假设有一副分辨率只有8个像素的一维图像对应的像素值分别为：IX=21 19 13 15 16 14 12 16可以用生成矢量空间中的哈尔基函数的线性组合表示请写出矢量空间的哈尔基函数。计算图像IX的哈尔小波变换系数写出计算过程。

一维哈尔基函数的矢量空间为：

$$ \begin{aligned} h_1 &= \frac{1}{\sqrt{2}} [1, 1, 0, 0, 0, 0, 0, 0] \ h_2 &= \frac{1}{\sqrt{2}} [0, 0, 1, 1, 0, 0, 0, 0] \ h_3 &= \frac{1}{\sqrt{2}} [0, 0, 0, 0, 1, 1, 0, 0] \ h_4 &= \frac{1}{\sqrt{2}} [0, 0, 0, 0, 0, 0, 1, 1] \ h_5 &= \frac{1}{\sqrt{2}} [1, -1, 0, 0, 0, 0, 0, 0] \ h_6 &= \frac{1}{\sqrt{2}} [0, 0, 1, -1, 0, 0, 0, 0] \ h_7 &= \frac{1}{\sqrt{2}} [0, 0, 0, 0, 1, -1, 0, 0] \ h_8 &= \frac{1}{\sqrt{2}} [0, 0, 0, 0, 0, 0, 1, -1] \end{aligned} $$

计算图像I(X)的哈尔小波变换系数，首先对图像进行一级变换：

$$ \begin{aligned} c_1 &= \frac{1}{\sqrt{2}} (21 + 19, 13 + 15, 16 + 14, 12 + 16) = [20, 14, 15, 14] \ c_2 &= \frac{1}{\sqrt{2}} (21 - 19, 13 - 15, 16 - 14, 12 - 16) = [1, -2, 0, -4] \ \end{aligned} $$

得到一级变换系数$c_1$和$c_2$。然后对$c_1$进行二级变换：

$$ \begin{aligned} c_{1,1} &= \frac{1}{\sqrt{2}} (20 + 14, 15 + 14) = [17, 14] \ c_{1,2} &= \frac{1}{\sqrt{2}} (20 - 14, 15 - 14) = [3, 0] \ \end{aligned} $$

对$c_2$进行二级变换：

$$ \begin{aligned} c_{2,1} &= \frac{1}{\sqrt{2}} (1 - 2, 0 - 4) = [-1, -2] \ c_{2,2} &= \frac{1}{\sqrt{2}} (1 + 2, 0 + 4) = [\frac{3}{\sqrt{2}}, \frac{4}{\sqrt{2}}] \ \end{aligned} $$

得到二级变换系数$c_{1,1}$、$c_{1,2}$、$c_{2,1}$、$c_{2,2}$。最后的哈尔小波变换系数为：

$$ \begin{aligned} W_1 &= [17, 14, 3, 0, -1, -2, \frac{3}{\sqrt{2}}, \frac{4}{\sqrt{2}}] \end{aligned} $$