I have been reading Image Processing from Gonzalez and Woods and in the chapter Image Transformation I have come across this equation

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

I am unable to understand this mathematical notation. Any help will be highly appreciated. Thanks


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 <= M-1; ++x)
        for (int y=0; y <= 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.


  1. Start with a starting total value of $0 .$

  2. Set $x$ to $0 .$

  3. Set $y$ to $0 .$

  4. 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 $f\left(0, 0\right) r\left(0, 0, u, v\right) .$
  5. Increment $y .$

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

    • If it's still less-than-or-equal-to $M - 1 ,$ then go back to Step (3).
  7. The final total is the calculated value for $T \left(u, v\right) .$

  • $\begingroup$ What is r(x,y,u,v) referring to? what are those extra parameters u and v? $\endgroup$ – Turing101 Dec 17 '19 at 13:42
  • $\begingroup$ Are we multiplying each pixel intensity alone with that function? any explanation showing how this works on a matrix level by taking 2 matrices will be of much help $\endgroup$ – Turing101 Dec 17 '19 at 13:44
  • $\begingroup$ @HIRAKMONDAL I'm not sure. The equation is very generic, so it's hard to guess what the variables might refer to without seeing the book. $u$ and $v$ may be coordinate positions for a pixel. You'll probably have to read where the book declares its notation to figure out what the symbols match up to. $\endgroup$ – Nat Dec 17 '19 at 13:49

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