The Numbers Σ J. Are Called The Singular Values Of Z

The trace statistics ʎ trace and the maximum Eigen statistics ʎ max were used and the results are presented in table 3 and 4 below.

| Z score (standardised value)-how many sds from the mean the value liesZ score = data value – mean Standard deviation

1) Divide matrices A and B in 4 sub-matrices of size N/2 x N/2 as shown in the below diagram.

The intuitive approach to the compressive sensing problem of recovering a sparse vector x ∈ RN from its measurement vector Φx=y ∈ Rm, where m < N, consists in the l0 -minimization problem

This representation can help to visualize the clusters in the high-dimensional spaces, or to automatically recognize them using relatively simple image processing techniques.

Hence the analyzed results are shown in fig.1 as old without using random mask generator and fig 2 with using random mask generator

Let R(k),E(k) and R(−k),E(−k) be the corresponding linear form and the residual and I − Ik be t set of other indices.

In the first step, the continuous problem is discretized only in time, resulting in a family of 1D linear stationary singularly perturbed problems depending on the time discretization parameter. In the second step, those problems are discretized in space. The main advantage of this technique is that it permits to analyze independently the contribution to the error of the time and space discretizations. Nevertheless, the method of \cite{221} only gives first order of uniform convergence in time variable. However, it is possible to combine this idea with the use of the Richardson extrapolation technique applied only to the time discretization. Then, the fully discrete method gives second order of uniform convergence in time and almost fourth order in space. So, the improved approximation is second order globally convergent in contrast with the first order proved in \cite{221}.

For a set of values κ1,...,κN solving local eigen value problem to detect the dominant scale (inverse of the eigen value) a offline space is constructed with basis

Then the following matrix lemma will be utilized to obtain the analytic eigenvalue solution [3]:

The QR transformation protects the upper Hessenberg type of the first matrix, and the workload on such a matrix is O(n2) per cycle as restricted to O(‪n3) on a general framework. As s tends to infinity, the original matrix merges to a structure where the eigenvalues are either segregated on the corner to corner or are eigenvalues of a 2 × 2 sub-matrix on the diagonal. Thus, we can see that the QR transformation reduces the complexity and the number of iterations. QR is one of the mostly used algorithms for finding the eigenvalues of a matrix, where Q represents an orthogonal matrix and R an upper-triangular one, R comes from right. ‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬

The DWT is carried out at the host picture to decompose the image into 4 non overlapping multi decision coefficient sets. The coefficients are:

Where, T denotes the matrix transpose, and the vectors u and v are the left-singular and right-singular vectors for $\sigma$, respectively~\cite{SVDRef}.

PCA is considered as special transformation which pro-duces the principal components (PCs) Known as eigenvectors

approximations used, inversion method explanation and other parameters explained using example of aki richards approximation