Table of Contents
Robotics: Perception Week 01 Quiz Answers
Quiz 1: Introduction
Q1. In the equation \dfrac{1}{f} = \dfrac{1}{a} + \dfrac{1}{b}f1=a1+b1, what does the ff stands for:
- Force
- Distance between image plane and lens
- Focal Length
- Distance between lens and object
Q2. If an object is originally in focus and then you start moving the image plane, what do you expect to happen:
- Image gets sharper
- f = a + bf=a+b
- \dfrac{1}{f} \neq \dfrac{1}{a} + \dfrac{1}{b}f1=a1+b1
- Image starts blurring
Q3. The size of the projection of an object increases as the object distance from the lens increases.
- True
- False
Q4. Parallel lines in the world remain always parallel after projection.
- True
- False
Q5. Parallel lines in the world remain parallel in the image plane when
- the lines are parallel to the image plane
- the lines are perpendicular to the image plane
Q6. A vanishing point in an image is the intersection of projections of parallel lines in the world. There is at most one vanishing point in an image
- True
- False
Q7. The two parameters that we can directly control using the bi-perspectograph construction are:
- Focal Length
- Distance from the objects
- Angle between image plane and world plane
- Height of the camera
Quiz 2: Vanishing Points
Q1. The School of Athens is a famous fresco by Raphael. Correct perspective projection is visible here. From the three specified points (A, B, or C), which is the vanishing point? (You need to use a ruler)
- A
- B
- C
Q2. From the three options (l_1, l_2, l_3l1,l2,l3), which is the horizon?
- l_1l1
- l_2l2
- l_3l3
Q3. In the following image, from the three options (l_1, l_2, l_3l1,l2,l3), which is the horizon?
- l_1l1
- l_2l2
- l_3l3
Q4. A vanishing point is always visible inside an image1 point
- True
- False
Q5. The horizon is the set of all directions to infinity for a plane1 point
- True
- False
Quiz 3: Perspective Projection
Q1. Assume you are given a line represented in the form 2x + 2y – 2\sqrt{2} = 02x+2y−22=0. Which set of parameters (\rho,\theta)(ρ,θ) gives the same line represented in the form \rho = x \cos\theta + y \sin \thetaρ=xcosθ+ysinθ:
- (-2,60^o)(−2,60o)
- (2,45^o)(2,45o)
- (1,30^o)(1,30o)
- (1,45^o)(1,45o)
Q2. The distance of a line to the origin is \rho=3ρ=3 and the norm direction of the line is \theta = \pi/4θ=π/4. Which of the following is/are valid equations for the line?
- x+y-3=0x+y−3=0
- x+y-3\sqrt{2} = 0x+y−32=0
- \sqrt{2}x-\sqrt{2}y-3=02x−2y−3=0
- \sqrt{2}x+\sqrt{2}y-3=02x+2y−3=0
Q3. What is the equation of the line passing through points with homogeneous coordinates (1,2,1)(1,2,1) and (-1,3,1)(−1,3,1)?
- -2x-y+5= 0−2x−y+5=0
- -2x+y+5=0−2x+y+5=0
- 2x+4y-10=02x+4y−10=0
- x+y+10=0x+y+10=0
Q4. The lines l_1=(1,1,0)l1=(1,1,0) and l_2=(-1,1,1)l2=(−1,1,1) instersect at the point with homogeneous coordinates:
- (0.5,-0.5,1)(0.5,−0.5,1)
- (-0.5,-0.5,1)(−0.5,−0.5,1)
- (1,1,1)(1,1,1)
- (1,-1,1)(1,−1,1)
Q5. Consider the lines y=1y=1 and y=2y=2 in 2D projective space (as previous s). What is the point of intersection in homogeneous coordinates?
- They do not intersect.
- (1,0,0)(1,0,0)
- (-1,0,0)(−1,0,0)
- (0,1,0)(0,1,0)
Quiz 4: Rotations and Translations
Q1. What is the determinant of a rotation matrix?
- -1 or +1
- -1, 0 or 1
- -1
- +1
Q2. What is the rotation { }^{c}R_{w}cRw such that \mathbf{X}_c = { }^{c}R_{w} \mathbf{X}_w +{ }^{c}T_{w} Xc=cRwXw+cTw for a point \mathbf{X}_wXw expressed in the world coordinate frame?
- { }^{c}R _{w}= ⎛⎝−1000−1000−1⎞⎠cRw=⎝⎜⎛−1000−1000−1⎠⎟⎞
- { }^{c}R _{w}= ⎛⎝−100001010⎞⎠cRw=⎝⎜⎛−100001010⎠⎟⎞
- { }^{c}R _{w}= ⎛⎝−10000−10−10⎞⎠cRw=⎝⎜⎛−10000−10−10⎠⎟⎞
- { }^{c}R _{w}= ⎛⎝0−10−10000−1⎞⎠cRw=⎝⎜⎛0−10−10000−1⎠⎟⎞
Q3. What is the corresponding translation { }^{c}T_{w}cTw?1 point
- { }^cT_{w} = (2,0,0)cTw=(2,0,0)
- { }^cT_{w} = (0,2,0)cTw=(0,2,0)
- { }^cT_{w} = (0,0,-2)cTw=(0,0,−2)
- { }^cT_{w} = (-2,0,0)cTw=(−2,0,0)
- { }^cT_{w} = (0,0,2)cTw=(0,0,2)
Q4. What is { }^wR_{c}wRc?
- { }^{w}R _{c}= ⎛⎝−100001010⎞⎠wRc=⎝⎜⎛−100001010⎠⎟⎞
- { }^wR_{c}= ⎛⎝−10000−10−10⎞⎠wRc=⎝⎜⎛−10000−10−10⎠⎟⎞
- { }^{w}R _{c}= ⎛⎝0−10−10000−1⎞⎠wRc=⎝⎜⎛0−10−10000−1⎠⎟⎞
- { }^{w}R _{c}= ⎛⎝−1000−1000−1⎞⎠wRc=⎝⎜⎛−1000−1000−1⎠⎟⎞
Q5. What is { }^wT_{c}wTc?
- { }^wT_{c} = (0,0,-2)wTc=(0,0,−2)
- { }^wT_{c} = (0,-2,0)wTc=(0,−2,0)
- { }^wT_{c} = (0,0,2)wTc=(0,0,2)
- { }^wT_{c} = (0,2,0)wTc=(0,2,0)
Q6. For the quadrotor configuration in the two images below (top view and side view), what is the transformation from the body (imu) coordinate system to the camera?
In particular, what is the rotation { }^{c}R_{b}cRb such that \mathbf{X}_c = { }^{c}R_{b} \mathbf{X}_b +{ }^{c}T_{b} Xc=cRbXb+cTb for a point \mathbf{X}_bXb expressed in the body coordinate frame?
[Top View] (Distance between origins on XY plane is 4cm)
[Side View] (Distance between origins along Z axis is 3cm)
- { }^cR_{b}= ⎛⎝⎜⎜⎜⎜⎜2√20−2√20−10−2√202√2⎞⎠⎟⎟⎟⎟⎟cRb=⎝⎜⎜⎜⎜⎛220−220−10−22022⎠⎟⎟⎟⎟⎞
- { }^cR_{b}= ⎛⎝⎜⎜⎜⎜⎜2√2−2√20−2√2−2√2000−1⎞⎠⎟⎟⎟⎟⎟cRb=⎝⎜⎜⎜⎜⎛22−220−22−22000−1⎠⎟⎟⎟⎟⎞
- { }^cR_{b}= ⎛⎝⎜⎜⎜⎜⎜2√20−2√2−2√20−2√20−10⎞⎠⎟⎟⎟⎟⎟cRb=⎝⎜⎜⎜⎜⎛220−22−220−220−10⎠⎟⎟⎟⎟⎞
- { }^cR_{b}= ⎛⎝⎜⎜⎜⎜⎜2√20−2√20−10−2√20−2√2⎞⎠⎟⎟⎟⎟⎟cRb=⎝⎜⎜⎜⎜⎛220−220−10−220−22⎠⎟⎟⎟⎟⎞
Q7. What is the corresponding translation { }^{c}T_{b}cTb?
- { }^cT_{b} = (0.03, 0, 0.04)cTb=(0.03,0,0.04)m
- { }^cT_{b} = (0.04, 0, 0.03)cTb=(0.04,0,0.03)m
- { }^cT_{b} = (-0.04, 0, -0.03)cTb=(−0.04,0,−0.03)m
- { }^cT_{b} = (-0.03, 0, -0.04)cTb=(−0.03,0,−0.04)m
Quiz 5: Dolly Zoom
Q1. Given Image 1, which of the four other images (2-5) would be the final result if we reduce the focal length?
[Image 1]
[Image 2]
[Image 3]
[Image 4]
[Image 5]
- Image 2
- Image 3
- Image 4
- Image 5
Q2. For the five images above, for which one do you think that the camera is the farthest away from the scene?
- Image 1
- Image 2
- Image 3
- Image 4
- Image 5
Quiz 6: Feeling of Camera Motion
Q1. You are given two images of a scene, before and after a change in the camera. Which transformation can produce this result?
[Before]
[After]
- Changing the focal length
- Movement of the camera on the vertical axis
- Movement of the camera on the horizontal axis
Q2. You are given two images of a scene, before and after a change in the camera. Which transformation can produce this result?
[Before]
[After]
- Movement of the camera on the vertical axis
- Rotation of the camera around the zz-axis
- Changing focal length
Q3. You are given two images of a scene, before and after a change in the camera. Which transformation can produce this result?
[Before]
[After]
- Increasing the focal length
- Rotation around the zz-axis of the camera
- Translation on the vertical axis of the camera
Quiz 7: How to Compute Intrinsics from Vanishing Points
Q1. In the image below, we can see the projections of three orthogonal vanishing points V_1,V_2, V_3V1,V2,V3 and the image center CC. Which of the following statements is always true?
- The image center is the centroid of the triangle formed by the projections of three orthogonal vanishing points.
- The image center is the orthocenter of the triangle formed by the projections of three orthogonal vanishing points.
- The image center is the circumcenter of the triangle formed by the projections of three orthogonal vanishing points.
- The image center is the incenter of the triangle formed by the projections of three orthogonal vanishing points.
Q2. Assume that the image center has been computed using the result of the previous question. Then, under which conditions can we compute the focal length from the image projections of three orthogonal vanishing points?
- At least two of the vanishing points are not at infinity.
- All of the vanishing points are at infinity.
- At least one of the vanishing points is not at infinity.
Quiz 8: Camera Calibration
Q1. The calibration procedure estimates:
- Focal length
- Image Center
- Radial distortion parameters
- All the above
Q2. Which two of the four images below suffer mostly from radial distortion effects?
A)
B)
C)
D)
- A
- B
- C
- D
Q3. For calibration you need to know the size of the checkerboard squares
- True
- False
Robotics: Perception Week 2 Quiz Answers
Quiz 1: Homogeneous Coordinates
Q1. The homogeneous coordinates of a point PP are (1,2,1)(1,2,1). Which of the following (homogeneous) coordinates represent the same point?
- (1,1,2)(1,1,2)
- (2,4,2)(2,4,2)
- (-0.5,-1,-0.5)(−0.5,−1,−0.5)
- (1,2,0)(1,2,0)
Q2. Given a square ABCD, with A = (0,0,1)A=(0,0,1) and C = (1,1,1)C=(1,1,1), the equation of the diagonal BD in \mathbb{P}^2P2 has the form l^Tx=0lTx=0 with ll equal to
Clarification: For this and following s, we use \mathbb{P}^2P2 to denote the real projective plane.
- (-1,-1,1)(−1,−1,1)
- (-1,2,1)(−1,2,1)
- (1,-1,1)(1,−1,1)
- (-1,1,1)(−1,1,1)
Q3. Determine the equation of the line in \mathbb{P}^2P2 through the points (a,0,1)(a,0,1) and (0,b,1)(0,b,1).
- ⎛⎝−b−aab⎞⎠
- ⎝⎜⎛−b−aab⎠⎟⎞
- ⎛⎝baab⎞⎠
- ⎝⎜⎛baab⎠⎟⎞
- ⎛⎝ab−ab⎞⎠
- ⎝⎜⎛ab−ab⎠⎟⎞
- ⎛⎝abab⎞⎠
- ⎝⎜⎛abab⎠⎟⎞
Q4. Determine the equation of the line in \mathbb{P}^2P2 through the points (a,b,c)(a,b,c) and (d,e,0)(d,e,0).
- ⎛⎝cecdae+bd⎞⎠
- ⎝⎜⎛cecdae+bd⎠⎟⎞
- ⎛⎝−cecdae−bd⎞⎠
- ⎝⎜⎛−cecdae−bd⎠⎟⎞
- ⎛⎝cecdae−bd⎞⎠
- ⎝⎜⎛cecdae−bd⎠⎟⎞
- ⎛⎝adbe0⎞⎠
- ⎝⎜⎛adbe0⎠⎟⎞
Q5. Determine the equation of the line in \mathbb{P}^2P2 through the points (a,b,0)(a,b,0) and (d,e,0)(d,e,0).
- ⎛⎝ae0bd⎞⎠
- ⎝⎜⎛ae0bd⎠⎟⎞
- ⎛⎝00ae+bd⎞⎠
- ⎝⎜⎛00ae+bd⎠⎟⎞
- ⎛⎝00ae−bd⎞⎠
- ⎝⎜⎛00ae−bd⎠⎟⎞
- ⎛⎝adbe0⎞⎠
- ⎝⎜⎛adbe0⎠⎟⎞
Quiz 2: Projective Transformations
Q1. What is the least number of non-collinear points required to estimate a projective transformation H:\mathbb{P}^2 \rightarrow \mathbb{P}^2H:P2→P2?
4
Q2. A projective transformation MM preserves the points (1,0,0)(1,0,0), (0,1,0)(0,1,0), and the origin of the coordinate system. However, it maps the point (1,1,1)(1,1,1) to the points (2,1,1)(2,1,1), meaning (2,1,1)^{T} = M (1,1,1)^{T} (2,1,1)T=M(1,1,1)T. Compute MM.
- M \sim ⎛⎝200010001⎞⎠M∼⎝⎜⎛200010001⎠⎟⎞
- M \sim ⎛⎝200010101⎞⎠M∼⎝⎜⎛200010101⎠⎟⎞
- M \sim ⎛⎝200010011⎞⎠M∼⎝⎜⎛200010011⎠⎟⎞
- M \sim ⎛⎝200010111⎞⎠M∼⎝⎜⎛200010111⎠⎟⎞
Q3. Find the projective transformation AA which will keep the points (0,0,1)(0,0,1) and (1,1,1)(1,1,1) fixed and will map point (1,0,1)(1,0,1) to (1,0,0)(1,0,0) and point (0,1,1)(0,1,1) to (0,1,0)(0,1,0)?
- A \sim ⎛⎝101011001⎞⎠A∼⎝⎜⎛101011001⎠⎟⎞
- A \sim ⎛⎝−10−10−1−1001⎞⎠A∼⎝⎜⎛−10−10−1−1001⎠⎟⎞
- A \sim ⎛⎝10−101−1001⎞⎠A∼⎝⎜⎛10−101−1001⎠⎟⎞
- A \sim ⎛⎝−1010−1100−1⎞⎠A∼⎝⎜⎛−1010−1100−1⎠⎟⎞
Q4. Find the projective transformation AA that maps the points (1,0,0)(1,0,0), (0,1,0)(0,1,0), (0,0,1)(0,0,1), and (1,1,1)(1,1,1) to the points (-2,0,1)(−2,0,1), (0,1,-1)(0,1,−1), (-1,2,-1)(−1,2,−1) and (-1,1,1)(−1,1,1), respectively.
- A \sim ⎛⎝⎜−2/301/305/3−5/3122⎞⎠⎟A∼⎝⎜⎛−2/301/305/3−5/3122⎠⎟⎞
- A \sim ⎛⎝⎜−2/301/305/3−5/31−21⎞⎠⎟A∼⎝⎜⎛−2/301/305/3−5/31−21⎠⎟⎞
- A \sim ⎛⎝⎜2/301/305/3−5/3121⎞⎠⎟A∼⎝⎜⎛2/301/305/3−5/3121⎠⎟⎞
- A \sim ⎛⎝⎜1/301/305/35/31−21⎞⎠⎟A∼⎝⎜⎛1/301/305/35/31−21⎠⎟⎞
Quiz 3: Vanishing Points
Q1. When the camera is zooming, do the vanishing points move?
- Yes
- No
Q2. What camera change would give the following result from Image 1 to Image 2
(Hint: Notice how the vanishing points change)
[Image 1]
[Image 2]
- Camera Translation
- Zooming
Q3. What camera change would give the following result from Image 1 to Image 2
(Hint: Notice if the vanishing points change)
[Image 1]
[Image 2]
- Camera Translation
- Zooming
Q4. The image of the rectangle-shaped facade of a building has two vanishing points, one at (-b,0)(−b,0) corresponding to horizontal lines and one at (0,h)(0,h) correspondng to the vertical lines. Which of the following transformations will map the facade to a rectangle. Assume that the origin (0,0)(0,0) and the point (1,1)(1,1) remain fixed.
4.z
h-b+bh 3,3 = bh
Quiz 4: Cross Ratios and Single View Metrology
Q1. For the image below, if AB = 12AB=12, BC = 4BC=4 and CD = 8CD=8, what is the cross ratio \mathcal{CR}(A,B,C,D)CR(A,B,C,D)?
2
Q2. For the same image as the previous question, if A’B’ = 5A′B′=5, B’C’ = 2B′C′=2 and C’D’ = 4C′D′=4, what is the cross ratio \mathcal{CR}(A’,B’,C’,D’)CR(A′,B′,C′,D′)? (Just Round it off to two decimals)
1.91
Q3. Is it possible that the image of A’B’C’D’A′B′C′D′ is the result of a perspective projection from ABCDABCD? (assume that the lengths are the same as those from the previous two questions)
- Yes
- No
Q4. If not, what should be the length A’B’A′B′, such that A’B’C’D’A′B′C′D′ is indeed the result of a perspective projection from ABCDABCD
- 4
- 6
- 7
- It is already a perspective projection
Robotics: Perception Week 3 Quiz Answers
Quiz 1: Visual Features
Q1. Features can be useful for
- Image retrieval
- Image based localization
- Scene reconstruction
- Panorama stitching
Q2. What properties of features are desirable?
- Descriptor variance.
- Descriptor invariance.
- Detection Invariance.
- Detection variance.
Q3. A scale space of an image can be build by
- convolving with gaussian filters and subsampling.
- subsampling.
- convolving the image with itself multiple times.
Q4. The scale of a feature is chosen by first convolving the corresponding image patch with Difference-of-Gaussian (DoG) filters and then, by taking the maximum response over all scales.
- False
- True
Q5. The SIFT detector is
- Scale but not rotation invariant.
- Scale and rotation invariant
- Rotation but not scale invariant
Q6. To compute the SIFT descriptor
- You compute a histogram of gradients in a 16 by 16 grid and rotate them to have the largest magnitude gradient oriented upwards
- You compute a histogram of colors around a point and rotate it so the brightest patch is up
Quiz 2: Singular Value Decomposition
Q1. If U \Sigma V^TUΣVT is an SVD for a given matrix AA then which if the following statements are true?
- UU and VV are orthogonal matrices
- UU and \SigmaΣ are orthogonal matrices
- UU is orthogonal and VV is diagonal
- UU is orthogonal and \SigmaΣ is diagonal
Q2. A symmetric real matrix has real eigenvalues and real singular values. Which of the following is true?
- All eigenvalues are nonnegative.
- Singular values are equal to the eigenvalues.
- All singular values are nonnegative.
Q3. The largest singular value of
- ⎡⎣−200001000⎤⎦
- ⎣⎢⎡−200001000⎦⎥⎤ is
2
Q4. Which of the following are valid SVD’s of the form U \Sigma V^TUΣVT for the matrix
A =
⎡⎣0020−10300⎤⎦
A=⎣⎢⎡0020−10300⎦⎥⎤
U=-1
U=1;-1;-1
Q5. Find the rank of the matrix
A =
⎡⎣⎢⎢3235437558377451⎤⎦⎥⎥
A=⎣⎢⎢⎢⎡3235437558377451⎦⎥⎥⎥⎤
4
Q6. Which of the following is true?
- The rank of matrix is equal to its largest singular value.
- The rank of a matrix is equal to the number of nonzero singular values.
- The rank of matrix has nothing to do with its singular values.
Q7. The minimizer of the fitting cost ||Ax||_2^2∣∣Ax∣∣22 with A \in \mathbb{R}^{m \times n}A∈Rm×n, \operatorname m > nm>n subject to ||x||_2=1∣∣x∣∣2=1 is
- The eigenvector of A^TAATA corresponding to the smallest eigenvalue.
- The eigenvector of A^TAATA corresponding to the largest eigenvalue.
- \mathbf{1}_n1n
Q8. Consider the points (0,-0.8), (1,0), (2.2,0.9), (2.9,2.1)(0,−0.8),(1,0),(2.2,0.9),(2.9,2.1). Which of the following lines best fits the given points?
- 0.59 x – 0.57y = 0.58 0.59x−0.57y=0.58
- x – y = 1x−y=1
- 0.58 x – 0.59y = 0.57 0.58x−0.59y=0.57
Quiz 3: RANSAC
Q1. Assume we have a case for RANSAC with 300 samples and 200 inliers. If we pick n = 10n=10 samples to build our model, what is the probability that we will build the correct model? (Use 3 decimals of precision)
0.017
Q2. For the same description, what is the probability that we won’t build a correct model after k = 100k=100 iterations? (Use 3 decimals of precision)
0.174
Q3. How many iterations will we need at least, in case the desired RANSAC success rate is p \geq 0.99p≥0.99?
264
Quiz 4: 3D-3D Pose
Q1. Find the rotation matrix RR such that ||A-RB||_F^2∣∣A−RB∣∣F2 is minimized, where
A = ⎡⎣1111−11−1−11−111⎤⎦, \qquad B = ⎡⎣−1.21310.0851−1.2334−1.4413−0.78580.55250.3470−1.65940.35500.5752−0.7885−1.4309⎤⎦A=⎣⎢⎡1111−11−1−11−111⎦⎥⎤,B=⎣⎢⎡−1.21310.0851−1.2334−1.4413−0.78580.55250.3470−1.65940.35500.5752−0.7885−1.4309⎦⎥⎤
- R = ⎡⎣−0.89410.1141−0.43300.43680.4355−0.78710.0988−0.8930−0.4392⎤⎦R=⎣⎢⎡−0.89410.1141−0.43300.43680.4355−0.78710.0988−0.8930−0.4392⎦⎥⎤
- R = ⎡⎣0.0623−0.01210.99800.3400−0.9399−0.03270.93840.3413−0.0545⎤⎦ R=⎣⎢⎡0.0623−0.01210.99800.3400−0.9399−0.03270.93840.3413−0.0545⎦⎥⎤
- R = ⎡⎣0.6580−0.71890.2242−0.7370−0.6759−0.00420.1545−0.1625−0.9745⎤⎦R=⎣⎢⎡0.6580−0.71890.2242−0.7370−0.6759−0.00420.1545−0.1625−0.9745⎦⎥⎤
2-0.8941
Quiz 5: Pose Estimation
Q1. What is the minimum number of point correspondences required for camera pose estimation given the perspective projections of points with known world coordinates?
3
Q2. What is the maximum number of solutions obtained from solving the P3P?
4
Q3. Assume that all points in the world lie on the plane Z_w = 0Zw=0. Let KK denote the camera calibration matrix. The transformation from the world frame to the camera frame reads R X_w + TRXw+T, where R=(r_1 \ r_2 \ r_3)R=(r1 r2 r3). Which of the following is the projective transformation from the world plane to the camera?
- K (r_1 \ r_2 \ T)K(r1 r2 T)
- K (r_1 \ T \ r_3)K(r1 T r3)
- (r_1 \ r_2 \ T)(r1 r2 T)
- (T\ r_2 \ r_3)(T r2 r3)
Q4. Assume that all points in the world lie on the plane Y_w = 0Yw=0. Let KK denote the camera calibration matrix. The transformation from the world frame to the camera frame reads R X_w + TRXw+T, where R=(r_1 \ r_2 \ r_3)R=(r1 r2 r3). Which of the following is the projective transformation from the world plane to the camera?
- K (r_1 \ r_2 \ T)K(r1 r2 T)
- K (r_1 \ r_3 \ T)K(r1 r3 T)
- K (r_1 \ T \ r_3)K(r1 T r3)
- K (r_2 \ r_3 \ T)K(r2 r3 T)
Robotics: Perception Week 4 Quiz Answers
Quiz 1: Epipolar Geometry
Q1. Let \widehat{} : \mathbb{R}^3 \rightarrow \mathbb{R}^{3 \times 3}:R3→R3×3 defined by
u =
⎡⎣u1u2u3⎤⎦
\mapsto\widehat{u} =
⎡⎣0u3−u2−u30u1u2−u10⎤⎦
u=⎣⎢⎡u1u2u3⎦⎥⎤↦u=⎣⎢⎡0u3−u2−u30u1u2−u10⎦⎥⎤
This is denoted [\cdot]_{\times}[⋅]× in the lectures, and you can consider it the cross product operator. We will be using this notation throughout the rest of the quiz.
Consider two images x_1x1, x_2x2 of the same point pp from two camera positions with relative pose (R,T) \in SE(3)(R,T)∈SE(3), where R \in SO(3)R∈SO(3) is the relative orientation and T \in \mathbb{R}^3T∈R3 is the relative position. Then, x_1,x_2x1,x2 always satisfy
- x_2^T \widehat{T}Rx_1 = 0 x2TTRx1=0
- x_2^T R x_1 = 0 x2TRx1=0
- x_2^T \widehat{T}x_1 = 0 x2TTx1=0
- x_2^T x_1 = 0 x2Tx1=0
Q2. If RR is a rotation matrix, which of the following properties hold?
- R^T \widehat{u} R = \widehat{R^T u}RTuR=RTu
- \widehat{u}^T = -\widehat{u}uT=−u
- u^T \widehat{u} = \vec{0}^TuTu=0T
- \widehat{u}u = \vec{0}uu=0
Q3. Let two cameras with poses g_1=(R_1,T_1) \in SE(3)g1=(R1,T1)∈SE(3) and g_2=(R_2,T_2) \in SE(3)g2=(R2,T2)∈SE(3). Note that the poses are such that a point X_wXw in the world frame is transformed to the frame of camera ii as X_{c,i} = R_i^T(X_w-T_i)Xc,i=RiT(Xw−Ti). Which of the following matrices are valid essential matrices, that is they satisfy x_1^T E x_2 = 0x1TEx2=0 for all point correspondences x_1 \leftrightarrow x_2x1↔x2 ? For convenience let T_{ij} \doteq T_j – T_iTij≐Tj−Ti and R_{ij} \doteq R_i^T R_jRij≐RiTRj.
- E = \widehat{R_1^T T_{12}} R_{12} E=R1TT12R12
- E = \widehat{R_1^T T_{21}} R_{12} E=R1TT21R12
- E = R_1^T \widehat{T_{21}} R_2 E=R1TT21R2
- E = R_1^T \widehat{T_{12}} R_2E=R1TT12R2
Q4. The relative pose between two views is (R,T) \in SE(3)(R,T)∈SE(3) where R=IR=I and TT corresponds to a translation of 11m in the direction of the zz-axis, which of the following is a valid essential matrix? Hint: use the fact that E = \widehat{T} RE=TR.
- E =
- ⎡⎣00−1000100⎤⎦
- E=⎣⎢⎡00−1000100⎦⎥⎤
- E =
- ⎡⎣010−100000⎤⎦
- E=⎣⎢⎡010−100000⎦⎥⎤
- E =
- ⎡⎣0000010−10⎤⎦
- E=⎣⎢⎡0000010−10⎦⎥⎤
Q5. The relative pose between two views is (R,T) \in SE(3)(R,T)∈SE(3) where R=IR=I and TT corresponds to a translation of 11m in the direction of the xx-axis, which of the following is a valid essential matrix? Hint: use the fact that E = \widehat{T} RE=TR.
- E =
- ⎡⎣00−1000100⎤⎦
- E=⎣⎢⎡00−1000100⎦⎥⎤
- E =
- ⎡⎣010−100000⎤⎦
- E=⎣⎢⎡010−100000⎦⎥⎤
- E =
- ⎡⎣0000010−10⎤⎦
- E=⎣⎢⎡0000010−10⎦⎥⎤
Q6. A nonzero matrix E \in \mathbb{R}^{3 \times 3}E∈R3×3 is a an essential matrix if and only if EE has a singular value deocmposition (SVD) E = U \Sigma V^TE=UΣVT with
- \Sigma = \textrm{diag}(\sigma, \sigma,0\}Σ=diag(σ,σ,0} for some \sigma<0σ<0 and U,V \in SO(3)U,V∈SO(3).
- \Sigma = \textrm{diag}(\sigma,0,0\}Σ=diag(σ,0,0} for some \sigma<0σ<0 and U,V \in SO(3)U,V∈SO(3).
- \Sigma = \textrm{diag}(\sigma, \sigma,0\}Σ=diag(σ,σ,0} for some \sigma>0σ>0 and U,V \in SO(3)U,V∈SO(3).
- \Sigma = \textrm{diag}(\sigma,0,0\}Σ=diag(σ,0,0} for some \sigma>0σ>0 and U,V \in SO(3)U,V∈SO(3).
Q7. Given a real matrix F \in \mathbb{R}^{3 \times 3}F∈R3×3 with SVD F = U\textrm{diag}(\lambda_1,\lambda_2,\lambda_3)V^TF=Udiag(λ1,λ2,λ3)VT with U,V \in SO(3)U,V∈SO(3), \lambda_1 \geq \lambda_2 \geq \lambda_3λ1≥λ2≥λ3, then the essential matrix that minimizes the error ||E-F||_F^2∣∣E−F∣∣F2 is given by
- E =U \textrm{diag}(\sigma,0,0)V^TE=Udiag(σ,0,0)VT with \sigma = (\lambda_1 + \lambda_2)/2σ=(λ1+λ2)/2.
- E =U \textrm{diag}(\sigma,0,0)V^TE=Udiag(σ,0,0)VT with \sigma = (\lambda_1 + \lambda_2+\lambda_3)/2σ=(λ1+λ2+λ3)/2.
- E =U \textrm{diag}(\sigma,\sigma,0)V^TE=Udiag(σ,σ,0)VT with \sigma = (\lambda_1 + \lambda_2+\lambda_3)/2σ=(λ1+λ2+λ3)/2.
- E =U \textrm{diag}(\sigma,\sigma,0)V^TE=Udiag(σ,σ,0)VT with \sigma = (\lambda_1 + \lambda_2)/2σ=(λ1+λ2)/2.
Q8. How many point correspondences are required to obtain an essential matrix using the linear algorithm?
- 4
- 5
- 6
- 8
Q9. Which of the following are valid essential matrices?
- E =
- ⎡⎣01/2√00010−1/2√0⎤⎦
- E=⎣⎢⎡01/200010−1/20⎦⎥⎤
- E =
- ⎡⎣01/2√01010−1/2√0⎤⎦
- E=⎣⎢⎡01/201010−1/20⎦⎥⎤
- E =
- ⎡⎣200010000⎤⎦
- E=⎣⎢⎡200010000⎦⎥⎤
- E =
- ⎡⎣100010000⎤⎦
- E=⎣⎢⎡100010000⎦⎥⎤
Q10. Suppose we know the camera motion always moves on a plane, say the XYXY– plane (i.e. translation with only x and y components and rotation only about the z-axis). The essential matrix E = \widehat{T}RE=TR has the special form
- E =
- ⎡⎣00c00dab0⎤⎦
- , \qquad a,b,c,d \in \mathbb{R}E=⎣⎢⎡00c00dab0⎦⎥⎤,a,b,c,d∈R
- E =
- ⎡⎣a000bc00d⎤⎦
- , \qquad a,b,c,d \in \mathbb{R}E=⎣⎢⎡a000bc00d⎦⎥⎤,a,b,c,d∈R
- E =
- ⎡⎣0bca0000d⎤⎦
- , \qquad a,b,c,d \in \mathbb{R}E=⎣⎢⎡0bca0000d⎦⎥⎤,a,b,c,d∈R
Q11. Now, assuming the same scenario as in the previous , which of the following solutions for (R,T)(R,T) in terms of a,b,c,da,b,c,d are valid? Assume that a^2+b^2=1a2+b2=1 and c^2+d^2=1c2+d2=1.
- T =
- ⎡⎣−ba0⎤⎦
- , \qquad R =
- ⎡⎣−bd−acad−bc0−ad+bc−bd−ac0001⎤⎦
- T=⎣⎢⎡−ba0⎦⎥⎤,R=⎣⎢⎡−bd−acad−bc0−ad+bc−bd−ac0001⎦⎥⎤
- T =
- ⎡⎣−ba0⎤⎦
- , \qquad R =
- ⎡⎣−bd+acad+bc0−ad+bc−bd−ac0001⎤⎦
- T=⎣⎢⎡−ba0⎦⎥⎤,R=⎣⎢⎡−bd+acad+bc0−ad+bc−bd−ac0001⎦⎥⎤
- T =
- ⎡⎣−ba0⎤⎦
- , \qquad R =
- ⎡⎣bd−acad+bc0−ad−bcbd−ac0001⎤⎦
- T=⎣⎢⎡−ba0⎦⎥⎤,R=⎣⎢⎡bd−acad+bc0−ad−bcbd−ac0001⎦⎥⎤
Q12. In general, given a normalized essential matrix, we get mm distinct poses (R,T)(R,T) and by enforcing the positive depth constraint, we end up with nn valid poses. Which of the following is true?
- (m,n) = (2,1)(m,n)=(2,1)
- (m,n) = (4,1)(m,n)=(4,1)
- (m,n) = (4,2)(m,n)=(4,2)
- (m,n) = (8,2)(m,n)=(8,2)
Quiz 2: Nonlinear Least Squares
Q1. Which of the following cost functions can be minimized in the framework of linear least squares? (Note the underscore on the norm refers to the p-norm)
- f(x) = ||x x^T – b||_2f(x)=∣∣xxT−b∣∣2
- f(x) = ||Ax – b||_2^2f(x)=∣∣Ax−b∣∣22
- f(x) = ||Ax – b||_1f(x)=∣∣Ax−b∣∣1
Q2. Consider the problem of minimizing f(x) = ||Ax-b||_2^2f(x)=∣∣Ax−b∣∣22, where the rank of AA is larger than the dimension of xx. Which of the following corresponds to the optimality condition?
- x=bx=b
- A^TAx=A^TbATAx=ATb
- x=AA^Tbx=AATb
Q3. Minimizing ||f(x)-b||^2∣∣f(x)−b∣∣2 is prone, in general, to the existence of local minima.
- True
- False
Q4. Examples of nonlinear least squares problems include
- Perspective-n-Point
- Triangulation
- Line fitting
Q5. Assume we want to minimize ||f(x)-b||_2^2∣∣f(x)−b∣∣22. Then, the (globally) optimal solution satisfies
- \dfrac{\partial f(x)}{\partial x}^T f(x)=\dfrac{\partial f(x)}{\partial x}^Tx∂x∂f(x)Tf(x)=∂x∂f(x)Tx
- \dfrac{\partial f(x)}{\partial x}^T x=\dfrac{\partial f(x)}{\partial x}^Tb∂x∂f(x)Tx=∂x∂f(x)Tb
- \dfrac{\partial f(x)}{\partial x}^T f(x) =\dfrac{\partial f(x)}{\partial x}^Tb∂x∂f(x)Tf(x)=∂x∂f(x)Tb
Q6. If a point satisfies the condition of the previous then it is globally optimal.
- True
- False
Quiz 3: 3D Velocities from Optical Flow
Q1. The equation of optical flow given in Lecture is:
u = \frac{1}{Z} (xVz−VxyVz−Vy) + (xy(1+y2)−(1+x2)−xyy−x) \Omega u=Z1(xVz−VxyVz−Vy)+(xy(1+y2)−(1+x2)−xyy−x)Ω
What does the VV in this equation represent?
- Angular Velocity
- Heading Direction
- Inverse Depth
Q2. What was the constraint \mathbf{I}(\mathbf{x}) = \mathbf{J}(\mathbf{x} + \mathbf{d}) I(x)=J(x+d) that we used to find the optical flow called?
- \mathbf{I}I-\mathbf{J}J Constancy Constraint
- Image Equality Constraint
- None of the above
Brightness Constancy Constraint
Q3. In trying to minimize \|\mathbf{I}(\mathbf{x}) – \mathbf{J}(\mathbf{x}+\mathbf{d}) \|∥I(x)−J(x+d)∥, we use which of the following items?
- Taylor Expansion \mathbf{J}(\mathbf{x} + \mathbf{d}) = \mathbf{J}(\mathbf{x}) + \frac{\delta \mathbf{J}(\mathbf{x})}{\delta \mathbf{x}} \mathbf{d}J(x+d)=J(x)+δxδJ(x)d
- Iterating to get incrementally closer the the optimal solution
- The second derivative of the image \frac{\delta^2 \mathbf{J}(\mathbf{x})}{\delta \mathbf{x}^2}δx2δ2J(x)
- The second moment matrix \frac{\delta \mathbf{J}(\mathbf{x})}{\delta \mathbf{x}}^T \frac{\delta \mathbf{J}(\mathbf{x})}{\delta \mathbf{x}}δxδJ(x)TδxδJ(x)
- The derivative of the second image \frac{\delta \mathbf{I}(\mathbf{x})}{\delta \mathbf{x}}δxδI(x)
Quiz 4: Bundle Adjustment
Q1. Bundle adjustment corresponds to minimization of
- 3D Error
- Reprojection Error
Q2. Bundle adjustment corresponds to optimization of a cost function with respect to
- Camera orientation
- Camera position
- 3D position of feature points
- All of the above
Q3. Assume that we want to minimize ||f(x)-b||_2^2∣∣f(x)−b∣∣22. A first order Taylor expansion of f(x)f(x) around the current value yields f(x+\Delta x) \approx f(x) + \dfrac{\partial f(x) }{\partial x} \Delta x f(x+Δx)≈f(x)+∂x∂f(x)Δx. Then, we update as x_{k+1} = x_k + \Delta xxk+1=xk+Δx where \Delta xΔx satisfies
- \dfrac{\partial f(x) }{\partial x}^T \dfrac{\partial f(x) }{\partial x} \Delta x = – \dfrac{\partial f(x) }{\partial x}^T f(x) ∂x∂f(x)T∂x∂f(x)Δx=−∂x∂f(x)Tf(x)
- \dfrac{\partial f(x) }{\partial x}^T \dfrac{\partial f(x) }{\partial x} \Delta x = \dfrac{\partial f(x) }{\partial x}^T (b-f(x)) ∂x∂f(x)T∂x∂f(x)Δx=∂x∂f(x)T(b−f(x))
- \dfrac{\partial f(x) }{\partial x}^T \dfrac{\partial f(x) }{\partial x} \Delta x = \dfrac{\partial f(x) }{\partial x}^T b ∂x∂f(x)T∂x∂f(x)Δx=∂x∂f(x)Tb
Q4. Which of the following tools are useful in a visual odometry framework
- Bundle adjustment over sliding window
- Key frame selection
- Visual loop closure when places are revisited
Q5. Select any answer that is an indispensable part of a structure from motion pipeline.
- Pairwise feature matching
- Image blending
- Triangulation of feature points
- Bundle adjustment
- Object detection
- Outlier rejection with RANSAC
- Essential matrix computation
Get all Quiz Answers of Robotics Specialization
Course 01: Robotics: Aerial Robotics Quiz Answers
Course 02: Robotics: Computational Motion Planning Quiz Answers
Course 03: Robotics: Mobility Quiz Answers
