Transformation Function: Gonzalez and Woods
Those ${\sum}\text{-signs}$ specify sigma notation.  They're basically like for loops where you keep summing up the value specified in the body.

The equation,
$$
T \left(u, v\right)
= 
\sum_{x=0}^{M-1}
{
  \sum_{y=0}^{N-1}
  {
    f \left(x, y\right)
    \,
    r \left(x, y, u, v\right)
  }
}
\,,
$$

translates into:

public double T(double u, double v)
{
    var sum = 0.0;

    for (int x=0; x &lt;= M-1; ++x)
    {
        for (int y=0; y &lt;= N-1; ++y)
        {
            sum = sum + (f(x, y) * r(x, y, u, v));
        }
    }

    return sum;
}




In other words, it defines some function, $T \left(u, v\right) ,$ that takes two input variables: $u$ and $v .$  You also need to know two other functions: $f \left(x, y\right)$ and $r\left(x, y, u, v\right) ,$ which means you'll need at least 2 more equations.

Then:


Start with a starting total value of $0 .$
Set $x$ to $0 .$
Set $y$ to $0 .$
Calculate $f\left(x, y\right) r\left(x, y, u, v\right) ,$ then add this value to the total.


On the first evaluation, $x=0$ and $y=0 ,$ so that's $f\left(0, 0\right) r\left(0, 0, u, v\right) .$

Increment $y .$


If it's still less-than-or-equal-to $N - 1 ,$ then go back to Step (4).

Increment $x .$


If it's still less-than-or-equal-to $M - 1 ,$ then go back to Step (3).

The final total is the calculated value for $T \left(u, v\right) .$