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

TL;DR

The paper addresses denoising by exploiting a patch-based manifold model, forming a geodesic distance Gramian over patches and denoising via a small set of its singular vectors. To overcome the $O(n^6)$ cost of full SVD on the Gramian, four efficient singular-vector approximation schemes—MCLA, ALB, PIME, and RSVD—are integrated into the Geodesic Gramian Denoising (GGD) framework, creating GGD-MCLA, GGD-ALB, GGD-PIME, and GGD-RSVD variants. Empirical results across standard images show RSVD typically offers the best PSNR with substantial speedups, while SSIM remains broadly similar across hybrids; the approach preserves edges and textures better than naive methods in many settings. The work demonstrates a robust, patch-based, manifold-aware denoising strategy that scales more favorably than exact SVD and points toward extensions to color images and video, potentially benefiting image restoration tasks where training data for deep models is limited.

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.

Figures (3)

• Figure 1: Computational time, in seconds ($s$), of GGD-SVD and four of its hybrid variants with respect to the length of one side of a square-shaped image, i.e, $n$ for an image of $n\times n$. First, we create six versions of the image Barbara with the sizes $n=50, 60, 70, 80, 90$, and $100$, and impose an additive Gaussian noise sample with relative percentage noise of 30%. We executed all five frameworks over these test images on a computer with the configuration of $11^{th}$ Generation Intel Core i7-1165G7, 4 cores each with 4.70 GHz, 8 GB DDR4 3200MHz RAM. (a) Computational time attained by each framework of denoising six test images where the plots of GGD-MCRA, GGD-ALB, GGD-RSVD, and GGD-PIME are overlapped. While (b) shows the first cropped version of (a) which limits the vertical axis at 1$s$, (c) shows the second cropped version of (a) which limits the vertical axis at 0.1$s$.
• Figure 2: Denoising images using GGD-SVD and four of its hybrid frameworks, GGD-MCLA, GGD-ALB, GGD-PIME, and GGD-RSVD. Three test images, namely, Barbara, cameraman, and mandrill, are imposed with three Gaussian relative noise levels, $\zeta=$ 20%, 30%, and 40%. Each noisy image is denoised using each method along with its best parameter combination $(\delta, \rho, L)$, shown in Table \ref{['tab:psnr']}, which is optimized with respect to PSNR.
• Figure 3: Denoising images using GGD-SVD and four of its hybrid frameworks, GGD-MCLA, GGD-ALB, GGD-PIME, and GGD-RSVD, with optimized parameters with respect to SSIM. Three test images, namely, Barbara, cameraman, and mandrill, are imposed with three Gaussian relative noise levels, $\zeta=$ 20%, 30%, and 40%. Each noisy image is denoised using each method along with its best parameter combination $(\delta, \rho, L)$, shown in Table \ref{['tab:ssim']}, which is optimized with respect to SSIM.

Theorems & Definitions (5)

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