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?


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