Mathematical derivation of Harris corner point detection

Introduction

This paper is mainly to deduce the mathematical formula of Harris Corner detection, and to understand the theoretical knowledge more deeply.

Pre-Knowledgerepresentation of the matrix equation of an ellipse

In high school textbooks, we learned about standard ellipses and their equations (as shown):

In fact, the matrix in the operation of a very broad, now the above standard equation is written in matrix form (convenient next processing):

the relationship between the elliptic half axis and the coefficient matrix

A nxn matrix can solve its eigenvalues, we solve the above coefficient matrix (including A, b), then we can get the relationship between the eigenvalues and the ellipse half axis (A, B), the process is as follows:

Harris Angle Point detection principle

The Harris algorithm is based on the self-correlation of the image grayscale in the window, sets a window, and moves in the image, calculating the autocorrelation coefficients of the image before and after the moving window.

The autocorrelation is computed as follows, (x, y) is the center position of the window, W (u,v) is the weight (generally takes the Gaussian function), L is the window, (u,v) represents the image position in the window:

The square is expanded and written as a matrix, with:

To return to a self-related expression:

Among them,.

The expression of autocorrelation function has been obtained through the derivation of the mathematical form above. It can be seen that this is also a matrix representation of an ellipse (non-standard ellipse), so the eigenvalues of the coefficient matrix M are related to the length of the ellipse's half-axis, which is the same as in the preliminary knowledge above.

Assuming that the characteristic value of M is λ 1, λ2, the following three cases are divided:

By the above, the eigenvalue can be calculated to determine whether it is a corner point.

Of course, this calculation is very large, because almost every point in the image needs to be computed at once, and an empirical formula is given below:

Detm represents the determinant of M, Tracem represents the trace of M, and R represents the corner response value. α is the empirical constant, which generally takes a value between 0.04 and 0.06.

Judging criteria: When R exceeds a set threshold, it can be considered a corner point;

In this way, a corner point in an image can be obtained, and a non-maximal value suppression operation can be performed in a 3x3 or 5x5 neighborhood.

Reference Documents

harrisc,stephens.m-A Combined Corner and Edge detector[j],1988.

Wang Yongming, Wang Guijin, image local invariance characteristics and description [m],2010.

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Mathematical derivation of Harris corner point detection