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QUIZ 4: Sensor Fusion, Triangulation

Due date: 2026-02-07 23:59.
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%✅ For a point in the field of view of the camera the sensor maps 3D point coordinates to 2D image coordinates. %❌ For a point in the field of view of the camera the sensor maps 3D point coordinates to the orientation of the ray from the point to the pinhole. %❌ The preimage is a single point in ℝ³. %❌ The preimage is a plane in ℝ³. %✅ The preimage is a ray in ℝ³, originating at the pinhole. %❌ The preimage consists of all points in the camera’s field of view. %❌ The preimage consists of all points outside the camera’s field of view. %❌ The preimage is empty. %❌ The observation uniquely determines the position of the point, 𝑝. %✅ The observation is the same for all points along the ray from the pinhole through 𝑝 (excluding the pinhole itself).

Question 1

Suppose 𝑋 = ℝ³, corresponding to the position of a point, 𝑝, in a 3D environment. Consider a sensor that is a pinhole camera with known position and orientation, as defined in lecture 5.
Suppose the point 𝑝 is observed in the image (i.e., it lies in the field of view of the camera). Which statements are TRUE?
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%❌ The preimage is a single point in ℝ³. %❌ The preimage is a plane in ℝ³. %❌ The preimage is a ray in ℝ³, originating at the pinhole. %❌ The preimage consists of all points in the camera’s field of view. %✅ The preimage consists of all points outside the camera’s field of view. %❌ The preimage is empty. %❌ The observation uniquely determines the position of the point, 𝑝. %✅ The observation is the same for all points along the ray from the pinhole through 𝑝 (excluding the pinhole itself).

Question 2

Using the same setup as in the previous question, suppose the point 𝑝 is not observed (i.e., it lies outside the field of view of the camera).
Which statements are TRUE?
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%✅ ℎ⁻¹(5) = { (𝑥,5) ∣ 𝑥 ∈ [−1,1] }. %❌ ℎ⁻¹(5) = { (𝑥,𝑡) ∣ 𝑥 ∈ [−1,1], 𝑡 ∈ [0,10]}. %❌ ℎ⁻¹(5) = [−1,1] x [0,10]. %✅ ℎ⁻¹(5) = [−1,1] x {5}. %✅ ℎ⁻¹(5) is a linear segment in 𝑍 = [−1,1] x [0,10]. %✅ ℎ⁻¹(5) ⊂ [−1,1] x [0,10]. %❌ ℎ⁻¹(5) ⊂ 𝑋. %❌ The sensor is bijective. %❌ ℎ(𝑧) = 5 uniquely determines the state 𝑥. %✅ ℎ reports time.

Question 3

Consider the state-time space 𝑍 defined as:
Z = X \times T, \qquad X = [-1,1], \qquad T = [0,10]. \\\
Define the sensor:
h(x,t)=t. \\\
Which statements are TRUE if the observation is ℎ(𝑧)=5?
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%✅ ℎ(𝑥) reports distance to a landmark at (1, 1). %❌ ℎ(𝑥₁, 𝑥₂) reports the minimum between 𝑥₁ and 𝑥₂. %❌ ℎ⁻¹(1) is a line. %✅ ℎ⁻¹(1) is a circle of radius 1 centered at (1,1). %❌ ℎ⁻¹(𝑦) is empty for all 𝑦 > 0. %✅ ℎ⁻¹(𝑦) is a circle of radius 𝑦 for all 𝑦 > 0. %❌ The sensor uniquely determines position, 𝑥. %✅ The observation remains the same if the sensor is rotated about (1, 1).

Question 4

Consider the preimages of a sensor ℎ : ℝ² → ℝ, defined as:
h(x)=\lVert x-(1,1)\rVert. \\\
Which statements are TRUE?
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%❌ ℎ is a proximity sensor. %✅ ℎ is a detection sensor. %❌ The sensor measures distance to 𝑉. %✅ The sensor detects whether or not the robot is in 𝑉. %✅ ℎ⁻¹(1) = 𝑉. %✅ ℎ⁻¹(0) = 𝑋 \ 𝑉. %❌ The sensor uniquely determines the state of the robot, 𝑥.

Question 5

Consider a point robot with the state space 𝑋 = ℝ². Define a region 𝑉 ⊂ 𝑋 and a sensor:
h(x)= \begin{cases} 1, & x\in V\\ 0, & x\notin V. \end{cases} \\\
Which statements are TRUE?
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%✅ Δ(1, 1) = ∅. %❌ Δ(𝑦₁, 𝑦₂) = ∅ ⟺ (𝑦₁ < 2) and (𝑦₂ < 2). %❌ Δ(𝑦₁, 𝑦₂) = ∅ ⟺ y₁ + y₂ < 4. %✅ Δ(y₁, y₂) = ∅ ⟺ (y₁ + y₂ < 4) or (|y₁ − y₂| > 4). %❌ Δ(0, 4) contains all points on the line between the two towers. %✅ Δ(0, 4) = {(0, 0)}. %✅ Δ(4, 0) = {(0, 4)}. %✅ Δ(𝑦₁, 𝑦₂) is a single point for any 𝑦₁, 𝑦₂, such that 𝑦₁ + 𝑦₂ = 4. %❌ Δ(2, 2) contains two points. %✅ Δ(2, 2) uniquely determines the state 𝑥.

Question 6

Let 𝑋 = ℝ² be a state space. There are two sensors. One sensor, ℎ₁, measures the distance to a landmark located at coordinates (0, 0). The other sensor, ℎ₂, measures the distance to a landmark located at coordinates (0, 4). Consider a spatial triangulation approach that uses these sensors to solve the problem of estimating the state 𝑥 ∈ 𝑋.
Define the intersection of two preimages, 𝑦₁ and 𝑦₂, as Δ(𝑦₁, 𝑦₂) = ℎ₁⁻¹(𝑦₁) ∩ ℎ₂⁻¹(𝑦₂). Which statements are TRUE?
Note: Symbol ⟺ denotes "if and only if".
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%✅ Δ(5, 2, 1) = (𝑋 × {5}) ∩ ({𝑥 : ∥𝑥∥ = 2} × 𝑇) ∩ (𝑉 × 𝑇). %✅ Δ(5, 2, 1) = ({𝑥 : ∥𝑥∥ = 2} ∩ 𝑉) × {5}. %❌ Δ(5, 2, 1) ⊂ 𝑋. %✅ If 𝑉 = {𝑥 ∣ 𝑥₂ > 4}, then Δ(5, 2, 1) = ∅. %✅ If 𝑉 = ℝ², then Δ(5, 2, 1) = {𝑥 : ∥𝑥∥ = 2} × {5}. %❌ Δ(𝑦₁, 𝑦₂, 𝑦₃) = (𝑋 × {𝑦₁}) ∩ ({𝑥 : ∥𝑥∥ = 𝑦₂} × 𝑇) ∩ (𝑉 × 𝑇). %✅ Δ(𝑦₁, 𝑦₂, 1) = ({𝑥 : ∥𝑥∥ = 𝑦₂} ∩ 𝑉) × {𝑦₁}, for all choices of 𝑉. %✅ Δ(𝑦₁, 𝑦₂, 𝑦₃) uniquely determines time t = 𝑦₁, for all choices of 𝑉. %❌ Δ(𝑦₁, 𝑦₂, 𝑦₃) uniquely determines 𝑥 for all choices of 𝑉.

Question 7

Consider the state–time space:
Z = X \times T, \quad X = \mathbb{R}^2, \quad T = [0,10]. \\\
Define three sensors on 𝑍:
  • Time sensor: ℎ₁(𝑥, 𝑡) = 𝑡.
  • Distance-to-landmark sensor: ℎ₂(𝑥, 𝑡) = ∥𝑥 − (0, 0)∥.
  • Detection sensor for a region 𝑉 ⊂ 𝑋:
h_3(x,t) = \begin{cases} 1, & x \in V, \\ 0, & x \notin V. \end{cases} \\\
For three observations, 𝑦₁, 𝑦₂, and 𝑦₃, define the 3-way triangulation as the intersection of three preimages:
\Delta(y_1, y_2, y_3) = h_1^{-1}(y_1) \cap h_2^{-1}(y_2) \cap h_3^{-1}(y_3). \\\
Which statements are TRUE?
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%❌ 𝐻₂⁻¹(5) is a circle. %✅ 𝐻₂⁻¹(5) = { 𝑥 ∈ ℝ² ∣ ∣∥𝑥 − (1, 1)∥ − 5∣ ≤ 𝜀 }. %✅ The set 𝐻₂⁻¹(5) is an annulus centered at (1, 1). %❌ 𝐻₂⁻¹(𝑦) is empty for all 𝑦 > 0. %✅ 𝐻₂⁻¹(𝑦) ⊃ ℎ₂⁻¹(𝑦). %❌ 𝐻₂⁻¹(𝑦) uniquely determines the position 𝑥. %✅ 𝐻₂(𝑥) = { 𝑦 ∈ [0, ∞) ∣ ∣𝑦 − ∥𝑥 − (1, 1)∥∣ ≤ 𝜀 }. %✅ The sensor ℎ₂ reports distance to the landmark at (1, 1) with bounded error.

Question 8

Consider the state space 𝑋 = ℝ².
  • The distance-to-landmark at (1, 1) sensor is defined as:
ℎ_1(𝑥) = ∥𝑥 − (1, 1)∥. \\\
  • Now suppose the sensor is affected by a nondeterministic disturbance with known bound 𝜀 > 0, with the resulting nondeterministic sensor:
h_2(x,d)= ∥x − (1,1)∥ + d, \,\,\,\, d \in [-\varepsilon, \varepsilon]. \\\
Let 𝐻₂⁻¹ denote the nondeterministic preimage of ℎ₂(𝑥, 𝑑). Which statements are TRUE?
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%✅ 𝐻₁⁻¹(𝑦₁) = 𝑋 × ([𝑦₁ − 𝜀₁, 𝑦₁ + 𝜀₁] ∩ 𝑇). %✅ 𝐻₂⁻¹(𝑦₂) = { (𝑥,𝑡) ∈ 𝑍 ∣ |‖𝑥‖ − 𝑦₂| ≤ 𝜀₂ }. %✅ Δ(𝑦₁, 𝑦₂, 1) = ({ 𝑥 ∣ |‖𝑥‖ − 𝑦₂| ≤ 𝜀₂ } ∩ 𝑉) × ([𝑦₁ − 𝜀₁, 𝑦₁ + 𝜀₁] ∩ 𝑇). %❌ Δ(𝑦₁, 𝑦₂, 𝑦₃) = { 𝑥 ∣ ‖𝑥‖ = 𝑦₂ } × {𝑦₁}. %❌ Δ(𝑦₁, 𝑦₂, 𝑦₃) ⊂ 𝑋. %✅ If 𝑉 = ℝ², then Δ(𝑦₁, 𝑦₂, 1) = { 𝑥 ∣ |‖𝑥‖ − 𝑦₂| ≤ 𝜀₂ } × ([𝑦₁ − 𝜀₁, 𝑦₁ + 𝜀₁] ∩ 𝑇). %✅ If 𝑉 = ∅, then Δ(𝑦₁, 𝑦₂, 1) = ∅. %❌ Δ(𝑦₁, 𝑦₂, 𝑦₃) is always a single point. %❌ Δ(𝑦₁, 𝑦₂, 𝑦₃) uniquely determines the state 𝑥 for all choices of 𝑉. %❌ Δ(𝑦₁, 𝑦₂, 𝑦₃) uniquely determines the time 𝑡.

Question 9

Consider the state-time space
Z = X \times T, \qquad X = \mathbb{R}^2, \qquad T = [0,10]. \\\
Now consider the time, distance, and detection sensors, as in Question 7, but assume that the time and distance sensors are affected by nondeterministic disturbances with known bounds. Let 𝜀₁ > 0 and 𝜀₂ > 0 denote the disturbance bounds for the time and distance sensors, respectively.
Define the following sensors on 𝑍:
  • Nondeterministic time sensor: ℎ₁(𝑥, 𝑡, 𝑑) = 𝑡 + 𝑑, 𝑑 ∈ [−𝜀₁, 𝜀₁].
  • Nondeterministic distance-to-landmark at (0,0) sensor: ℎ₂(𝑥, 𝑡, 𝑑) = ‖𝑥‖+ 𝑑, 𝑑 ∈ [−𝜀₂, 𝜀₂].
  • Detection sensor for a region 𝑉 ⊂ 𝑋 (no disturbance):
h_3(x,t) = \begin{cases} 1, & x \in V, \\ 0, & x \notin V. \end{cases} \\\
For observations 𝑦₁, 𝑦₂, 𝑦₃, define the 3-way triangulation as the intersection of the corresponding preimages:
\Delta(y_1, y_2, y_3) = H_{1}^{-1}(y_1) \cap H_{2}^{-1}(y_2) \cap h_3^{-1}(y_3). \\\
Which statements are TRUE?
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%❌ The sensor uniquely determines whether 𝑥 ∈ 𝑉. %❌ 𝑃(𝑥 ∈ 𝑉 ∣ 𝑦 = 1) = 1/2. %✅ 𝑃(𝑥 ∈ 𝑉 ∣ 𝑦 = 1) = 9/10. %❌ 𝑃(𝑥 ∉ 𝑉 ∣ 𝑦 = 1) = 9/10. %✅ 𝑃(𝑥 ∉ 𝑉 ∣ 𝑦 = 1) = 1/10. %❌ 𝑃(𝑥 ∈ 𝑉 ∣ 𝑦 = 1) = 1. %❌ The observation provides no information about the state.

Question 10

Consider a point robot with the state space 𝑋 = ℝ². Let 𝑉 ⊂ 𝑋 be a detection region.
The robot is equipped with a probabilistic detection sensor that outputs 𝑦 = ℎ(𝑥) ∈ {0, 1}, where 𝑦 = 1 indicates 𝑥 ∈ 𝑉 and 𝑦 = 0 indicates 𝑥 ∉ 𝑉.
The sensor model is:
P(y=1 \mid x\in V)=\tfrac{9}{10}, \qquad P(y=0 \mid x\in V)=\tfrac{1}{10}, \\\ \\\ P(y=1 \mid x\notin V)=\tfrac{1}{10}, \qquad P(y=0 \mid x\notin V)=\tfrac{9}{10}. \\\
Assume the prior:
P(x\in V)=\tfrac{1}{2}. \\\
Suppose the observation is 𝑦 = 1. Which statements are TRUE?
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Authors

Anna LaValle, Steven M. LaValle.
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