Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Computer VisionSeptember 2006 L1.1© 2006 by Davi GeigerImage Measurements and DetectionComputer VisionSeptember 2006 L1.2© 2006 by Davi GeigerImages are Intensity Surfaces and Edges are Surface DiscontinuitiesI1(x,y)I(x,y)Computer VisionSeptember 2006 L1.3© 2006 by Davi GeigerFor each pixel (x,y) we evaluate the accumulation of the intensity along eight (8) directions,                       and three (3) different scales, s= 3, 5, 7sdIssyxI0)sin,cos(1),,,(~10)sin,cos(1),,,(~siyiyxixIssyxIMeasurements:Intensity Accumulation, ),,,(~syxIs=3 pixelss=7 pixels                                                                                                                                            / /4/Discrete approximationyxComputer VisionSeptember 2006 L1.4© 2006 by Davi Geiger10)sin,cos(1),,,(~siyiyxixIssyxI1 y2 x2 y1 xMeasurements:Intensity Accumulation, ),,,(~syxIwhere, and for        and for             ),(1),,,(~1)sin,cos(11),,,(~11yxIssyxIssyiyxixIssyxIsicRemoving the center pixel:),,(~syxIc                                                                                                                                            / /4/Computer VisionSeptember 2006 L1.5© 2006 by Davi GeigerIn order to obtain measure for any orientation, we use linear combinations of the results obtained for the 8 base directions, in the spirit of steerable filters. For a given angle we assign the two base directions that are closest to So for orientation we write Measurements:Intensity Accumulation, ),,,(~syxIyx4   ),0,,(~cos4coscos),4,,(~cos4cos4cos),,,(~syxIsyxIsyxI      angles base 8 theare'wherecoscos),,,(~cos),,,(~cos),,,(~7,...,1,0,1111ssyxIsyxIsyxIiIfiiiiiiiiiExampleComputer VisionSeptember 2006 L1.6© 2006 by Davi GeigerMeasurements: Some Problems on valueaccumulati average actual thefrom different very isn combinatiolinear the?),,,(~syxI   ),0,,(~cos4coscos),4,,(~cos4cos4cos),,,(~syxIsyxIsyxI                                                                                                                                  / /4 /Reminder: The linear combination in this case is given byDark grey values areComputer VisionSeptember 2006 L1.7© 2006 by Davi Geiger                                                                                                                                                  /4s=3s=5s=5s=7s=7s=3Measurements: Intensity Accumulation II, ),,,(ˆsyxI)),,,1(~),,1,(~),,1,(~),,,(~(41syxIsyxIsyxIsyxIiiii)),,1,(~),,,1(~),,,1(~),,,(~(41syxIsyxIsyxIsyxIiii)),,1,1(~),,1,(~),,,1(~),,,(~(41syxIsyxIsyxIsyxIiii),,,(ˆsyxIifor i  for i     for i  In order to obtain more robust measures, for an angle we can average the value of along   to obtain),,,(~syxI),,,(ˆsyxIComputer VisionSeptember 2006 L1.8© 2006 by Davi GeigerMeasurements: Derivatives),,,(ˆsyxID.),,,(ˆ),,,(ˆsin),,,(ˆcos),,,(ˆ)sin,(cos),,,(ˆ),,,(ˆ),,,(ˆsyxIsyxIysyxIxsyxIsyxIsyxIsyxIDcc--Taking the derivative along the angle i.e., in a direction )sin,(cosˆWe have now a bank of eight (8) distinct oriented filters at different scales, namely { ; 