Jasper den Ouden: 20200814
0.1 The model
Real image k −> R_{k} where k indexes values.
The seen value is defined as:
Where f_{kl} is presumed known the factor how much k pixel of R contributes to l pixel of V.
To be clearer; the model has R_{k} which is an image we expect there to be out there. k indexes that image, in a more specific example later, it will be a higher resolution raster image with k=(i,j) just two image coordinates.
Also in the later specific example V_{l} is the real value as measured by a pixel of a camera, and v_{l} the measured value, off due to random variation. l=(i,j,t) with i and j image coordinates and t the time, or frame number. The camera has a lower resolution than the real image in the model.
I was mistaken that weighed averages are of images scaled and translated over where the perfect choice under the assumptions. See below about that.
The actual measuring is also part of the model,V_{l} is the value, and v_{l} the actually measured number. Yielding independent gaussian solutions, which means you can multiply them for the combined probabilty;
P(k−>R_{k}) ∝   exp  (V_{l} − v_{l})^{2} 

2σ_{l}^{2} 

= exp    (V_{l} − v_{l})^{2} 

2σ_{l}^{2} 

 
0.2 Processing
Can easily apply the maximum likelyhood to that sum; we want the all the real image values of index n
0 =    (V_{l} − v_{l})^{2} 

2σ_{l}^{2} 
 =
2    (V_{l} − v_{l})dV_{l}/dR_{n} =
2    (V_{l} − v_{l}) 
Rearrange the sum to both sides:
It is a linear equation to solve to get R_{k},
0.3 My mistake about disjoint areas for V_{l}
I initially thought disjoint areas on l didn’t follow the linear equation. Here is how i defined it.(once i went to the math of it)
And that D(l) is disjoint; l1≠ l2 => D(l1) ∩ D(l2) = ∅ in the sum:
The factors are only nonzero if both n and k are in the same disjoint area, and otherwise they’re one. I.e. we can sum over the area instead;
   =
  R_{k}    
with l_{2} the one value with n ∈ D(l_{2}) 
So my mistake: basically i thought you could select just k=n there. I.e. that if k, n ∈ D(l) then k=n but that’s obviously not true, but not when doing in my head, apparently. (if it is just R_{n} instead of the sum, it’s the weighed average)
This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.