Closed Form Solution and Statistical Performance Analyses for Regularized Least Squares Estimation of Retinal Oxygen Tension

Abstract

For improved estimation of oxygen tension in retinal blood vessels, regularization of least squares estimation method was proposed earlier and it was shown to be very effective. However, closed form solutions for the estimation, and bias and variance of the estimator were not provided and comprehensive statistical analyses were not done. In this chapter, we derive the closed form solution for the regularized least squares estimation, bias and variance of the regularized least squares estimator and with the help of the closed form solutions, statistical performance analyses of the estimator are realized for different values of estimation parameters.

Appendix

For 3 × 3 regularization window size, $\varvec{K}$ is formed as follows:

First, we assume that the regularization window has coefficients as:

$$ \left| {\begin{array}{*{20}c} q & p & q \\ p & l & p \\ q & p & q \\ \end{array} } \right|, = \frac{a}{{4*\left( {b + c} \right) + a}},\quad p = \frac{b}{{4*\left( {b + c} \right) + a}},\quad q = \frac{c}{{4*\left( {b + c} \right) + a}} $$

(A.1)

where l, p and q denote weight of pixel to itself, to direct adjacent pixels and to cross adjacent pixels, respectively. In order to have mean of the regularization window coefficients be one, we normalize these coefficients.

After defining l, p and q, we form the $\varvec{K}$ as follows:

$$ \varvec{K}(j,k) = \left\{ {\begin{array}{*{20}l} {l\;\;if\;\;j = k} \hfill \\ {p\;\;if\;\;j = k \pm 1} \hfill \\ {p\;\;if\;\;j = k \pm M} \hfill \\ {q\;\;if\;\;j = k + M \pm 1} \hfill \\ {q\;\;if\;\;j = k - M \pm 1} \hfill \\ {0\;\;otherwise} \hfill \\ \end{array} } \right\}, $$

(A.2)

where K(j, k) denotes weight coefficient of the k-th pixel on j-th pixel in the image, and M denotes the number of rows. For the regularization window 5 × 5 size, the same approach described above can be followed.