Graph spectral image smoothing using the heat kernel

Fan Zhang, Edwin R. Hancock

Research output: Contribution to journalArticlepeer-review

Abstract

A new method for smoothing both gray-scale and color images is presented that relies on the heat diffusion equation on a graph. We represent the image pixel lattice using a weighted undirected graph. The edge weights of the graph are determined by the Gaussian weighted distances between local neighboring windows. We then compute the associated Laplacian matrix (the degree matrix minus the adjacency matrix). Anisotropic diffusion across this weighted graph-structure with time is captured by the heat equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigensystem with time. Image smoothing is accomplished by convolving the heat kernel with the image, and its numerical implementation is realized by using the Krylov subspace technique. The method has the effect of smoothing within regions, but does not blur region boundaries. We also demonstrate the relationship between Our method, standard diffusion-based PDEs, Fourier domain signal processing and spectral Clustering. Experiments and comparisons on standard images illustrate the effectiveness of the method. (C) 2008 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)3328-3342
Number of pages15
JournalPattern recognition
Volume41
Issue number11
DOIs
Publication statusPublished - Nov 2008

Keywords

  • image smoothing
  • heat kernel
  • graph Laplacian
  • graph spectra
  • heat equation
  • anisotropic diffusion
  • denoising
  • low-pass filtering
  • KRYLOV SUBSPACE APPROXIMATIONS
  • MATRIX EXPONENTIAL OPERATOR
  • NONLINEAR DIFFUSION
  • ANISOTROPIC DIFFUSION
  • EDGE-DETECTION
  • SCALE-SPACE
  • MULTIVALUED IMAGES
  • RESTORATION
  • SEGMENTATION
  • ENHANCEMENT

Cite this