Math 3360: Mathematical ImagingLecture4:Singular ValueDecompsitionThe Chinese University of Hong Kong
Math 3360: Mathematical Imaging The Chinese University of Hong Kong Lecture 4: Singular Value Decompsition
Recap:Main ideaFordetails,pleaserefertoSupplementarynote2!StackingoperatorStacking operator:Convert2D image to a column vectorDefinition2.1:Define07:0←(n-l)NxNzeromatriaVn=1and N=←Nx Nidentitymatriaroun00(N-n)NxNzeromatrir:0Let f eI (N x N image). We define the stacking operator on f as :NSf=F=NfVnn=
Recap: Main idea ◼ Stacking operator For details, please refer to Supplementary note 2!
SVDFordetails,pleaserefertoSupplementarynote2!SVDSingular ValueDecomposition (SVD)Let g bean image (canbe general m xn image)Assume gTg is of rank r.Then g can be written asg=UA1/2VTwhere U E Mmxm and V E Mnxn are orthogonal matrices (UUT = UTU = I andVVT-VTV=I)and A1/2is a diagonal n x nmatrix.Animagecanbedecomposedas:g = UA1/2VT =Za,ui=1Eigen-image
SVD ◼ SVD ◼ An image can be decomposed as: Eigen-image For details, please refer to Supplementary note 2!
ExampleofSvDdecompsitionofanimageExample2.1:SvDdecompositionofanimageShow the different stages of the SVD of the following image:/255255255255255255255255255255255100255255100100255255100150255150150100255255255100150200150100g=2552552551001501501501002552552551001001002552552552552552555025525525550505050255255255255
Example of SVD decompsition of an image Example 2.1: SVD decomposition of an image
Example of SvD decompsition of animageExample2.1:SvDdecompositionofanimageTheimagelookslike:
Example of SVD decompsition of an image Example 2.1: SVD decomposition of an image The image looks like: