机器视觉代写|图像处理作业代写Image Processing代考|Ice Edge Detection

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

The gradient, which is the first-order derivative, has a direction toward the most rapid change in intensity. The gradient of a digital image with pixel value $f(x, y)$ is defined as the vector:
$$\nabla f=\left[\begin{array}{l} G_{x} \ G_{y} \end{array}\right]=\left[\begin{array}{l} \frac{\partial f}{\partial x} \ \frac{\partial f}{\partial y} \end{array}\right]$$
and the gradient magnitude is given by:
$$|\nabla f|=\sqrt{G_{x}^{2}+G_{y}^{2}}=\sqrt{\left(\frac{\partial f}{\partial x}\right)^{2}+\left(\frac{\partial f}{\partial y}\right)^{2}}$$

while the direction of the gradient vector is given by the angle:
$$\theta=\angle f=\arctan \left(\frac{G_{x}}{G_{y}}\right)$$
with respect to the $x$-axis, where for implementation we use the arctan() function for correct quadrant mapping.

For computational efficiency, the gradient magnitude is sometimes approximated by using the squared gradient magnitude:
$$\nabla f \approx G_{x}^{2}+G_{y}^{2}$$
$$\nabla f \approx\left|G_{x}\right|+\left|G_{y}\right|$$
where these two approximations also preserve the relative changes in intensity scales.
The gradient of an image can be used for the detection of edges in the image; it requires the calculation of the partial derivatives $G_{x}$ and $G_{y}$ at every pixel location in the image. To directly estimate the partial derivatives $G_{x}$ and $G_{y}$ is one of the key issues in this method. The discrete approximation of partial derivatives over a neighborhood about a point is required. For example, it is a common and simple way to form the running difference of pixels along rows and columns of the image, which gives the approximation:
\begin{aligned} &\frac{\partial f}{\partial x}(x, y) \approx f(x+1, y)-f(x, y) \ &\frac{\partial f}{\partial y}(x, y) \approx f(x, y+1)-f(x, y) \end{aligned}
To implement the derivatives over an entire image, the edge detector, which is a local image processing method designed to detect edge pixels, filters the image with convolution kernels. So, the Equations $4.6 \mathrm{a}$ and $4.6 \mathrm{~b}$ can then be implemented for all pertinent values of $x$ and $y$ by filtering $f(x, y)$ with the simple 1-dimensional convolution kernels shown in Figure 4.1.

机器视觉代写|图像处理作业代写Image Processing代考|LAPLACIAN

Similar to the first-order derivative, the second-order derivative, which is the Laplacian of the image, is defined as:
$$\nabla^{2} f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}$$
The second-order derivative along the $x$ direction can be approximated by differentiating Equation $4.6 \mathrm{a}$ with respect to $x$, e.g.:
\begin{aligned} \frac{\partial^{2} f}{\partial x^{2}}(x, y) & \approx \frac{\partial G_{x}(x, y)}{\partial x} \ &=\frac{\partial f(x+1, y)}{\partial x}-\frac{\partial f(x, y)}{\partial x} \ & \approx[f(x+2, y)-f(x+1, y)]-[f(x+1, y)-f(x, y)] \ &=f(x+2, y)-2 f(x+1, y)+f(x, y) \end{aligned}

Since this approximation is centered about the pixel $(x+1, y)$, however, we replace $x$ with $x-1$ and obtain the result:
$$\frac{\partial^{2} f}{\partial x^{2}}(x, y) \approx f(x+1, y)+f(x-1, y)-2 f(x, y)$$
This is the desired approximation to the second partial derivative centered about the pixel $(x, y)$. Similarly,
$$\frac{\partial^{2} f}{\partial y^{2}}(x, y) \approx f(x, y+1)+f(x, y-1)-2 f(x, y)$$
Combining Equations $4.11$ and $4.12$ two equations into a single operator according to Equation $4.9$ gives an approximation of the Laplacian:
$$\nabla^{2} f(x, y)=f(x-1, y)+f(x+1, y)+f(x, y-1)+f(x, y+1)-4 f(x, y)$$
This expression simply measures the weighted differences between a pixel and its 4-neighbors, and it can be implemented by using the kernel in Figure 4.4(a).

Sometimes it is desired to give more weight to the center pixels in the neighborhood, and Equation $4.13$ can be extended to include the diagonal terms, for instance, using the kernel in Figure 4.4(b).

机器视觉代写|图像处理作业代写Image Processing代考|MORPHOLOGICAL EDGE DETECTION

Morphology refers to geometrical characteristics related to the form and structure of objects, such as size, shape, and orientation. In image processing, mathematical morphology involves geometric analysis of shapes and textures in images based on some simple mathematical concepts from set theory. It is used to extract image components that are useful in representation and description of region shapes, such as boundaries, skeletons, convex hull, etc.

Morphological operators work with an image and a structuring element. The structuring element is a small set or subimage used to probe the given image for specific properties. It is also known as a kernel, and can be represented as a matrix of 0 s and Is. Values of 1 in the matrix indicate the points that belong to the structuring element, while values of 0 indicate otherwise. The structuring element has a desired shape, such as square, rectangle, disk, diamond, etc. The origin of a structuring element identifies the pixel of interest (the pixel being processed), and it must be clearly specified. The origin is typically at the center of gravity; however, it could be located at any desired position of the structuring element. Figure $4.7$ shows examples of different structuring elements of various sizes with their origins highlighted in the corresponding geometric centers.

图像处理代考

∇F=[GX G是]=[∂F∂X ∂F∂是]

|∇F|=GX2+G是2=(∂F∂X)2+(∂F∂是)2

θ=∠F=反正切⁡(GXG是)

∇F≈GX2+G是2

∇F≈|GX|+|G是|

∂F∂X(X,是)≈F(X+1,是)−F(X,是) ∂F∂是(X,是)≈F(X,是+1)−F(X,是)

机器视觉代写|图像处理作业代写Image Processing代考|LAPLACIAN

∇2F=∂2F∂X2+∂2F∂是2

∂2F∂X2(X,是)≈∂GX(X,是)∂X =∂F(X+1,是)∂X−∂F(X,是)∂X ≈[F(X+2,是)−F(X+1,是)]−[F(X+1,是)−F(X,是)] =F(X+2,是)−2F(X+1,是)+F(X,是)

∂2F∂X2(X,是)≈F(X+1,是)+F(X−1,是)−2F(X,是)

∂2F∂是2(X,是)≈F(X,是+1)+F(X,是−1)−2F(X,是)

∇2F(X,是)=F(X−1,是)+F(X+1,是)+F(X,是−1)+F(X,是+1)−4F(X,是)

有限元方法代写

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MATLAB代写

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