Moore-Penrose Pseudoinverse

Definition: Moore-Penrose Pseudoinverse, or simply the Moore-Penrose inverse, is a mathematical operation that produces an inverse of a matrix that may not have a conventional inverse. For a square matrix A, the inverse $A^{-1}$ exists if and only if the determinant of A is non-zero. However, many matrices encountered in practice are not invertible. The pseudoinverse is a generalization of the inverse that can be used for any matrix, regardless of its invertibility (not a square or a full rank matrix).In image reconstruction, to recover the optical properties of the voxels, the computation of the Moore-Penrose Pseudoinverse $A^+$ is required.Since fNIRS measurements contain different types of noise, $A^+$ is formulated including Tikhonov regularization to aid in smoothing the noise, as follows: $A^{+} = A^{T} {(A A^{T} + \alpha \mathbf{I})}^{-1}$ ,

$$A^+ = A^T (AA^T + \alpha \mathbb{1})^{-1} $ where $\alpha$ is the regularization parameter and $\mathbb{1}$ is the iden$ty matrix. It should be noted that Moore-Penrose's is not the only pseudo inverse. Category: Math Related terms: Image reconstruction Abbreviation or symbol: $

A^+

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References:

https://doi.org/10.1017/S0305004100030401

https://doi.org/10.1063/5.0015512

https://doi.org/10.1117%2F1.JBO.26.5.056001

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