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# 数学代写|图像处理代写Digital image processing代考|MATH345 Neighborhood Relations

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## 数学代写|图像处理代写Digital image processing代考|Neighborhood Relations

An important property of discrete images is their neighborhood relations since they define what we will regard as a connected region and therefore as a digital object. A rectangular grid in two dimensions shows the unfortunate fact, that there are two possible ways to define neighboring pixels (Fig. 2.4a, b). We can regard pixels as neighbors either when they have a joint edge or when they have at least one joint corner. Thus a pixel has four or eight neighbors and we speak of a 4-neighborhood or an 8-neighborhood.

Both types of neighborhood are needed for a proper definition of objects as connected regions. A region or an object is called connected when we can reach any pixel in the region by walking from one neighboring pixel to the next. The black object shown in Fig. 2.4c is one object in the 8-neighborhood, but constitutes two objects in the 4-neighborhood. The white background, however, shows the same property. Thus we have either two connected regions in the 8-neigborhood crossing each other or two separated regions in the 4-neighborhood. This inconsistency can be overcome if we declare the objects as 4-neighboring and the background as 8-neighboring, or vice versa.

These complications occur not only with a rectangular grid. With a triangular grid we can define a 3-neighborhood and a 12-neighborhood where the neighbors have either a common edge or a common corner, respectively (Fig. 2.3a). On a hexagonal grid, however, we can only define a 6-neighborhood because pixels which have a joint corner, but no joint edge, do not exist. Neighboring pixels always have one joint edge and two joint corners. Despite this advantage, hexagonal grids are hardly used in image processing, as the imaging sensors generate pixels on a rectangular grid. The photosensors on the retina in the human eye, however, have a more hexagonal shape [193].

## 数学代写|图像处理代写Digital image processing代考|Discrete Geometry

The discrete nature of digital images makes it necessary to redefine elementary geometrical properties such as distance, slope of a line, and coordinate transforms such as translation, rotation, and scaling. These quantities are required for the definition and measurement of geometric parameters of object in digital images.

In order to discuss the discrete geometry properly, we introduce the grid vector that represents the position of the pixel. The following discussion is restricted to rectangular grids. The grid vector is defined in 2-D, 3-D, and 4-D spatiotemporal images as
$$\boldsymbol{r}{m, n}=\left[\begin{array}{c} n \Delta x \ m \Delta y \end{array}\right], \boldsymbol{r}{l, m, n}=\left[\begin{array}{c} n \Delta x \ m \Delta y \ l \Delta z \end{array}\right], \boldsymbol{r}_{k, l, m, n}=\left[\begin{array}{c} n \Delta x \ m \Delta y \ l \Delta z \ k \Delta t \end{array}\right]$$
To measure distances, it is still possible to transfer the Euclidian distance from continuous space to a discrete grid with the definition
$$d_e\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|=\left[\left(n-n^{\prime}\right)^2 \Delta x^2+\left(m-m^{\prime}\right)^2 \Delta y^2\right]^{1 / 2} .$$
Equivalent definitions can be given for higher dimensions. In digital images two other metrics have often been used. The city block distance
$$d_b\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\left|n-n^{\prime}\right|+\left|m-m^{\prime}\right|$$
gives the length of a path, if we can only walk in horizontal and vertical directions (4-neighborhood). In contrast, the chess board distance is defined as the maximum of the horizontal and vertical distance
$$d_c\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\max \left(\left|n-n^{\prime}\right|,\left|m-m^{\prime}\right|\right) .$$

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## 数学代写|图像处理代写数字图像处理代考|离散几何

$$\boldsymbol{r}{m, n}=\left[\begin{array}{c} n \Delta x \ m \Delta y \end{array}\right], \boldsymbol{r}{l, m, n}=\left[\begin{array}{c} n \Delta x \ m \Delta y \ l \Delta z \end{array}\right], \boldsymbol{r}_{k, l, m, n}=\left[\begin{array}{c} n \Delta x \ m \Delta y \ l \Delta z \ k \Delta t \end{array}\right]$$

$$d_e\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|=\left[\left(n-n^{\prime}\right)^2 \Delta x^2+\left(m-m^{\prime}\right)^2 \Delta y^2\right]^{1 / 2} .$$

$$d_b\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\left|n-n^{\prime}\right|+\left|m-m^{\prime}\right|$$

$$d_c\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\max \left(\left|n-n^{\prime}\right|,\left|m-m^{\prime}\right|\right) .$$

## MATLAB代写

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