How to search for patterns on a gameboard?

Assume a gameboard with two kinds of pieces.

Let the task be to find a common patterns of pieces, namely the patterns that lead to winning.

The problem is general, but the game in question is Gomoku, i.e. tic-tac-toe on a 15x15 board with requirement of having five pieces in a row. But the similar problem would be the mouse moves recognition when concerning screen as an array of pixels.

To let any data mining model solve the problem of finding pattern that can lead to a win, the problem of identity of patterns must be solved. That is, it must be programmed that a given pattern moved left/right/down/up or rotated is still the same pattern. Like in an example of mouse moves, it is unimportant where a move starts on a screen.

My question is how to to solve this issue of identity efficiently.