Slide 1AdministriviaPhysical parameters of image formationPhysical parameters of image formationPerspective and artPerspective projection equationsHomogeneous coordinatesPerspective Projection MatrixPerspective projection & calibrationPerspective projection & calibrationSlide 11Intrinsic parametersIntrinsic parametersIntrinsic parametersIntrinsic parametersIntrinsic parameters, homogeneous coordinatesExtrinsic parameters: translation and rotation of camera frameSlide 18Other ways to write the same equationCalibration targetCamera calibrationCamera calibrationCamera calibrationCamera calibrationSlide 25Recall, perspective effects…Perspective effectsPerspective effectsPerspective effectsProjection propertiesWeak perspectiveOrthographic projectionSlide 33Slide 342D3DOther types of projection360 degree field of view…Tilt-shifttilt, shiftTilt-shift perspective correctionSlide 42Rotating sensor (or object)PhotofinishPhysical parameters of image formationPinhole size / apertureAdding a lensPinhole vs. lensCameras with lensesHuman eyeThin lensThin lens equationFocus and depth of fieldFocus and depth of fieldFocus and depth of fieldDepth from focusField of viewField of view depends on focal lengthField of view depends on focal lengthVignettingVignettingChromatic aberrationChromatic aberrationPhysical parameters of image formationEnvironment mapBDRFDiffuse / LambertianForeshorteningSpecular reflectionPhongPhysical parameters of image formationDigital camerasHistorical contextSlide 74Digital SensorsResolutionDigital imagesSlide 78Color sensing in digital camerasSlide 80Physical parameters of image formationSummarySlide 83Next timeC280, Computer VisionProf. Trevor [email protected] 2: Image FormationAdministrivia•We’re now in 405 Soda…•New office hours: Thurs. 5-6pm, 413 Soda.•I’ll decide on waitlist decisions tomorrow.•Any Matlab issues yet?•Roster…Physical parameters of image formation•Geometric–Type of projection–Camera pose•Optical–Sensor’s lens type–focal length, field of view, aperture•Photometric–Type, direction, intensity of light reaching sensor–Surfaces’ reflectance properties•Sensor–sampling, etc.Physical parameters of image formation•Geometric–Type of projection–Camera pose•Optical–Sensor’s lens type–focal length, field of view, aperture•Photometric–Type, direction, intensity of light reaching sensor–Surfaces’ reflectance properties•Sensor–sampling, etc.Perspective and art•Use of correct perspective projection indicated in 1st century B.C. frescoes•Skill resurfaces in Renaissance: artists develop systematic methods to determine perspective projection (around 1480-1515)Durer, 1525RaphaelK. GraumanPerspective projection equations•3d world mapped to 2d projection in image planeForsyth and PonceCamera frameImage planeOptical axisFocal lengthScene / world pointsScene pointImage coordinates‘’‘’Homogeneous coordinatesIs this a linear transformation?Trick: add one more coordinate:homogeneous image coordinateshomogeneous scene coordinatesConverting from homogeneous coordinates•no—division by z is nonlinearSlide by Steve SeitzPerspective Projection Matrixdivide by the third coordinate to convert back to non-homogeneous coordinates•Projection is a matrix multiplication using homogeneous coordinates:'/10'/10000100001fzyxzyxf)','(zyfzxfSlide by Steve SeitzComplete mapping from world points to image pixel positions?Perspective projection & calibration•Perspective equations so far in terms of camera’s reference frame….•Camera’s intrinsic and extrinsic parameters needed to calibrate geometry.Camera frameK. GraumanPerspective projection & calibrationCamera frameIntrinsic:Image coordinates relative to camera Pixel coordinatesExtrinsic:Camera frame World frameWorld frameWorld to camera coord. trans. matrix(4x4)Perspectiveprojection matrix(3x4)Camera to pixel coord. trans. matrix (3x3)=2Dpoint(3x1)3Dpoint(4x1)K. GraumanIntrinsic parameters: from idealized world coordinates to pixel valuesForsyth&PoncezyfvzxfuPerspective projectionW. FreemanIntrinsic parameterszyvzxu But “pixels” are in some arbitrary spatial unitsW. FreemanIntrinsic parameterszyvzxu Maybe pixels are not squareW. FreemanIntrinsic parameters00 vzyvuzxuWe don’t know the origin of our camera pixel coordinatesW. FreemanIntrinsic parameters00 )sin()cot( vzyvuzyzxuMay be skew between camera pixel axesv u v u vuvuuvv)cot()cos()sin(W. FreemanppC K Intrinsic parameters, homogeneous coordinates00 )sin()cot( vzyvuzyzxuuv1cot() u00sin()v00 0 1000xyz1Using homogenous coordinates,we can write this as:or:In camera-based coordsIn pixelsW. FreemanExtrinsic parameters: translation and rotation of camera frametpRpCWWCWC Non-homogeneous coordinatesHomogeneous coordinatesptRpWCWCWC1000||W. FreemanCombining extrinsic and intrinsic calibration parameters, in homogeneous coordinatesForsyth&Ponce ptRKpWCWCW ppCK pMpW IntrinsicExtrinsicptRpWCWCWC1000||World coordinatesCamera coordinatespixels0 0 0 1W. FreemanOther ways to write the same equation1.........1321zWyWxWTTTpppmmmvupMpW PmPmvPmPmu3231pixel coordinatesworld coordinatesConversion back from homogeneous coordinates leads to:W. FreemanCalibration targethttp://www.kinetic.bc.ca/CompVision/opti-CAL.htmlFind the position, ui and vi, in pixels, of each calibration object feature point.Camera
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