Joyqat's ideas

使用高斯回波模型的测厚方法

1. 取信号的极大值

   1. 若极大值y_M小于阈值则跳过该极大值
   2. 取极大值点N_M附近$2N$点进行模型拟合
   \left \lgroup \begin{array}{l} \bm n = [N_M - N, N_M + N) \\ \bm t = \bm n / f_s \\ \bm y = Samples(\bm n) \end{array} \right .

2. 使用优化的高斯-牛顿迭代法优化模型参数

   1. 使用模型
   \left \lgroup \begin{array}{l} \bm\theta = [\alpha, \tau, f_c, \phi, \beta] \\ s(\bm t;\theta) = \beta e^{-\alpha(\bm t - \tau)^2}\cos[2\pi f_c(\bm t - \tau) + \phi] \end{array} \right .
   2. 取初始向量
   \left \lgroup \begin{array}{l} \bm\theta^{(0)} = [\alpha^{(0)}, \tau^{(0)}, f_c^{(0)}, \phi^{(0)}, \beta^{(0)}] \\ \alpha^{(0)} = 10 \\ \tau^{(0)} = \underset{t}{\arg\max}(\bm y(t))=N_M / f_s \\ f_c^{(0)} = 4 \\ \phi^{(0)} = 0 \\ \beta^{(0)} = \max(\bm y(t)) = y_M \end{array} \right .
   3. 使用以下迭代过程
   \bm\theta^{(k+1)} = \theta^{(k)} + cM(\bm t;\bm\theta^{(k)})(\bm y - s(\bm t;\bm\theta^{(k)})), c=2\sqrt{\frac{2}{\pi}}\frac{\sqrt{\alpha^{(k)}}}{f_s\beta^{(k)}}

   其中f_s为采样频率(MHz)，再定义

   \left \lgroup \begin{array}{l} \bm T = \bm t - \tau \\ \bm A = e^{-\alpha\bm T^2} \\ \bm \Phi = 2\pi f_c \bm T + \phi \\ \bm f = \bm A \cos \Phi \\ \bm g = \bm A \sin \Phi \end{array} \right .

   则s(\bm t;\bm\theta^{(k)})) = \beta\bm f，且：

   \left \lgroup \begin{array}{l} \bm{M_1} = 2\alpha\bm f - 2\alpha\cdot4\alpha\bm T^2\bm f \\ \bm{M_2} = 2\bm T\bm f \\ \bm{M_3} = -2\alpha \bm T\bm g / \pi \\ \bm{M_4} = 4\pi f_c\bm T\bm f - \bm g \\ \bm{M_5} = 1.5\beta\bm f - 2\alpha\beta\bm T^2\bm f \end{array} \right .

   有：

   \left \lgroup \begin{array}{l} \alpha^{(k+1)} = \alpha^{(k)} + \bm{M_1} \cdot (\bm y - \beta\bm f) \\ \tau^{(k+1)} = \tau^{(k)} + \bm{M_2} \cdot (\bm y - \beta\bm f)\\ f_c^{(k+1)} = f_c^{(k)} + \bm{M_3} \cdot (\bm y - \beta\bm f) \\ \phi^{(k+1)} = \phi^{(k)} + \bm{M_4} \cdot (\bm y - \beta\bm f) \\ \beta^{(k+1)} = \beta^{(k)} + \bm{M_5} \cdot (\bm y - \beta\bm f) \end{array} \right .

   迭代终止条件可取：

   ||\bm\theta^{(k+1)} - \bm\theta^{(k)}|| < tolerance

   或者：

   \max|\bm\theta^{(k+1)} - \bm\theta^{(k)}| < tolerance

   在测厚应用中，由于在迭代最终阶段，\tau随迭代次数变化很小，终止条件也可以取:

   |\tau^{(k+1)}-\tau^{(k)}| < tolerance

   一般可取tolerance = 1\times10^{-6}

3. 使用得到的多个回波模型的参数\bm\tau=[\tau_1, \tau_2, ..., \tau_n]计算回波延时(TOF)，再根据声速计算厚度即可。