Public interest in land law

Observation Model Introduction

• The observation model $p(y|x)$ gives the probability of a measurement $y$ given a true system state $x$.
• The specific observation model depends on the sensor type, such as a camera or LiDAR sensor.

Camera Observation Model

• Cameras measure the intensity of light reflected off objects.
• Light intensity depends on object material, illumination, and the relative pose of the camera and object.
• Cameras project the 3D world onto a 2D image plane using perspective projection.
• The perspective projection equations are:
  • $u = f_x \frac{X}{Z} + c_x$
  • $v = f_y \frac{Y}{Z} + c_y$
  • $(X, Y, Z)$ is the 3D point
  • $(u, v)$ is the 2D point
  • $f_x$ and $f_y$ are focal lengths in the x and y directions
  • $c_x$ and $c_y$ are principal point coordinates.
• Focal lengths and principal point coordinates are intrinsic camera parameters.
• Camera calibration estimates the intrinsic parameters.
• Perspective projection is a non-linear transformation, making the observation model non-linear.

LiDAR Observation Model

• LiDAR sensors measure the distance to objects using laser beams and measuring return time.
• Distance is calculated by $d = \frac{c \Delta t}{2}$, where $c$ is the speed of light and $\Delta t$ is the time for the laser beam to return.
• The LiDAR measures the horizontal angle ($\theta$) and the vertical angle ($\phi$) of the laser beam.
• 3D position is calculated as:
  • $X = d \cos(\theta) \cos(\phi)$
  • $Y = d \cos(\theta) \sin(\phi)$
  • $Z = d \sin(\theta)$
• LiDAR sensors also measure angles, allowing for 3D position calculation.
• Typically, noise is modeled as a Gaussian distribution
• LiDAR is more accurate but also more expensive than camera observation models.

Vector Functions of a Real Variable Introduction

• A vector function of a real variable assigns a vector to each real number in an interval.
• Notation: $\vec{r}(t) = (x(t), y(t), z(t)) = x(t)\vec{i} + y(t)\vec{j} + z(t)\vec{k}$
  • $x(t)$, $y(t)$, and $z(t)$ are real-valued component functions of $t$.

Limit of a Vector Function

• The limit of $\vec{r}(t)$ as $t$ approaches $t_0$ is the vector $\vec{L}$ composed of the limits of its component functions.
• $\lim_{t \to t_0} \vec{r}(t) = (\lim_{t \to t_0} x(t), \lim_{t \to t_0} y(t), \lim_{t \to t_0} z(t))$, provided component limits exist.

Continuity of a Vector Function

• $\vec{r}(t)$ is continuous at $t_0$ if and only if:
  • $\vec{r}(t_0)$ exists
  • $\lim_{t \to t_0} \vec{r}(t)$ exists
  • $\lim_{t \to t_0} \vec{r}(t) = \vec{r}(t_0)$
• Each of its component functions must be continuous at $t_0$.

Derivative of a Vector Function

• The derivative is found by differentiating each component function.
• $\frac{d\vec{r}}{dt} = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t} = (\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt})$, provided component derivatives exist.

Geometric Interpretation of the Derivative

• The derivative of $\vec{r}(t)$ at $t_0$ is a tangent vector to the curve described by $\vec{r}(t)$ at $\vec{r}(t_0)$.
• The tangent line at $\vec{r}(t_0)$ passes through $\vec{r}(t_0)$ and is parallel to the tangent vector $\frac{d\vec{r}}{dt}(t_0)$.

Differentiation Rules

• For differentiable vector functions $\vec{r}(t)$ and $\vec{s}(t)$, and a differentiable scalar function $f(t)$:
  • $\frac{d}{dt}(\vec{r}(t) + \vec{s}(t)) = \frac{d\vec{r}}{dt} + \frac{d\vec{s}}{dt}$
  • $\frac{d}{dt}(f(t)\vec{r}(t)) = f'(t)\vec{r}(t) + f(t)\frac{d\vec{r}}{dt}$
  • $\frac{d}{dt}(\vec{r}(t) \cdot \vec{s}(t)) = \frac{d\vec{r}}{dt} \cdot \vec{s}(t) + \vec{r}(t) \cdot \frac{d\vec{s}}{dt}$
  • $\frac{d}{dt}(\vec{r}(t) \times \vec{s}(t)) = \frac{d\vec{r}}{dt} \times \vec{s}(t) + \vec{r}(t) \times \frac{d\vec{s}}{dt}$
  • $\frac{d}{dt}(\vec{r}(f(t))) = \frac{d\vec{r}}{df} \cdot f'(t)$

Integrals of Vector Functions

• The integral is found by integrating each component function.
• $\int \vec{r}(t) dt = (\int x(t) dt, \int y(t) dt, \int z(t) dt)$