Image Formation and Camera Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

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  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Image Formation and Camera Models

    Allen Y. Yang

    Berkeley EE 225b

    Feb 28th, 2007

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    1 Images and Projection Models Introduction Perspective Projection Orthographic Projection

    2 Camera Models Imaging through a Pinhole

    3 Camera Intrinsic Parameters Lense Distortions Modeling Camera Parameters

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Representation of Images

    I : Σ ⊂ R2 → R+; (x , y) 7→ I (x , y).

    This Lecture

    1 How are images captured from 3-D world to 2-D?

    2 Camera projection model?

    3 Image formation?

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Perspective Projection

    A modern camera projects 3-D world into 2-D image plane through perspective projection:

    Properties of perspective projection: Foreshortening: Distance objects are smaller.

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Perspective Projection

    Horizon line:

    Vanishing points: Parallel lines in 3-D intersect at a point in the image plane.

    How many vanishing points in these images?

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Correct perspective projections are visible in paintings.

    (a) 1st Century B.C., Pompeii (b) “School of Athens”, Raphael, 1518

    More reading: “Perspective” in wikipedia.com.

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Perspective Projection and Illusions

    (c) Necker cube (d) Escher waterfall (e) Ames room

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Orthographic Projection Model

    Orthographic projection:

    The difference between perspective and orthographic was illustrated in Christian artwork:

    (f) Perugino Fresco, Vatican (g) Birth of the Virgin, Ukraine

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Pinhole camera model (Camera Obscura)

    Mo-Zi (5th century BC) −→ Aristotle (300 BC) −→ Da Vinci (1490) −→ Kepler (17th century)

    Figure: Pinhole camera model.

    Let p = [X , Y , Z ]T ∈ R3, and its image x = [x , y ]T ∈ R2:

    x : X = y : Y = −f : Z

    Hence, x = −f X Z

    , y = −f Y Z

    .

    Question: What is the projection model for orthographic projection?

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Frontal Camera Model

    The pinhole camera model is inconvenience that the focal length f is negative.

    Figure: Frontal pinhole camera model.

    x = [ x

    y

    ] =

    f

    Z

    [ X Y

    ] .

    In homogeneous coordinates:

    Z

    xy 1

     = f 0 0 00 f 0 0

    0 0 1 0

     

    X Y Z 1

     . Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Distortions from Physical Lenses

    Pinhole camera model assumes

    1 Perfect pinhole camera lenses.

    2 Image x = I (p) ∈ R2 be measured in infinite accuracy. 3 Principal point is at the center of the image.

    Physical camera lenses give us

    1 Distorted imaging projections.

    (a) Fish- eye

    (b) Nor- mal/Portrait

    (c) Tele- photo

    2 Finite resolutions defined by the sensing devices in digital cameras.

    3 Offset between image center and optical center.

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    Camera Intrinsic Parameters

    Figure: Transformation from image coordinates to pixel coordinates.

    Modeling the camera distortion

    1 Transform from image coordinates (e.g., in metric units) to pixel coordinates.[ xs ys

    ] =

    [ sx 0 0 sy

    ] [ x y

    ] .

    2 Translate the image origin. x ′ = xs + ox ; y

    ′ = ys + oy .

    Allen Y. Yang Image Formation and Camera Models

  • Outline Images and Projection Models Camera Models Camera Intrinsic Parameters

    In homogeneous coordinates:

    x′ =

    x ′y ′ 1

     = sx 0 ox0 sy oy

    0 0 1

     xy 1

     . Camera Intrinsic Matrix:

    K . =

    sx 0 ox0 sy oy 0 0 1

     . Complete transformation from 3-D to 2-D pixel coordinates:

    Z

    x ′y ′ 1

     = sx 0 ox0 sy oy

    0 0 1

     f 0 0 00 f 0 0 0 0 1 0

     

    X Y Z 1

     =

    fsx 0 ox0 fsy oy 0 0 1

     1 0 0 00 1 0 0 0 0 1 0

     

    X Y Z 1

     . In short hand,

    λx′ = KΠ0X.

    K is called calibration matrix, Π0 is called (perspective) projection matrix.

    Allen Y. Yang Image Formation and Camera Models

    Outline Images and Projection Models Introduction Perspective Projection Orthographic Projection

    Camera Models Imaging through a Pinhole

    Camera Intrinsic Parameters Lense Distortions Modeling Camera Parameters

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