Onto 3DVector ProjectionVector Cross ProductCoordinate System: Definitions3-D Camera CoordinatesGoing from 2-D to 3-D3-D Scaling3-D Rotations3-D Euler Rotation Matrices3-D Rotation MatricesCoordinate System ConversionSlide 12Slide 13Change of Coordinates: Special Case of Same AxesChange of Coordinates: Special Case of Same Origin3-D Rigid Transformations3-D Transformations: Arbitrary Change of CoordinatesRigid Transformations: Homogeneous CoordinatesCamera Projection MatrixCamera Projection Matrix: Image OffsetsFactoring the Camera MatrixCamera Calibration MatrixDealing with World CoordinatesCombining Intrinsic & Extrinsic ParametersSkew ignoredApplicationsCamera MatrixLinear SystemMatrix Form of Linear SystemSolving Linear SystemsFitting LinesSlide 32Example: Fitting a LineSlide 34Homogeneous Systems of EquationsLine-Fitting as a Homogeneous SystemExample: Homogeneous Line-FittingCamera CalibrationA Vision Problem: Estimating PA Calibration TargetEstimating P: The Direct Linear Transformation (DLT) AlgorithmDLT Camera Matrix Estimation: PreliminariesDLT Camera Matrix Estimation: Step 1DLT Camera Matrix Estimation: Step 2DLT Camera Matrix Estimation: Step 3What We Just DidDLT Camera Matrix Estimation: Step 4Slide 48DLT Camera Matrix Estimation: Step 5Slide 50Slide 51Slide 52Slide 53Slide 54Gold Standard algorithmSlide 56Slide 57Radial distortionSlide 59Slide 60Recovery of world positionComputer Vision : CISC4/689Onto 3D•Coordinate systems•3-D homogeneous transformations–Translation, scaling, rotation•Changes of coordinates–Rigid transformationsComputer Vision : CISC4/689Vector Projection•The projection of vector a onto u is that component of a in the direction of uComputer Vision : CISC4/689Vector Cross Product•Definition: If a = (xa, ya, za)T and • b = (xb, yb, zb)T, then: c = a X bc is orthogonal to both a and b from HillComputer Vision : CISC4/689•Let x = (x, y, z)T be a point in 3-D space (R3). What do these values mean?•A coordinate system in Rn is defined by an origin o and n orthogonal basis vectors –In R3, positive direction of each axis X, Y, Z is indicated by unit vector i, j, k, respectively, where k = i X j (in a right-handed system)–Coordinate is length of projection of vector from origin to point onto axis basis vector—e.g., x = x ¢ i Coordinate System: DefinitionsxoComputer Vision : CISC4/6893-D Camera Coordinates•Right-handed system•From point of view of camera looking out into scene:+X right, {X left+Y down, {Y up+Z in front of camera, {Z behindComputer Vision : CISC4/689Going from 2-D to 3-D•Points: Add z coordinate •Transformations: Become 4 x 4 matrices with extra row/column for z component—e.g., translation:Computer Vision : CISC4/6893-D ScalingComputer Vision : CISC4/6893-D Rotations•In 2-D, we are always rotating in the plane of the image, but in 3-D the axis of rotation itself is a variable•Three canonical rotation axes are the coordinate axes X, Y, Z•These are sometimes referred to •in aviation terms: pitch, yaw or heading, and roll, respectivelyfrom Hillfrom HillPitch is the angle that its longitudinal axis (running from tail to nose and along n) makes with horizontal plane.Computer Vision : CISC4/6893-D Euler Rotation Matrices•Similar to 2-D rotation matrices, but with coordinate corresponding to rotation axis held constant•E.g., a rotation about the X axis of µ radians:Computer Vision : CISC4/6893-D Rotation Matrices•General form is:•Properties–RT = R-1–Preserves vector lengths, angles between vectors–Upper-left block R3£3 is orthogonal matrix•Rows form orthonormal basis (as do columns): Length = 1, mutually orthogonal•So R3£3 x projects point x onto unit vectors represented by rows of R3£3Computer Vision : CISC4/689•Camera coordinates C: Origin at center of camera, Z axis pointed in viewing direction•World coordinates W: Arbitrary origin, axes–Way to specify camera location, orientation (aka pose) in same frame as scene objects (we like to move camera to world, so as to convert world coordinates into camera coordinates) • Cx, Wx,: Same point in different coordinatesCoordinate System ConversionComputer Vision : CISC4/689•Camera coordinates C: Origin at center of camera, Z axis pointed in viewing direction•World coordinates W: Arbitrary origin, axes–Way to specify camera location, orientation (aka pose) in same frame as scene objects• Cx, Wx,: Same point in different coordinatesCoordinate System ConversionComputer Vision : CISC4/689•Camera coordinates C: Origin at center of camera, Z axis pointed in viewing direction•World coordinates W: Arbitrary origin, axes–Way to specify camera location, orientation (aka pose) in same frame as scene objects• Cx, Wx,: Same point in different coordinatesCoordinate System ConversionComputer Vision : CISC4/689Change of Coordinates: Special Case of Same Axes•Distinct origins, parallel basis vectors: If B is world, Ax (camera) can be obtained by Bx (world) minusits CG.Computer Vision : CISC4/689Change of Coordinates: Special Case of Same Origin•Just need to rotate basis vectors so that they are aligned•Rotation matrix is projection of basis vectors in new frame ia ib ja 0 ka 0 ia 0 ja jb ka 0 ia 0 ja 0 ka kbCheck by multing(ib 0 0), etc.i.e, take A coordinate systemAs (1 0 0), (0 1 0), (0 0 1)Computer Vision : CISC4/6893-D Rigid Transformations•Combination of rotation followed by translation without scaling•“Moves” an object from one 3-D position and orientation (pose) to anotherT R MComputer Vision : CISC4/6893-D Transformations: Arbitrary Change of Coordinates •A rigid transformation can be used to represent a general change in the coordinate system that “expresses” a point’s locationComputer Vision : CISC4/689•Points in one coordinate system are transformed to the other as follows:• takes the camera to the world origin, transforming world coordinates to camera coordinates•If A is camera and B is world, inverse translationand inverse rotationRigid Transformations: Homogeneous CoordinatesComputer Vision : CISC4/689Camera Projection Matrix•Using homogeneous coordinates, we can describe perspective projection as the result of multiplying by a 3 x 4 matrix P: (by the rule for converting between homo-geneous and regular coordinates—this is perspective division)Computer Vision : CISC4/689Camera Projection Matrix: