<P> be the class of all component distributions . Then the convex hull K of J defines the class of all finite mixture of distributions in J: </P> <Dl> <Dd> K = (p (⋅): p (⋅) = ∑ i = 1 n a i f i (⋅; θ i), a i> 0, ∑ i = 1 n a i = 1, f i (⋅; θ i) ∈ J ∀ i, n) (\ displaystyle K = \ left \ (p (\ cdot): p (\ cdot) = \ sum _ (i = 1) ^ (n) a_ (i) f_ (i) (\ cdot; \ theta _ (i)), a_ (i)> 0, \ sum _ (i = 1) ^ (n) a_ (i) = 1, f_ (i) (\ cdot; \ theta _ (i)) \ in J \ \ forall i, n \ right \)) </Dd> </Dl> <Dd> K = (p (⋅): p (⋅) = ∑ i = 1 n a i f i (⋅; θ i), a i> 0, ∑ i = 1 n a i = 1, f i (⋅; θ i) ∈ J ∀ i, n) (\ displaystyle K = \ left \ (p (\ cdot): p (\ cdot) = \ sum _ (i = 1) ^ (n) a_ (i) f_ (i) (\ cdot; \ theta _ (i)), a_ (i)> 0, \ sum _ (i = 1) ^ (n) a_ (i) = 1, f_ (i) (\ cdot; \ theta _ (i)) \ in J \ \ forall i, n \ right \)) </Dd> <P> K is said to be identifiable if all its members are unique, that is, given two members p and p ′ in K, being mixtures of k distributions and k ′ distributions respectively in J, we have p = p ′ if and only if, first of all, k = k ′ and secondly we can reorder the summations such that a = a ′ and ƒ = ƒ ′ for all i . </P>

What is gaussian mixture model in image processing