Persistent Homology-Guided Frequency Filtering for Image Compression

TL;DR

Persistent Homology-Guided Frequency Filtering for Image Compression introduces a method that couples PH with the DFT to identify image frequencies tied to robust topological features. The approach computes persistence diagrams for grayscale images, ranks Fourier frequencies by a topological-impact score using Wasserstein distance, and uses selected frequencies for PH-based compression with Gaussian smoothing. The study compares PH compression with JPEG across six metrics and demonstrates competitive visual quality while improving topological fidelity at moderate to high frequency retention, though with larger file sizes. The work suggests potential benefits for topology-aware preprocessing in noisy imaging and for improving classification tasks without heavy augmentation.

Abstract

Feature extraction in noisy image datasets presents many challenges in model reliability. In this paper, we use the discrete Fourier transform in conjunction with persistent homology analysis to extract specific frequencies that correspond with certain topological features of an image. This method allows the image to be compressed and reformed while ensuring that meaningful data can be differentiated. Our experimental results show a level of compression comparable to that of using JPEG using six different metrics. The end goal of persistent homology-guided frequency filtration is its potential to improve performance in binary classification tasks (when augmenting a Convolutional Neural Network) compared to traditional feature extraction and compression methods. These findings highlight a useful end result: enhancing the reliability of image compression under noisy conditions.

Figures (8)

• Figure 1: Oriented $k$-simplices with $k = 0, 1, 2, 3$. Adapted from edelsbrunner2010computational.
• Figure 2: Diagram of the filtration of simplicial complexes used to compute the persistent homology of a point cloud.
• Figure 3: Persistence diagram and Persistence Barcodes generated from a Swiss roll dataset using scikit-learnscikit-learn. In the persistence diagrams, the blue points represent generators of $H_0$ cycles and the orange points represent generators of $H_1$ cycles, with $x$-coordinates as birth radius and $y$-coordinates as the death radius. Features further from the diagonal are more persistent. In the persistence barcodes, the blue lines are generators of $H_0$ cycles and the orange lines are generators of $H_1$ cycles. The beginning of the line on the left represents the radius at the time of birth, and the end of the line on the right represents the radius at the time of death.
• Figure 4: Binary image highlights foreground (value 1) and background (value 0) pixels. The cubical complex is built by placing cells around the foreground region, enabling topological analysis of the image’s structure through persistent homology. dongjin_figure_2023
• Figure 5: Diagram of the FFT and how it computes frequency from time where the colored lines represent different input frequencies and the black line represents the output frequency.