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CS 691 Computational Photography Instructor: Gianfranco Doretto Image Warping

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Image Transformations 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 Transformations TT f f g g Image Filtering: change range of image g(x) = T(f(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) Let’s represent T as a matrix: p’ = Mp T p = (x,y)p’ = (x’,y’)

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Scaling Scaling a coordinate means multiplying each of its components by a scalar Uniform scaling means this scalar is the same for all components: x 2

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Non-uniform scaling: different scalars per component: Scaling X x 2, Y x 0.5

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Scaling Scaling operation: Or, in matrix form: scaling matrix S What’s inverse of S?

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2-D Rotation θ (x, y) (x’, y’) x’ = x cos(θ) - y sin(θ) y’ = x sin(θ) + y cos(θ)

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2-D Rotation x = r cos ( ) y = r sin ( ) x’ = r cos ( + θ) y’ = r sin ( + θ) Trig Identity… x’ = r cos( ) cos(θ) – r sin( ) sin(θ) y’ = r sin( ) cos(θ) + r cos( ) sin(θ) Substitute… x’ = x cos(θ) - y sin(θ) y’ = x sin(θ) + y cos(θ) θ (x, y) (x’, y’)

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2-D Rotation This is easy to capture in matrix form: Even though sin(θ) and cos(θ) are nonlinear functions of θ, –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 –θ –For rotation matrices R

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2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Identity? 2D Scale around (0,0)?

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2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Rotate around (0,0)? 2D Shear?

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2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis? 2D Mirror over (0,0)?

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2x2 Matrices What types of transformations can be represented with a 2x2 matrix? 2D Translation? Only linear 2D transformations can 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

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Consider a different Basis v =(v i,v j ) u=(u i,u j ) p uv j =(0,1) i =(1,0) p ij p ij =5i+4j = (5,4)p uv = 4u+3v = (4,3)

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Linear Transformations as Change of Basis Any linear transformation is a change of basis!!! j =(0,1) i =(1,0) p ij = 4u+3v = (p i,p j ) ? p i = 4u i +3v i p j = 4u j +3v j v =(v i,v j ) u=(u i,u j ) p uv p ij p uv = (4,3) p uv p ij jj ii jj ii vu vu vu vu 3 4

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What’s the inverse transform? How can we change from any basis to any basis? What if the basis are orthonormal? And orthogonal? v =(v i,v j ) u=(u i,u j ) p uv j =(0,1) i =(1,0) p ij = (5,4) p ij p uv = (p u,p v ) = ?= p u u + p v v p ij p uv jj ii jj ii vu vu vu vu 4 5

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Projection onto orthonormal basis v =(v i,v j ) u=(u i,u j ) p uv j =(0,1) i =(1,0) p ij = (5,4) p ij p uv = (u·p ij, v·p ij ) p ij p uv ji ji ji ji vv uu vv uu 4 5 NOTE: The inverse is just the transpose!

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Homogeneous Coordinates Q: How can we represent translation as a 3x3 matrix?

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Homogeneous Coordinates Represent coordinates in 2 dimensions with a 3-vector

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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 or a vector –(0, 0, 0) is not allowed Convenient coordinate system to represent many useful transformations 1 2 1 2 (2,1,1) or (4,2,2)or (6,3,3) x y

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Homogeneous Coordinates Q: How can we represent translation as a 3x3 matrix? A: Using the rightmost column:

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Translation Example of translation t x = 2 t y = 1 Homogeneous Coordinates

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Basic 2D Transformations Basic 2D transformations as 3x3 matrices Translate RotateShear Scale

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Matrix Composition Transformations can be combined by matrix multiplication p’ = T(t x,t y ) R( ) S(s x,s y ) p Does the order of multiplication matter?

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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 –Model change of basis Will the last coordinate w always be 1?

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Projective Transformations Projective transformations … –Affine transformations, and –Projective warps Properties 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

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2D image transformations These transformations are a nested set of groups Closed under composition and inverse is a member

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Recovering Transformations What if we know f and g and want to recover the transform T? –willing to let user provide correspondences How many do we need? xx’ T(x,y) yy’ f(x,y)g(x’,y’) ?

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Translation: # correspondences? How many Degrees of Freedom? How many correspondences needed for translation? What is the transformation matrix? xx’ T(x,y) yy’ ?

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Euclidian: # correspondences? How many DOF? How many correspondences needed for translation+rotation? xx’ T(x,y) yy’ ?

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Affine: # correspondences? How many DOF? How many correspondences needed for affine? xx’ T(x,y) yy’ ?

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Projective: # correspondences? How many DOF? How many correspondences needed for projective? xx’ T(x,y) yy’ ?

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Slide Credits This set of sides also contains contributions kindly made available by the following authors –Alexei Efros –Richard Szelisky

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