Test Latex equations

ch2.md

@@ -167,4 +167,12 @@ Where, ![](http://latex.codecogs.com/gif.latex?\\bm{v}_{k,j}) is the noise in th
 
 Readers may wonder that the function ![](http://latex.codecogs.com/gif.latex?f,h) we used do not seem to specify what the motion and observations exactly are. Besides, what are ![](http://latex.codecogs.com/gif.latex?\\bm{x}), ![](http://latex.codecogs.com/gif.latex?\\bm{y}), ![](http://latex.codecogs.com/gif.latex?\\bm{z}) here? In fact, according to the actual movement of Little Carrot and the type of sensor it carries, there are different ways for **parameterization**. What is parameterization then? For example, suppose the Little Carrot moves in a plane, then its pose (footnote: In this book, we use the word "pose" to refer to "position" plus "orientation".) is described by two position values and one angle, i.e. ![](http://latex.codecogs.com/gif.latex?\\bm{x}_k=[x,y,\theta]_k^\mathrm{T}). At the same time, the motion sensor can measure the amount of change in the position and angle of Little Carrot at any time step interval ![](http://latex.codecogs.com/gif.latex?\\bm{u}_k=[\\Delta{x},\\Delta{y},\\Delta\\theta]_k^\\mathrm{T}). Then, the motion equation can be specified as 
 
-![](http://latex.codecogs.com/gif.latex?{\\left[\\begin{array}{l}x\\\y\\\\\theta\\end{array}\\right]_k}={\\left[\\begin{array}{l}x\\\y\\\\\theta\\end{array}\\right]_{k-1}}+{\\left[\\begin{array}{l}\\Delta{x}\\\\\Delta{y}\\\\\Delta\\theta\\end{array}\\right]_k}+w_k)
+![](http://latex.codecogs.com/gif.latex?{\\left[\\begin{array}{l}x\\\y\\\\\theta\\end{array}\\right]_k}={\\left[\\begin{array}{l}x\\\y\\\\\theta\\end{array}\\right]_{k-1}}+{\\left[\\begin{array}{l}\\Delta{x}\\\\\Delta{y}\\\\\Delta\\theta\\end{array}\\right]_k}+w_k)
+
+This is a simple linear relationship. However, not all of the sensors can directly measure the displacement and angle changes, so there are other forms of more complex equations of motion, then we may need to carry out dynamic analysis. On the observation equation, for example, a small radish carrying a two-dimensional laser sensor. We know that when a laser sensor observes a 2D punctuation, two quantities can be measured: the distance between the road sign and the radish body is $ r $ and the angle $$ \ phi $$. The observation data is $ \ bm {y} = [p_x, p_y] ^ \ mathrm {T} $ (for the sake of simplicity, omitting the subscript), the observation data is $ \ bm {z} = [r, \ phi] ^ \ Mathrm {T} $, then the observation equation is as follows:
+
+最后，我们在一个图片类别的evidence中加入偏置(bias)，加入偏置的目的是加入一些与输入独立无关的信息。所以图片类别的evidence为
+
+$$ evidence\_{i}=\sum \_{j}W\_{ij}x\_{j}+b\_{i} $$
+
+其中，\\( W\_i \\) 和 \\( b\_i \\) 分别为类别 \\( i \\) 的权值和偏置。