Chapter 2 Local Image Descriptors

2.1 corner point

Corner point: moving in any direction, there will be obvious grayscale changes.

Translating the image window [u, v] produces a grayscale change E(u, v)

$E(u,v)=\sum_{x,y}^{}w(x,y)[I(x+u,y+v)-I(x,y)]^{2}$

w(x,y) is the window function, I(x+u,y+v) is the image grayscale after translation, and I(x,y) is the image grayscale

How to solve I(x+u,y+v), and E(u,v)?

Therefore, for the local small movement amount [u, v], the following expression can be approximated:

$E(u, v) \cong[u, v] \quad M\left[\begin{array}{l} u \\ v \end{array}\right]$
Where M is a 2×2 matrix, which can be obtained from the derivative of the image:

$M=\sum_{x, y} w(x, y)\left[\begin{array}{cc} I_{x}^{2} & I_{x} I_{y} \\ I_{x} I_{y} & I_{y}^{2} \end{array}\right]$

Image Variations Caused by Window Movement: Eigenvalue Analysis of Real Symmetric Matrix M

$E(u, v) \cong[u, v] M\left[\begin{array}{l} u \\ v \end{array}\right]$
Record M eigenvalues as λ1, λ2
$M=\sum_{x, y} w(x, y)\left[\begin{array}{cc} I_{x}^{2} & I_{x} I_{y} \\ I_{x} I_{y} & I_{y}^{2} \end{array}\right]$ Ix, Iy as the gradient in the corresponding direction

Physical meaning: Classify image points by the size of the two eigenvalues of M

If the color texture inside the window is close (flat area), the Ix, Iy values are generally small, and there is no obvious grayscale change in all directions.
If the inside of the window is at the corner position, the Iy value of the horizontal line in the middle of the image block is larger, and the Ix value is smaller. There is an obvious grayscale change when moving in the y direction, but not in the x direction.
If the inside of the window is at the corner position, the Ix value of the vertical line in the middle of the image block is larger, and the Iy value is smaller. There is an obvious grayscale change when moving in the x direction, but not in the y direction.

R = detM - k(traceM)²
detM = λ1λ2
traceM = λ1+λ2

R is only related to the eigenvalues of M
Corner point: R is a large positive number
Margins: R is negative for large values
Flat region: R for fractional values

Summary: Corner calculation process

Threshold the corner response function R: R>threshold
Extract local maxima of R

Sometimes in order to eliminate the impression of parameter k, the quotient can also be used to calculate the response:

$\begin{array}{c} R=\frac{\operatorname{det} M}{(\operatorname{trace} M)^{2}} \\ \operatorname{det} M=\lambda_{1} \lambda_{2} \\ \operatorname{trace} M=\lambda_{1}+\lambda_{2} \end{array}$

"Computer Vision" set of learning notes for large online courses [1]

Chapter 2 Local Image Descriptors

2.1 corner point

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