COMS 482: unofficial class blog

Autofocus, Foreground and background differentiation:

Given a 2d array of pixels with scores a_i representing the likelyhood that pixel i is in the foreground, b_i representing the likelyhood that i is in the background, and p_ij representing a penalty on adjacent pixels who are not both in the foreground or both in the backgroun, we can partition the pixels into two sets, A and B, such that we maximize:

q(A, B) = sum(a_i in A) + sum(b_j in B) - sum(p_ij adjacent and not in same set)

This is equivalent to minimizing:

q’(A, B) = sum(b_i in A) + sum(a_j in B) + sum(p_ij adjacent and not in same set)

Graph G = (V,E):

    1) A vertex for each pixel, s, t
    2) An edge from s to each pixel j with capacity a_j
    3) An edge from each pixel i with capacity b_i to t
    4) For adjacent pixels i, j an edge from i->j and j->i of capacity p_ij

Imagine that you pay a_j for j in B, b_i for i in A, and p_ij for adjacent pixels. The min-cut (max flow) is the minimization of these sets.

Protein Sequence Design: [Kleinberg ‘99]

The protein folding problem: given a sequence of amino acids, we want to predict the protein structure. From the other direction, given a particular protein fold, what is the best sequence of amino acids to fit it? If we reduce the set of amino acids to hydrophilic / polar (hydrophobic), we have a reasonable problem.

Let H be the set of amino acids that are hydrophilic, and a < 0, B > 0. Then:

q(H) = a*(# of h->h contacts) + B * sum(exposed surface of residue) for all i in H

We can rewrite q(H) as:

q(H) = a*(# of all contacts) + |a| * (# of contacts not h->h) + B * sum(exposed surface of residue) for all i in H

Our goal is to minimize

q(H) = |a|*(# of contacts not h->h) + sum(S_i) for all i in H

Graph G = (V,E):

    1) A vertex i for each amino acid, s, t, and c_ij for all contacts
    2) An edge s->c_ij for all i, j with capacity |a|
    3) An edge c_ij->i,j with infinite capacity
    4) An edge i->t with capacity B - S_i

The cost of the cut:

sum(B*S(i)) for all i in A + sum(|a|) for all c_ij with i or j in B

This entry was posted on Wednesday, March 9th, 2005 at 1:02 pm and is tagged with 2d array, protein sequence, sequence of amino acids, protein structure, sequence design, pij, backgroun, likelyhood, max flow, cij, autofocus, minimization, vertex, foreground and background, amino acid, differentiation, aj, pixel, bj, partition. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback.

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