Accessing Maximum Value from a Singular Value Decomposed Matrix

Question

I have read few image enhancement papers, where an image is enhanced in transformed domain using either DCT (discrete cosine transform) or DWT (discrete wavelet transform) with SVD (singular value decomposition). An example is given in Enhancement of low contrast satellite images using discrete cosine transform and singular value decomposition, 2011.

In these works, once the image matrix is factored into three matrices (say $U$, $S$, and $V$) using the SVD, then only the maximum value of matrix S is taken for processing. Can anyone please tell that why only the maximum value is taken for processing?

Answer 1

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).

Answer 2

[EDIT: some code made available] A common framework for (multivariate) image processing is to suppose that its useful features (edges, textures, spectral correlation) contain redundancy, while the noise is pretty uncorrelated. A major preprocessing task consists in decorrelating or compacting the image features prior to processing, hoping to separate them from unwanted noises better in some transformed domain, or by projection onto a set of vectors $v_i$.

One holy grail in image $I$ decomposition is to find the best $M$-term approximation, so that a norm $\|I-\sum_1^M \lambda_i v_i\|$ is minimal, when $M$ is small with respect to the size of the image. Such a problem is generally very difficult. So many methods try to find the best $v_i$s or $\lambda_i$ (or both) under some assumptions, or using optimization methods.

Two main types of linear transformations are usually considered:

1. adaptive transformations, dependent on image characteristics
2. fixed transformations, independent of images, but generally designed to capture features of image classes or models

SVD, PCA, Karhunen-Loève transformations belong to Type 1. Fourier, DCT and wavelets belong to Type 2. However, they are somehow related, and under some assumptions, Fourier derives from PCA, and DCT is a good approximation to Karhunen-Loève. They are all related to minimum squared norm formulations, provide "orthogonal" vectors $v_i$, and very often, one considers that the energy of the projection coefficients $\lambda_i$ onto the $v_i$s are good enough representations for practical purposes, less efficient than best $M$-terms, but easier to compute.

Now, the bigger the $\lambda_i^2$, the more it contributes to the energy of the image. Resultingly, as long as the vectors have the same norm, it is a good bet to select the most important projection coefficients $|\lambda_i$| (in magnitude), to provide a sparse representation of the image.

Then, depending on the direction (spatial, spectral), fixed and adapted transformations can be combined, such as PCA (or other learnt linear transformations) and wavelets. Three additional sources on multivariate denoising for you:

• Multivariate denoising using wavelets and principal component analysis, 2006, Computational Statistics & Data Analysis
• A Nonlinear Stein Based Estimator for Multichannel Image Denoising, 2008, IEEE Transactions on Image Processing (the decorrelating part is discrete, but dwells in Section III.B) [EDIT: Matlab code is now available in $M$-band 2D dual-tree (Hilbert) wavelet multicomponent image denoising]
• Hyperspectral Image Compression Using JPEG2000 and Principal Component Analysis, 2007, IEEE Geosciences Remote Sensing Letters