Projective Geometry Conversion Euclidean -> Homogenous ... Projective Geometry • Axioms of...

Click here to load reader

  • date post

    10-Oct-2020
  • Category

    Documents

  • view

    5
  • download

    0

Embed Size (px)

Transcript of Projective Geometry Conversion Euclidean -> Homogenous ... Projective Geometry • Axioms of...

  • 1

    CS252A, Fall 2017

    Computer Vision I

    Image Formation and Cameras

    Computer Vision I CSE 252A Lecture 4

    CS252A, Fall 2017

    Computer Vision I

    Announcements •  HW0 is due on today •  HW1 will be assigned tomorrow. •  Read Chapters 1 & 2 of Forsyth & Ponce •  Everyone is off waitlist •  Welcome extension students – send email to

    Bolun Zhang [email protected] •  (Subset of?) Final exam can be use for CSE

    MS Comprehensive Exam [Pending]

    CS252A, Fall 2017

    Computer Vision I

    Projective Geometry •  Axioms of Projective Plane

    1.  Every two distinct points define a line 2.  Every two distinct lines define a point (intersect

    at a point) 3.  There exists three points, A,B,C such that C

    does not lie on the line defined by A and B. •  Different than Euclidean (affine) geometry •  Projective plane is “bigger” than affine

    plane – includes “line at infinity”

    Projective Plane

    Affine Plane = + Line at Infinity

    CS252A, Fall 2017

    Computer Vision I

    Conversion Euclidean -> Homogenous -> Euclidean In 2-D •  Euclidean -> Homogenous:

    (x, y) -> λ(x,y,1)= (λx, λy, λ)

    •  Homogenous -> Euclidean: (x, y, w) -> (x/w, y/w)

    In 3-D •  Euclidean -> Homogenous:

    (x, y, z) -> λ(x ,y, z, 1) •  Homogenous -> Euclidean:

    (x, y, z, w) -> (x/w, y/w, z/w)

    X

    Y (x,y)

    (x,y,1)

    1 W

    CS252A, Fall 2017

    Computer Vision I

    The equation of projection: Euclidean & Homogenous Coordinates

    Cartesian coordinates:

    (x,y,z)→( f x z , f y

    z )

    U V W

    ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟

    =

    1 0 0 0 0 1 0 0 0 0 1 f 0

    ⎜ ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟ ⎟

    X Y Z T

    ⎜ ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟ ⎟

    Homogenous Coordinates and Camera matrix

    CS252A, Fall 2017

    Computer Vision I

    Affine Camera Model

    •  Take perspective projection equation, and perform Taylor series expansion about some point (x0,y0,z0).

    •  Drop terms that are higher order than linear. •  Resulting expression is affine camera model

    ⎥ ⎥ ⎥

    ⎢ ⎢ ⎢

    0

    0

    0

    z y x

    Appropriate in Neighborhood About (x0,y0,z0)

  • 2

    CS252A, Fall 2017

    Computer Vision I

    Simplified Camera Models Perspective Projection

    Scaled Orthographic Projection

    Affine Camera Model

    Orthographic Projection

    Approximation: Taylor Series Expansion about (X0, Y0, Z0)

    Case of affine where X0= Y0 = 0

    Case of scaled orthograhic where f/Z0 = 1

    CS252A, Fall 2017

    Computer Vision I

    Projective Transformation Mapping from Plane to Plane

    •  3 x 3 linear transformation of homogenous coordinates

    •  Points map to points •  Lines map to lines

    ⎥ ⎥ ⎥

    ⎢ ⎢ ⎢

    ⎥ ⎥ ⎥

    ⎢ ⎢ ⎢

    =

    ⎥ ⎥ ⎥

    ⎢ ⎢ ⎢

    3

    2

    1

    333231

    232221

    211211

    3

    2

    1

    x x x

    aaa aaa aaa

    u u u

    X U

    CS252A, Fall 2017

    Computer Vision I

    ^

    Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”

    Mapping from a Plane to a Plane under Perspective is given by a projective transform H

    x’ = Hx H is a 3x3 matrix, x, x’ are 3x1 vectors of homogenous coordinates

    î ĵ

    CS252A, Fall 2017

    Computer Vision I

    The equation of projection

    Cartesian coordinates:

    (x,y,z)→( f x z , f y

    z )

    U V W

    ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟ =

    1 0 0 0 0 1 0 0 0 0 1 f 0

    ⎜ ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟ ⎟

    X Y Z T

    ⎜ ⎜ ⎜ ⎜

    ⎟ ⎟ ⎟ ⎟

    Homogenous Coordinates and Camera matrix

    CS252A, Fall 2017

    Computer Vision I

    ^

    x = xî + yĵ +Oπ

    3D point coordinates on plane π

    As a matrix equation

    X = î ĵ Oπ ⎡ ⎣⎢

    ⎤ ⎦⎥

    x y 1

    ⎢ ⎢ ⎢

    ⎥ ⎥ ⎥

    In homogenous coordinates

    x ' y ' w '

    ⎢ ⎢ ⎢

    ⎥ ⎥ ⎥ =

    1 0 0 0 0 1 0 0 0 0 1 f 0

    ⎢ ⎢ ⎢ ⎢

    ⎥ ⎥ ⎥ ⎥

    î 0

    ĵ 0

    Oπ 1

    ⎢ ⎢

    ⎥ ⎥

    x y 1

    ⎢ ⎢ ⎢

    ⎥ ⎥ ⎥ = H

    x y 1

    ⎢ ⎢ ⎢

    ⎥ ⎥ ⎥

    And taking perspective projection onto π’

    î ĵX = î0

    ĵ 0

    Oπ 1

    ⎢ ⎢

    ⎥ ⎥

    x y 1

    ⎢ ⎢ ⎢

    ⎥ ⎥ ⎥

    x’ = Hx

    4x3

    CS252A, Fall 2017

    Computer Vision I

    Application of Homography: OCRs, scan,…

    Homography: Can be estimated from four points.

    P1

    P3

    P4

    P2

    (0,0) (0,1)

    (1,1) (1,0)

  • 3

    CS252A, Fall 2017

    Computer Vision I Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”

    Planar Homography

    x =H1X X’ =H2X

    x’ = H2X = H2(H1-1 x) = (H2H1-1)x

    Application: Two photos of a white board CS252A, Fall 2017

    Computer Vision I Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”

    Planar Homography: Pure Rotation

    x’ = H2X = H2(H1-1 x) = (H2H1-1)x

    Application: Panoramas

    CS252A, Fall 2017

    Computer Vision I

    Application: Panoramas and image stitching

    All images are warped to central image CS252A, Fall 2017

    Computer Vision I

    Lenses

    CS252A, Fall 2010 Computer Vision I

    Beyond the pinhole Camera Getting more light – Bigger Aperture

    CS252A, Fall 2010 Computer Vision I

    Pinhole Camera Images with Variable Aperture

    1mm

    .35 mm

    .07 mm

    .6 mm

    2 mm

    .15 mm

  • 4

    CS252A, Fall 2010 Computer Vision I

    Limits for pinhole cameras

    CS252A, Fall 2010 Computer Vision I

    The reason for lenses

    CS252A, Fall 2010 Computer Vision I

    Thin Lens

    O

    •  Rotationally symmetric about optical axis. •  Spherical interfaces.

    Optical axis

    CS252A, Fall 2010 Computer Vision I

    Thin Lens: Center

    O

    •  All rays that enter lens along line pointing at O emerge in same direction.

    F

    CS252A, Fall 2010 Computer Vision I

    Thin Lens: Focus

    O

    Parallel lines pass through the Focal Point, F

    F

    CS252A, Fall 2010 Computer Vision I

    Thin Lens: Image of Point

    O

    All rays passing through lens and starting at P converge upon P’

    F

    P

    P’

  • 5

    CS252A, Fall 2010 Computer Vision I

    Thin Lens: Image of Point

    O F

    P

    P’ z’

    f

    z

    1 z ' − 1 z = 1 f

    Where z, z’, and f are the z-coordinates of P, P, and F. i.e., z is negative.

    j k

    i

    CS252A, Fall 2010 Computer Vision I

    Thin Lens: Image Plane

    O F

    P

    P’

    Image Plane

    Q’

    Q

    A price: Whereas the image of P is in focus, the image of Q isn’t.

    CS252A, Fall 2010 Computer Vision I

    Thin Lens: Aperture

    O

    P

    P’

    Image Plane •  Smaller Aperture -> Less Blur •  Pinhole -> No Blur

    CS252A, Fall 2010 Computer Vision I

    Field of View

    O Field of View

    Im age Plane

    f

    – Telephoto lens: small field of view (large f) – Wide angle lens: large field of view(small f) – Zoom lens: variable f using compound lens

    CS252A, Fall 2010 Computer Vision I

    Deviations from the lens model Deviations from this ideal are aberrations Two types

    2. chromatic

    1. geometrical ❑  spherical aberration ❑  astigmatism ❑  distortion ❑  coma

    Aberrations are reduced by combining lenses

    Compound lenses CS252A, Fall 2010 Computer V