SIFT - Taylor Expansion
Consider one-dimensional function $f(x)$.

The first order taylor expansion is $f(x_0+h) \approx f(x_0) + f'(x_0)h$

The second order taylor exapnsion is $f(x_0+h) \approx f(x_0) + f'(x_0)h + \frac 1 2 f''(x_0)h^2$

Now we expand three-dimensional function.
$$ D(\mathbf{x_0}+\mathbf h) \approx D(\mathbf{X_0}) + \bigg(\frac{ \partial D}{\partial \mathbf x}\bigg)^T\bigg|_{\mathbf x=\mathbf x_0}\mathbf h + \frac 1 2 \mathbf h^TH(\mathbf x)\mathbf h $$
, where
$$
\frac {\partial D}{\partial \mathbf x} = \begin{bmatrix} \frac {\partial D}{\partial x} \\ \frac {\partial D}{\partial y} \\ \frac {\partial D}{\partial \sigma}\end{bmatrix} = 
\begin{bmatrix} \frac {D(x+1,y,\sigma) - D(x-1,y,\sigma)}{2} \\ \frac {D(x,y+1,\sigma)-D(x,y-1,\sigma}{2} \\ \frac {D(x,y,\sigma+1) - D(x,y,\sigma-1)}{2} \end{bmatrix}
$$
$$ H(\mathbf x) = \begin{bmatrix} D_{xx} &amp; D_{xy} &amp; D_{x\sigma} \\
D_{yx} &amp; D_{yy} &amp; D_{y\sigma} \\
D_{\sigma x} &amp; D_{\sigma y} &amp; D_{\sigma\sigma} \end{bmatrix} $$

Additionally, $D_{xy}$
$$= \frac { \frac{D(x+1,y+1,s)-D(x-1,y+1,s)}{2} - \frac{D(x+1,y-1,s)-D(x-1,y-1,s)}{2} } 2 $$

Let's go back.
$$D' = \bigg ( \frac{ \partial D}{ \partial \mathbf x} \bigg ) ^T + H(\mathbf x) \mathbf h = 0 $$ $$\mathbf h = - H^{-1}(\mathbf x) \bigg ( \frac{ \partial D }{\partial \mathbf x} \bigg ) ^T $$

Assume that $\mathbf{x_0}$ is discrete feature point. Iteratively we can get $\mathbf h$.

If $\mathbf{x_0}$ is the closest point to $\mathbf h$, then we select $\mathbf{x_0}+\mathbf h$. (Imagine that 3-dimensional space)

My english ability is poor and there may be some errors.