What is second order accuracy?

On a discrete grid, the non-maximum suppression stage can be implemented by estimating the gradient direction using first-order derivatives, then rounding off the gradient direction to multiples of 45 degrees, and finally comparing the values of the gradient magnitude in the estimated gradient direction. A more refined approach to obtain edges with sub-pixel accuracy is by detecting zero-crossings of the second-order directional derivative in the gradient direction (Lindeberg 1998)