### 机器视觉代写|图像处理作业代写Image Processing代考|BILINEAR INTERPOLATION

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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

## 机器视觉代写|图像处理作业代写Image Processing代考|BILINEAR INTERPOLATION

The bilinear interpolation, also called first-order interpolation, calculates the intensity value for any point $(u, v)$ in the input image by using a low-degree polynomial of the form:
$$f(u, v)=\sum_{m=0}^{1} \sum_{n=0}^{1} a_{m n} u^{m} v^{n}$$
where the function $f$ gives the intensity value at $(u, v), a_{m n}(m, n=0,1)$ are coefficients determined by the four nearest neighbors.

When the intensity values of the four nearest neighbors are known, the general idea of the bilinear interpolation is to use linear interpolations along the $x$ – and $y$ directions to determine the intensity value at $(u, v)$. As exemplified in Figure $2.24, P$ denotes the interpolated point for which an intensity value must be calculated, $(u, v)$

are its coordinates mapped from the output image by Equation $2.33$, and $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are its four nearest neighbors in the input image with the coordinates $(i, j)$, $(i, j+1),(i+1, j)$, and $(i+1, j+1)$, respectively. The bilinear interpolation first interpolates linearly along the $x$-direction to find the values at $Q_{1}$ and $Q_{2}$ :
\begin{aligned} &f\left(Q_{1}\right)=(j+1-v) f\left(P_{1}\right)+(v-j) f\left(P_{2}\right) \ &f\left(Q_{2}\right)=(j+1-v) f\left(P_{3}\right)+(v-j) f\left(P_{4}\right) \end{aligned}
then interpolates linearly along $y$-direction to obtain the value of $P$ :
\begin{aligned} f(P)=&(i+1-u) f\left(Q_{1}\right)+(u-i) f\left(Q_{2}\right) \ =&(i+1-u)\left[(j+1-v) f\left(P_{1}\right)+(v-j) f\left(P_{2}\right)\right] \ &+(u-i)\left[(j+1-v) f\left(P_{3}\right)+(v-j) f\left(P_{4}\right)\right] \ =&(i+1-u)(j+1-v) f\left(P_{1}\right)+(i+1-u)(v-j) f\left(P_{2}\right) \ &+(u-i)(j+1-v) f\left(P_{3}\right)+(u-i)(v-j) f\left(P_{4}\right) \end{aligned}
which gives:
$$f(P)=[i+1-u \quad u-i]\left[\begin{array}{cc} f\left(P_{1}\right) & f\left(P_{2}\right) \ f\left(P_{3}\right) & f\left(P_{4}\right) \end{array}\right]\left[\begin{array}{c} j+1-v \ v-j \end{array}\right]$$

## 机器视觉代写|图像处理作业代写Image Processing代考|BICUBIC INTERPOLATION

The bicubic interpolation, also called third-order interpolation, calculates the intensity value of any point $(u, v)$ in the input image by reconstructing a surface among its four nearest neighbors based on their intensity values, the derivatives in both $x$ – and $y$-directions, and the cross derivatives.

Similar to the bilinear interpolation, the bicubic interpolation calculates the intensity value for a point $(u, v)$ by fitting a cubic polynomial:
$$f(u, v)=\sum_{m=0}^{3} \sum_{n=0}^{3} a_{m n} u^{m} v^{n}$$
where $a_{m n}(m, n=0,1,2,3)$ are coefficients determined by its $4 \times 4$ nearest neighbors in the input image, that is, the four nearest neighbors of the point $(u, v)$ (empty circles as seen in Figure 2.25), and their horizontal, vertical, and diagonal neighboring pixels (black dots as seen in Figure 2.25). The latter are used to calculate the first-order derivatives in both $x$ – and $y$-directions and the cross derivative at each of the four nearest neighbors of point $(u, v)$. Then 8 first-order derivatives in both the $x$ – and $y$ directions and 4 cross derivatives, together with 4 intensity values at the four nearest neighbors of point $(u, v)$ give a linear system of 16 equations to determine the 16 coefficients of $a_{m n}$ in Equation $2.39$ [122].

## 机器视觉代写|图像处理作业代写Image Processing代考|Bicubic interpolation

Instead of directly calculating the solution of this linear system, typically by some matrix inversion, an alternative approach is to use a cubic convolution interpolation kernel that is composed of piecewise cubic polynomials defined on the subintervals $(-2,-1),(-1,0),(0,1)$, and $(1,2)[78]$. Assume the coordinates of the four nearest neighbors of point $(u, v)$ in the input image are $(i, j),(i, j+1),(i+1, j)$, and $(i+$ $1, j+1)$. Then the interpolated pixel intensity may be expressed in the compact form [121]:
$$f(u, v)=\sum_{m=-1}^{2} \sum_{n=-1}^{2} f(u+m, v+n) r_{c}{(m+i-u)} r_{c}{-(n+j-v)}$$
36
Sea Ice Image Processing with MATLAB
where $r_{c}(x)$ denotes a bicubic interpolation function, given by :
$$r_{c}(x)= \begin{cases}(a+2)|x|^{3}-(a+3)|x|^{2}+1, & \text { if } 0 \leq|x| \leq 1 \ a|x|^{3}-5 a|x|^{2}+8 a|x|-4 a, & \text { if } 1<|x| \leq 2 \ 0, & \text { if }|x|>2\end{cases}$$
where $a$ is the weighting factor that can be used as a tuning parameter to obtain a best visual interpolation result [118].

Compared with the bilinear interpolation, the bicubic interpolation method extends the influence of more neighboring pixels, and it takes not only the intensity values but also the intensity derivatives into account. Therefore, this method can produce more clear result than the bilinear interpolation method; however, at the expense of more computational complexity.

## 机器视觉代写|图像处理作业代写Image Processing代考|BILINEAR INTERPOLATION

F(在,在)=∑米=01∑n=01一种米n在米在n

F(问1)=(j+1−在)F(磷1)+(在−j)F(磷2) F(问2)=(j+1−在)F(磷3)+(在−j)F(磷4)

F(磷)=(一世+1−在)F(问1)+(在−一世)F(问2) =(一世+1−在)[(j+1−在)F(磷1)+(在−j)F(磷2)] +(在−一世)[(j+1−在)F(磷3)+(在−j)F(磷4)] =(一世+1−在)(j+1−在)F(磷1)+(一世+1−在)(在−j)F(磷2) +(在−一世)(j+1−在)F(磷3)+(在−一世)(在−j)F(磷4)

F(磷)=[一世+1−在在−一世][F(磷1)F(磷2) F(磷3)F(磷4)][j+1−在 在−j]

## 机器视觉代写|图像处理作业代写Image Processing代考|BICUBIC INTERPOLATION

F(在,在)=∑米=03∑n=03一种米n在米在n

## 机器视觉代写|图像处理作业代写Image Processing代考|Bicubic interpolation

F(在,在)=∑米=−12∑n=−12F(在+米,在+n)rC(米+一世−在)rC−(n+j−在)
36使用 MATLAB进行

rC(X)表示双三次插值函数，由 给出：
rC(X)={(一种+2)|X|3−(一种+3)|X|2+1, 如果 0≤|X|≤1 一种|X|3−5一种|X|2+8一种|X|−4一种, 如果 1<|X|≤2 0, 如果 |X|>2

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## MATLAB代写

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