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Advanced Multimedia. Warping & Morphing Tamara Berg. Reminder. Project u pdates next week. Prepare a 10 minute presentation and submit a 4 paragraph summary to cseise364@gmail.com by midnight Monday, April 22. Image Warping. http://www.jeffrey-martin.com.

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## Advanced Multimedia

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**Advanced Multimedia**Warping & Morphing Tamara Berg**Reminder**• Project updates next week. • Prepare a 10 minute presentation and submit a 4 paragraph summary to cseise364@gmail.com by midnight Monday, April 22.**Image Warping**http://www.jeffrey-martin.com Slides from: Alexei Efros & Steve Seitz**f**f f T T x x x f x Image Transformations • image filtering: change range of image • g(x) = T(f(x)) image warping: change domain of image g(x) = f(T(x))**T**T Image Transformations • image filtering: change range of image • g(x) = T(f(x)) f g image warping: change domain of image g(x) = f(T(x)) f g**Parametric (global) warping**• Examples of parametric warps: aspect rotation translation perspective cylindrical affine**T**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 p = (x,y) p’ = (x’,y’)**Scaling**• Scaling a coordinate means multiplying each of its components by a scalar • Uniform scaling means this scalar is the same for all components: 2**X 2,Y 0.5**Scaling • Non-uniform scaling: different scalars per component:**Scaling**• Scaling operation: • Or, in matrix form: scaling matrix S What’s inverse of S?**Scaling**• Scaling operation: • Or, in matrix form: scaling matrix S What’s inverse of S?**(x’, y’)**(x, y) 2-D Rotation x’ = x cos() - y sin() y’ = x sin() + y cos()**(x’, y’)**(x, y) 2-D Rotation x = r cos (f) y = r sin (f) x’ = r cos (f + ) y’ = r sin (f + ) f**(x’, y’)**(x, y) 2-D Rotation x = r cos (f) y = r sin (f) x’ = r cos (f + ) y’ = r sin (f + ) Trig Identity… x’ = r cos(f) cos() – r sin(f) sin() y’ = r sin(f) cos() + r cos(f) sin() f**(x’, y’)**(x, y) 2-D Rotation x = r cos (f) y = r sin (f) x’ = r cos (f + ) y’ = r sin (f + ) Trig Identity… x’ = r cos(f) cos() – r sin(f) sin() y’ = r sin(f) cos() + r cos(f) sin() Substitute… x’ = x cos() - y sin() y’ = x sin() + y cos() f**2-D Rotation**• This is easy to capture in matrix form: • What is the inverse transformation? R**2-D Rotation**• This is easy to capture in matrix form: • What is the inverse transformation? • Rotation by –q • For rotation matrices R**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Identity?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Identity?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Identity? 2D Scale around (0,0)?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Identity? 2D Scale around (0,0)?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Rotate around (0,0)? 2D Shear?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis? 2D Mirror over (0,0)?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis? 2D Mirror over (0,0)?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Translation?**2x2 Matrices**• What types of transformations can be represented with a 2x2 matrix? 2D Translation? NO! Only linear 2D transformations can be represented with a 2x2 matrix**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**Homogeneous Coordinates**• Q: How can we represent translation as a 3x3 matrix?**Homogeneous Coordinates**• Homogeneous coordinates • represent coordinates in 2 dimensions with a 3-vector**Homogeneous Coordinates**• Q: How can we represent translation as a 3x3 matrix?**Homogeneous Coordinates**• Q: How can we represent translation as a 3x3 matrix? • A: Using the rightmost column:**Translation**• Example of translation Homogeneous Coordinates tx = 2ty= 1**Translation**• Example of translation Homogeneous Coordinates tx = 2ty= 1**Basic 2D Transformations**• Basic 2D transformations as 3x3 matrices Translate Scale Rotate Shear**Matrix Composition**• Transformations can be combined by matrix multiplication p’ = T(tx,ty) R(Q) S(sx,sy) p**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**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**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? ? T(x,y) y y’ x x’ f(x,y) g(x’,y’)**Translation: # correspondences?**• How many correspondences needed for translation? ? T(x,y) y y’ x x’**Translation: # correspondences?**• How many correspondences needed for translation? • How many Degrees of Freedom? ? T(x,y) y y’ x x’**Translation: # correspondences?**• How many correspondences needed for translation? • How many Degrees of Freedom? • What is the transformation matrix? ? T(x,y) y y’ x x’**Euclidian: # correspondences?**• How many correspondences needed for translation+rotation? • How many DOF? ? T(x,y) y y’ x x’**Affine: # correspondences?**• How many correspondences needed for affine? • How many DOF? ? T(x,y) y y’ x x’**Affine: # correspondences?**• How many correspondences needed for affine? • How many DOF? ? T(x,y) y y’ x x’**Projective: # correspondences?**• How many correspondences needed for projective? • How many DOF? ? T(x,y) y y’ x x’**Example: warping triangles**B’ • Given two triangles: ABC and A’B’C’ in 2D (12 numbers) • Need to find transform T to transfer all pixels from one to the other. • How can we compute the transformation matrix: B ? T(x,y) C’ A C A’ Source Destination**Example: warping triangles**B’ • Given two triangles: ABC and A’B’C’ in 2D (12 numbers) • Need to find transform T to transfer all pixels from one to the other. • How can we compute the transformation matrix: B ? T(x,y) C’ A C A’ Source Destination cp2transform in matlab! input – correspondences. output – transformation.**Image warping**• Given a coordinate transform (x’,y’)=T(x,y) and a source image f(x,y), how do we compute a transformed image g(x’,y’)=f(T(x,y))? T(x,y) y y’ x x’ f(x,y) g(x’,y’)

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