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MMPersistence: A mathematical morphology-oriented software library for computing persistent homology on cubical complexes

The paper introduces MMPersistence, a software library that integrates mathematical morphology operations with persistent homology on cubical complexes to extract multiscale topological features from digital images by leveraging diverse structuring elements for richer geometric and topological analysis.

Original authors: Chuan-Shen Hu

Published 2026-02-18

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Chuan-Shen Hu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a black-and-white photograph. To a computer, this image is just a grid of numbers (pixels). To a mathematician, it's a landscape of shapes, holes, and connections.

For a long time, the best way for computers to "understand" the shape of an image was a method called Persistent Homology. Think of this like watching a time-lapse video of a snowball melting. As the snowball shrinks, you can count how many separate ice chunks exist and how many holes (like a donut shape) appear and disappear. This tells you about the global structure of the image.

However, this traditional method has a flaw: it's obsessed with intensity. If you have a tiny, bright speck of dust on a photo, the computer thinks, "Wow, that's a huge, important feature!" because the pixel value is extreme. But if you have a large, faint gray cloud, the computer might ignore it because the pixel values are "boring" and average. It's like judging a book solely by the brightness of its cover, ignoring the story inside.

Enter: MMPersistence (The "Morphology" Library)

The paper introduces a new software tool called MMPersistence. It combines the shape-analysis power of Persistent Homology with a classic image-processing technique called Mathematical Morphology.

To understand this, let's use a creative analogy: The Cookie Cutter.

1. The Old Way (Sublevel-Set Filtration)

Imagine you have a tray of cookies (your image). The old method looks at the cookies based on how "dark" they are. It says, "Let's only look at the darkest cookies first, then the slightly lighter ones."

The Problem: If you sprinkle a single grain of black pepper (noise) on a white cookie, the computer gets confused. It thinks that tiny grain is a massive, important structure because it's the darkest thing there. It distorts the whole picture.

2. The New Way (MMPersistence)

MMPersistence introduces a Cookie Cutter (called a Structuring Element). Instead of just looking at darkness, it physically tries to "cut" shapes out of the image.

Erosion (The Shrinker): Imagine pressing a cookie cutter down and removing everything that doesn't fit inside it. This shrinks the white parts of your image. If a tiny speck of noise is smaller than your cutter, it gets wiped out completely. The computer learns: "This speck wasn't a real structure; it was just noise."
Dilation (The Expander): Imagine the cookie cutter pushing the edges of the white parts outward. This helps connect nearby islands of white that are separated by a thin gap.
Opening & Closing: These are combinations of shrinking and expanding. "Opening" is like smoothing out a bumpy cookie edge by shrinking it and then expanding it back. It removes small bumps but keeps the main shape.

How It Works in Practice

The library takes your image and runs it through a series of these "cookie cutters" of increasing sizes.

The Process: It starts with a tiny cutter, then a medium one, then a big one. At each step, it counts the holes and shapes.
The Result: Instead of just seeing "a hole," the computer now sees "a hole that is 5 pixels wide" or "a hole that is 10 pixels wide."
The Magic: Because it uses these physical cutters, it ignores the tiny, noisy specks (the pepper grain) that would have confused the old method. It focuses on the actual shape and size of the objects.

Why Does This Matter?

The authors tested this on images with "salt noise" (random white or black specks).

Old Method: The noise made the computer think the image was totally different. The "fingerprint" of the image changed drastically.
MMPersistence: The computer looked at the image with its cookie cutters, ignored the tiny noise, and said, "Hey, this is still the same image!" The "fingerprint" remained stable.

The Big Picture

Think of MMPersistence as a new pair of glasses for data scientists.

Old Glasses: Only saw how bright or dark things were.
MMPersistence Glasses: See the size, shape, and thickness of things.

This is huge for fields like medical imaging (looking at cell structures), where you need to know if a network of cells is connected or broken, regardless of how bright the image is. It also helps in AI, giving robots a better, more human-like understanding of the shapes in the world around them.

In short: MMPersistence is a software tool that helps computers understand images not just by their colors, but by their actual physical shapes, making them much better at ignoring noise and seeing the real story.

1. Problem Statement

Traditional Persistent Homology (PH) applied to digital images typically relies on sublevel-set filtrations. In this approach, a grayscale image is treated as a function where pixel intensity determines the filtration order. While effective, this method has significant limitations:

Intensity Bias: It prioritizes topological structures formed by pixels with extreme intensity values (very bright or very dark). Structures composed of pixels with moderate intensities, even if they occupy large spatial regions, may be deemed insignificant due to their "averaged" birth-death information.
Sensitivity to Noise: The method is highly sensitive to salt-and-pepper noise. A single pixel with an extreme intensity value can create a topological feature with a disproportionately long lifespan in the Persistence Diagram (PD), distorting the true geometric and spatial characteristics of the image.
Lack of Geometric Context: Conventional PH captures global topology (connected components, loops) but often fails to encode specific local geometric properties, such as the physical size, thickness, or shape of structures, which are crucial for image analysis.

2. Methodology

The paper proposes MMPersistence, a Python library that integrates Mathematical Morphology (MM) with Persistent Homology to address these limitations. The core methodology involves constructing filtrations based on morphological operations rather than pixel intensity thresholds.

Core Concepts

Mathematical Morphology (MM): Utilizes set-theoretic operations (Erosion, Dilation, Opening, Closing) driven by a Structuring Element (SE). The library primarily uses a sequence of nested square SEs ( $S_1 \subseteq S_2 \subseteq \dots \subseteq S_n$ ).
Cubical Complexes: Digital images are represented as cubical complexes where black pixels correspond to closed unit squares.
Filtration Construction:
- Binary Images: The library constructs filtrations by applying sequences of MM operations.
  - Erosion/Opening Filtration: As the SE size increases, white regions shrink, and black regions expand. This creates an increasing sequence of topological spaces ( $\mathcal{X}_f \subseteq \mathcal{X}_{\epsilon_{B_1}(f)} \subseteq \dots$ ).
  - Dilation/Closing Filtration: As the SE size increases, black regions shrink, and white regions (holes) expand/connect.
- Grayscale Images: The framework extends to grayscale images by first applying a threshold to create a binary image, then applying the MM-based filtration. This creates a bifiltration (parameterized by both threshold and SE size), though the current implementation focuses on aggregating results across multiple thresholds.

Software Implementation

Language: Python.
Dependencies: Built upon the GUDHI library for cubical complex PH computation and Persim (Scikit-TDA) for visualization.
Operations: Supports Erosion, Dilation, Opening, and Closing filtrations for both binary and grayscale images.

3. Key Contributions

MMPersistence Library: The release of an open-source Python library that automates the construction of MM-based filtrations and the computation of PH on cubical complexes.
Geometric Interpretation of Persistence Intervals:
- Erosion/Opening: The "death" values in the 1st PD (loops) correspond to the thickness or size of the holes/structures. Unlike sublevel-set PH, where holes are indistinguishable if they have similar intensity ranges, MM-based PH distinguishes them by their physical dimensions.
- Dilation/Closing: These operations capture the connectivity and spatial sparseness of structures. The lifespan of a feature in the PD reflects how far apart adjacent holes are before they merge.
Robustness to Noise: The paper demonstrates that MM-based PH is significantly more robust to salt-and-pepper noise than conventional sublevel-set PH. Extreme intensity noise pixels are often filtered out by morphological operations before topological analysis, preventing them from creating spurious long-lived features.
Unified Framework: It establishes a unified approach for analyzing digital images that combines the local geometric sensitivity of MM with the global topological insight of TDA.

4. Results

The authors validated the library using synthetic and real-world image examples:

Binary Image Analysis:
- In images with rectangular holes of varying sizes, the erosion-based PD correctly encoded the widths of the holes as death values.
- In images with irregular curves, the PD distinguished between the original holes (birth at 0) and artifacts generated by the erosion process (birth > 0), providing a quantitative measure of curve "thickness."
- Dilation-based filtrations successfully quantified the spatial arrangement of holes; holes that were far apart resulted in longer persistence intervals (longer lifespans) compared to clustered holes.
Grayscale Image Analysis (Noise Robustness):
- A comparison was made between a clean image and a "salt-noise" corrupted image.
- Conventional PH: The bottleneck distance between the PDs of the clean and noisy images was high ( $\approx 0.4353$ ), indicating a drastic change in topological features due to noise.
- MM-based PH: Using an opening filtration, the bottleneck distance dropped significantly ( $\approx 0.091$ ). The noise artifacts were treated as small, short-lived features near the diagonal of the PD, while the true structural features remained stable.
Cluster Formation: The study observed that PIs with similar death values tend to form clusters in the PD, reflecting the geometric grouping of structures in the image.

5. Significance

Enhanced Image Analysis: MMPersistence provides a richer, more interpretable representation of image data by encoding spatial and morphological characteristics (size, shape, connectivity) that are lost in intensity-based PH.
Noise Resilience: The framework offers a principled way to filter out high-frequency noise (like salt-and-pepper) without losing critical topological information, making it highly suitable for biomedical and microscopy imaging where noise is prevalent.
Deep Learning Integration: By providing stable and differentiable vectorizations (as noted in related work cited), MMPersistence serves as a bridge for integrating topological priors into deep learning architectures, potentially improving model interpretability and reliability.
Open Science: The release of the library lowers the barrier for researchers to apply advanced morphological-topological analysis to 2D image data, fostering further research in Topological Data Analysis (TDA) and computational topology.

In summary, MMPersistence shifts the paradigm of persistent homology in image processing from a purely intensity-driven approach to a geometry- and shape-driven approach, offering superior robustness and richer structural insights for digital image analysis.