Geometric transformation of the image

Geometric transformation refers to the translation, rotation, enlargement and reduction of images in image processing, these simple transformations, and gray-scale interpolation in transformation. In actual use, it can also be used to eliminate aberrations in the optical system, registration between images and images, and the like.

First, the linear coordinate transformation

In the geometric transformation of the image, various forms of coordinate transformation can be used. That is, the coordinates are converted from the XY system to the UV system. As shown in Figure 3.26, the relationship between them can be expressed as:

If only a linear transformation system is used, that is, in linear space, the above equation can be expressed as:

In the formula: At that time, it means translation a in the x direction and b in the y direction.

At that time, it was the rotation centered on the origin.

At the time, it represented an enlargement or reduction of the image.

Therefore, Equation (3-57) comprehensively represents the coordinate transformation when any two combinations of coordinate systems are combined.

In actual calculations, the Hermite transformation is most often used for coordinate transformation. The Hermitian transformation is to set the current coordinate as (x,y) and the transformed coordinate as (u,v), which can be generally expressed by the following formula:

If the above formula is divided into enlargement, reduction, translation and rotation, it can be rewritten as:

among them,

And λ is a scaling factor, λ>1 indicates amplification, and λ<1 indicates reduction. When x-direction and y-direction magnification and reduction magnification are different, λ is replaced by the following matrix:

Second, gray interpolation

As can be seen from the foregoing, digital image processing can only transform the values ​​of coordinate grid points (discrete points). The new coordinate values ​​generated after the coordinate transformation are not coincident with the grid point values. Therefore, the gray value of the non-grid points needs to be converted into the gray value of the grid points by an interpolation method. This algorithm is called gray-scale interpolation.

As shown in Figure 3.27, there are three ways of grayscale interpolation:

Nearest neighbor method

The nearest neighbor method is an algorithm that sets the gray value of the grid point in the nearest u--v coordinate system to the (u, v) gray value of the non-phase node (u, v). As shown in Figure 3.27(a), its deficiency is the jaggedness of the thin-line target boundary.

2. Linear interpolation

Linear interpolation refers to the interpolation of the gray values ​​of the four grid points around (u, v) as shown in Figure 3.27(b). The relationship is:

3. Three-time interpolation

The cubic interpolation method refers to a high-accuracy algorithm that uses a cubic polynomial to interpolate the 16 grid points around (u, v).

Where (u,v) denotes grid points around (u,v).

The interpolation function C(x) is shown in Figure 3.37(c) and is defined as: