# Understanding Symmetric Smoothing Filters: A Gaussian Mixture Model Perspective

arXiv: https://arxiv.org/abs/1601.00088 · HTML: https://mlsanity.com/paper/1601.00088/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jan 2016.

## Built on by (most cited first, 2 of 2)

- [Graph Signal Processing for Heterogeneous Change Detection Part I: Vertex Domain Filtering](https://mlsanity.com/paper/2208.01881/) (2022)
- [Graph Signal Processing for Heterogeneous Change Detection Part II: Spectral Domain Analysis](https://mlsanity.com/paper/2208.01905/) (2022)

## Abstract (arXiv)

Many patch-based image denoising algorithms can be formulated as applying a smoothing filter to the noisy image. Expressed as matrices, the smoothing filters must be row normalized so that each row sums to unity. Surprisingly, if we apply a column normalization before the row normalization, the performance of the smoothing filter can often be significantly improved. Prior works showed that such performance gain is related to the Sinkhorn-Knopp balancing algorithm, an iterative procedure that symmetrizes a row-stochastic matrix to a doubly-stochastic matrix. However, a complete understanding of the performance gain phenomenon is still lacking. In this paper, we study the performance gain phenomenon from a statistical learning perspective. We show that Sinkhorn-Knopp is equivalent to an Expectation-Maximization (EM) algorithm of learning a Gaussian mixture model of the image patches. By establishing the correspondence between the steps of Sinkhorn-Knopp and the EM algorithm, we provide a geometrical interpretation of the symmetrization process. This observation allows us to develop a new denoising algorithm called Gaussian mixture model symmetric smoothing filter (GSF). GSF is an extension of the Sinkhorn-Knopp and is a generalization of the original smoothing filters. Despite its simple formulation, GSF outperforms many existing smoothing filters and has a similar performance compared to several state-of-the-art denoising algorithms.