# Image denoising as a conditional expectation

arXiv: https://arxiv.org/abs/2505.21546 · HTML: https://mlsanity.com/paper/2505.21546/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 59 references that are in the map.
- First posted on arXiv 24 May 2025.

## Builds on (most cited first, 5 of 5)

- [Image Segmentation Using Deep Learning: A Survey](https://mlsanity.com/paper/2001.05566/) (2020)
- [Understanding Deep Learning Techniques for Image Segmentation](https://mlsanity.com/paper/1907.06119/) (2019)
- [Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation](https://mlsanity.com/paper/2006.02037/) (2020)
- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)
- [Conditional expectation using compactification operators](https://mlsanity.com/paper/2306.10592/) (2023)

## Abstract (arXiv)

All techniques for denoising involve a notion of a true (noise-free) image, and a hypothesis space. The hypothesis space may reconstruct the image directly as a grayscale valued function, or indirectly by its Fourier or wavelet spectrum. Most common techniques estimate the true image as a projection to some subspace. We propose an interpretation of a noisy image as a collection of samples drawn from a certain probability space. Within this interpretation, projection based approaches are not guaranteed to be unbiased and convergent. We present a data-driven denoising method in which the true image is recovered as a conditional expectation. Although the probability space is unknown apriori, integrals on this space can be estimated by kernel integral operators. The true image is reformulated as the least squares solution to a linear equation in a reproducing kernel Hilbert space (RKHS), and involving various kernel integral operators as linear transforms. Assuming the true image to be a continuous function on a compact planar domain, the technique is shown to be convergent as the number of pixels goes to infinity. We also show that for a picture with finite number of pixels, the convergence result can be used to choose the various parameters for an optimum denoising result.