# Quantify the variance of pixel measurements

Say $$E_j$$ is an estimator of the value $$I_j=\int f_j\:{\rm d}\lambda$$ of the $$j$$th pixel given by the path space integral of the corresponding measurement contribution function $$f_j$$. $$f_j$$ is of the form $$f_j=h_jf$$, where $$h_j$$ is essentially the image reconstruction filter of the pixel $$j$$ (as described, for example, here).

If we want to quantify the "variance" of the estimates $$E_j$$ over all pixels $$j$$, which notion of variance should we consider? Since $$f_j$$ is 3-dimensional, the natural choice would be given by the covariance matrix of $$E_j$$. Another option would be to take the variance $$\operatorname{Var}\left[\left\|E_j\right\|^2\right]$$ of the squared Euclidean norm $$\left\|E_j\right\|$$ of $$E_j$$ (which is the trace of the covariance matrix), but this might not contain enough information.

Since I want to minimize the variances over all $$j$$, the covariance matrices are too complicated to handle.

So, my question is: Can we somehow control the supremum of all variances in terms of a scalar-valued variance? Maybe in terms of the luminance?