Question

Given a grid (or table) with x*y cells. Each cell contains a value. Most of these cells have a value of 0, but there may be a “hot spot” somewhere on this grid with a cell that has a high value. The neighbours of this cell then also have a value > 0. As farer away from the hot spot as lower the value in the respective grid cell.

So this hot spot can be seen as the top of a hill, with decreasing values the farer we are away from this hill. At a certain distance the values drop to 0 again.

Now I need to determine the cell within the grid that represents the grid’s center of gravity. In the simple example above this centroid would simply be the one cell with the highest value. However it’s not always that simple:

1. the decreasing values of neighbour cells around the hot spot cell may not be equally distributed, or a “side of the hill” may fall down to 0 sooner than another side.

2. there is another hot spot/hill with values > 0 elsewehere within the grid.

I could think that this is kind of a typical problem. Unfortunately I am no math expert so I don’t know what to search for (at least I have not found an answer in Google).

Any ideas how can I solve this problem?

Thanks in advance.

Answer 1

You are looking for the “weighted mean” of the cell values. Assuming each cell has a value z(x,y), then you can do the following

zx = sum( z(x, y) ) over all values of y
zy = sum( z(x, y) ) over all values of x

meanX = sum( x * zx(x)) / sum ( zx(x) )
meanY = sum( y * zy(y)) / sum ( zy(y) )

I trust you can convert this into a language of your choice…

Example: if you know Matlab, then the above would be written as follows

zx = sum( Z, 1 ); % sum all the rows
zy = sum( Z, 2 ); % sum all the columns

[ny nx] = size(Z); % find out the dimensions of Z

meanX = sum((1:nx).*zx) / sum(zx);
meanY = sum((1:ny).*zy) / sum(zy);

This would give you the meanX in the range 1 .. nx : if it’s right in the middle, the value would be (nx+1)/2. You can obviously scale this to your needs.

EDIT: one more time, in “almost real” code:

// array Z(N, M) contains values on an evenly spaced grid
// assume base 1 arrays

zx = zeros(N);
zy = zeros(M);

// create X profile:
for jj = 1 to M
  for ii = 1 to N
    zx(jj) = zx(jj) + Z(ii, jj);
  next ii
next jj

// create Y profile:
for ii = 1 to N
  for jj = 1 to M
    zy(ii) = zy(ii) + Z(ii, jj);
  next jj
next ii

xsum = 0;
zxsum = 0;
for ii = 1 to N
  zxsum += zx(ii);
  xsum += ii * zx(ii);
next ii
xmean = xsum / zxsum;

ysum = 0;
zysum = 0;
for jj = 1 to M
  zysum += zy(jj);
  ysum += jj * zy(ii);
next jj
ymean = ysum / zysum;