Can the Kalman achieve low mean square errors in these two mentioned scenarios.
Gaussian noise can be reduced using a spatial filter, though when smoothing an image, an undesirable outcome may result in the blurring of fine-scaled image edges and details because they also correspond to blocked high frequencies.
I think the KF will work OK for most noise distributions, so long as the errors are zero mean, and the distribution is symmetric around the mean. But it won't be optimal anymore, so you may as well use a different (simpler) smoothing filter without so many statistical assumptions.
As noise is considered to be AC with zero mean - regardless of the probability distribution - Kalman filters do what's expected regarding noise.
It is more about the applicability to your 'problem': if noise is low compared to the signal, other filters may be more applicable. The same applies eg. if you have 'prior knowledge' about your signal. Digging a bit deeper (starting eg. here: https://en.wikipedia.org/wiki/Kalman_filter) may lead you to alternatives.
I have uploaded a tutorial on the "(simpler) smoothing filter[s]" that I mentioned in my earlier answer.
https://www.researchgate.net/publication/304508893_A_tutorial_on_non-Bayesian_target-tracking_Steady-state_Kalman_alpha-beta_and_fading-memory_polynomial_filters
Technical Report A tutorial on non-Bayesian target-tracking: Steady-state Kal...
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