Persistent Homology-Guided Frequency Filtration

• Persistent homology-guided frequency filtration integrates discrete Fourier analysis with topological data analysis to selectively retain image features based on their structural robustness.
• It employs a pipeline that computes persistence diagrams and uses the 1-Wasserstein distance to rank frequency components, ensuring retention of salient global shapes.
• This technique enhances image compression and feature selection in noisy settings, achieving competitive performance with standards like JPEG by preserving key topological properties.

Persistent homology-guided frequency filtration is a principled image compression and feature selection methodology that integrates discrete Fourier analysis with persistent homology. By ranking and filtering frequency components according to the robustness of the topological structures they preserve in the image, this approach prioritizes features critical for downstream tasks in noisy environments, such as classification. The method enables targeted retention of image content salient to global shape and topology, and provides competitive compression performance against established standards like JPEG (Chintapalli et al., 8 Dec 2025).

1. Mathematical Foundations

At its core, persistent homology-guided frequency filtration merges frequency-domain representation with topological data analysis. The discrete Fourier transform (DFT) maps a grayscale image $f(x,y)$ of size $M \times N$ to complex coefficients $F(u,v)$, encoding the amplitude and phase for each spatial frequency. The fast Fourier transform (FFT) algorithm performs this in $O(MN \log(MN))$ time.

Persistent homology provides topological signatures through filtrations on cubical complexes formed from the image grid. Sublevel sets $$K_\alpha = \{ \text{pixels with intensity} \leq \alpha \}$$ yield a sequence of nested complexes. Homology groups $H_k(K_\alpha)$ capture $k$-dimensional holes (connected components, loops, voids) with Betti numbers $\beta_k(\alpha)$ quantifying counts at each dimension. Persistent features are described via pairs $(b_i,d_i)$, corresponding to their birth and death in the filtration; the multiset $Dg_k(f)$ forms the persistence diagram, with lifetimes %%%%10%%%% quantifying feature robustness.

2. Algorithmic Pipeline

The method follows a structured pipeline for image analysis and compression:

1. Preprocessing: Convert RGB image to grayscale, resize to standard grid (e.g., $128 \times 128$).
2. Topological Signature Computation: Generate the cubical complex and compute the full persistence diagram $Dg_{\text{full}} = \bigcup_k Dg_k(f)$.
3. Fourier Spectrum Analysis: Apply FFT to obtain $F(u,v)$ for all frequency indices.
4. Frequency Ranking: For each $(u,v)$ frequency:
   • Zero all coefficients except $F(u,v)$ and its conjugate.
   • Inverse FFT reconstructs $f_{u,v}(x,y)$ from a single frequency.
   • Compute $Dg_{u,v}$, the persistence diagram of this reconstruction.
   • Calculate the 1-Wasserstein distance $W_1(Dg_{\text{full}}, Dg_{u,v})$, quantifying loss in topological fidelity.
   • Assign an importance score:

   $$\text{score}(u,v) = \frac{W_1(Dg_{\text{full}}, Dg_{u,v})}{\sqrt{u^2 + v^2}}$$

   This favors low-frequency components reflecting the $1/f$ spectral property of natural images. - Sort $(u,v)$ indices by descending importance.

5. Frequency Selection: Retain top $K = \alpha MN$ frequencies by score or select indices by persistence threshold $\tau$ (i.e., lifetimes $\geq \tau$).
6. Reconstruction & Smoothing: Form $\hat F(u,v)$ with selected frequencies, set others to zero, and recover compressed image $\hat f(x,y)$ via inverse FFT. Optionally, apply Gaussian smoothing $G_\sigma (\sigma=1)$.

3. Central Formulations

The compression procedure exploits explicit connections between persistence modules and optimization objectives:

• Persistence Module: For $f: X \to \mathbb{R}$, sublevel filtration $X_\alpha = f^{-1}((-\infty, \alpha])$ induces $Dg_k(f) = \{(b_i, d_i)\} \subset \mathbb{R}^2$.
• Frequency Filtration by Threshold: $$S_\tau = \{(u,v): \text{pers}(u,v) = (d_i - b_i) \geq \tau\}$$.
• Compression–Fidelity Trade-off: For given frequency budget $K$:

$D_{\text{topo}} = W_1(Dg_{\text{full}}, Dg_S)$

where $S$ is the selected frequency subset, and $Dg_S$ is its persistence signature.

4. Pseudocode Overview

A high-level pseudocode for the methodology encapsulates preprocessing, persistence computation, spectral analysis, scoring, selection, and reconstruction:

1. f ← RGB_to_grayscale(f_original) 2. f ← resize(f, M, N) 3. Dg_full ← compute_cubical_PH(f) 4. F ← FFT2D(f) 5. For each (u,v) in [0..M−1]×[0..N−1]: F_uv_only ← zero all F except (u,v) and conjugate f_uv ← IFFT2D(F_uv_only) Dg_uv ← compute_cubical_PH(f_uv) W ← Wasserstein₁(Dg_full, Dg_uv) score(u,v) ← W / sqrt(u² + v² + ε) 6. Select: K ← floor(α ⋅ M ⋅ N) S ← top-K by score(u,v) # or S ← {(u,v): score(u,v) ≥ τ} 7. F̂ ← zeros_like(F); set F̂[u,v] ← F[u,v] for S and conjugates 8. hat_f ← real(IFFT2D(F̂)) 9. hat_f ← GaussianSmooth(hat_f, σ=1) 10. Return hat_f

5. Quantitative Performance and Evaluation Metrics

Performance is benchmarked against JPEG compression using six established metrics over datasets including Tiny-ImageNet, CIFAR-10, and STL-10:

Metric	Description	Outcome
Mean Squared Error (MSE)	$MSE = (1/N) \sum (I_1[i] - I_2[i])^2$	PH matches JPEG at $\sim$40% retention
SSIM	Structural similarity index, $[0,1]$	SSIM$_{\text{PH}} > 0.95$ at $\sim$40% retention
Wasserstein Distance ($W_1$)	1-Wasserstein between diagrams	PH outperforms JPEG above $\sim$50% retention
Bottleneck Distance ($d_B$)	Bottleneck metric on diagrams	Topological distance approaches zero by 80%
Betti Number Distance	$L^1$ difference of Betti curves	Monitors homology structure loss
Final File Size (KB)	Disk cost for compressed file	PH’s file-size curve non-monotonic

Key observations include robust topological fidelity with >50% retained frequencies, and file sizes for PH compression peaking at intermediate retentions due to mid-frequency noise but decreasing again at high retention; JPEG sizes increase near-linearly with quality. For a representative jellyfish image: at 30% retention, PH yields 7.4 KB (SSIM$_{\text{PH}}$=0.94, $W_1$=0.15) versus JPEG’s 1.4 KB (SSIM=0.96, $W_1$=0.30); at 90%, both methods achieve SSIM$>$0.99, but PH maintains lower topological distance ($W_1$=0.02) (Chintapalli et al., 8 Dec 2025).

6. Applications and Implications

Persistent homology-guided frequency filtration finds utility in:

• CNN Preprocessing: Retained frequencies transmit robust topological information (loops, components) useful for binary classification under noise, improving reliability relative to JPEG-based compression.
• Noisy Environments: Selective filtering suppresses topologically weak frequencies, filtering out structural noise rather than applying blanket low- or high-pass thresholds.
• Real-Time and Embedded Hardware: Approximating Wasserstein distance (e.g., via Betti curves or persistence landscapes) may facilitate hardware-friendly implementations.
• Denoising and Compression Extension: Two-stage compression is feasible, with PH-guided selection followed by entropy coding (e.g., Huffman encoding) to exploit retained coefficient sparsity.

Early experiments suggest PH-filtered inputs can improve CNN accuracy in noisy or perturbed scenarios. The methodology provides a bridge between frequency-space filtering and topological analysis, enabling direct control over shape fidelity in image compression, and holds promise for further applications in domains sensitive to global image structure rather than strictly pixel-level similarity (Chintapalli et al., 8 Dec 2025).