Efficient Image Denoising by Low-Rank Singular Vector Approximations of Geodesics' Gramian Matrix

TL;DR

This paper introduces a manifold-based image denoising method that leverages low-rank approximations of singular vectors of geodesic Gramian matrices, offering an efficient alternative to traditional SVD-based techniques.

Contribution

It proposes a novel denoising framework that uses low-rank singular vector approximations of geodesic Gramian matrices, reducing computational complexity compared to SVD-based methods.

Findings

The proposed method achieves comparable denoising performance with reduced computational time.

Efficient approximation techniques successfully reveal prominent singular vectors without full SVD.

The framework effectively partitions images into patches for localized denoising.

Abstract

With the advent of sophisticated cameras, the urge to capture high-quality images has grown enormous. However, the noise contamination of the images results in substandard expectations among the people; thus, image denoising is an essential pre-processing step. While the algebraic image processing frameworks are sometimes inefficient for this denoising task as they may require processing of matrices of order equivalent to some power of the order of the original image, the neural network image processing frameworks are sometimes not robust as they require a lot of similar training samples. Thus, here we present a manifold-based noise filtering method that mainly exploits a few prominent singular vectors of the geodesics' Gramian matrix. Especially, the framework partitions an image, say that of size $n \times n$, into $n^2$ overlapping patches of known size such that one patch is centered at each pixel. Then, the prominent singular vectors, of the Gramian matrix of size $n^2 \times n^2$ of the geodesic distances computed over the patch space, are utilized to denoise the image. Here, the prominent singular vectors are revealed by efficient, but diverse, approximation techniques, rather than explicitly computing them using frameworks like Singular Value Decomposition (SVD) which encounters $\mathcal{O}(n^6)$ operations. Finally, we compare both computational time and the noise filtration performance of the proposed denoising algorithm with and without singular vector approximation techniques.