A square with side length $\frac{1}{2}$ contains five congruent shaded squares. Four squares are in the corners, aligned with the outer square's axes. The fifth square is at the center, rotated 45 degrees. The vertices of the central square touch the inner boundaries of the cross-shaped white region. The common side length is $\frac{a-\sqrt{2}}{b}$, where $a$ and $b$ are positive integers. What is $a+b$?