06 Warping

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    Image Warping

    15-463: Computational PhotographyAlexei Efros, CMU, Fall 2005

    Some slides from Steve Seitz

    http://www.jeffrey-martin.com

    http://www.jeffrey-martin.com/http://www.jeffrey-martin.com/
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    Image Warping

    image filtering: change range of image

    g(x) = T(f(x))

    f

    x

    T

    f

    x

    f

    x

    T

    f

    x

    image warping: change domain of image

    g(x) = f(T(x))

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    Image Warping

    T

    T

    f

    f g

    g

    image filtering: change range of image

    g(x) = h(T(x))

    image warping: change domain of image

    g(x) = f(T(x))

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    Parametric (global) warping

    Examples of parametric warps:

    translation rotation aspect

    affine perspective cylindrical

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    Parametric (global) warping

    Transformation T is a coordinate-changing machine:

    p =T(p)

    What does it mean that T is global? Is the same for any point p

    can be described by just a few numbers (parameters)

    Lets represent T as a matrix:p =M*p

    T

    p = (x,y) p = (x,y)

    =

    yx

    yx M

    ''

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    Scaling

    Scaling a coordinate means multiplying each of its components bya scalar

    Uniform scaling means this scalar is the same for all components:

    2

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    Non-uniform scaling: different scalars per component:

    Scaling

    X 2,

    Y 0.5

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    Scaling

    Scaling operation:

    Or, in matrix form:

    byyaxx

    ==''

    =

    y

    x

    b

    a

    y

    x

    0

    0

    '

    '

    scaling matrix S

    Whats inverse of S?

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    2-D Rotation

    (x, y)

    (x, y)

    x = x cos() - y sin()y = x sin() + y cos()

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    2-D Rotation

    x = r cos ()y = r sin ()x = r cos ( + )y = r sin ( + )

    Trig Identityx = r cos() cos() r sin() sin()y = r sin() sin() + r cos() cos()

    Substitute

    x = x cos() - y sin()y = x sin() + y cos()

    (x, y)

    (x, y)

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    2-D Rotation

    This is easy to capture in matrix form:

    Even though sin() and cos() are nonlinear functions of, x is a linear combination of x and y

    y is a linear combination of x and y

    What is the inverse transformation?

    Rotation by For rotation matrices, det(R) = 1 so

    ( ) ( )

    ( ) ( )

    =

    y

    x

    y

    x

    cossin

    sincos

    '

    '

    TRR =1

    R

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    2x2 Matrices

    What types of transformations can berepresented with a 2x2 matrix?

    2D Identity?

    yy xx =='' = yxyx 10 01''

    2D Scale around (0,0)?

    ysy

    xsx

    y

    x

    *'

    *'

    =

    =

    =

    y

    x

    s

    s

    y

    x

    y

    x

    0

    0

    '

    '

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    2x2 Matrices

    What types of transformations can berepresented with a 2x2 matrix?

    2D Rotate around (0,0)?

    yxy yxx *cos*sin' *sin*cos' += =

    =

    yx

    yx

    cossinsincos

    ''

    2D Shear?

    yxshy

    yshxx

    y

    x

    +=

    +=

    *'

    *'

    =

    y

    x

    sh

    sh

    y

    x

    y

    x

    1

    1

    '

    '

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    2x2 Matrices

    What types of transformations can berepresented with a 2x2 matrix?

    2D Mirror about Y axis?

    yy xx = ='' = yxyx 10 01''

    2D Mirror over (0,0)?

    yyxx

    ==

    ''

    =

    y

    x

    y

    x

    10

    01

    '

    '

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    2x2 Matrices

    What types of transformations can berepresented with a 2x2 matrix?

    2D Translation?

    y

    x

    tyytxx

    += +=''

    Only linear 2D transformationscan be represented with a 2x2 matrix

    NO!

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    All 2D Linear Transformations

    Linear transformations are combinations of Scale,

    Rotation,

    Shear, and Mirror

    Properties of linear transformations: Origin maps to origin

    Lines map to lines

    Parallel lines remain parallel Ratios are preserved

    Closed under composition

    =

    y

    x

    dc

    ba

    y

    x

    '

    '

    =

    yx

    lk

    ji

    hg

    fedcba

    yx

    ''

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    Linear Transformations as Change of Basis

    Any linear transformation is a basis!!! Whats the inverse transform? How can we change from any basis to any basis?

    What if the basis are orthogonal?

    j =(0,1)

    i =(1,0)

    p

    p=4i+3j = (4,3) p=4u+3vpx=4ux+3vxpy=4uy+3vy

    v =(vx,vy)

    u=(ux,uy)

    p

    pp

    =

    =

    yy

    xx

    yy

    xx

    vu

    vu

    vu

    vu

    3

    4'

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    Homogeneous Coordinates

    Q: How can we represent translation as a 3x3matrix?

    y

    x

    tyy

    txx

    +=+=

    '

    '

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    Homogeneous Coordinates

    Homogeneous coordinates represent coordinates in 2dimensions with a 3-vector

    1

    coordsshomogeneou

    y

    x

    y

    x

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    Homogeneous Coordinates

    Q: How can we represent translation as a 3x3matrix?

    A: Using the rightmost column:

    = 10010

    01

    y

    x

    t

    t

    ranslationT

    y

    x

    tyy

    txx

    +=

    +=

    '

    '

    T l i

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    Translation

    Example of translation

    ++

    =

    =

    11100

    10

    01

    1

    '

    '

    y

    x

    y

    x

    ty

    tx

    y

    x

    t

    t

    y

    x

    tx = 2

    ty = 1

    Homogeneous Coordinates

    H C di t

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    Homogeneous Coordinates

    Add a 3rd coordinate to every 2D point (x, y, w) represents a point at location (x/w, y/w)

    (x, y, 0) represents a point at infinity

    (0, 0, 0) is not allowed

    Convenientcoordinate system torepresent many

    usefultransformations

    1 2

    1

    2(2,1,1) or (4,2,2) or (6,3,3)

    x

    y

    B i 2D T f ti

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    Basic 2D Transformations

    Basic 2D transformations as 3x3 matrices

    =

    1100

    0cossin

    0sincos

    1

    '

    '

    y

    x

    y

    x

    =

    1100

    10

    01

    1

    '

    '

    y

    x

    t

    t

    y

    x

    y

    x

    =

    1100

    01

    01

    1

    '

    '

    y

    x

    sh

    sh

    y

    x

    y

    x

    Translate

    Rotate Shear

    =

    1100

    00

    00

    1

    '

    '

    y

    x

    s

    s

    y

    x

    y

    x

    Scale

    Affi T f ti

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    Affine Transformations

    Affine transformations are combinations of

    Linear transformations, and

    Translations

    Properties of affine transformations:

    Origin does not necessarily map to origin

    Lines map to lines

    Parallel lines remain parallel

    Ratios are preserved

    Closed under composition

    Models change of basis

    =

    w

    yx

    fedcba

    w

    yx

    100

    ''

    P j ti T f ti

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    Projective Transformations

    Projective transformations

    Affine transformations, and

    Projective warpsProperties of projective transformations:

    Origin does not necessarily map to origin

    Lines map to lines

    Parallel lines do not necessarily remain parallel

    Ratios are not preserved

    Closed under composition

    Models change of basis

    =

    w

    yx

    ihg

    fedcba

    w

    yx

    '

    ''

    Matrix Composition

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    Matrix Composition

    Transformations can be combined bymatrix multiplication

    =

    w

    yx

    sysx

    tytx

    w

    yx

    1000000

    1000cossin0sincos

    1001001

    '''

    p = T(tx,ty) R() S(sx,sy) p

    2D image transformations

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    2D image transformations

    These transformations are a nested set of groupsClosed under composition and inverse is a member

    Image warping

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    Image warping

    Given a coordinate transform (x,y) =h(x,y) and asource image f(x,y), how do we compute atransformed image g(x,y) =f(T(x,y))?

    x x

    T(x,y)

    g(x,y)

    f(x,y)

    y y

    Forward warping

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    f(x,y) g(x,y)

    Forward warping

    Send each pixel f(x,y) to its corresponding location

    (x,y) =T(x,y) in the second image

    x x

    T(x,y)

    Q: what if pixel lands betweentwo pixels?

    y y

    Forward warping

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    f(x,y) g(x,y)

    Forward warping

    Send each pixel f(x,y) to its corresponding location

    (x,y) =T(x,y