In the SIFT parse (a) establish a Gaussian pyramid , we get a difference of Gaussian pyramid;
Detecting scale-space extrema DOG
SIFT key points are local extreme points of DOG space thereof. Compared to the center point of adjacent dots 3X3X3 detects whether it is a maximum or minimum values of neighboring points of the image field and field scale.
(1) In order to ensure that we are not advanced type noise threshold binarization;
as S and n, the number you want to extract image feature; (n) how many layers S represents each extraction
(2) find extreme point difference in the pyramid
Feature points are local extreme points of DOG space thereof. In order to find extreme points DoG function to each pixel and all its neighboring points compared to see whether it is larger or smaller than the adjacent points of its domain and image scale domain. Feature points are local extreme points of DOG space thereof. As shown above, the intermediate point and the detection of its scales with eight adjacent and vertically adjacent points of 26 points of comparison 9 × 2 dots corresponding to the scale to ensure a two-dimensional scale space and image space are detected extremum point.
Because pixels are discrete, and the scale space is discrete (longitudinal dimension to k times), it is possible to find extreme point is a point next to the real extremal point, as shown below:
Requiring precise positioning extrema points; using known methods discrete spatial point interpolated continuous space is called the extreme points of the sub-pixel interpolation (Sub-pixel Interpolation).
(3) precise positioning of extreme points
To enhance the stability of the critical point, the need for scale space DoG function curve fitting. DoG function expansion using the formula (fit function) in the scale space is Taylor:
The method of derivation : a pixel difference value is replaced by the derivative of neighbor
May refer to: SIFT algorithm theory explain (c)
Extreme points the equation corresponding to the detected extreme point is:
(4) low contrast extrema discarded
Wherein 
, representing the relative shift amount of the center of the interpolation, when it is shifted in either dimension is greater than 0.5 (i.e., x or y, or 
), the interpolation means has been shifted to the center of its adjacent points, it is necessary to change the current the key points of the position. Interpolation while repeatedly until convergence on a new position; it is also possible to set the number of iterations beyond the image boundary or beyond the scope of this point should be removed at this time, 5 iterations performed in Lowe. In addition, 
too small point susceptible to interference noise becomes unstable, it will be 
less than a certain experience (Lowe paper 0.03, use 0.04 / S achieved when Rob Hess et al) extreme point deleted. Meanwhile, to obtain accurate position of the feature point (home position plus the offset fitting) and scale () in the process.
Corresponding to the formula: T take 0.04, n and S, like number of image you want to extract feature; (n-) S represents a number of layers each extraction
(5) removing an edge effect ( the Hessian matrix )
Specific Hessian matrix mathematical formula can refer to: https: //blog.csdn.net/qq_33854260/article/details/69808970 and https://blog.csdn.net/qq_39521554/article/details/78895869
Its relationship with the image of: https: //blog.csdn.net/lwzkiller/article/details/55050275
H (x, y) is the difference of Gaussian pyramid of x, y of the second derivative, and
Then we need to know the matrix trace (Tr (H)) and are characteristic roots (beta] and [alpha]) and matrices calculated determinant (Det (H));
First, we know that in OpenCV - Harris, Shi Tomas, custom, sub-pixel corner detection
Harris characteristic value variation in the focus detection can be detected edge;
Similarly, since we can detect an edge, the edge points can be eliminated; and desired characteristic root α and β two values are very similar, and no edge effects; it is desirable to r is smaller than a certain threshold;
So that α is the maximum eigenvalue beta] is the smallest eigenvalue is the minimum value of the formula wherein the two values are equal, increases as the. The larger the value is, the larger the ratio of two characteristic values, i.e., the larger the gradient value in one direction, while the smaller the value of the gradient in the other direction, and the edge is precisely the case. So in order to eliminate the edge in response points, need to make the ratio is less than a certain threshold, therefore, in order to detect whether a primary curvature at a certain threshold r, only detected
(4-7)
When established formula (4-7) will retain the key point, on the contrary removed.
Original link: https: //blog.csdn.net/qq_37374643/article/details/88606351
Reference: SIFT algorithm principle Detailed