3D Camera Models Navab

download 3D Camera Models Navab

of 56

  • date post

    04-Jun-2018
  • Category

    Documents

  • view

    214
  • download

    0

Embed Size (px)

Transcript of 3D Camera Models Navab

  • 8/13/2019 3D Camera Models Navab

    1/56

    3D Computer Vision II

    ReminderCamera Models

    Nassir Navabbased on a course given at UNC by Marc Pollefeys & the book Multiple View Geometry by Hartley & Zisserman

    October 27, 2009

  • 8/13/2019 3D Camera Models Navab

    2/56

    Outline Camera Models

    # Geometric Parameters of a Finite Camera# Projective Camera Model

    !# Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray!# Decomposition of Camera Matrix

    # Cameras at Infinity# Other Camera Models (Pushbroom and Line Cameras)

    23D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    3/56

    Pinhole Camera Model

    # Mapping between 3D world and 2D image# Central projection# Models are described in matrices with particular properties

    33D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    4/56

    Homogeneous Coordinates

    43D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    5/56

    Central Projection

    53D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    6/56

    are the coordinates of the principal point

    Principal Point Offset

    where

    principal point(perpendicular intersection point of

    principal axis and image plane)

    63D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    7/56

    is called camera calibration

    matrix

    Principal Point Offset

    where

    73D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    8/56

    Inhomogeneous

    coordinates

    where represents the point in

    world coordinates

    represents the same point in

    camera coordinates

    represents the coordinates of the camera

    origin in the world coordinate frame

    Camera Rotation and Translation

    83D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    9/56

    Homogeneous

    coordinates

    projection to image plane from camera coordinates

    projection to image plane from world coordinates

    Camera Rotation and Translation

    3D Computer Vision II - Camera Models 9

  • 8/13/2019 3D Camera Models Navab

    10/56

    where

    projection matrix of a general

    pinhole camera with 9 DOF

    intrinsic camera parameters with 3 DOF

    extrinsic camera parameters with each 3 DOF

    (camera orientation, position in world coordinates)

    Extrinsic and Intrinsic Parameters

    3D Computer Vision II - Camera Models 10

  • 8/13/2019 3D Camera Models Navab

    11/56

    No explicit

    camera center

    where

    from

    Camera Rotation and Translation

    3D Computer Vision II - Camera Models 11

  • 8/13/2019 3D Camera Models Navab

    12/56

    CCD Cameras Non-Square Pixels

    number of pixels per unit distance

    4 DOF

    10 DOF

    123D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    13/56

    Skew Parameter

    skew parameter

    5 DOF

    finite projective camera with11 DOF

    133D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    14/56

    non-singular

    Finite Projective Camera Summary

    projection matrix

    11 DOF (5+3+3)

    143D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    15/56

    decompose projection matrix P in K,R,C

    Finite Projective Camera Decomposition of P

    non-singular 3x3 matrix (8 DOF)

    QR matrix

    decomposition

    153D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    16/56

    Finite Projective Camera Summary

    # Camera matrices Pare identical with the set of homogeneous 3x4matrices for which the left 3x3 sub-matrix is non-singular

    # If rank(P) = 3, but rank(M) < 3, then camera at infinity# if rank(P) < 3 the matrix mapping will be a line or a point and not a

    plane (not a 2D image)

    where

    163D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    17/56

    Outline Camera Models

    # Geometric Parameters of a Finite Camera# Projective Camera Model

    !# Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray!# Decomposition of Camera Matrix

    # Cameras at Infinity# Other Camera Models (Pushbroom and Line Cameras)

    173D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    18/56

    Camera Anatomy

    # Camera center# Column vectors# Principal plane# Axis plane# Principal point# Principal ray

    183D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    19/56

    P has a 1D null-space

    Finite cameras:

    Infinite cameras:

    Camera Center

    we will prove that the 4-vector C is the camera center

    points on a line through A and C

    since

    All 3D points on the line are mapped on the same 2D image point, and

    thus the line is a ray through the camera center

    193D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    20/56

    Column vectors are the image points which project the

    axis directions (X,Y,Z) and the origin

    Column Vectors

    Example for the image

    of the y-axis

    is the image of the world origin

    203D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    21/56

    Row Vectors

    Represent geometrically particular world planes.

    row vectors

    column vectors

    213D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    22/56

    Row Vectors of the Projection Matrix

    p1is defined by the camera center and the

    line x=0 on the image.

    p2is defined by the camera center and the

    line y=0 on the image.

    Example p2

    respectively for p1

    223D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    23/56

    Principal Plane

    Plane through camera center and

    parallel to the image plane.

    points Xare imaged on the line at

    infinity

    if X is on the principle plane

    especially

    233D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    24/56

    The line through camera center and

    perpendicular to principal plane is the

    principal axis.

    The intersection of the principal

    axis with the image plane is the

    principal point.

    normal direction to the principal plane

    principal point

    where

    third row of Mand

    Principal Point

    3D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    25/56

    Ambiguity that principal axis points

    towards the front of the camera (positive

    direction)

    direction unaffected by scaling

    since

    Principal Axis Vector

    towards the front of the camera

    253D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    26/56

    Maps a point in space on the image plane

    Forward Projection

    Vanishing points

    Only M affects the projection of vanishing points

    263D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    27/56

  • 8/13/2019 3D Camera Models Navab

    28/56

    w can be interpreted

    as the dot product of

    the ray CX with the

    principal ray direction

    (PC=0)

    If ,

    then m3is a unit vector pointing in positive axis direction

    Suppose . Then

    Depth of Points

    283D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    29/56

    Depth of Points: Examples

    293D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    30/56

    Outline Camera Models

    # Geometric Parameters of a Finite Camera# Projective Camera Model

    !# Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray!#

    Decomposition of Camera Matrix

    # Cameras at Infinity# Other Camera Models (Pushbroom and Line Cameras)

    303D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    31/56

    Finding the camera center C

    numerically: find right null-space by SVD of P

    Algebraically:

    where

    Camera Matrix Decomposition

    313D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    32/56

    Finding the camera center C

    Any plane !going through C will be a linear combination of the three

    planes defined by the rows of P. Therefore:

    where

    Camera Matrix Decomposition

    323D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    33/56

    Finding the camera orientation and internal parameters

    using RQ decomposition

    Q R=( )-1= -1 -1QR

    Decompose

    Ambiguity removed by enforcing positive diagonal entries

    Camera Matrix Decomposition

    333D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    34/56

    1

    arctan(1/s)

    for CCD/CMOS, always s=0

    Image from image, s"0 possible

    (non coinciding principal axis)

    resulting camera:

    When is Skew Non-zero?

    where H is a 3x3 homography

    343D Computer Vision II - Camera Models

  • 8/13/2019 3D Camera Models Navab

    35/56

    General projective interpretation

    # Meaningful decomposition in K,R,t requiresEuclidean image