Watershed Algorithm Proof

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Let f ∈ C(D) have minima mkk∈I , for some index
set I. The catchment basin CB(mi) of a minimum mi is defined as the set of points x ∈ D
which are topographically closer to mi than to any other regional minimum mj :
CB(mi) = ∀j ∈ Ii : f(mi) + Tf (x, mi) < f(mj ) + Tf (x, mj )



The watershed of f is the set of points which do not belong to any catchment basin:
Wshed(f) = D ∩ (⋃_(i∈I)▒〖CB(mi))c〗



Let W be some label, W 6∈ I. The watershed transform of f is a mapping λ : D → I ∪ W,
such that λ(p) = i if p ∈ CB(mi), and λ(p) = W if p ∈ Wshed(f).



Can someone please explain this and give the proof of this. Thank You.




matlab computational-mathematics image-processing




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    Let f ∈ C(D) have minima mkk∈I , for some index
    set I. The catchment basin CB(mi) of a minimum mi is defined as the set of points x ∈ D
    which are topographically closer to mi than to any other regional minimum mj :
    CB(mi) = ∀j ∈ Ii : f(mi) + Tf (x, mi) < f(mj ) + Tf (x, mj )



    The watershed of f is the set of points which do not belong to any catchment basin:
    Wshed(f) = D ∩ (⋃_(i∈I)▒〖CB(mi))c〗



    Let W be some label, W 6∈ I. The watershed transform of f is a mapping λ : D → I ∪ W,
    such that λ(p) = i if p ∈ CB(mi), and λ(p) = W if p ∈ Wshed(f).



    Can someone please explain this and give the proof of this. Thank You.




    matlab computational-mathematics image-processing




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let f ∈ C(D) have minima mkk∈I , for some index
      set I. The catchment basin CB(mi) of a minimum mi is defined as the set of points x ∈ D
      which are topographically closer to mi than to any other regional minimum mj :
      CB(mi) = ∀j ∈ Ii : f(mi) + Tf (x, mi) < f(mj ) + Tf (x, mj )



      The watershed of f is the set of points which do not belong to any catchment basin:
      Wshed(f) = D ∩ (⋃_(i∈I)▒〖CB(mi))c〗



      Let W be some label, W 6∈ I. The watershed transform of f is a mapping λ : D → I ∪ W,
      such that λ(p) = i if p ∈ CB(mi), and λ(p) = W if p ∈ Wshed(f).



      Can someone please explain this and give the proof of this. Thank You.




      matlab computational-mathematics image-processing




      share|cite|improve this question











      Let f ∈ C(D) have minima mkk∈I , for some index
      set I. The catchment basin CB(mi) of a minimum mi is defined as the set of points x ∈ D
      which are topographically closer to mi than to any other regional minimum mj :
      CB(mi) = ∀j ∈ Ii : f(mi) + Tf (x, mi) < f(mj ) + Tf (x, mj )



      The watershed of f is the set of points which do not belong to any catchment basin:
      Wshed(f) = D ∩ (⋃_(i∈I)▒〖CB(mi))c〗



      Let W be some label, W 6∈ I. The watershed transform of f is a mapping λ : D → I ∪ W,
      such that λ(p) = i if p ∈ CB(mi), and λ(p) = W if p ∈ Wshed(f).



      Can someone please explain this and give the proof of this. Thank You.





      matlab computational-mathematics image-processing





      share|cite|improve this question










      share|cite|improve this question




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      asked Jul 24 at 6:01









      Methmal Fonseka

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