Quantum Nondecimated Wavelet Transform: Theory, Circuits, and Applications

The nondecimated or translation-invariant wavelet transform (NDWT) is a central tool in classical multiscale signal analysis, valued for its stability, redundancy, and shift invariance. This paper develops two complementary quantum formulations of the NDWT that embed these classical properties coherently into quantum computation. The first formulation is based on the epsilon-decimated interpretation of the NDWT and realizes all circularly shifted wavelet transforms simultaneously by promoting the shift index to a quantum register and applying controlled circular shifts followed by a wavelet analysis unitary. The resulting construction yields an explicit, fully unitary quantum representation of redundant wavelet coefficients and supports coherent postprocessing, including quantum shrinkage via ancilla-driven completely positive trace preserving maps. The second formulation is based on the Hadamard test and uses diagonal phase operators to probe scale-shift wavelet structure through interference, providing direct access to shift-invariant energy scalograms and multiscale spectra without explicit coefficient reconstruction. Together, these two approaches demonstrate that redundancy and translation invariance can be exploited rather than avoided in the quantum setting. Applications to denoising, feature extraction, and spectral scaling illustrate how quantum NDWTs provide a flexible and physically meaningful foundation for multiscale quantum signal processing.

A Method for Detecting Murmurous Heart Sounds based on Self-similar Properties

A heart murmur is an atypical sound produced by the flow of blood through the heart. It can be a sign of a serious heart condition, so detecting heart murmurs is critical for identifying and managing cardiovascular diseases. However, current methods for identifying murmurous heart sounds do not fully utilize the valuable insights that can be gained by exploring intrinsic properties of heart sound signals. To address this issue, this study proposes a new discriminatory set of multiscale features based on the self-similarity and complexity properties of heart sounds, as derived in the wavelet domain. Self-similarity is characterized by assessing fractal behaviors, while complexity is explored by calculating wavelet entropy. We evaluated the diagnostic performance of these proposed features for detecting murmurs using a set of standard classifiers. When applied to a publicly available heart sound dataset, our proposed wavelet-based multiscale features achieved comparable performance to existing methods with fewer features. This suggests that self-similarity and complexity properties in heart sounds could be potential biomarkers for improving the accuracy of murmur detection.

Early Detection of Ovarian Cancer by Wavelet Analysis of Protein Mass Spectra

Accurate and efficient detection of ovarian cancer at early stages is critical to ensure proper treatments for patients. Among the first-line modalities investigated in studies of early diagnosis are features distilled from protein mass spectra. This method, however, considers only a specific subset of spectral responses and ignores the interplay among protein expression levels, which can also contain diagnostic information. We propose a new modality that automatically searches protein mass spectra for discriminatory features by considering the self-similar nature of the spectra. Self-similarity is assessed by taking a wavelet decomposition of protein mass spectra and estimating the rate of level-wise decay in the energies of the resulting wavelet coefficients. Level-wise energies are estimated in a robust manner using distance variance, and rates are estimated locally via a rolling window approach. This results in a collection of rates that can be used to characterize the interplay among proteins, which can be indicative of cancer presence. Discriminatory descriptors are then selected from these evolutionary rates and used as classifying features. The proposed wavelet-based features are used in conjunction with features proposed in the existing literature for early stage diagnosis of ovarian cancer using two datasets published by the American National Cancer Institute. Including the wavelet-based features from the new modality results in improvements in diagnostic performance for early-stage ovarian cancer detection. This demonstrates the ability of the proposed modality to characterize new ovarian cancer diagnostic information.

Robust Wavelet-based Assessment of Scaling with Applications

A number of approaches have dealt with statistical assessment of self-similarity, and many of those are based on multiscale concepts. Most rely on certain distributional assumptions which are usually violated by real data traces, often characterized by large temporal or spatial mean level shifts, missing values or extreme observations. A novel, robust approach based on Theil-type weighted regression is proposed for estimating self-similarity in two-dimensional data (images). The method is compared to two traditional estimation techniques that use wavelet decompositions; ordinary least squares (OLS) and Abry-Veitch bias correcting estimator (AV). As an application, the suitability of the self-similarity estimate resulting from the the robust approach is illustrated as a predictive feature in the classification of digitized mammogram images as cancerous or non-cancerous. The diagnostic employed here is based on the properties of image backgrounds, which is typically an unused modality in breast cancer screening. Classification results show nearly 68% accuracy, varying slightly with the choice of wavelet basis, and the range of multiresolution levels used.