Editing Scale-invariant feature transform (section)

==== Interpolation of nearby data for accurate position ====
First, for each candidate keypoint, interpolation of nearby data is used to accurately determine its position. The initial approach was to just locate each keypoint at the location and scale of the candidate keypoint.<ref name="Lowe1999" /> The new approach calculates the interpolated location of the extremum, which substantially improves matching and stability.<ref name=Lowe2004 /> The interpolation is done using the quadratic [[Taylor expansion]] of the Difference-of-Gaussian scale-space function, <math>D \left( x, y, \sigma \right)</math> with the candidate keypoint as the origin. This Taylor expansion is given by:

:<math>D(\textbf{x}) = D + \frac{\partial D}{\partial \textbf{x}}^T\textbf{x} + \frac{1}{2}\textbf{x}^T \frac{\partial^2 D}{\partial \textbf{x}^2} \textbf{x}</math>

where D and its derivatives are evaluated at the candidate keypoint and <math>\textbf{x} = \left( x, y, \sigma \right)^T</math> is the offset from this point. The location of the extremum, <math>\hat{\textbf{x}}</math>, is determined by taking the derivative of this function with respect to <math>\textbf{x}</math> and setting it to zero. If the offset <math>\hat{\textbf{x}}</math> is larger than <math>0.5</math> in any dimension, then that's an indication that the extremum lies closer to another candidate keypoint. In this case, the candidate keypoint is changed and the interpolation performed instead about that point. Otherwise the offset is added to its candidate keypoint to get the interpolated estimate for the location of the extremum. A similar subpixel determination of the locations of scale-space extrema is performed in the real-time implementation based on hybrid pyramids developed by Lindeberg and his co-workers.<ref name="Lindenberg2003" />