[Paper Review] Non-decimated Complex Wavelet Spectral Tools with Applications

Taewoon Kong, Brani Vidaković|arXiv (Cornell University)|Feb 4, 2019

Image and Signal Denoising Methods30 references4 citations

TL;DR

This paper introduces a non-decimated complex wavelet transform (NDWT_c) that combines structural redundancy from non-decimated wavelets and componential redundancy from complex wavelet coefficients to enhance signal and image classification. By leveraging both modulus and phase information in wavelet spectra, the method significantly improves classification accuracy—particularly through phase-based features—outperforming standard wavelet transforms and even deep learning models in specific biomedical applications with lower computational cost.

ABSTRACT

In this paper we propose spectral tools based on non-decimated complex wavelet transforms implemented by their matrix formulation. This non-decimated complex wavelet spectra utilizes both real and imaginary parts of complex-valued wavelet coefficients via their modulus and phases. A structural redundancy in non-decimated wavelets and a componential redundancy in complex wavelets act in a synergy when extracting wavelet-based informative descriptors. In particular, we suggest an improved way of separating signals and images based on their scaling indices in terms of spectral slopes and information contained in the phase in order to improve performance of classification. We show that performance of the proposed method is significantly improved when compared with procedures based on standard versions of wavelet transforms or on real-valued wavelets. It is also worth mentioning that the matrix-based non-decimated wavelet transform can handle signals of an arbitrary size and in 2-D case, rectangular images of possibly different and non-dyadic dimensions. This is in contrast to the standard wavelet transforms where algorithms for handling objects of non-dyadic dimensions requires either data preprocessing or customized algorithm adjustments. To demonstrate the use of defined spectral methodology we provide two examples of application on real-data problems: classification of visual acuity using scaling in pupil diameter dynamic in time and diagnostic and classification of digital mammogram images using the fractality of digitized images of the background tissue. The proposed tools are contrasted with the traditional wavelet based counterparts.

Motivation & Objective

• To develop a wavelet-based spectral tool that enhances classification performance for signals and images with intrinsic self-similarity.
• To address limitations of standard wavelet transforms, such as lack of translation invariance and insufficient phase utilization.
• To create a computationally efficient, matrix-based implementation capable of handling arbitrary-sized 1D and 2D data, including non-dyadic dimensions.
• To demonstrate the superiority of phase-based descriptors derived from non-decimated complex wavelets in real-world biomedical classification tasks.
• To provide a reproducible, open-source MATLAB demo for broader research adoption and validation.

Proposed method

• The method employs a matrix formulation of the non-decimated complex wavelet transform (NDWT_c), enabling efficient computation on arbitrary-sized signals and images.
• It extracts wavelet spectra using both the modulus and phase of complex-valued coefficients at multiple scales, capturing both energy and directional information.
• Phase statistics—specifically the average phase at the finest scale—are used as discriminative features, enhancing classification beyond amplitude-based spectral slopes.
• The approach avoids decimation, preserving full signal resolution and enabling translation invariance through structural redundancy.
• The method is applied to 1D time series (pupil diameter dynamics) and 2D images (mammograms), using spectral slopes and phase features for classification.
• A comparative evaluation is conducted against standard WT, WT_c, NDWT, and CNNs, using accuracy, specificity, sensitivity, and computation time as metrics.

Experimental results

Research questions

• RQ1 Can combining structural redundancy (non-decimation) and componential redundancy (complex wavelets) lead to improved wavelet-based spectral descriptors for classification?
• RQ2 Does phase information in complex wavelet coefficients significantly enhance classification performance compared to magnitude-only features?
• RQ3 How does the NDWT_c method perform relative to standard wavelet transforms and deep learning models like CNNs in low-data, self-similar image classification tasks?
• RQ4 Can the proposed method handle arbitrary-sized 1D and 2D data without preprocessing or dyadic constraints?
• RQ5 What is the computational cost of NDWT_c relative to state-of-the-art models like CNNs in real biomedical applications?

Key findings

• The NDWT_c method achieved the highest classification accuracy in both 1D and 2D applications, with the best performance observed when using only phase-based features from the finest scale.
• In the 1D pupil diameter study, the method based on phase averages of NDWT_c achieved 90.0% accuracy and 90.0% specificity, outperforming magnitude-only and combined feature methods.
• In the 2D mammogram classification, the phase-based NDWT_c method reached 87.5% accuracy and 85.7% specificity, significantly improving upon standard WT_c and NDWT.
• The computation time for extracting features using NDWT_c on a 1024×1024 image was approximately 17 minutes, which is substantially lower than the 15 hours required by a CNN for the same task.
• Despite lower computational cost, NDWT_c outperformed the CNN in terms of specificity and sensitivity, suggesting its strength in capturing fine-scale, self-similar features critical in medical imaging.
• The study confirms that phase information in complex wavelets is highly discriminative, especially when combined with non-decimated transforms, and should be prioritized in feature extraction.