From 5bde0be5d2b8a2640fd8ab039f31e189acb1eec6 Mon Sep 17 00:00:00 2001
From: Henry Moore <henry.moore@picknik.ai>
Date: Fri, 1 Nov 2024 17:52:18 +0000
Subject: [PATCH] switch to using math mode instead of $$

---
 doc/Constraints.md | 22 ++++++++++++++++------
 1 file changed, 16 insertions(+), 6 deletions(-)

diff --git a/doc/Constraints.md b/doc/Constraints.md
index 9ea083cb..74662726 100644
--- a/doc/Constraints.md
+++ b/doc/Constraints.md
@@ -15,7 +15,9 @@ documentation is highly recommended.
 
 The nonlinear least-squares system used by the Ceres Solver is written as:
 
-$$ \mathrm{arg\ min}_{x} \left(\frac{1}{2} \sum_i \rho_i \left(||f(x_{i_1},...,x_{i_k})||^2\right)\right)$$
+```math
+\mathrm{arg\ min}_{x} \left(\frac{1}{2} \sum_i \rho_i \left(||f(x_{i_1},...,x_{i_k})||^2\right)\right)
+```
 
 In Ceres Solver parlance, &rho;() is called a "loss function". f() is called a "cost function", which accepts one or
 more inputs, x. And the inputs, x, are called "parameter blocks". The "parameter blocks" themselves may be
@@ -43,7 +45,9 @@ The "cost function" is the main component of the Constraint object. It is respon
 minimized by the Ceres Solver optimizer. The cost function must implement some sort of equation to generate a score
 for arbitrary input values. In its most generic form, that equation is written simply as:
 
-$$ \begin{bmatrix} r_1 \\ ... \\ r_i\end{bmatrix} = f\left(\begin{bmatrix}x_{1_1} \\ ... \\ x_{1_j}\end{bmatrix}, ..., \begin{bmatrix}x_{n_1} \\ ... \\ x_{n_k}\end{bmatrix}\right)$$
+```math
+\begin{bmatrix} r_1 \\ ... \\ r_i\end{bmatrix} = f\left(\begin{bmatrix}x_{1_1} \\ ... \\ x_{1_j}\end{bmatrix}, ..., \begin{bmatrix}x_{n_1} \\ ... \\ x_{n_k}\end{bmatrix}\right) 
+```
 
 
 where f() is the cost function, x<sub>1</sub> through x<sub>n</sub> are the input Variables, each of which may contain
@@ -59,7 +63,9 @@ An observation model, sometimes called a sensor model, predicts a sensor measure
 of the system Variables. The cost is then computed as the difference between the predicted sensor measurement and the
 actual sensor measurement, normalized by the measurement uncertainty.
 
-$$ \begin{bmatrix} r_1 \\ ... \\ r_i\end{bmatrix} = \left(\begin{bmatrix}z_1 \\ ... \\ z_i\end{bmatrix} - h\left(\begin{bmatrix}x_{1_1} \\ ... \\ x_{1_j}\end{bmatrix}, ..., \begin{bmatrix}x_{n_1} \\ ... \\ x_{n_k}\end{bmatrix}\right)\right) \cdot \Sigma ^{-\frac{1}{2}}$$
+```math
+\begin{bmatrix} r_1 \\ ... \\ r_i\end{bmatrix} = \left(\begin{bmatrix}z_1 \\ ... \\ z_i\end{bmatrix} - h\left(\begin{bmatrix}x_{1_1} \\ ... \\ x_{1_j}\end{bmatrix}, ..., \begin{bmatrix}x_{n_1} \\ ... \\ x_{n_k}\end{bmatrix}\right)\right) \cdot \Sigma ^{-\frac{1}{2}}
+```
 
 where z is the sensor measured, h() is the sensor prediction function, and &Sigma; is the covariance matrix. Within
 the least-squares minimization, the entire cost function will get squared. By dividing by the square root of the
@@ -73,7 +79,9 @@ A state transition model, sometimes called a motion model, predicts the value of
 current estimates of the system Variables. This is generally used to enforce a physical model of the system, such
 as known vehicle kinematics.
 
-$$ \begin{bmatrix} r_1 \\ ... \\ r_i\end{bmatrix} = \left(\begin{bmatrix}x_{t_1} \\ ... \\ x_{t_i}\end{bmatrix} - f\left(\begin{bmatrix}x_{{t-1}_1} \\ ... \\ x_{{t-1}_i}\end{bmatrix}\right)\right) \cdot \Sigma ^{-\frac{1}{2}}$$
+```math
+\begin{bmatrix} r_1 \\ ... \\ r_i\end{bmatrix} = \left(\begin{bmatrix}x_{t_1} \\ ... \\ x_{t_i}\end{bmatrix} - f\left(\begin{bmatrix}x_{{t-1}_1} \\ ... \\ x_{{t-1}_i}\end{bmatrix}\right)\right) \cdot \Sigma ^{-\frac{1}{2}}
+```
 
 where x<sub>t</sub> is the current Variable estimate for time _t_, x<sub>t-1</sub> is the current Variable estimate
 for time _t-1_, f() is the state prediction function that implements the desired kinematic or dynamic model
@@ -246,10 +254,12 @@ modeled this way. Our cost function will follow the "observation model", where t
 predict the sensor measurement, and the cost will be the different between the measured and the prediction normalized
 by the measurement uncertainty.
 
-$$ \begin{bmatrix} \mathrm{cost}_1 \\ \mathrm{cost}_2 \\ \mathrm{cost}_3\end{bmatrix}
+```math
+\begin{bmatrix} \mathrm{cost}_1 \\ \mathrm{cost}_2 \\ \mathrm{cost}_3\end{bmatrix}
 = \left(\begin{bmatrix}z_x \\ z_y \\ z_{yaw}\end{bmatrix}
 - \begin{bmatrix}position_x \\ position_y \\ orientation_{yaw}\end{bmatrix}\right)
-\cdot \Sigma ^{-\frac{1}{2}}$$
+\cdot \Sigma ^{-\frac{1}{2}}
+```
 
 We will make use of Ceres Solver's automatic derivative system to compute the Jacobians. For that to work, we must
 implement the cost function equation as a functor object (has an `operator()` method). To compute the cost, our