Recovering true data from multiple noisy versions I am trying to find if there is any way to get the true data from multiple noisy versions, but the true data has a peculiar property.
Problem Statement
Consider a matrix $F=[f_1, f_2, ... , f_n]$ where $f_i$ is 1984-dimensional. Now, each $f_i$ can be expressed as $f_i=H^i+n_i$. Here $H^i$ is the true (unknown) data and $n_i$ is the embedded noise. The peculiar property is that each $H^i$ has high correlation with each other (in other words, all $H^i$ are similar to each other). I want to build a filter or a vector which when multiplied with $f_i$, will give me $H^i$ or approximated $H^i$ i.e. noise should reduce.
My thoughts
I understand we should place some extra assumptions on the noise type, like Gaussian etc. In my case, I don't think my noise is Gaussian. Each feature vector $f_i$ basically represents an image patch. So, noise can be random, but I am open to modelling the noise as Gaussian and see what happens. I think a machine learning approach will work better. I had two thoughs:


*

*Learn a vector such that when multiplied with the data it should minimize $||w^Tf_i-w^Tf_{i+1}||$ should be minimized such that (say) $||w||\le1$.

*Add up a subset of $f_i$ so that by adding up different $n_i$, it will approach Gaussian (?) and I assume there are methods to recover data when noise is Gaussian.
I am open to any other ideas as long as I get the desired result. 
You can edit tags if necessary.
 A: I would try linear regression, at least as my first approach.  To put it in strictly matrix terms, define $F_{-i}$ as $F$ with the $i^{th}$ column removed.  Then construct the matrix $P_i = F_{-i}(F_{-i}^TF_{-i})^{-1}F_{-i}^T$ (the projection matrix.)  Your estimate of $f_i$ would then be $\hat{f_i} = P_if_i$.  
This minimizes the Euclidean distance between $f_i$ and $\hat{f_i}$ where $\hat{f_i}$ is located in the space spanned by the columns of $F_{-i}$.  Since the $H^i$ are highly correlated, there should be some explanatory power in $F_{-i}$, but how much depends upon the variance of the noise vectors relative to the variance of the $H^i$ themselves.
This wouldn't work well if the distribution of your noise has fat tails relative to the Gaussian distribution, but I don't know enough about your topic area to make any statement about how likely that is.
Edit: Fixed the typo in line 5.
A: There are some interesting and useful methods for automatic noise estimation and removal using multiresolution support and wavelets. I have used this to clean Gaussian noise from images to great effect. There are also techniques that are applicable to Poisson noise. The key is to estimate the level of noise and follow that up with a wavelet approach to remove noise of thae predicted level.
Granted, this doesn't use the fact that there may be high correlation between the $H^i$ but in my experience the algorithms described in the references should work well
Another possibility is to normalise the $f_i$. As @whuber mentions, in the case of no noise and perfect correlation, the normalised $f_i$'s should be equal. Ask yourself what would happen in the case of perfect correlation between the $H^i$ but in the presence of noise. Can you predict the level (or distribution) of noise under these assumptions? And what happens when you add the additional assumption that the $H_i$ are not perfectly correlated (but are highly correlated). 
A: Let's say that $F_{ti}=f_i(t)$, where $t\in [1,1984]$, then I'd simply use the average $\hat H_i=\frac{1}{1984}\sum_{t=1}^{1984} f_i(t)$.
