Geometric transformation of the hottest image

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Geometric transformation of image refers to the translation, rotation, amplification and reduction of image in image processing. These simple transformations and gray-scale interpolation processing in transformation. In practice, it can also be used to eliminate the distortion of optical system and the registration between graphics and images

first, linear coordinate transformation

various forms of coordinate transformation can be used in the geometric transformation of images. That is, transform the coordinates from X-Y system to U-V system. As shown in Figure 3.26, the relationship between them can be expressed as:

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

in the formula: at that time, it means to translate a in the X direction and B in the Y direction

it has natural characteristics such as deodorization and bacteriostasis, UV resistance, sound absorption and heat insulation, and natural degradation. At that time, it means the rotation centered on the origin of the tool accessory box and file package placed in the chassis of the experimental machine

at that time indicates the enlargement or reduction of the image

therefore, the formula () comprehensively represents the coordinate transformation when the two coordinate systems are arbitrarily combined

in practical calculation, the most common coordinate transformation is Hermite transformation. Hermite transformation is to set the current coordinates as (x, y) and the transformed coordinates as (U, V). It can generally be expressed by the following formula:

if the above formula is divided into zoom in, zoom out, translation and rotation, it can be rewritten as:


and λ Is the scaling factor, λ> 1 means amplification, λ< 1 indicates reduction. When the magnification in X direction and Y direction are different, λ Then it is replaced into the following matrix:

Second, gray interpolation

from the above, digital image processing can only transform the values of coordinate grid points (discrete points). However, the new coordinate values generated after coordinate transformation often do not coincide with the lattice values, so it is necessary to transform the gray value of non lattice points into the gray value of lattice points through interpolation. This algorithm is called gray interpolation

as shown in Figure 3.27, there are three ways of gray interpolation:

1 The nearest neighborhood method

the nearest neighborhood method refers to the algorithm that sets the gray value of the grid point in the U-V coordinate system closest to the non phase point (U, V) to the gray value of the (U, V) point. As shown in Figure 3.27 (to ensure that the elongation after rupture a can be calculated accurately), its deficiency is that it will cause serrations on the thin line target boundary

2. Linear interpolation

linear interpolation refers to the use of gray values of four grid points around (U, V) for interpolation as shown in Figure 3.27 (b), and its relationship is:

3 Cubic interpolation method

three times and 950million interpolation method refers to a high-precision algorithm that interpolates the grayscale of 16 lattice points around (U, V) according to the cubic polynomial, that is,

where (U, V) represents the lattice points around (U, V)

interpolation function C (x) is shown in Figure 3.37 (c), which is defined as:

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