I would like some help interpreting the following formula, equation 1 from this paper: https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/strela.pdf

$\hat{X} = \sum\limits_{m}\frac{\beta_{m}^{2} }{\beta_{m}^{2} + \sigma^{2}} \langle X, g_{m} \rangle g_{m}$

It is used to perform a wiener filtering in the wavelet domain in a block-based approach. Suppose $X$ is an $n\times n$ block of wavelet co-efficents to be denoised (just a 2D matrix). In the paper the covariance matrix of the clean signal in $X$ is estimated (for denoising), as a matrix $C$. The formula above uses the matrix $C$ to filter $X$ to produce a clean estimate $\hat{X}$. $\beta_{m}$ and $g_{m}$ are eigenvalues and eigenvectors of $C$, and $\sigma$ is the known standard deviation of noise.

So: $\frac{\beta_{m}^{2} }{\beta_{m}^{2} + \sigma^{2}}$ is a scalar value.

And $\langle X, g_{m} \rangle$ is the inner product of the 2D $n\times n$ matrix $X$ and the vector $g_{m}$ of length $n$, which produces a vector of length $n$. (http://mathinsight.org/matrix_vector_multiplication).

The outer product of the vectors $\langle X, g_{m} \rangle$ and $g_{m}$, both of length $n$, produces an $n\times n$ dimensional matrix. The result of this is then multiplied by the scalar $\frac{\beta_{m}^{2} }{\beta_{m}^{2} + \sigma^{2}}$.

Is my understanding of the operations correct up to this point?

Finally, does $\sum\limits_{m}$ mean the summation of this calculation over all ($n$ linearly independent) pairs of eigenvalues and eigenvectors for the matrix $C$?

Thanks in advance for any help or suggestions.


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