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PSU CSE/EE 486 - Camera Projection II

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CSE486, Penn StateRobert CollinsLecture 13: Camera Projection IIReading: T&V Section 2.4CSE486, Penn StateRobert CollinsRecall: Imaging GeometryVUWObject of Interestin World CoordinateSystem (U,V,W)CSE486, Penn StateRobert CollinsImaging GeometryZfCamera Coordinate System (X,Y,Z). • Z is optic axis• Image plane located f unitsout along optic axis• f is called focal lengthXYCSE486, Penn StateRobert CollinsImaging GeometryVUWZForward Projection onto image plane.3D (X,Y,Z) projected to 2D (x,y)yxXYCSE486, Penn StateRobert CollinsImaging GeometryVUWZyOur image gets digitizedinto pixel coordinates (u,v)xXYuvCSE486, Penn StateRobert CollinsImaging GeometryVUWZyWorld CoordinatesCameraCoordinatesImage (film)CoordinatesPixelCoordinatesuvxXYCSE486, Penn StateRobert CollinsForward ProjectionUVWXYZxyuvWorldCoordsCameraCoordsFilmCoordsPixelCoordsWe want a mathematical model to describehow 3D World points get projected into 2DPixel coordinates.Our goal: describe this sequence of transformations by a big matrix equation!CSE486, Penn StateRobert CollinsIntrinsic Camera ParametersUVWXYZxyuvWorldCoordsCameraCoordsFilmCoordsPixelCoordsAffine TransformationCSE486, Penn StateRobert CollinsIntrinsic parameters• Describes coordinate transformation between film coordinates (projected image) and pixel array• Film cameras: scanning/digitization• CCD cameras: grid of photosensorsstill in T&V section 2.4CSE486, Penn StateRobert CollinsIntrinsic parameters (offsets)film plane(projected image)xy(0,0)u (col)v (row)(0,0)pixel arrayyoxoxouZXfyovZYfoxand oycalled image center or principle pointCSE486, Penn StateRobert CollinsIntrinsic parametersfilm planexy(0,0)u (col)v (row)(0,0)pixel arrayyoxosometimes one or more coordinate axes are flipped (e.g. T&V section 2.4)xouZXfyovZYfCSE486, Penn StateRobert CollinsanalogIntrinsic parameters (scales)pixel arraysampling determines how many rows/cols in the imageC cols x R rowsfilmscanning resolutionCCDresampleCSE486, Penn StateRobert CollinsEffective Scales: sxand syO.Camps, PSUxouZXfyovZYf1sx1syNote, since we have different scale factors in x and y,we don’t necessarily have square pixels!Aspect ratio is sy/ sxCSE486, Penn StateRobert CollinsPerspective projection matrix 100010/000/'''ZYXoosfsfzyxyxyxO.Camps, PSUAdding the intrinsic parameters into theperspective projection matrix:xouZXf1sxyovZYf1syuz’x’vz’y’To verify:CSE486, Penn StateRobert CollinsNote:Sometimes, the image and the camera coordinate systemsSometimes, the image and the camera coordinate systemshave opposite orientations: [the book does it this way] have opposite orientations: [the book does it this way] 100010/000/'''ZYXoosfsfzyxyxyxyyxxsovZYfsouZXf)()(CSE486, Penn StateRobert CollinsNote 2 10001000000'''ZYXffzyx100a13a12a11a23a22a21w’v’u’In general, I like to think of the conversion asa separate 2D affine transformation from filmcoords (x,y) to pixel coordinates (u,v):u = MintPC= MaffMprojPCMprojMaffCSE486, Penn StateRobert CollinsHuh?Did he just say it was “a fine” transformation?No, it was “affine” transformation, a type of2D to 2D mapping defined by 6 parameters.More on this in a moment...CSE486, Penn StateRobert CollinsSummary : Forward ProjectionUVWXYZxyuvWorldCoordsCameraCoordsFilmCoordsPixelCoordsMextMprojMaffMUVWuv342414332313312212312111mmmmmmmmmmmmMintUVWXYZuvMextCSE486, Penn StateRobert CollinsSummary: Projection EquationWorld to cameraPerspectiveprojectionFilm planeto pixelsMextMprojMaffMintMCSE486, Penn StateRobert CollinsLecture 13/14: Intro to Image MappingsCSE486, Penn StateRobert Collinsfrom R.SzeliskiImage MappingsOverviewCSE486, Penn StateRobert CollinsGeometric Image Mappingsimagetransformed imageGeometrictransformation(x,y)(x’,y’)x’ = f(x, y, {parameters})y’ = g(x, y, {parameters})CSE486, Penn StateRobert CollinsLinear Transformationsimagetransformed imageGeometrictransformation(x,y)(x’,y’)x’y’1xy1(Can be written as matrices)M(params)=CSE486, Penn StateRobert CollinsTranslation0 110txtyxyx’y’equations matrix formtransformCSE486, Penn StateRobert CollinsScale0 110xyx’y’equations matrix formtransform0SS0CSE486, Penn StateRobert CollinsRotation0 110xyx’y’equations matrix formtransform)CSE486, Penn StateRobert CollinsEuclidean (Rigid)0 110xyx’y’equations matrix formtransform)txtyCSE486, Penn StateRobert CollinsPartitioned Matriceshttp://planetmath.org/encyclopedia/PartitionedMatrix.htmlCSE486, Penn StateRobert CollinsPartitioned Matrices2x1 2x2 2x12x11x1 1x2 1x11x1equation formmatrix formCSE486, Penn StateRobert CollinsAnother Example (from last time)1WVU1ZYXtx1000r13r12r11r23r22r21r33r32r31tytzPC= R PW+ TPC1=R T0 1PW13x11x13x3 3x11x31x1 1x13x1CSE486, Penn StateRobert CollinsSimilarity (scaled Euclidean)0 110xyx’y’equations matrix formtransform)txtySCSE486, Penn StateRobert CollinsAffine0 110xyx’y’equations matrix formtransformCSE486, Penn StateRobert CollinsProjective0 110xyx’y’equations matrix formtransformNote!CSE486, Penn StateRobert CollinsSummary of 2D TransformationsCSE486, Penn StateRobert CollinsSummary of 2D TransformationsEuclideanCSE486, Penn StateRobert CollinsSummary of 2D TransformationsSimilarityCSE486, Penn StateRobert CollinsSummary of 2D TransformationsAffineCSE486, Penn StateRobert CollinsSummary of 2D TransformationsProjectiveCSE486, Penn StateRobert CollinsSummary of 2D Transformationsfrom


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