Computer Vision: Image Alignment Raquel Urtasun TTIChicago Jan 24, 2013 RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 1/44 Readings Chapter 2.1, 3.6, 4.3 and 6.1 of Szeliski’s book Chapter 1 of Forsyth & Ponce RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 2/44 What did we see in class last week? RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 3/44 What is the geometric relationship between these images? !" [Source: N. Snavely] RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 4/44 What is the geometric relationship between these images? Very important for creating mosaics! [Source: N. Snavely] RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 4/44 Image Warping Image filtering: change range of image g(x)=h(f(x)) !" %" $" #" #" Image warping: change domain of image g(x)=f(h(x)) !" %" $" #" #" [Source: R. Szeliski] RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 5/44 Parametric (global) warping !" !!"!#$%&'! !"!"!#$(%&('! Transformation T is a coordinate-changing machine: p(cid:48) =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) [Source: N. Snavely] RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 6/44 Forward and Inverse Warping Forward Warping: Send each pixel f(x) to its corresponding location (x(cid:48),y(cid:48))=T(x,y) in g(x(cid:48),y(cid:48)) Inverse Warping: Each pixel at the destination is sampled from the original image RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 7/44 All 2D Linear Transformations Linear transformations are combinations of Scale, Rotation Shear Mirror (cid:20)x(cid:48)(cid:21) (cid:20)a b(cid:21)(cid:20)x(cid:21) = y(cid:48) c d y [Source: N. Snavely] RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 8/44 Parallel lines remain parallel Ratios are preserved Closed under composition (cid:20)x(cid:48)(cid:21) (cid:20)a b(cid:21)(cid:20)e f(cid:21)(cid:20)i j(cid:21)(cid:20)x(cid:21) = y(cid:48) c d g h k l y What about the translation? [Source: N. Snavely] All 2D Linear Transformations Properties of linear transformations: Origin maps to origin Lines map to lines RaquelUrtasun (TTI-C) ComputerVision Jan24,2013 9/44