TBSD: Texture Basis Integrated Smooth Decomposition

• TBSD is a method that decomposes images into smooth backgrounds, quasi-periodic textures, and sparse anomaly components, enabling precise anomaly detection.
• It employs a two-stage pipeline—texture basis learning with convex optimization and anomaly detection via soft-thresholding—to robustly separate normal textures from defects.
• TBSD demonstrates superior performance over methods like RPCA, SSD, and FFT by achieving higher true-positive rates and lower false-positive rates on both simulated and real datasets.

Texture Basis Integrated Smooth Decomposition (TBSD) is a methodology for decomposing images with complex textured backgrounds into smooth, texture, and anomaly components, optimally designed for anomaly detection tasks in high-dimensional, textured images with smooth backgrounds and potentially sparse anomalies. The distinguishing feature of TBSD is the explicit integration of quasi-periodic texture models into the decomposition, which robustly separates normal texture from true anomalies and reduces misidentification rates relative to prior approaches (Song et al., 23 Dec 2025).

1. Mathematical Formulation and Decomposition Model

Let $Y\in\mathbb{R}^{m\times n}$ represent the observed image. TBSD decomposes $Y$ as follows: $Y = C_{bg} + C_{tex} + C_a + e$ where:

• $C_{bg}$: low-rank, smooth background,
• $C_{tex}$: high-rank, quasi-periodic texture,
• $C_a$: sparse anomaly component,
• $e$: i.i.d. noise.

TBSD operates via a two-stage pipeline:

• Texture Basis Learning: Estimates texture bases from defect-free images.
• Anomaly Detection: Uses learned bases to decompose new images and isolate anomalies.

Texture Basis Learning

Given defect-free $Y$ and a smooth basis $B$ (e.g., tensor-product B-splines), coefficients $\theta$ and residual texture $\widehat{C}_{tex}$ are estimated by minimizing: $$\min_{\theta, C_{tex}} \|Y - B\theta - C_{tex}\|_2^2 + \lambda\,\theta^\top R\,\theta + \gamma\,\|C_{tex}\|_1 \quad \text{s.t.} \; C_{bg}=B\theta,\, C_a \equiv 0$$ with $R$ a discrete roughness matrix (for smoothness). Updates use block coordinate descent with closed-form iterates: $$\theta \leftarrow (B^\top B + \lambda R)^{-1}B^\top (Y - C_{tex}),\quad C_{tex} \leftarrow S_{\gamma/2}(Y - B\theta)$$ Here, $S_\tau$ is the entrywise soft-thresholding operator.

Post extraction, $\widehat{C}_{tex}$ is decomposed into directional subspaces, yielding an overcomplete dictionary $B_t = \left\{ B_t^{(d)} \right\}$, where $d$ indexes dominant texture directions identified through statistical tests.

Anomaly Detection

For a new (possibly faulty) image $\dot{Y}$, using learned $B$ and $B_t$, the following is solved: $$\min_{\theta, \theta_t, C_a} \|\dot{Y} - B\theta - B_t\theta_t - C_a\|_2^2 + \lambda \theta^\top R \theta + \gamma \|\theta_t\|_1 + \eta \|C_a\|_1$$ with iterative updates for $\theta$, $\theta_t$, and $C_a$, ultimately yielding the anomaly map $\widehat{C}_a$.

2. Quasi-Periodicity in Texture Modeling

TBSD formally characterizes image textures via quasi-periodicity. For a finite 1D signal $S$, periodicity is defined by strict repetition: $s_{i+T} = s_i$. Quasi-periodicity generalizes this, permitting small bounded deviations: $$S \text{ is quasi-periodic of mode } S_{\tilde{T}} \text{ and tolerance } \sigma \text{ if for segments } S^{\tilde{t}_m}, \|S^{\tilde{t}_m} - S_{\tilde{T}}\|_2 \leq \sigma$$ 2D quasi-periodicity is constructed via orientations of 1D quasi-periodic samples. Linear combinations preserve quasi-periodicity, given appropriate coefficient and tolerance bounds (via Cauchy–Schwarz).

Detection of dominant directions exploits the difference in standard deviation of line samples (transverse vs. along ridges). These directions, $D$, define the axes for dictionary construction, ensuring texture bases capture quasi-repetitive structure.

3. Algorithmic Workflow

The TBSD methodology proceeds through distinct algorithmic stages:

a. Low-Rank Decomposition (Algorithm 1)

• Input: defect-free $Y$, smooth basis $B$, penalties $\lambda, \gamma$.
• Alternate minimization:
  1. Update $\theta$ via ridge regression on background.
  2. Refine $\widehat{C}_{tex}$ via soft-thresholding.

• Convergence due to strict convexity.

b. Direction Detection (Algorithm 2; LSERA + std-dev test)

• Rotate a sampling line by increments $\Delta\theta$ through $[0, \pi]$.
• For each angle $\alpha$, compute $$\Delta\sigma = \mathrm{std}(S_\alpha) - \mathrm{std}(S_{\alpha+\pi/2})$$.
• Select directions where $\Delta\sigma$ exceeds threshold $\phi$, yielding $D$.

c. Mode-Clustering (Algorithm 3)

• For each $d \in D$, trace in direction $d$, extract segments of length up to $K$, and cluster patterns.
• Cluster centroids become dictionary atoms in $B_t$.

Each subproblem (ridge, $\ell_1$-soft-thresholding, density clustering) is convex, ensuring global convergence to a stationary point via block-coordinate descent.

4. Anomaly Detection Pipeline and Decision Rules

The detection pipeline is structured as follows:

1. Project $\dot{Y}$ onto the background, removing $B\theta$.
2. Reconstruct textures using sparse coding over $B_t$:

$R = \dot{Y} - B\theta - B_t\theta_t$

1. Apply soft-thresholding to the residual to obtain $C_a = S_{\eta/2}(R)$.
2. Pixels with $\widehat{C}_a(i,j) \neq 0$ are labeled as anomalous. The fraction of anomalous pixels $\rho_a$ may be used for image-level decisions versus a global threshold $\phi_A$.

5. Benchmarking and Quantitative Performance

TBSD has been empirically evaluated across both simulated and real data settings. Key metrics are true-positive rate (TPR) and false-positive rate (FPR). The principal benchmarks are Robust PCA (RPCA), Sum of Squared Differences (SSD), and Fast Fourier Transform (FFT) baselines.

Results Summary

Dataset	Method	TPR	FPR
Simulation	TBSD	0.43	0.09
	RPCA	0.28	0.25
	SSD	0.29	0.075
	FFT	0.066	0.10
Real (MVTec)	TBSD	0.58	0.022
TBSD (best 5)	TBSD	0.76±0.08	0.024±0.010
	RPCA	0.33	0.246
	SSD	0.43	0.242
	FFT	0.46	0.082

TBSD consistently achieves superior TPR and lower FPR, particularly notable on real-world MVTec wood patches, where it reduces mis-detection of quasi-periodic textures and improves anomaly recall (Song et al., 23 Dec 2025).

6. Practical Considerations and Applications

Key characteristics of TBSD in practical deployments include:

• Training Data Requirements: Robust dictionary learning is achieved with small sets (5–10) of defect-free patches.
• Robustness: The method accommodates moderate variations in texture spacing, orientation, and illumination due to its quasi-periodic modeling.
• Parameter Tuning: Recommended ranges—
  • $\lambda$ (background smoothness): 0.05–0.2,
  • $\gamma$ (texture sparsity): 0.1–0.3,
  • $\eta$ (anomaly sparsity): 0.01–0.1,
  • $\Delta\theta$ (direction increments): $1^\circ–5^\circ$.
  • Degradation with parameter deviation is gradual.
• Application Domains: Wood-grain inspection, steel-rolling defect detection, anomaly detection in 3D printed surfaces, and non-destructive testing of aviation composites.
• Methodological Distinction: TBSD augments classical smooth + sparse decompositions with a learned quasi-periodic texture prior, nearly eliminating misclassification of normal texture and yielding state-of-the-art detection performance.

7. Context and Significance

TBSD addresses the limitations of prior anomaly detection methods on textured images, notably their higher rates of misidentification and dependence on extensive, structured datasets. The incorporation of a learned quasi-periodic texture prior enables efficient operation with small training sets and robust discrimination between quasi-periodic normal textures and true anomalies. The approach is theoretically justified via the properties of quasi-periodic signals and empirically validated with benchmark beating results on both synthetic and real-world datasets. This suggests a significant advance in industrial applications where textured backgrounds complicate conventional defect detection workflows (Song et al., 23 Dec 2025).