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===Connection with Shannon entropy===
The Shannon entropy <math>H</math> of a system is defined as
:<math>H:=\sum_{S_{i}\in\Omega_{S,V}}P(S_{i})\log_{2}\left(\frac{1}{P(S_{i})}\right)</math>.
For <math>U\in\Psi_{V}</math>, <math>R\in\Psi_{S}</math> and <math>A_{S_{i}}:=\{S_{j}\in\Omega_{S,V}\colon d(R,R\{U,S_{j}\})\leq d(R,R\{U,S_{i}\})\}</math> the following equalities holds
:<math>\sum_{S_{i}\in\Omega_{S,V}}P(S_{i})\log_{2}\left(\frac{1}{P(S_{i}\mid A_{S_{i}})}\right)</math>
<math>=\sum_{S_{i}\in\Omega_{S,V}}P(S_{i})\log_{2}\left(\frac{\sum_{S_{j}\in A_{S_{i}}}P(S_{j})}{P(S_{i})}\right)=H+\sum_{S_{i}\in\Omega_{S,V}}P(S_{i})\log_{2}\left(\sum_{S_{j}\in A_{S_{i}}}P(S_{j}) \right)</math> (1).
The expression on the left is similar in form to the definition of Shannon entropy. The middle expression reveals the value to be similar to that of <math>efe(R,U,P)</math> when the probabilities in the argument of the logarithm are comparable. Indeed, <math>efe(R,U,P)</math> is an approximation of (1). The expression on the right of (1) shows the mathematical connection to Shannon entropy; the first term is the Shannon entropy <math>H</math> of the system and, with consideration of the log function, the second term has a negative value between <math>-H</math> and 0.
Expected Float Entropy Minimisation (view source)
Revision as of 12:48, 16 March 2021
, 12:48, 16 March 2021Subsection removed due to new research showing that EFE is actually a rather special function and is not generally an approximation to the expressions that were shown in this subsection.
: <math>efe(R,U,P)=\min_{R'\in\Psi_{S},\,U'\in\Psi_{V}}efe(R',U',P)</math>
are the weighted relations of interest. For example, when the theory is applied to digital photographs, U gives the relationships between colours and R gives the relationships that determine the geometry of the field of view.
===Connection with ideas in topology===
