Image Warping (Szeliski Sec 2..2) http://www.jeffre-martin.com CS94: Image Manipulation & Computational Photograph Aleei Efros, UC Berkele, Fall 7 Some slides from Steve Seitz
Image Transformations image filtering: change range of image g() T(f()) f T f image warping: change domain of image f g() f(t()) T f
Image Transformations image filtering: change range of image g() T(f()) f T g image warping: change domain of image f g() f(t()) T g
Parametric (global) warping Eamples of parametric warps: translation rotation aspect affine perspective clindrical
Parametric (global) warping T p (,) p (, ) Transformation T is a coordinate-changing machine: p T(p) What does it mean that T is global? Is the same for an point p can be described b just a few numbers (parameters) Let s represent a linear T as a matri: p Mp M
Scaling Scaling a coordinate means multipling each of its components b a scalar Uniform scaling means this scalar is the same for all components: 2
Scaling Non-uniform scaling: different scalars per component: X 2, Y.5
Scaling Scaling operation: Or, in matri form: b a b a scaling matri S What s inverse of S?
2-D Rotation (, ) (, ) q cos(q) - sin(q) sin(q) + cos(q)
2-D Rotation q f (, ) (, ) r cos (f) r sin (f) r cos (f + q) r sin (f + q) Trig Identit r cos(f) cos(q) r sin(f) sin(q) r sin(f) cos(q) + r cos(f) sin(q) Substitute cos(q) - sin(q) sin(q) + cos(q)
2-D Rotation This is eas to capture in matri form: Even though sin(q) and cos(q) are nonlinear functions of q, is a linear combination of and is a linear combination of and What is the inverse transformation? Rotation b q For rotation matrices ( q ) sin( q ) ( ) ( ) cos - sin q cos q R - T R R
22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Identit? 2D Scale around (,)? s s * * s s
22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Rotate around (,)? * cos * sin * sin * cos + - - cos sin sin cos 2D Shear? sh sh + + * * sh sh
22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Mirror about Y ais? - - 2D Mirror over (,)? - - - -
22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Translation? + t + t NO! Onl linear 2D transformations can be represented with a 22 matri
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 a c b d a c be d g f i h k j l
Consider a different Basis j (,) v (v,v ) q p u(u,u ) i (,) q4i+3j (4,3) p4u+3v
Linear Transformations as Change of Basis v (v,v ) j (,) p uv p ij p uv (4,3) p 4u +3v p 4u +3v u(u,u ) p ij 4u+3v An linear transformation is a basis!!! i (,) u p ij v 4 u v p uv u v 3 u v
What s the inverse transform? j (,) v (v,v ) p ij p uv u(u,u ) p ij (5,4) p uv i (,) p u + p v u u v v - 5 4 u u v v p uv (p,p )? - p ij How can we change from an basis to an basis? What if the basis are orthogonal?
Projection onto orthogonal basis j (,) v (v,v ) p ij p uv i (,) p ij (5,4) - u p uv v 5 u v 4 u v p uv (u p ij, v p ij ) u v p ij u(u,u )
Homogeneous Coordinates : How can we represent translation as a 33 matri? + + t t
Homogeneous Coordinates Homogeneous coordinates represent coordinates in 2 dimensions with a 3-vector ¾¾ homogeneous ¾¾¾¾ coords
Homogeneous Coordinates Add a 3rd coordinate to ever 2D point (,, w) represents a point at location (/w, /w) (,, ) represents a point at infinit (,, ) is not allowed 2 (2,,) or (4,2,2) or (6,3,3) Convenient coordinate sstem to represent man useful transformations 2
Homogeneous Coordinates : How can we represent translation as a 33 matri? A: Using the rightmost column: t t Translation t t + +
Translation Eample of translation + + t t t t t 2 t Homogeneous Coordinates
Basic 2D Transformations Basic 2D transformations as 33 matrices - cos sin sin cos t t sh sh Translate Rotate Shear s s Scale
Matri Composition Transformations can be combined b matri multiplication ø ö ç ç è æ - w s s t t w cos sin sin cos p T(t,t ) R() S(s,s ) p Does the order of multiplication matter?
Affine Transformations Affine transformations are combinations of Linear transformations, and Translations Properties of affine transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition Models change of basis w a d b e c f w Will the last coordinate w alwas be?
Projective Transformations Projective transformations Affine transformations, and Projective warps Properties of projective transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines do not necessaril remain parallel Ratios are not preserved Closed under composition Models change of basis w a d g b e h c f i w
2D image transformations These transformations are a nested set of groups Closed under composition and inverse is a member