Accessing Maximum Value from a Singular Value Decomposed Matrix
The SVD Decompose the image into the (One way to look at it) many matrices.
For instance, given an Image $ I $ its SVD is given by:

$$ I = U S {V}^{T} = \sum_{i=1}^{\textrm{rank}(I)} {s}_{i} {u}_{i} {v}_{i}^{T} $$

As you can see, the image is decomposed into linear combination of matrices (Given by $ {u}_{i} {v}_{i}^{T} $) where the weight of the $ i $ -th matrix is given by $ {s}_{i} $ which is the $ i $-th Singular Value (Usually they are organized in descending order).

The dominant features are usually goes with the large Singular Values (Moreover, the smallest one usually describe noise).
I guess methods want to focus on the large dominant features and avoid dealing with noise hence it makes sense to deal only with the dominant matrix in this decomposition.

By the way, I'd experiment with trying to work on the next ones as well.
In addition, on the recompositon process I'd neglect the last few to have "Simple Denoising".

One should be aware that the SVD of the whole image might be time and memory consuming when dealing with large images.

Moreover, it makes sense to stack few image patches into large matrix and then apply the SVD.
This might extract what is common to all patches (Linear Dimensionality Reduction).