MATLAB Computing distance from a point to a set of points

Question

Consider a matrix A:

A = magic(5)

17    24     1     8    15
23     5     7    14    16
 4     6    13    20    22
10    12    19    21     3
11    18    25     2     9

I have to compute the following formula: w_ij = ||I(i) - I(j)|| ^ 2 from point A(1,1) to its neighborhood i.e. A(1:2, 1:2). Now I don't understand well what this formula stand for since it is not specified. Is this the Euclidean distance?

I tried

norm(A(1, 1) - A(1:2, 1:2))

But that gives me a scalar. I'm expecting a vector of 4 elements. Can you help me?

Answer 1

You can see that formula in context on page 4 of http://www.cs.berkeley.edu/~malik/papers/SM-ncut.pdf (equation 11). In that paper they use F for intensity where I assume you have I. Since your intensities are scalars, you just want to take the square of their differences.

You want to calculate a weight matrix that calculates the affinity of any entry in A to any other entry in A. Because your A has 25 entries, your weight matrix will be 25x25.

Since you are only worried about the brightness this is easy:

len = length(A(:));
W = zeros(len);
for i = 1:len
    for j = 1:len
       W(i,j) = (A(i) - A(j))^2;
    end
end

Now if you want to look up the weight between A(1,1) and A(1,2) you can do it like this:

i = sub2ind(size(A), 1, 1)
j = sub2ind(size(A), 1, 2)
W(i, j)

But if you set r=1 (according to the NCuts formula) then you might want something like this:

sigma= 10;
r = 1;
A = magic(3);
siz = size(A);
len = length(A(:));
W = zeros(len);
for i = 1:len
    for j = 1:len
       [xi,yi] = ind2sub(siz,i);
       [xj,yj] = ind2sub(siz,j);
       if((xi-xj)^2 + (yi-yj)^2) > r^2
           W(i,j) = 0;
       else
           W(i,j) = exp(-(A(i) - A(j))^2 / sigma^2);
       end
    end
end

A11 = sub2ind(siz, 1, 1)
A12 = sub2ind(siz, 1, 2)
W(A11, A12)

Answer 2

I'm pretty sure there's a built-in function for this, but implementing one by hand isn't too difficult:

% some sample data
A = magic(5);


% generic distance calculator
C = cell(size(A));
for jj = 1:size(A,1)
    for ii = 1:size(A,2)

        % define the edges
        left   = max(1, ii-1);
        right  = min(size(A,2), ii+1);
        top    = max(1, jj-1);
        bottom = min(size(A,1), jj+1);

        % extract the neighborhood
        N = A(left:right, top:bottom);

        % compute the squared distance
        C{ii,jj} = (N-A(ii,jj)).^2;

    end
end

Now I also don't understand your formula...I just assumed it means the squared numerical distance between the two points, but you'll have to give me more context before I can fully understand the problem...