than single--basis wavelet transforms. The number of levels of decomposition is $3$ for three of the four signals, with one level of decomposition showing clearest disbalance for the Bumps signal. For HeaviSine and Blocks signals, $\bm W_1=\bm W_2$ so only $\bm W_1$ is shown.
• Figure 2: Product transform matrices $\bm W_1 \bm W_2$ (chosen respectively for each signal) exhibit lower AMSE over $200$ simulations for Donoho--Johnstone signals compared to single-basis. Kronecker product performs similarly or better than single-basis on two of the four signals.
• Figure 3: Comparison of Kronecker, similarity and product transforms against single--basis transforms for the Doppler signal ($N=1024$); $\bm W_1 =$ complex DAUB 6, $\bm W_2 =$ real Haar. After calculating MSE over $200$ simulations, the product transform $\bm W_1 \bm W_2$ and Kronecker $\bm W_k \otimes \bm W_l$ produce the lowest average MSE and variance. These composite matrices have superior performance to the single--basis methods. Table \ref{['tab:table1']} contains the exact values.
• Figure 4: Side--by--side boxplots displaying distributions of MSEs ($200$ simulations) for a combined signal of four Donoho--Johnstone signals. The red bar corresponds to the median MSE value, which due to symmetry is close to the mean. Applying the adaptive wavelet matrix results in a lower average MSE than single--basis transforms (Table \ref{['tab:table2']}).
• Figure 5: Comparison of single--base (Symm4) and product--base (Symm4 $\circ$ Coif3) denoising across noise levels $\sigma$. Negative $\Delta$MSE favors the product transform.