Unit 2: Camera Geometry and 2-D Projective Geometry - Practice Quiz

1 What is the simplest camera model that describes the geometric mapping from 3D space to a 2D image without using lenses?

2 In the pinhole camera model, what is the distance between the pinhole (camera center) and the image plane called?

3 Why are lenses typically used in real cameras instead of simple pinholes?

4 What is a common optical distortion introduced by lenses where straight lines in the real world appear curved in the image?

5 What does 'CCD' stand for in the context of digital cameras?

6 In a CCD camera, the continuous image is sampled into a discrete grid of elements. What are these individual elements called?

7 A general projective camera maps 3D world points to 2D image points. What are the dimensions of the projection matrix used for this transformation?

8 Which assumption characterizes an affine camera model (such as an orthographic camera)?

9 What is the primary goal of camera calibration?

10 Which of the following is considered an INTRINSIC parameter of a camera?

11 In 2-D planar geometry, what geometric entity is represented by the equation ?

12 What conceptual geometric elements are added to Euclidean space to form a projective space?

13 In a 2D projective space, where do two parallel lines intersect?

14 What are the homogeneous (projective) coordinates for a 2D Euclidean point ?

15 How do you convert the homogeneous point back to 2D Euclidean coordinates, assuming ?

16 What is a homography?

17 What is the size of a homography matrix used for 2D projective transformations?

18 Although a 2D homography matrix has 9 elements, how many degrees of freedom does it actually possess?

19 Which geometric transformation is predominantly used to map and align overlapping images into a single panorama in image stitching?

20 What is the purpose of image rectification in computer vision?

21 In a pinhole camera with focal length , a point in 3D space is projected onto the image plane at . If the object moves twice as far from the camera along the Z-axis, what happens to its projected image coordinates?

22 If a pinhole camera's principal point is offset by from the origin of the image coordinates, how is the projection of a 3D point modified?

23 How does decreasing the aperture size (increasing the f-number) affect a camera with a lens?

24 In a CCD camera, if the sensor elements are not perfectly rectangular, which intrinsic parameter accounts for this geometric distortion?

25 A general finite projective camera is represented by a matrix . How many degrees of freedom does this matrix have?

26 The camera center (in homogeneous coordinates) of a general projective camera matrix can be found by solving which of the following equations?

27 Which of the following is a defining characteristic of an affine camera model compared to a general projective camera?

28 An orthographic projection is a special case of an affine camera. What happens to the apparent size of an object as its depth (distance from the camera) increases in this model?

29 When using the Direct Linear Transformation (DLT) algorithm for camera calibration, what is the minimum number of known 3D-2D point correspondences required to uniquely solve for the general projective camera matrix?

30 Which of the following scene features is primarily used to estimate and correct radial distortion in a camera?

31 In 2D planar geometry, a line is defined by . In homogeneous coordinates, what is the intersection point of two distinct, non-parallel lines and ?

32 What does the homogeneous coordinate represent in 2D planar geometry?

33 If a 3D Euclidean point is mapped into a projective space, what is the dimension of the resulting projective space , and how many coordinates are used to represent the point?

34 Given a 3D point in homogeneous coordinates , what is its equivalent representation in standard 3D Euclidean coordinates?

35 In 2D projective coordinates, what is the mathematical formulation for the line passing through two points and ?

36 A 2D homography is a projective transformation mapping points from one plane to another. How many degrees of freedom does a general 2D homography matrix have?

37 Which geometric property is strictly preserved under a general homography (projective) transformation?

38 Under which specific condition can a 2D homography perfectly relate two overlapping images of a general, non-planar 3D scene taken by a camera?

39 In perspective correction, to remove projective distortion and map an image back to an affine frame, where must the image of the line at infinity be mapped?

40 How can vanishing points be practically used to rectify a perspective image of a building facade?

41 A projective camera matrix is given by , where is a non-singular matrix and is a 3-vector. What is the geometrical interpretation of the 1D right null-space of ?

42 Suppose a pinhole camera is modified such that the pinhole is shifted by a vector parallel to the image plane, while the image plane itself remains fixed. How does the new camera calibration matrix relate to the original matrix ?

43 In the presence of radial distortion modeled by , straight lines in the 3D world that do not pass through the optical axis project to curves. If a camera exhibits severe barrel distortion (), what happens to the curvature of the projected line as its shortest distance from the principal point increases?

44 For a CCD camera, the skew parameter in the intrinsic matrix is non-zero. If the angle between the and pixel axes on the sensor array is , what is the exact expression for ?

45 Let be a projection matrix of a general projective camera. To determine the intrinsic parameters and rotation , we decompose the left submatrix . Which decomposition is strictly required, and what constraint must be enforced on ?

46 Given two distinct pixels and on the image plane of a calibrated camera with intrinsic matrix , the angle between their corresponding back-projected 3D rays is given by . What does the matrix represent?

47 Consider an affine camera matrix . Which of the following mathematical properties strictly differentiates an affine camera from a general projective camera regarding the camera center?

48 A generic affine camera maps 3D points to 2D image points using a matrix. How many degrees of freedom does a general affine camera matrix have?

49 When calibrating a camera using the Direct Linear Transformation (DLT) algorithm, we stack point correspondence equations to form . If all 3D calibration points happen to lie on a single 3D plane, what is the rank of the matrix (assuming noise-free correspondences), and what consequence does this have?

50 In Zhang's camera calibration method, each image of a planar checkerboard pattern provides constraints on the Image of the Absolute Conic (IAC) denoted by . If and are the first two columns of the homography mapping the pattern to the image, which of the following equations correctly expresses the orthogonal constraints of the checkerboard axes?

51 Four distinct collinear points have a cross-ratio . If a projective transformation permutes these points such that their new cross ratio is equal to , which permutation of the points occurred?

52 A non-degenerate conic is transformed under a 2D projective transformation represented by an invertible matrix . If a point maps to , how does the dual conic , which represents the envelope of tangent lines to , transform?

53 In , the line at infinity is . What is the geometric locus of points defined by the intersection of an arbitrary circle with in this projective space?

54 Consider a 3D projective space . Which of the following best describes the properties of the Absolute Dual Quadric ?

55 Two parallel lines on the Euclidean plane are given by and . When represented in homogeneous coordinates in , what is the exact homogeneous coordinate of their intersection point?

56 In , a line can be represented by a Plücker matrix , which is a skew-symmetric matrix. If two 3D points in homogeneous coordinates are and , how is constructed from and , and what is its rank?

57 Suppose a camera captures a planar scene , meaning points on the plane satisfy . If the camera moves by a rotation and translation , with an intrinsic matrix , the transformation between the old image coordinates and new image coordinates is given by a homography . What is the closed-form expression for ?

58 A projective homography matrix is estimated between two images taken by the same camera undergoing purely rotational motion around its camera center. Let the rotation angle be . If the intrinsic matrix is known, what are the eigenvalues of the conjugate matrix up to a common non-zero scale factor?

59 When performing image stitching on a sequence of overlapping images taken by a freely translating and rotating camera, a single 2D projective homography generally fails to perfectly align an image pair. However, perfect alignment via homography is theoretically guaranteed regardless of scene depth variations under which specific condition?

60 In stereo vision, image rectification applies 2D projective homographies and to a pair of images to make corresponding epipolar lines collinear and horizontal. If is the epipole in the first unrectified image, what is the strict algebraic constraint on to achieve standard horizontal epipolar lines?