Geometric Transformations: 
           Warping, Registration, 
                  MorphingMorphing
                    Yao Wang
             Polytechnic University, Brooklyn, NY 11201
           With contribution from Zhu Liu, Onur Guleryuz, and 
                   Partly based on
           AA. KK. JJainain, FFundamentalsundamentals ooff DigitalDigital ImageImage ProcessingProcessing
                                Lecture Outline
          • Introduction
          • ImageImage deformationdeformation modelmodel
          • Image warping
          • Image registration
          • ImageImage morphingmorphing
           Geometric Transformation      EL512 Image Processing                       2
            What is Geometric Transformation?
           • So far, the image processing operations 
              wewe havehave discusseddiscussed mmodifyodify tthehe colorcolor valuesvalues
              of pixels in a given image
           • WithWith geometricgeometric ttransformation,ransformation, wewe modifymodify 
              the positions of pixels in a image, but keep 
              their colors unchanged
               –To create special effects
               –To register two images taken of the same 
                  scene at different times
               –To morph one image to another
            Geometric Transformation        EL512 Image Processing                           3
         How to define a geometric transformation?
           •  Let (u, v) represent the image coordinate in an original 
              image, and (x, y) in a deformed (or warped) image. We 
              useuse aa functionfunction pairpair toto relaterelate correspondingcorresponding ppixelsixels iinn tthehe 
              two images:
               – Forward mapping:           xx(u,v), or xx(u)
                                            yy  yy((uu,vv))
                                            
               – Inverse mapping:            u u(x,y)
                                                     , or  uu(x)
                                            vv((x,, yy))
                                            
           •  Let f(u, v) or f(u) denote the original image and g(x, y) or 
              g(g(xx)) tthehe deformeddeformed iimage.mage. ThenThen theythey areare rrelatedelated by:by:
                     g(x,y) f(u(x,y),v(x,y)),  or  g(x) f(u(x))
                      f (u,v)  g(x(u,v), y(u,v))   f(u) g(x(u))
                                                    
            Geometric Transformation        EL512 Image Processing                           4