A.The point of intersection of two epipolar lines in 3D space
B.The intersection of the baseline connecting the two camera centers with the image plane
C.The 3D point being projected onto the image planes
D.The focal point of the camera lens
Correct Answer: The intersection of the baseline connecting the two camera centers with the image plane
Explanation:
An epipole is the point where the line connecting the optical centers of the two cameras (the baseline) intersects the image plane. It represents the projection of one camera's center onto the other camera's image plane.
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2Which geometric entity contains the two camera centers and the observed 3D point?
epipolar geometry
Easy
A.Epipolar plane
B.Principal plane
C.Focal plane
D.Image plane
Correct Answer: Epipolar plane
Explanation:
The epipolar plane is formed by three points: the optical center of the first camera, the optical center of the second camera, and the 3D point being observed.
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3What is the line formed by the intersection of the epipolar plane and the image plane called?
epipolar geometry
Easy
A.Horizon line
B.Epipolar line
C.Principal axis
D.Baseline
Correct Answer: Epipolar line
Explanation:
An epipolar line is the straight line of intersection between the epipolar plane and the camera's image plane. The projection of a 3D point must lie on this line in the second image.
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4What are the dimensions of the fundamental matrix ?
fundamental matrix
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The fundamental matrix is a matrix that relates corresponding points in stereo images in homogeneous coordinates.
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5The fundamental matrix maps a point in one image to which geometric entity in the second image?
fundamental matrix
Easy
A.An epipolar plane
B.An epipole
C.An epipolar line
D.A single corresponding point
Correct Answer: An epipolar line
Explanation:
If is a point in the first image, the product yields a vector that represents the equation of the corresponding epipolar line in the second image.
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6What is the rank of a standard fundamental matrix?
fundamental matrix
Easy
A.3
B.2
C.4
D.1
Correct Answer: 2
Explanation:
The fundamental matrix has a rank of 2. This rank deficiency is because its determinant is zero, mathematically expressing that all epipolar lines intersect at a single point (the epipole).
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7What is the minimum number of point correspondences required to compute the fundamental matrix using the 8-point algorithm?
normalized 8-point algorithm
Easy
A.7
B.9
C.8
D.5
Correct Answer: 8
Explanation:
The 8-point algorithm requires a minimum of 8 corresponding point pairs between two images to solve for the 9 elements of the fundamental matrix up to scale.
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8What is the primary purpose of data normalization in the normalized 8-point algorithm?
normalized 8-point algorithm
Easy
A.To align the two cameras parallel to each other
B.To improve numerical stability and accuracy of the estimation
C.To convert color images to grayscale
D.To remove optical distortion from the lenses
Correct Answer: To improve numerical stability and accuracy of the estimation
Explanation:
Normalizing the image coordinates (translating the centroid to the origin and scaling) prevents numerical instability caused by the large scale differences in the matrix equations, yielding much more accurate results.
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9In algebraic minimization algorithms for estimating the fundamental matrix, which mathematical tool is typically used to find the least squares solution to ?
algebric minimization algorithm
Easy
A.Fourier Transform
B.Laplacian of Gaussian
C.Singular Value Decomposition (SVD)
D.Hough Transform
Correct Answer: Singular Value Decomposition (SVD)
Explanation:
SVD is used to solve systems of linear equations of the form . The solution is the eigenvector corresponding to the smallest singular value of .
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10Algebraic minimization minimizes an algebraic error (like ). Why is this sometimes less preferred than geometric minimization?
algebric minimization algorithm
Easy
A.It requires thousands of point correspondences
B.It does not have a direct physical or geometric meaning in the image space
C.It only works for video sequences, not still images
D.It cannot be solved using linear algebra
Correct Answer: It does not have a direct physical or geometric meaning in the image space
Explanation:
Algebraic error is a mathematical construct useful for simple linear solutions but does not measure the actual pixel distance error in the image, unlike geometric minimization.
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11In fundamental matrix estimation, what does the geometric distance typically measure?
geometric distance computation
Easy
A.The distance between the two camera centers
B.The perpendicular distance from points to their corresponding epipolar lines
C.The distance from the object to the camera lens
D.The focal length of the cameras
Correct Answer: The perpendicular distance from points to their corresponding epipolar lines
Explanation:
Geometric distance measures how far an observed image point deviates from its theoretical position, which is the corresponding epipolar line.
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12What is the Sampson error used for in stereo vision?
geometric distance computation
Easy
A.It measures the lens distortion of the camera
B.It defines the distance between the two optical centers
C.It is used to calculate the focal length of a camera
D.It is a first-order approximation of the geometric error
Correct Answer: It is a first-order approximation of the geometric error
Explanation:
The Sampson error provides a computationally efficient, first-order approximation of the true geometric distance (reprojection error) without needing to fully triangulate the 3D points.
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13How many degrees of freedom does a general rigid camera motion (rotation and translation) have in 3D space?
camera motion
Easy
A.9
B.4
C.3
D.6
Correct Answer: 6
Explanation:
Rigid camera motion in 3D has 6 degrees of freedom: 3 for translation (along the X, Y, and Z axes) and 3 for rotation (pitch, yaw, and roll).
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14If a camera undergoes pure forward translation, what is the point in the image where all motion vectors seem to originate?
camera motion
Easy
A.Optical Center
B.Focus of Expansion (FOE)
C.Epipole
D.Vanishing Point
Correct Answer: Focus of Expansion (FOE)
Explanation:
Under pure forward translation, the optical flow vectors radiate outward from a single point in the image plane known as the Focus of Expansion (FOE).
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15Which motion model only accounts for horizontal and vertical shifts between image frames?
motion models
Easy
A.Translational model
B.Affine model
C.Perspective model
D.Quadratic model
Correct Answer: Translational model
Explanation:
The translational motion model is the simplest 2D motion model, accounting only for shifts along the X and Y axes (2 parameters).
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16How many parameters are required to fully define a 2D affine motion model?
motion models
Easy
A.2
B.8
C.4
D.6
Correct Answer: 6
Explanation:
A 2D affine motion model uses 6 parameters to account for translation (2 parameters), rotation, scale, and shear.
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17What does optical flow represent in computer vision?
optical flow
Easy
A.The apparent motion of pixels between consecutive image frames
B.The speed at which the camera shutter closes
C.The amount of light entering the camera aperture
D.The flow of data from the camera sensor to the processor
Correct Answer: The apparent motion of pixels between consecutive image frames
Explanation:
Optical flow is the pattern of apparent motion of image objects, surfaces, and edges in a visual scene caused by the relative motion between an observer and a scene.
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18What is the primary assumption used to derive the standard optical flow equation?
optical flow
Easy
A.Zero camera rotation
B.Rigid body structure
C.Brightness constancy
D.Color constancy
Correct Answer: Brightness constancy
Explanation:
The brightness constancy assumption states that the pixel intensity of an object remains constant as it moves from one frame to the next: .
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19What is the primary goal of the linear triangulation method?
Linear triangulation method
Easy
A.To segment an image into triangles for processing
B.To determine the 3D coordinates of a point from its 2D projections in multiple images
C.To calibrate the intrinsic parameters of a camera
D.To calculate the optical flow between two frames
Correct Answer: To determine the 3D coordinates of a point from its 2D projections in multiple images
Explanation:
Triangulation is the process of intersecting lines of sight from multiple camera views to estimate the 3D position of a point.
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20The linear triangulation method reformulates the projection equations into a system of the form . How is the solution for the 3D point typically found?
Linear triangulation method
Easy
A.By taking the cross product of the rows of
B.By finding the eigenvector associated with the smallest eigenvalue of using SVD
C.By simply adding all the elements of matrix
D.By setting to the origin
Correct Answer: By finding the eigenvector associated with the smallest eigenvalue of using SVD
Explanation:
To solve the homogeneous system in the presence of noise, we use Singular Value Decomposition (SVD) and extract the right singular vector corresponding to the smallest singular value.
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21In a stereo vision setup, if the epipole of the second camera is located at infinity, what does this imply about the physical camera configuration?
epipolar geometry
Medium
A.The two cameras are undergoing pure rotation with no translation.
B.The image planes of the two cameras are perfectly parallel to the translation vector.
C.The focal lengths of both cameras are zero.
D.The optical axes of the two cameras are orthogonal.
Correct Answer: The image planes of the two cameras are perfectly parallel to the translation vector.
Explanation:
When the translation vector between two cameras is parallel to their image planes, the epipoles (the projection of one camera center onto the other's image plane) are pushed to infinity. This is a common property of rectified stereo rigs.
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22Given a point in the first image, its corresponding epipolar line in the second image is defined by the fundamental matrix . Which of the following correctly represents this relationship?
epipolar geometry
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The fundamental matrix maps a point in the first image to its corresponding epipolar line in the second image via the equation . The corresponding point must lie on this line, leading to the condition .
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23Which of the following describes the necessary mathematical properties of a valid Fundamental Matrix ?
fundamental matrix
Medium
A.It is an orthogonal matrix where .
B.It has rank 3 and its determinant is non-zero.
C.It must be symmetric and positive definite.
D.It has rank 2 and its determinant is strictly 0.
Correct Answer: It has rank 2 and its determinant is strictly 0.
Explanation:
A fundamental matrix maps points to lines, which means it has a null space (specifically, the epipole). Because of this null space, it is a singular matrix with a rank of exactly 2, so its determinant is 0.
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24If is the epipole in the second image and is the fundamental matrix, which of the following equations holds true?
fundamental matrix
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The epipole in the second image lies on all epipolar lines . Therefore, for all , which implies . Thus, is the left null-vector of .
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25How many degrees of freedom does a fundamental matrix possess?
fundamental matrix
Medium
A.7
B.9
C.8
D.5
Correct Answer: 7
Explanation:
A matrix has 9 elements. The fundamental matrix is defined up to scale (reducing it to 8) and is constrained to have a determinant of 0 (reducing it by 1 more). Thus, it has 7 degrees of freedom.
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26Why is the normalization step crucial in the normalized 8-point algorithm for computing the fundamental matrix?
normalized 8-point algorithm
Medium
A.It automatically filters out outliers and mismatched feature points.
B.It converts the fundamental matrix into an essential matrix.
C.It improves the condition number of the linear system, preventing numerical instability and reducing noise sensitivity.
D.It ensures the resulting fundamental matrix has exactly rank 2 without needing Singular Value Decomposition (SVD).
Correct Answer: It improves the condition number of the linear system, preventing numerical instability and reducing noise sensitivity.
Explanation:
Image coordinates are often large (e.g., in hundreds or thousands of pixels). Formulating the matrix with these raw coordinates leads to a highly ill-conditioned system. Normalization scales and centers the data, drastically improving numerical stability.
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27In the data normalization step of the normalized 8-point algorithm, what is the standard target mean and average distance from the origin for the transformed image points?
normalized 8-point algorithm
Medium
A.Mean at and maximum distance of 1.
B.Mean at and average distance of 1.
C.Mean at and average distance of .
D.Mean at and average distance of .
Correct Answer: Mean at and average distance of .
Explanation:
Hartley's normalization standardizes the points such that their centroid is at the origin and the root-mean-square distance of the points from the origin is . This makes the average coordinate magnitude roughly 1.
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28When estimating the fundamental matrix, algebraic minimization involves formulating a system . How is the non-trivial least-squares solution subject to typically found?
algebric minimization algorithm
Medium
A.By calculating the inverse of matrix .
B.By performing a Cholesky decomposition on .
C.By taking the eigenvector corresponding to the smallest eigenvalue of .
D.By taking the eigenvector corresponding to the largest eigenvalue of .
Correct Answer: By taking the eigenvector corresponding to the smallest eigenvalue of .
Explanation:
To minimize subject to , we use Singular Value Decomposition (SVD) or eigenvalue decomposition. The solution is the right singular vector of associated with the smallest singular value, which is equivalent to the eigenvector of with the smallest eigenvalue.
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29What is a major limitation of strictly minimizing the algebraic error when estimating a fundamental matrix, before applying any rank enforcement?
algebric minimization algorithm
Medium
A.The resulting matrix will always be the identity matrix.
B.It requires exactly 7 points and cannot handle overdetermined systems.
C.The resulting matrix may not have rank 2, thus failing to represent valid epipolar geometry.
D.It intrinsically assumes the cameras are fully calibrated.
Correct Answer: The resulting matrix may not have rank 2, thus failing to represent valid epipolar geometry.
Explanation:
Directly solving yields a matrix that minimizes the algebraic error but does not enforce the constraint. A subsequent step (like setting the smallest singular value to zero) is required to enforce the rank 2 constraint.
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30In robust fundamental matrix estimation, the Sampson error is often used instead of the exact Symmetric Epipolar Distance. Why is this?
geometric distance computation
Medium
A.The Sampson error is exactly equal to the reprojection error in 3D.
B.The Sampson error requires no camera calibration.
C.The Sampson error provides a excellent first-order approximation to the geometric error while being much cheaper to evaluate.
D.The Sampson error enforces the rank 2 constraint automatically.
Correct Answer: The Sampson error provides a excellent first-order approximation to the geometric error while being much cheaper to evaluate.
Explanation:
The true geometric distance involves finding the closest points on the epipolar lines that perfectly satisfy the epipolar constraint, which requires solving a high-degree polynomial. The Sampson error approximates this distance using a first-order Taylor expansion, making it highly efficient.
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31The Symmetric Epipolar Distance for a given point correspondence and a fundamental matrix measures which of the following?
geometric distance computation
Medium
A.The angular error between the two optical axes.
B.The distance between the camera centers in 3D space.
C.The Euclidean distance between and in the image plane.
D.The sum of squared perpendicular distances from to and from to .
Correct Answer: The sum of squared perpendicular distances from to and from to .
Explanation:
The Symmetric Epipolar Distance evaluates how well a point correspondence satisfies the epipolar constraint geometrically by summing the orthogonal distance from point to epipolar line in the first image, and point to line in the second image.
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32If a camera undergoes pure rotational motion around its optical center, how can the relationship between the two resulting views be best described?
camera motion
Medium
A.The transformation depends entirely on the depth of the scene points.
B.The fundamental matrix has a rank of 3.
C.The transformation can be completely modeled by a 2D homography, independent of scene depth.
D.Epipolar lines converge at the center of the image.
Correct Answer: The transformation can be completely modeled by a 2D homography, independent of scene depth.
Explanation:
Under pure rotation (no translation), parallax is eliminated. Because there is no baseline, depth cannot be recovered, and the mapping between the two image planes is a simple projective transformation (homography).
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33In the presence of forward translational camera motion along the optical axis, where is the Focus of Expansion (FOE) located in the image?
camera motion
Medium
A.At the epipole in the image plane.
B.Outside the bounds of the image plane completely.
C.At the vanishing line of the ground plane.
D.At infinity, making all flow vectors parallel.
Correct Answer: At the epipole in the image plane.
Explanation:
The Focus of Expansion is the point from which all optical flow vectors appear to radiate when moving forward. Geometrically, it coincides with the epipole, which is the intersection of the translation vector with the image plane.
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34Which parameterized 2D motion model accounts for translation, rotation, scale, and shear, requiring exactly 6 parameters?
motion models
Medium
A.Translational model
B.Similarity (Scaled rigid) model
C.Projective (Homography) model
D.Affine motion model
Correct Answer: Affine motion model
Explanation:
The 2D affine motion model maps coordinates using a transformation matrix (4 parameters for rotation, scale, and shear) and a 2D translation vector (2 parameters), totaling 6 parameters.
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35When applying a local translational motion model for tracking features in a small window, what fundamental assumption is being made?
motion models
Medium
A.All pixels in the window share the exact same displacement vector .
B.The scene depth varies rapidly within the window.
C.The pixels in the window undergo distinct affine transformations.
D.The optical flow vectors strictly point towards the epipole.
Correct Answer: All pixels in the window share the exact same displacement vector .
Explanation:
A local translational model assumes that the image patch is small enough that spatial deformations (rotation, scale) are negligible, meaning all pixels within that small neighborhood move by the same uniform 2D shift .
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36The brightness constancy constraint equation for optical flow is given by . What is the primary mathematical reason this equation cannot be solved for at a single pixel?
optical flow
Medium
A.It is a non-linear equation requiring iterative optimization.
B.The temporal derivative is always zero in static environments.
C.It represents a single linear equation with two unknowns, creating an underdetermined system.
D.Image gradients and are undefined at object boundaries.
Correct Answer: It represents a single linear equation with two unknowns, creating an underdetermined system.
Explanation:
The brightness constancy equation only gives one constraint for the two variables and . This leads to the 'aperture problem', where only the flow component parallel to the spatial gradient can be uniquely determined from a single pixel.
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37How does the Lucas-Kanade optical flow method overcome the aperture problem?
optical flow
Medium
A.By assuming the flow vector is constant over a local neighborhood of pixels and solving a least-squares problem.
B.By utilizing a multi-scale image pyramid to detect large motions.
C.By adding a global smoothness constraint across the entire image.
D.By applying a Gaussian blur to eliminate image noise before processing.
Correct Answer: By assuming the flow vector is constant over a local neighborhood of pixels and solving a least-squares problem.
Explanation:
Lucas-Kanade is a local method. It assumes that a small window of pixels moves with the same velocity . This creates an overdetermined system of equations (one for each pixel in the window) which can be solved via least-squares.
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38Unlike the Lucas-Kanade method, the Horn-Schunck method calculates optical flow by:
optical flow
Medium
A.Tracking discrete corner features using a local translational model.
B.Ignoring the brightness constancy assumption entirely.
C.Introducing a global smoothness penalty term to solve for a dense flow field.
D.Using epipolar geometry to restrict flow vectors to 1D lines.
Correct Answer: Introducing a global smoothness penalty term to solve for a dense flow field.
Explanation:
Horn-Schunck is a global optical flow algorithm. It overcomes the aperture problem by adding a regularization term that penalizes variations (gradients) in the flow field across the entire image, thereby yielding dense flow.
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39In the Direct Linear Transformation (DLT) approach to linear triangulation, a 3D point is reconstructed from projections and by setting up . How are the equations for matrix originally derived?
Linear triangulation method
Medium
A.Using the cross product relation to eliminate the unknown scale factor.
B.By converting the fundamental matrix into an essential matrix.
C.Using the dot product of the camera projection matrices and .
D.By minimizing the reprojection error directly in the image plane.
Correct Answer: Using the cross product relation to eliminate the unknown scale factor.
Explanation:
In projective geometry, the image point is parallel to the projected 3D point up to a scale factor. The cross product of parallel vectors is zero, so provides linear equations in the coordinates of independent of depth.
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40Why is the Linear Triangulation (DLT) method often considered to produce mathematically suboptimal 3D point estimates in the presence of noise?
Linear triangulation method
Medium
A.It restricts the 3D points to lie strictly on the baseline connecting the camera centers.
B.It assumes the cameras are completely uncalibrated, which introduces scaling ambiguities.
C.It minimizes an algebraic error in projective space rather than a physically meaningful geometric reprojection error.
D.It requires exactly 3 camera views to function properly.
Correct Answer: It minimizes an algebraic error in projective space rather than a physically meaningful geometric reprojection error.
Explanation:
DLT solves by minimizing . This algebraic error does not directly correspond to the 2D pixel distance in the image plane (the geometric reprojection error). For high accuracy, DLT results are often refined using non-linear geometric optimization.
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41Suppose two cameras are set up such that the second camera's optical center is derived from the first by a pure translation strictly along the optical Z-axis (forward motion). Which of the following statements correctly describes the resulting epipolar geometry?
epipolar geometry
Hard
A.The epipolar lines form a set of concentric circles around the principal point in both image planes.
B.The epipole in each image is located at the principal point, and the epipolar lines radiate outward from the center like a starburst.
C.The fundamental matrix evaluates to the identity matrix, and no unique epipolar lines can be determined.
D.The epipoles are located at infinity, and the epipolar lines are parallel to each other in both image planes.
Correct Answer: The epipole in each image is located at the principal point, and the epipolar lines radiate outward from the center like a starburst.
Explanation:
In pure forward translation along the Z-axis, the baseline intersects the image planes exactly at the principal points. These intersection points are the epipoles (Focus of Expansion). Since all epipolar lines must pass through the epipole, they radiate outward from the principal point.
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42When computing epipolar geometry between two uncalibrated views, what is the consequence of all observed 3D points lying perfectly on a single 3D plane?
epipolar geometry
Hard
A.The fundamental matrix is uniquely determined because the planar constraint removes the scale ambiguity.
B.The fundamental matrix becomes strictly orthogonal and can be estimated using only 4 point correspondences.
C.The points are related by a 2D homography, causing a degeneracy where the fundamental matrix cannot be uniquely determined from point correspondences alone.
D.The epipoles shift to infinity, rendering the epipolar lines strictly parallel regardless of the camera rotation.
Correct Answer: The points are related by a 2D homography, causing a degeneracy where the fundamental matrix cannot be uniquely determined from point correspondences alone.
Explanation:
If all 3D points are coplanar, their projections in the two views are related by a homography (). This forms a degenerate configuration for the 8-point algorithm, leading to an infinite family of valid fundamental matrices () unless additional constraints are applied.
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43Let be the fundamental matrix relating two views. Under what specific physical condition does become a skew-symmetric matrix ()?
fundamental matrix
Hard
A.When the two cameras have identical intrinsic parameters and differ by a pure rotation around the Y-axis.
B.When both cameras share the exact same optical center but have different focal lengths.
C.When the relative transformation between the cameras consists of a pure translation along the X-axis only.
D.When the two cameras have identical intrinsic parameters and differ by a pure translation.
Correct Answer: When the two cameras have identical intrinsic parameters and differ by a pure translation.
Explanation:
If the cameras are identical () and differ by pure translation (), the Essential matrix is . The Fundamental matrix is . For general pure translation, is skew-symmetric, meaning and for all , implying auto-epipolarity.
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44A valid fundamental matrix must have rank 2. In the presence of noise, an estimated matrix is often rank 3. To enforce the rank-2 constraint minimally in the Frobenius norm, one performs Singular Value Decomposition . If with , what is the correct updated matrix ?
fundamental matrix
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
By the Eckart-Young-Mirsky theorem, the closest rank-2 approximation of a matrix in the Frobenius norm is obtained by setting the smallest singular value (here ) to zero while keeping the others unchanged.
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45In the normalized 8-point algorithm, an isotropic scaling transformation is applied such that the RMS distance of the points from the origin is . If the unnormalized design matrix has a condition number of , what is the primary numerical failure mode if normalization is omitted before solving via SVD?
normalized 8-point algorithm
Hard
A.The solution will heavily bias the epipole towards the center of the image, irrespective of the true camera motion.
B.The algorithm will always return a rank-3 matrix, making it impossible to enforce the epipolar constraint.
C.The singular vector corresponding to the smallest singular value will be dominated by noise in the translation components, ignoring the rotation.
D.The large variations in pixel coordinate magnitudes will cause the singular values to span orders of magnitude, making the null-space estimation highly susceptible to numerical round-off errors.
Correct Answer: The large variations in pixel coordinate magnitudes will cause the singular values to span orders of magnitude, making the null-space estimation highly susceptible to numerical round-off errors.
Explanation:
Without normalization, elements of (which contain terms like , , and $1$) will have vastly different magnitudes (e.g., vs $1$). This poor conditioning makes the SVD numerically unstable, as floating-point precision cannot reliably differentiate the true null space from numerical noise.
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46Assume and are image correspondences, and and are the normalizing transformations for the respective images. If is the rank-2 fundamental matrix computed from the normalized coordinates, how is the final, unnormalized fundamental matrix correctly obtained?
normalized 8-point algorithm
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Normalized coordinates are and . The epipolar constraint is . Substituting the normalization yields . Thus, .
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47When estimating a homography or fundamental matrix, the Direct Linear Transform (DLT) minimizes an algebraic error subject to . What is the primary theoretical drawback of minimizing this algebraic error compared to the geometric (reprojection) error?
algebric minimization algorithm
Hard
A.The algebraic error does not correspond to a physically meaningful quantity in the image space and implicitly weights points differently based on their spatial coordinates.
B.The algebraic error function has multiple local minima, preventing a closed-form solution.
C.Algebraic minimization cannot handle more than the minimum number of required points (e.g., exactly 8 for fundamental matrices).
Correct Answer: The algebraic error does not correspond to a physically meaningful quantity in the image space and implicitly weights points differently based on their spatial coordinates.
Explanation:
The algebraic error minimized by DLT is an artificial construct lacking physical interpretation in pixel distances. Without normalization, it unfairly up-weights points with larger pixel coordinates, causing the estimate to be biased, unlike geometric error which minimizes true image-space distances.
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48To avoid the computational cost of full nonlinear Bundle Adjustment, the Sampson approximation is often used to estimate geometric error for the fundamental matrix. For a point correspondence , the Sampson error is defined as , where is the squared norm of the gradient. What is the correct expression for the denominator ?
geometric distance computation
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The Sampson error approximates the squared geometric distance to the epipolar lines. The denominator evaluates the magnitude of the gradients of the algebraic constraint with respect to the coordinates in both images, dropping the homogeneous scale coordinate (index 3). Thus, .
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49Consider a bundle adjustment problem for 3D reconstruction from two calibrated cameras minimizing the total reprojection error. Let the first camera's pose be fixed at the origin (). If there are observed 3D points, what is the exact dimensionality of the parameter space that needs to be optimized?
geometric distance computation
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Since the first camera is fixed to establish a global coordinate system, we must optimize the 3D coordinates of the points (3 parameters each, yielding ) and the pose of the second camera (3 for rotation and 3 for translation). Total dimensionality is .
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50Decomposing the Essential matrix yields four possible pose configurations for . Which geometric constraint is definitively used to disambiguate and select the single true physical camera motion?
camera motion
Hard
A.The planarity constraint, ensuring all points lie on the same side of the principal plane.
B.The Cheirality constraint, ensuring that the triangulated 3D points have positive depth in both camera frames.
C.The epipolar constraint, ensuring for all correspondences.
D.The orthogonality constraint, ensuring rather than .
Correct Answer: The Cheirality constraint, ensuring that the triangulated 3D points have positive depth in both camera frames.
Explanation:
All four solutions mathematically satisfy the epipolar constraint and have proper rotation matrices (det R = +1). The only way to find the physically realizable solution is the Cheirality constraint: triangulating a point and ensuring its Z-coordinate (depth) is strictly positive in both the first and second camera coordinate systems.
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51If a camera undergoes a pure rotation with zero translation (), what occurs to the Fundamental matrix and the process of 3D reconstruction?
camera motion
Hard
A. evaluates to a matrix of zeros, epipolar geometry is undefined, and 3D depth cannot be reconstructed from the two views.
B.The Fundamental matrix becomes exactly the identity matrix, and 3D depth can be directly extracted from the rotational disparities.
C. reduces to a 2D homography matrix, allowing reconstruction up to a single unknown scale factor.
D.The epipoles remain fixed at the principal point, but the epipolar lines become concentric circles.
Correct Answer: evaluates to a matrix of zeros, epipolar geometry is undefined, and 3D depth cannot be reconstructed from the two views.
Explanation:
Because , if translation , then , leading to . Epipolar geometry collapses because the baseline is zero, meaning parallax is absent. Without parallax, triangulation is impossible, and depth cannot be reconstructed.
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52When analyzing planar scenes, a camera motion can induce a transformation described by a 2D Homography (8 degrees of freedom) or an Affine transformation (6 degrees of freedom). Under what specific combination of motion and camera models does the exact true image transformation degrade strictly from a Homography to an Affine transformation?
motion models
Hard
A.When the scene contains high depth relief and the camera undergoes a large rotation around the optical axis.
B.When the camera uses an affine (weak perspective) projection model, and observes any planar surface under arbitrary 3D affine transformation.
C.When the camera uses a pure perspective projection model, and the motion is a pure translation along the Z-axis.
D.When the camera is perfectly calibrated, and the planar surface is exactly perpendicular to the optical axis.
Correct Answer: When the camera uses an affine (weak perspective) projection model, and observes any planar surface under arbitrary 3D affine transformation.
Explanation:
An affine camera model (e.g., orthographic or weak perspective) preserves parallelism. When an affine camera observes a planar scene, the mapping between two views is strictly an affine transformation (6 DOF). A true perspective camera observing a plane yields a full homography (8 DOF).
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53Consider the degrees of freedom (DoF) associated with essential and fundamental matrices. How many continuous degrees of freedom exist in the parameter space of a valid Essential Matrix , and why?
motion models
Hard
A.5 DoF: 3 for camera rotation and 2 for translation direction, because the overall translation scale is inherently unrecoverable from images.
B.7 DoF: 3 for rotation, 3 for translation, and 1 for overall scale.
C.6 DoF: 3 for rotation and 3 for full 3D translation.
D.8 DoF: A 3x3 matrix has 9 elements, minus 1 for arbitrary scale.
Correct Answer: 5 DoF: 3 for camera rotation and 2 for translation direction, because the overall translation scale is inherently unrecoverable from images.
Explanation:
An Essential matrix represents relative pose up to scale. Camera rotation provides 3 DoF. The translation vector provides 3 DoF, but because monocular reconstruction has scale ambiguity, the translation magnitude is unknown, leaving 2 DoF for direction. Total = 3 + 2 = 5 DoF.
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54In the Lucas-Kanade optical flow formulation, flow is solved using the structure tensor . Let be the eigenvalues of with . Which of the following eigenvalue conditions indicates the classical 'aperture problem' where flow can only be computed in one direction?
optical flow
Hard
A. and
B. and
C. and
D. and
Correct Answer: and
Explanation:
If and , the local neighborhood represents an edge. Motion can be reliably estimated perpendicular to the edge but not parallel to it. This rank-1 degeneracy is the exact manifestation of the aperture problem.
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55The fundamental Optical Flow Constraint Equation, , relies on a first-order Taylor expansion. For large pixel displacements, this linear approximation fails catastrophically. Which algorithmic strategy is standardly used to effectively resolve this large-motion failure without changing the continuous Taylor formulation?
optical flow
Hard
A.Switching from Lucas-Kanade to the Horn-Schunck method to enforce global smoothness over large distances.
B.Replacing the L2 norm with an L1 norm in the Lucas-Kanade objective function.
C.Using a coarse-to-fine Gaussian image pyramid to ensure the motion is sub-pixel at the coarsest scale.
D.Applying a median filter over the flow field to remove outliers.
Correct Answer: Using a coarse-to-fine Gaussian image pyramid to ensure the motion is sub-pixel at the coarsest scale.
Explanation:
To handle large motions that violate the small-displacement assumption of the Taylor expansion, image pyramids are used. The image is down-sampled (coarse scale) so that the large displacement becomes small (e.g., < 1 pixel). The flow is estimated at the coarse scale and then propagated and refined at finer scales.
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56When formulating the Direct Linear Transformation (DLT) for triangulating a 3D point from an observation and camera matrix , we use the cross product . Which of the following represents the two linearly independent equations extracted for building the measurement matrix ?
Linear triangulation method
Hard
A. and
B. and
C. and
D. and
Correct Answer: and
Explanation:
The cross product yields three equations: , , and . The third is a linear combination of the first two. Therefore, the first two equations are standardly used to form the rows of matrix A.
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57Assume 3D points are triangulated via DLT from two completely uncalibrated cameras, estimating the projection matrices strictly from the Fundamental matrix without intrinsic parameters. What is the inherent ambiguity of the resulting 3D point cloud?
Linear triangulation method
Hard
A.It is completely arbitrary and preserves neither lines nor incidence properties.
B.It is accurate up to an affine transformation.
C.It is accurate up to an unknown similarity transformation (scale, rotation, translation).
D.It is accurate up to an arbitrary 4x4 projective transformation.
Correct Answer: It is accurate up to an arbitrary 4x4 projective transformation.
Explanation:
Without camera intrinsics (uncalibrated stereo), one can only compute a projective reconstruction. The scene is reconstructed up to a 3D projective transformation (, where is a non-singular 4x4 matrix), which preserves straight lines and incidence but distorts parallelism, angles, and distances.
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58In a canonical stereo rig where the baseline is strictly parallel to the -axes of both identical image planes, what is the exact algebraic form of the fundamental matrix ?
epipolar geometry
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
For a standard rectified stereo pair, translation is strictly along the X-axis () and rotation is Identity. . The epipolar lines are perfectly horizontal (parallel to the scanlines).
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59The Horn-Schunck method for optical flow minimizes an energy functional containing a data term and a spatial smoothness term. If the weight of the smoothness term approaches zero, what happens to the estimated flow field?
optical flow
Hard
A.It becomes strictly rigid, allowing only purely translational optical flow.
B.It uniformly converges to the global average motion of the entire image.
C.The flow vectors uniformly rotate 90 degrees to align with the image gradients.
D.The flow becomes undefined anywhere the image gradient is zero, and suffers strictly from the aperture problem along edges.
Correct Answer: The flow becomes undefined anywhere the image gradient is zero, and suffers strictly from the aperture problem along edges.
Explanation:
As , the Horn-Schunck functional depends solely on the data term . Without the global smoothness constraint to propagate information, the system is underdetermined. Flow cannot be computed in flat regions (gradients are zero) and only normal flow can be computed at edges (aperture problem).
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60When constructing the transformation matrix for the normalized 8-point algorithm, we set as the centroid of the points. To achieve an RMS distance of from the origin, what must the scalar be, assuming is the original distance of point from the centroid?
normalized 8-point algorithm
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
We want the average squared distance from the origin to be 2. Therefore, . Solving for gives , which yields . Note: computing it using average distance is a common variation, but based strictly on RMS, the squared formulation is exact.