From c5f7477a987ccaa333dc48c997bfe8530ebe099a Mon Sep 17 00:00:00 2001
From: Henry Moore <henry.moore@picknik.ai>
Date: Fri, 1 Nov 2024 13:10:50 -0600
Subject: [PATCH] Switch to latex for equations (#8)

* switch to latex for equations

* update to use math mode instead of 3299563
---
 doc/Constraints.md                            |  25 +++++++++++++-----
 doc/constraints-generic_cost_equation.gif     | Bin 2292 -> 0 bytes
 doc/constraints-least_squares_equation.gif    | Bin 1706 -> 0 bytes
 ...constraints-observation_model_equation.gif | Bin 3525 -> 0 bytes
 doc/constraints-pose_2d_prior_equation.gif    | Bin 4202 -> 0 bytes
 ...raints-state_transition_model_equation.gif | Bin 3039 -> 0 bytes
 6 files changed, 18 insertions(+), 7 deletions(-)
 delete mode 100644 doc/constraints-generic_cost_equation.gif
 delete mode 100644 doc/constraints-least_squares_equation.gif
 delete mode 100644 doc/constraints-observation_model_equation.gif
 delete mode 100644 doc/constraints-pose_2d_prior_equation.gif
 delete mode 100644 doc/constraints-state_transition_model_equation.gif

diff --git a/doc/Constraints.md b/doc/Constraints.md
index 263c3e12b..4d74a0bbf 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:
 
-![least squares equation](constraints-least_squares_equation.gif)
+```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
@@ -42,8 +44,9 @@ are acceptable.
 The "cost function" is the main component of the Constraint object. It is responsible for computing the cost to be
 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:
-
-![generic cost equation](constraints-generic_cost_equation.gif)
+```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
 multi-dimensional data, and r<sub>i</sub> are one or more dimensions of the computed costs. In Ceres Solver notation,
@@ -57,8 +60,9 @@ one or more outputs. However, in practice there are two common forms for cost fu
 An observation model, sometimes called a sensor model, predicts a sensor measurement based on the current estimates
 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.
-
-![observation model equation](constraints-observation_model_equation.gif)
+```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
@@ -72,7 +76,9 @@ 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.
 
-![state transition model equation](constraints-state_transition_model_equation.gif)
+```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
@@ -245,7 +251,12 @@ 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.
 
-![2D pose prior cost equation](constraints-pose_2d_prior_equation.gif)
+```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}} 
+```
 
 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
diff --git a/doc/constraints-generic_cost_equation.gif b/doc/constraints-generic_cost_equation.gif
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