Wavelet Transform

Unlike the Fourier transform, which loses all spatial information, the wavelet transform preserves spatial localization. It reveals where specific frequency components occur in an image, making it ideal for analyzing heterogeneous tumors where texture varies across the region of interest. This is achieved by convolving the image with a small, localized oscillating waveform called the mother wavelet.

The transform applies a series of high-pass and low-pass filters iteratively, breaking the image into detail coefficients and approximation coefficients.

• Approximation coefficients: Represent the coarse, low-frequency background.
• Detail coefficients: Capture fine edges and textures in horizontal, vertical, and diagonal directions. This creates a hierarchical pyramid of images at progressively coarser resolutions.

The choice of mother wavelet is critical and application-specific. Key families include:

• Haar: A simple, discontinuous step function effective for detecting sharp edges.
• Daubechies (db-N): Compact support wavelets with varying vanishing moments, widely used for texture classification.
• Coiflets: Near-symmetric wavelets with scaling functions that also have vanishing moments.
• Biorthogonal: Allows linear phase filtering, preserving edge locations precisely.

After decomposition, radiomic features are extracted from the resulting sub-bands rather than the original image. This captures scale-specific heterogeneity.

• LHL Sub-band: Low-pass filtered horizontally, high-pass vertically—captures horizontal edges at a specific scale.
• HHH Sub-band: High-pass filtered in all directions—captures fine, chaotic noise-like textures. Statistical measures like energy and entropy are then calculated on these sub-bands.

The standard Discrete Wavelet Transform (DWT) involves downsampling, which makes it shift-variant—a slight shift in the ROI changes the coefficients. The Stationary Wavelet Transform (SWT) omits decimation, preserving translation invariance at the cost of redundancy. For radiomics, SWT is often preferred because it produces a consistent feature map regardless of minor segmentation variations.

The Image Biomarker Standardisation Initiative (IBSI) provides strict guidelines for wavelet-based feature extraction to ensure reproducibility. This includes specifications for:

• Boundary handling: How to pad image edges during convolution.
• Filter normalization: Ensuring energy preservation across scales.
• Decomposition levels: Standardizing the number of filter iterations based on physical voxel dimensions rather than arbitrary pixel counts.

A mathematical operation that convolves a signal with a scaled and translated mother wavelet across all possible scales and positions. Unlike the discrete version, CWT produces a highly redundant, high-resolution time-frequency representation. In radiomics, CWT is used to generate scalogram images that visualize how frequency content evolves across an image, revealing subtle textural transitions at tumor boundaries that fixed-scale filters miss.

A computationally efficient implementation that applies dyadic scaling and shifting—scaling by powers of two—to decompose an image into approximation and detail coefficients. DWT uses quadrature mirror filter banks to iteratively split low-frequency components, producing a multi-resolution pyramid. In medical imaging, DWT is the workhorse for radiomic feature extraction because it provides non-redundant, compact representations ideal for machine learning inputs.

The process of splitting an image into four frequency channels at each decomposition level:

• LL (Approximation): Low-pass horizontal, low-pass vertical—captures global intensity trends
• LH (Horizontal Detail): Low-pass horizontal, high-pass vertical—detects horizontal edges
• HL (Vertical Detail): High-pass horizontal, low-pass vertical—detects vertical edges
• HH (Diagonal Detail): High-pass in both directions—captures corner and fine texture information

Radiomic features are extracted from each sub-band independently, yielding multi-scale texture signatures that characterize tissue heterogeneity at different spatial frequencies.

Also called the undecimated wavelet transform, SWT omits the downsampling step after filtering, producing shift-invariant coefficients at each scale. This property is critical for radiomics because it ensures that small translations of the ROI do not alter extracted features—a major limitation of standard DWT. The trade-off is increased computational redundancy, but the gain in test-retest reproducibility makes SWT preferred for multi-center imaging studies where precise alignment cannot be guaranteed.

A generalization of DWT that decomposes both approximation and detail coefficients at each level, creating a complete binary tree of sub-bands. While DWT only splits the LL channel, WPT explores the full frequency spectrum, enabling adaptive selection of the best basis for a given classification task. In radiomics, WPT is used when the most discriminative texture information resides in mid-frequency or high-frequency bands that standard DWT would not further decompose.