Wavelet Transform

Wavelet transforms decompose an image into a series of approximation (low-frequency) and detail (high-frequency) coefficients. This creates a hierarchical representation where:

• Approximation sub-bands capture global intensity variations and coarse structures
• Horizontal, vertical, and diagonal detail sub-bands isolate edge information at specific orientations Each decomposition level halves the resolution, enabling texture analysis at progressively coarser scales. In radiomics, this allows the quantification of tissue patterns ranging from fine granularity to broad architectural distortion.

Unlike the Fourier transform, which provides only global frequency information, wavelets offer simultaneous spatial and frequency localization. This is achieved through compactly supported basis functions that are localized in both domains. Key implications for medical imaging:

• Textural features can be attributed to specific anatomical locations
• Localized high-frequency anomalies, such as microcalcifications or spiculations, are preserved
• The transform adapts to abrupt intensity changes without introducing ringing artifacts This dual localization makes wavelets particularly effective for analyzing heterogeneous tumor regions where texture varies spatially.

The discrete wavelet transform is implemented using a cascade of quadrature mirror filters: a low-pass scaling filter and a high-pass wavelet filter. The process follows a structured decomposition:

• Convolution with the low-pass filter produces approximation coefficients
• Convolution with the high-pass filter yields detail coefficients
• Downsampling by a factor of two follows each filtering operation Common filter families used in radiomics include Daubechies, Coiflet, and Symlet wavelets, each offering different trade-offs between compact support and smoothness. The choice of wavelet basis influences the sensitivity of extracted texture features.

After wavelet decomposition, standard texture matrices are computed on each sub-band to quantify patterns at specific scales:

• GLCM features (contrast, correlation, energy) capture second-order statistics within each frequency band
• GLRLM features (run percentage, gray-level non-uniformity) measure directional texture coarseness
• First-order statistics (mean, skewness, kurtosis) describe the intensity distribution of wavelet coefficients This multi-scale approach dramatically expands the radiomic feature space. A single ROI can yield hundreds of wavelet-derived features, capturing textural signatures invisible at the original resolution. The Image Biomarker Standardisation Initiative (IBSI) provides standardized definitions for wavelet-based radiomic features.

The standard decimated wavelet transform reduces image size at each level through downsampling, which sacrifices shift invariance. The undecimated wavelet transform (also called stationary wavelet transform) addresses this limitation:

• Omits the downsampling step, preserving the original image dimensions at every decomposition level
• Produces translation-invariant representations, meaning small shifts in the ROI do not alter feature values
• Generates redundant, overcomplete representations that improve texture classification stability In radiomics, undecimated transforms are preferred when precise spatial correspondence is critical, such as in delta-radiomics studies tracking temporal changes. The trade-off is increased computational cost and memory requirements.

Wavelet-based radiomic features have demonstrated prognostic and predictive value across multiple cancer types:

• Non-small cell lung cancer: Wavelet features from CT scans predict overall survival and distant metastasis
• Glioblastoma: Multi-scale texture analysis of MRI sub-bands correlates with genetic markers like MGMT methylation status
• Breast cancer: Wavelet decomposition of mammograms improves the characterization of lesion margins and architectural distortion
• Colorectal cancer: Wavelet-filtered CT texture features differentiate KRAS-mutant from wild-type tumors These applications leverage the ability of wavelets to isolate biologically meaningful textural patterns associated with tumor aggressiveness and treatment response.

A second-order statistical method that quantifies texture by calculating the frequency of specific pairs of pixel intensities occurring at a defined spatial offset. While the Wavelet Transform isolates frequency sub-bands, the GLCM analyzes the spatial relationship of intensities within those sub-bands to derive features like contrast, homogeneity, and correlation.

• Captures directional textural patterns
• Defined by distance (d) and angle (θ) parameters
• Generates Haralick texture features for tumor classification

A texture matrix quantifying the size of homogeneous connected regions of identical voxel intensity, independent of their rotational orientation. After a Wavelet Transform decomposes an image into high-frequency (detail) and low-frequency (approximation) components, the GLSZM can be applied to each sub-band to measure structural coarseness.

• Rotation-invariant texture analysis
• Features include Zone Percentage (ZP) and Large Zone Emphasis (LZE)
• Effective for characterizing necrotic vs. viable tumor regions

The process of binning continuous voxel intensity values into a finite number of discrete gray levels. This is a critical preprocessing step for both Wavelet Transform analysis and texture matrix calculation. The number of bins directly impacts the sensitivity and reproducibility of derived features.

• Fixed bin width vs. fixed bin count methodologies
• IBSI guidelines recommend consistent discretization parameters
• Insufficient bins can mask subtle textural variations in wavelet sub-bands

An independent international collaboration providing consensus-based reference values and standardized nomenclature for radiomic feature computation, including those derived from Wavelet Transforms. IBSI ensures that wavelet-based features extracted using different software packages yield interoperable and reproducible results.

• Defines mathematical definitions for wavelet decomposition parameters
• Provides digital phantoms for validation of feature extraction algorithms
• Essential for multi-center clinical trial harmonization

A statistical batch-effect correction method adapted from genomics to remove non-biological technical variance in radiomic features across different imaging scanners. Wavelet-derived features are particularly sensitive to scanner-specific noise profiles and reconstruction kernels, making ComBat essential for harmonizing multi-institutional datasets.

• Estimates and removes additive and multiplicative batch effects
• Preserves biological variability while eliminating scanner-induced variance
• Applied post-extraction to wavelet sub-band features