  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.
