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Mathematical morphology operations are reframed using persistent homology (PH). This topological data analysis approach enables automated image denoising and structure analysis, yielding results comparable to deep learning methods.

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Area of Science:

  • Image processing
  • Topological data analysis
  • Mathematical morphology

Background:

  • Mathematical morphology provides established image processing techniques applicable across diverse scientific fields.
  • Persistent homology (PH) is a key tool in topological data analysis (TDA) for uncovering shape and structure in data.

Purpose of the Study:

  • To reframe morphological operations within the framework of persistent homology.
  • To demonstrate the utility of PH for analyzing image topology and geometry.
  • To develop an automated, unsupervised image denoising method.

Main Methods:

  • Viewing morphological operations as a multiparameter filtration.
  • Applying persistent homology to extract topological and geometric information from images.
  • Developing an automated denoising algorithm for various image types and noise levels.

Main Results:

  • Morphological operations naturally form multiparameter filtrations suitable for PH analysis.
  • The PH framework allows for automated optimization of image structure study and rendering.
  • The developed unsupervised denoising method achieves comparable performance to supervised deep learning techniques.

Conclusions:

  • Persistent homology offers a powerful lens for understanding and enhancing mathematical morphology.
  • This approach enables effective, automated image denoising across different image types and noise conditions.
  • The integration of TDA with image processing opens new avenues for data analysis in science and engineering.