Cepstrum Texture Features Analysis

• Cepstrum-based texture features are signal-derived descriptors that use Fourier analysis to isolate periodic structures and repetitive patterns in images.
• They involve transforming data into the cepstral domain, followed by masking and statistical extraction (e.g., GLCM, Haralick metrics) for improved classification and segmentation.
• Advanced variants such as graph cepstrum and interferometric approaches enhance robustness, offering complementary solutions to conventional spatial-frequency methods.

Cepstrum-based texture features are a class of signal-derived descriptors that leverage the Fourier-based cepstral domain for quantitative texture analysis in images and multidimensional signals. Theoretical and applied developments in this area focus on transforming input data into a domain where periodic patterns, repetitive structures, and certain types of spatial relationships become more salient and tractable for classification, segmentation, or retrieval. These features are increasingly relevant in medical imaging, acoustic scene analysis, interferometric microscopy, and content-based image retrieval due to their ability to capture harmonic structures and robustly characterize fine-scale periodicities.

1. Mathematical Formulation of the Cepstrum in Texture Analysis

Let $x(t)$ denote a real or complex-valued 2D spatial signal (e.g., an image). The cepstrum of $x(t)$, denoted $C_x$, is defined as

$$C_x = \mathrm{Re} \left\{ \mathcal{F}^{-1} \left[ \log \left| \mathcal{F} \{ x(t) \} \right| \right] \right\}$$

where $\mathcal{F}$ is the Fourier transform, $\log|\cdot|$ denotes the log-magnitude spectrum, and $\mathcal{F}^{-1}$ is the inverse Fourier transform. By converting convolution into addition via the logarithmic property, the cepstral domain uniquely separates periodic (harmonic) structures, manifesting as impulses in $C_x$ when strong periodicities are present. This is particularly valuable for detecting structural repetitions, regular spatial modulations, or quasi-periodic patterns in textured images.

2. Feature Engineering with Cepstral Representations

A key innovation in recent work involves the extraction of classical statistical features from the 2D cepstral image, rather than the original spatial image. The pipeline, as implemented for melanoma detection in dermoscopic images (Miller et al., 31 Aug 2025), follows:

• Compute the 2D cepstrum of each image (and, optionally, across multiple color channels or color spaces).
• Mask the resultant cepstrum to focus on regions of interest (e.g., restricting to lesion areas via segmentation).
• Calculate Gray-Level Co-occurrence Matrix (GLCM) statistics from the masked cepstrum for multiple orientations (0°, 45°, 90°, 135°).
• Derive a set of Haralick texture statistics—contrast, correlation, energy, entropy, homogeneity, and others—supplemented by ad-hoc measures such as the matrix Trace, which encodes directionality.

A summary of feature derivation steps is provided below.

Step	Operation	Output
1. Cepstrum Calculation	$\mathcal{F}^{-1}\{\log(|\mathcal{F}\{x(t)\}|)\}$	2D cepstral image
2. Segmentation/Masking	Binary mask applied to lesion/background	Masked cepstral region
3. GLCM Construction	Pixel-pair co-occurrence for each direction	Four GLCM matrices
4. Statistical Extraction	Haralick + Trace metrics from each GLCM, summarized directionally	Feature vector

These features expand the discriminatory space beyond purely spatial or color descriptors and are found to be complementary in ensemble machine learning models (e.g., XGBoost) for melanoma classification, improving accuracy and F1 scores when fused with handcrafted features (Miller et al., 31 Aug 2025).

3. Graph Cepstrum: Topology-Aware Cepstral Features

A distinct approach within cepstrum-based feature extraction is the graph cepstrum, developed for spatial information extraction from distributed microphone arrays in acoustic scene analysis (Imoto, 2018). Here, sensor outputs are organized as a log-amplitude vector $q_\tau \in \mathbb{R}^N$ at time $\tau$ with $N$ microphones:

$$q_\tau = [\log \overline{a}_{\tau,1}, \ldots, \log \overline{a}_{\tau,N}]^T$$

Spatial relationships (e.g., synchronization, proximity) are modeled via a graph with adjacency matrix $A$. The unweighted Laplacian $L = D - A$ (with $D$ as the degree matrix) admits an eigendecomposition $L = U\Lambda U^T$. The graph cepstrum feature is given by the inverse graph Fourier transform:

$e_\tau = Uq_\tau$

This construction respects microphone connectivity and produces features resilient to synchronization errors and nonuniform array topologies. Empirical evidence demonstrates enhanced acoustic scene classification robustness compared to standard spatial cepstrum or time-difference feature sets, particularly under partial synchronization (Imoto, 2018).

4. Cepstrum in Interferometric Microscopy and Imaging

In quantitative phase imaging (QPI)—especially interferometric microscopy—cepstrum-based algorithms have been employed to decouple amplitude and phase without imposing classical reference constraints. The Spatial-Shifting Cepstrum (SSC) algorithm (Rubio-Oliver et al., 17 Jan 2025) processes two interferograms per direction (one static, one shifted by one pixel):

• Calculate the cross-correlation of two interferometric fields in the Fourier domain.
• Apply the log operator to transform convolution into addition.
• Engineer a 1-pixel spatial shift, translating into a known phase ramp in the frequency domain.
• Through weighted cepstral subtraction,

$$\widetilde{O}_1(u,v) = \frac{\widetilde{U}(u,v) - \widetilde{U}_x(u,v)}{1 - e^{-2\pi i u x_0}}$$

isolate the object spectrum (with $\widetilde{U}$ the cross-correlation’s cepstrum and $x_0$ the shift). Ambiguities at zero-frequency are mitigated by performing the operation in both horizontal and vertical directions and combining results via a masking strategy.

Experimental validation reveals that this approach enables retrieval of three independent fields of view from four holograms, tripling the traditional effective FOV, and achieving phase accuracy comparable to established digital holographic microscope (DHM) systems (Rubio-Oliver et al., 17 Jan 2025). Such complete phase and amplitude recovery is critical for quantifying subtle textural and refractive index variations in calibrated targets and biological tissue.

5. Comparative Insights with Spatial-Frequency Domain Alternatives

Cepstrum-based texture features are not unmatched in the spatial-frequency analysis landscape. Comprehensive reviews (Baaziz et al., 2010) report that while cepstral analysis excels in highlighting periodic and harmonic structures, alternative representations such as Discrete Wavelet Transform (DWT), Gabor Wavelets (GWT), Dual-Tree Complex Wavelet Transform (DT-CWT), and Contourlets offer finer control over spatial locality, orientation selectivity, and directional sensitivity. These methods facilitate multi-scale, multi-orientation decompositions, exhibiting varying degrees of computational efficiency, robustness to rotation, and redundancy.

Key comparison criteria include:

• Periodicity Extraction: Cepstrum naturally isolates harmonic spatial patterns.
• Multi-Scale/Orientation: Gabor and wavelet-based methods provide explicit control.
• Rotational Invariance: DT-CWT and statistical models over transform coefficients improve rotation robustness.
• Computational Overhead: Energy-based feature vectors from all transforms are lightweight; statistical modeling (e.g., GGD-KL divergence) increases discrimination at the expense of resource usage.

A plausible implication is that cepstrum-based features offer maximum benefit in domains where periodic or echo-like textures are prominent and in settings tolerant of the global frequency analysis inherent to the cepstrum, whereas spatial-frequency decompositions afford localized and directionally-adaptive texture characterization (Baaziz et al., 2010).

6. Applications and Implications Across Fields

Cepstrum-based texture features have been successfully deployed or proposed in several applications:

• Medical Imaging: Improved melanoma discrimination via cepstral GLCM features on dermoscopic data; interpretable measures such as cepstral entropy and directional statistics provide non-redundant diagnostic clues (Miller et al., 31 Aug 2025).
• Acoustic Scene Analysis: Graph cepstrum features enable robust scene classification in environments with partially synchronized or variably located sensors, relevant for both surveillance and ambient monitoring (Imoto, 2018).
• Interferometric Microscopy: Accurate and reference-agnostic phase retrieval in QPI, expanding effective imaging area and facilitating cost-effective adaptation on existing microscopes (Rubio-Oliver et al., 17 Jan 2025).
• Texture Analysis in Retrieval Systems: Although not explicitly deployed as of (Baaziz et al., 2010), cepstrum-based features are well-positioned to complement energy-based and statistical modeling approaches in CBIR.

A forward-looking view suggests increased integration of cepstrum-based and spatial-frequency domain features (either at feature or model-level fusion), particularly as multi-domain texture analysis matures and domain-specific constraints—such as geometric invariance or reference beam generation—are relaxed.

7. Open Problems and Future Directions

While cepstrum-based features provide demonstrated value, several methodological and practical challenges persist:

• Parameter Selection: Choice of masking, directionality measures, and summary statistics in cepstral GLCM extraction remain ad hoc and may require further standardization for transferability across applications (Miller et al., 31 Aug 2025).
• Graph Construction: Robustness to inaccurate or variable microphone connectivity models is critical in graph cepstrum deployment for real-world sensor networks (Imoto, 2018).
• Zero-Ambiguity Handling: In Cepstrum-based interferometric algorithms, resolving singularities at critical spatial frequencies necessitates careful design of offset strategies and measurement redundancy (Rubio-Oliver et al., 17 Jan 2025).
• Fusion with Deep Learning: Integration of cepstrum-domain features with modern deep learning backbones, especially for resource-limited or explainability-driven deployments, is a suggested area of future exploration (Miller et al., 31 Aug 2025).
• Multi-class and Multi-modal Classification: Scaling from binary discrimination (e.g., melanoma vs. nevus) to multi-class paradigms, and extending cepstral features to additional imaging modalities, represent open research avenues (Miller et al., 31 Aug 2025).

In summary, cepstrum-based texture features, through innovations in their mathematical construction, feature engineering, and emerging application frameworks, offer a complementary and often uniquely sensitive tool for texture analysis in diverse domains.