Camera Projection Matrix - University Of Minnesota
Camera Projection Matrix
Camera Model
Camera Model (1st Person Coordinate)CameraGround plane
Camera Model (1st Person Coordinate)CameraRecall camera projection matrix:XCGround plane u f v 1 fpx X p y Y 1 Z Camera intrinsic parameter: metric space to pixel space
Camera Model (1st Person Coordinate)CameraRecall camera projection matrix:XCGround plane u f v 1 px X fK p y Y 1 Z Camera intrinsic parameter: metric space to pixel space
Camera Model (1st Person Coordinate)CameraRecall camera projection matrix:XC u f v 1 2D image (pix)Ground planepx X fK p y Y 1 Z 3D world (metric)
Camera Model (1st Person Coordinate)CameraRecall camera projection matrix:Origin at cameraXC u f v 1 2D image (pix)Ground planepx X fK p y Y 1 Z 3D world (metric) u X K -1 v Y XC 1 Z
Camera Model (1st Person Coordinate)CameraRecall camera projection matrix:Origin at cameraX C1X C2 Where is origin? u f v 1 2D image (pix)Ground planepx X fK p y Y 1 Z 3D world (metric) u 1 X 1 K -1 v 1 Y1 X C1 1 Z 1 u 2 X 2 K -1 v 2 Y2 X C2 1 Z 2
Camera Model (3rd Person Coord. World Coord.)CameraRecall camera projection matrix:X3D worldOrigin at world coordinateGround plane u f v 1 2D image (pix)px X fK p y YX 1 Z 3D world (metric)
Point Rotation2D rotationX (x ,y ) x y World
Point Rotation2D rotationX 1 ( x 1, y 1 )X (x ,y ) x 1 cos y 1 sin World sin x cos y
Coordinate Transform (Rotation)2D coordinate transform:X (x ,y ) x y World
Coordinate Transform (Rotation)2D coordinate transform:X (x,y )XC ( x C , y C )Camera World x C cos sin x ? y C sin cos y
Coordinate Transform (Rotation)2D coordinate transform:X (x,y )XC ( x C , y C )Camera x C cos y C sin sin x cos y Coordinate transformation: Inverse of point rotationWorld
Coordinate Transform (Rotation)2D coordinate transform:X (x,y )XC ( x C , y C )Camera World x C cos y C sin cos det sin sin x cos y sin 22 cos sin 1 cos
Coordinate Transform (Rotation)2D coordinate transform:X (x,y )XC ( x C , y C )Camera x C cos rX sin x y C sin cos y rXWorldrX : x axis of camera seen from the world
Coordinate Transform (Rotation)2D coordinate transform:X (x,y )XC ( x C , y C )rYCamera x C cos rX sin x y C sin rY cos y rXWorldrX : x axis of camera seen from the worldrY : y axis of camera seen from the world
Coordinate Transform (Rotation)2D coordinate transform:X (x,y )XC ( x C , y C )r2Camera x C cos y r1 C sin sin x r2 cos y r1Worldr1 : x axis of world seen from the camerar2 : y axis of world seen from the camera
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldXOrigin at world coordinateGround planeX ( x , y ,z )World rx1 rx2 rx 3 X C ry1 ry2? ry 3 X C R W X r r z1 z 2 rz 3
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldXOrigin at world coordinateGround planeX ( x , y ,z )XC ( x C , y C ,z C )Camera rx1 rx 2 rx 3 X C ry1 ry?2 ry 3 X C R W X r r z1 z 2 rz 3
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldXOrigin at world coordinateGround planeX ( x , y ,z )XC ( x C , y C ,z C )Camera rx1 rx 2 rx 3 X C ry1 ry 2 ry 3 X C R W X r r z1 z 2 rz 3
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldXOrigin at world coordinateGround planeX ( x , y ,z )XC ( x C , y C ,z C )Camera rx1 rx 2 rx 3 X C ry1 ry 2 ry 3 X C R W X r r z1 z 2 rz 3 Degree of freedom?
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldX rx1 rx2 rx 3 X C ry1 ry2 ry 3 X C R W X r r z1 z 2 rz 3 Degree of freedom?Origin at world coordinateCGround planeR W SO(3) Orthogonal matrixX ( x , y ,z )XC ( x C , y C ,z C )CameraWorld CRW TC R W I3 , det C R W 1
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldX rx1 rrx2X rx 3 X C ry1 rry2Y ry 3 X C R W X r rr r z1 zZ2 z 3 Origin at world coordinateCGround planeX ( x , y ,z )XC ( x C , y C ,z C )R W SO(3)rXrYCameraWorld Orthogonal matrix Right hand rulerZ rX rYCRW TC R W I3 , det C R W 1
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RWX3D world rx1 rrx2X rx 3 X C ry1 rry2Y ry 3 X C R W X r rr r z1 zZ2 z 3 Origin at world coordinateCGround planeR W SO(3) Orthogonal matrix Right hand rulerX : camera x axis seen fromthe world coord.CameraWorldrXrYrZ rX rYCRW TC R W I3 , det C R W 1
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldX rx1 rrx2X rx 3 X C ry1 rry2Y ry 3 X C R W X r rr r z1 zZ2 z 3 Origin at world coordinateCGround planerZrY : camera y axis seen fromthe world coord.rX : camera x axis seen fromthe world coord.CameraWorldR W SO(3) Orthogonal matrix Right hand rulerXrYrZ rX rYCRW TC R W I3 , det C R W 1
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RWX3D world rx1 rx2 rx 3 X C rry11 rry22 ry33 X C R W X r r z1 z 2 rz 3 Origin at world coordinateCGround planeR W SO(3) Orthogonal matrix Right hand ruler1r2CameraWorldr1 : world x axis seen fromthe camera coord.r3 r1 r2CRW TC R W I3 , det C R W 1
Coordinate Transform (Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldX rx1 rx2 rx 3 X C rry11 rry22 ry33 X C R W X r r z1 z 2 rz 3 Origin at world coordinateCGround planeR W SO(3) Orthogonal matrix Right hand ruler3r1r2 : world y axis seen fromthe camera coord.r2CameraWorldr1 : world x axis seen fromthe camera coord.r3 r1 r2CRW TC R W I3 , det C R W 1
Camera Projection (Pure Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldXOrigin at world coordinateGround planer3r2 : world y axis seen fromthe camera coord.CameraWorldr1 : world x axis seen fromthe camera coord. rx1 rx2 rx 3 X C rry11 rry22 ry33 X C R W X r r z1 z 2 rz 3 Camera projection of world point: u f v 1 f px X C fK p y YC 1 Z C fp x rx1 rx2 rx 3 X p y ry1 ry2 ry 3 Y 1 rz1 rz2 rz 3 Z
Camera Projection (Pure Rotation)CameraC1Coordinate transformation from world to camera:RW3D worldXOrigin at world coordinateGround planer3r2 : world y axis seen fromthe camera coord.CameraWorldr1 : world x axis seen fromthe camera coord. rx1 rx2 rx 3 X C rry11 rry22 ry33 X C R W X r r z1 z 2 rz 3 Camera projection of world point: u f v 1 f px X C fK p y YC 1 Z C p x rx1 rx2 rx 3 X f K p y ry1 ry2 ry 3 Y 1 rz1 rz2 rz 3 Z
Euclidean Transform Rotation TranslationCameraCRW , tC3D worldXOrigin at world coordinateGround planeX (x , y ,z )World
Euclidean Transform Rotation TranslationCameraCRW ,tC3D worldXOrigin at world coordinateGround planeXC ( x C , y C ,z C )X (x , y ,z )CameraWorld
Euclidean Transform Rotation Translation2D coordinate transform:X (x,y )XC ( x C , y C )Camera x C cos y C sin sin x cos y Coordinate transformation: Inverse of point rotationWorld
Euclidean Transform Rotation Translation2D coordinate transform:X (x,y ) x C cos y C sin WorldXC ( x C , y C )TranslationCamerasin x cos y
Euclidean Transform Rotation Translation2D coordinate transform:X (x,y )XC ( x C , y C ) x C cos y C sin sin x t x cos y t y WorldTranslation t x t y Camera t x : the location of world coordinate t seen from camera coord. y
Euclidean Transform Rotation TranslationCameraCRW ,tC3D worldXOrigin at world coordinateGround planeXC ( x C , y C ,z C )X (x , y ,z )CameraWorld
Euclidean Transform Rotation TranslationCameraCCoordinate transformation from world to camera:RW ,tC3D worldXOrigin at world coordinateGround planeXC ( x C , y C ,z C )X (x , y ,z )CtCameraWorld rx1 rx 2 rx 3 t x X X C C R W X C t ry1 ry 2 ry 3 t y 1 r r z1 z2 rz 3 t z where C t is the world orgin seen from camera.
Euclidean Transform Rotation TranslationCameraCCoordinate transformation from world to camera:RW ,tC3D worldXOrigin at world coordinateGround planeXC ( x C , y C ,z C )X (x , y ,z )CtCameraWorld rx1 rx 2 rx 3 t x X X C C R W X C t ry1 ry 2 ry 3 t y 1 r r z1 z2 rz 3 t z where C t is the world orgin seen from camera.
Geometric InterpretationCameraCCoordinate transformation from world to camera:RW ,tC3D worldXOrigin at world coordinate rx1 rx 2 rx 3 t x X X C C R W X C t ry1 ry 2 ry 3 t y 1 r r z1 z2 rz 3 t z where C t is the world orgin seen from camera.Ground planeRotate and then, translate.XC ( x C , y C ,z C )X (x , y ,z )CtCameraWorld
Geometric InterpretationCameraCCoordinate transformation from world to camera:RW ,tC3D worldXOrigin at world coordinate rx1 rx 2 rx 3 t x X X C C R W X C t ry1 ry 2 ry 3 t y 1 r r z1 z2 rz 3 t z where C t is the world orgin seen from camera.Ground planeRotate and then, translate.XC ( x C , y C ,z C )X (x , y ,z )cf) Translate and then, rotate.CCameraWorld rx1 rx 2 rx 3 1-C x X X C C R W X C ry1 ry 2 ry 3 1-C y 1 r r 1 -C z z1 z2 rz 3 where C is the camera location seen from world.
Camera Projection MatrixCameraCCoordinate transformation from world to camera:RW ,tC3D worldXOrigin at world coordinate rx1 rx 2 rx 3 t x X X C C R W X C t ry1 ry 2 ry 3 t y 1 r r z1 z2 rz 3 t z Camera projection of world point: u f v 1 Ground planeXC ( x C , y C ,z C )X (x , y ,z ) f CCameraWorldpx X C fK p y YC 1 Z C X p x rx1 rx 2 rx 3 t x Y fK p y ry1 CrRy 2W ry 3 C tty Z 1 rz1 rz2 rz 3 t z 1
Image Projection -r C 0 0 r 0 -1 0 R 0 0 -1 1 0 0
Image Projection -r C 0 0 r r 0 -1 0 R 0 0 -1 1 0 0
Image Projectionr r -r C 0 0 0 -1 0 R 0 0 -1 1 0 0 r cos C r sin 0 -sin R 0 -cos cos 0-sin 0 -1 0
RotateCamera.mImage ProjectionK [200 0 100;0 200 100;0 0 1];radius 5;theta 0:0.02:2*pi;for i 1 : length(theta)camera offset [radius*cos(theta(i)); radius*sin(theta(i)); 0];camera center camera offset center of mass';rz [-cos(theta(i)); -sin(theta(i)); 0];ry [0 0 -1]';rx [-sin(theta(i)); cos(theta(i)); 0];R [rx'; ry'; rz'];C camera center;P K * R * [ eye(3) -C];proj [];for j 1 : size(sqaure point,1)u P * [sqaure point(j,:)';1];proj(j,:) u'/u(3);endend
Geometric InterpretationCameraCRW ,tCCamera projection of world point:3D world u f v 1 X p x rx1 rx 2 rx 3 t x X Y CC P fK p y ry1 rRy 2 ry 3 tty W Z 1 1 rz1 rz2 rz 3 t z 1 Ground plane u p11 p12 v p 21 p 22 1 p 31 p 32p13p 23p 33 X p14 Yp 24 Z p 34 1 What does each number mean?
Geometric InterpretationCameraCRW ,tCCamera projection of world point:3D worldGround planeXX 0 0 1 u f v 1 X p x rx1 rx 2 rx 3 t x X Y CC P fK p y ry1 rRy 2 ry 3 tty W Z 1 1 rz1 rz2 rz 3 t z 1 u p11 p12 v p 21 p 22 1 p 31 p 32p13p 23p 33 p14 0p 24 0 p 34 1 What is point at infinity in world x direction?
Geometric InterpretationCameraCRW ,tCCamera projection of world point:3D worldXX Ground planeuX 0 0 1 u f v 1 X p x rx1 rx 2 rx 3 t x X Y CC P fK p y ry1 rRy 2 ry 3 tty W Z 1 1 rz1 rz2 rz 3 t z 1 u p11 p12 v p 21 p 22 1 p 31 p 32p13p 23p 33 p14 0p 24 0 p 34 1 What is point at infinity in world x direction?This point is at infinite but finite in image.
Geometric InterpretationCameraCRW ,tCCamera projection of world point:3D worldXX Ground planeuX 0 0 1 u f v 1 X p x rx1 rx 2 rx 3 t x X Y CC P fK p y ry1 rRy 2 ry 3 tty W Z 1 1 rz1 rz2 rz 3 t z 1 u p11 p12 v p 21 p 22 1 p 31 p 32p13p 23p 33 p14 0p 24 0 p 34 1 p11X p14 p11 X p X pp 313134u limp 21X p 24 p 21 X p X pp 313134v lim
Geometric InterpretationCameraCRW ,tCCamera projection of world point: uX p11 p 21 p 31 3D worldXX Ground planeuX 0 0 1 u f v 1 X p x rx1 rx 2 rx 3 t x X Y CC P fK p y ry1 rRy 2 ry 3 tty W Z 1 1 rz1 rz2 rz 3 t z 1 u p11 p12 v p 21 p 22 1 p 31 p 32p13p 23p 33 p14 0p 24 0 p 34 1 p11X p14 p11 X p X pp 313134u limp X p 24 p 21v lim 21 X p X pp 313134 uX u p11 v p 21 1 p 31
Geometric InterpretationCameraCRW ,t uY C p12 p 22 p 32 3D worlduY Camera projection of world point: u f v 1 X p x rx1 rx 2 rx 3 t x X Y CC P fK p y ry1 rRy 2 ry 3 tty W Z 1 1 rz1 rz2 rz 3 t z 1 Ground planeuX u p11 p12 v p 21 p 22 1 p 31 p 32p13p 23p 33 0 p14 p 24 0 p 34 1 p12Y p14 p12 X p Y pp 323234u limp Y p 24 p 22v lim 22 X p Y pp 323234 uY u p12 v p 22 1 p 32
Geometric InterpretationCameraCRW ,tC uZ p13 p 23 p 33 Camera projection of world point:3D world u f v 1 X p x rx1 rx 2 rx 3 t x X Y CC P fK p y ry1 rRy 2 ry 3 tty W Z 1 1 rz1 rz2 rz 3 t z 1 Ground plane u p11 p12 v p 21 p 22 1 p 31 p 32p13p 23p 33 0 p14 0p 24 p 34 1 p13 Z p14 p13 X p Z pp 333334u limp Z p 24 p 23v lim 23 X p Z pp 333334 uZ u p13 v p 23 1 p 33
PracticeWimg3840 1224pixWccd0.0048WH21603840p x img p y img 1080pix 1920pix2222f fm 0.00153PredictVanishingPoint.m 0.0070 C 0.7520 -0.2738 -0.8496 0.0498 0.5731 R -0.3216 -0.8203 -0.4067 0.4180 -0.5299 0.6835
PracticeWimg3840 1224pixWccd0.0048WH21603840p x img p y img 1080pix 1920pix2222f fm 0.00153PredictVanishingPoint.m 0.0070 C 0.7520 -0.2738 P KR I3 -0.8496 0.0498 0.5731 R -0.3216 -0.8203 -0.4067 0.4180 -0.5299 0.6835 -0.2374 -0.9565 2.0138 1.2723 C 0.0578 -1.5763 0.2404 1.2508 0.0004 -0.0005 0.0007 0.0006
PracticeWimg3840 1224pixWccd0.0048WH21603840p x img p y img 1080pix 1920pix2222f fm 0.00153PredictVanishingPoint.m 0.0070 C 0.7520 -0.2738 P KR I3 -0.8496 0.0498 0.5731 R -0.3216 -0.8203 -0.4067 0.4180 -0.5299 0.6835 -0.2374 -0.9565 2.0138 1.2723 C 0.0578 -1.5763 0.2404 1.2508 0.0004 -0.0005 0.0007 0.0006 R [-0.8496 0.0498 0.5731-0.3216 -0.8203 -0.40670.4180 -0.5299 0.6835];u x -567.8239138.2813C [0.00700.7520-0.2738];u y 1.0e 03 *1.80502.9748P K * R * [eye(3) -C]u x P(1:2,1)/P(3,1)u y P(1:2,2)/P(3,2)u z P(1:2,3)/P(3,3)u z 1.0e 03 *2.94630.3517
PracticeWimg3840 1224pixWccd0.0048WH21603840p x img p y img 1080pix 1920pix2222f fm 0.00153 0.0070 C 0.7520 -0.2738 P KR I3 -0.8496 0.0498 0.5731 R -0.3216 -0.8203 -0.4067 0.4180 -0.5299 0.6835 -0.2374 -0.9565 2.0138 1.2723 C 0.0578 -1.5763 0.2404 1.2508 0.0004 -0.0005 0.0007 0.0006 R [-0.8496 0.0498 0.5731-0.3216 -0.8203 -0.40670.4180 -0.5299 0.6835];u x -567.8239138.2813C [0.00700.7520-0.2738];u y 1.0e 03 *1.80502.9748P K * R * [eye(3) -C]u x P(1:2,1)/P(3,1)u y P(1:2,2)/P(3,2)u z P(1:2,3)/P(3,3)u z 1.0e 03 *2.94630.3517
PracticeWimg3840 1224pixWccd0.0048WH21603840p x img p y img 1080pix 1920pix2222f fm 0.00153PredictVanishingPoint.m 0.0070 C 0.7520 -0.2738 P KR I3 -0.8496 0.0498 0.5731 R -0.3216 -0.8203 -0.4067 0.4180 -0.5299 0.6835 -0.2374 -0.9565 2.0138 1.2723 C 0.0578 -1.5763 0.2404 1.2508 0.0004 -0.0005 0.0007 0.0006 R [-0.8496 0.0498 0.5731-0.3216 -0.8203 -0.40670.4180 -0.5299 0.6835];u x -567.8239138.2813C [0.00700.7520-0.2738];u y 1.0e 03 *1.80502.9748P K * R * [eye(3) -C]u x P(1:2,1)/P(3,1)u y P(1:2,2)/P(3,2)u z P(1:2,3)/P(3,3)u z 1.0e 03 *2.94630.3517
Inverse of Camera Projection MatrixCamerau u v KX C 1 K -1uXCGround plane3D world
Inverse of Camera Projection MatrixCameraCu u v KX C 1 K R W u CC-1XGround plane3D world K C R W X C t K CR W ( X C )
Inverse of Camera Projection MatrixCameraCu u v KX C 1 K R W u CC-1XGround plane3D world K C R W X C t K CR W ( X C ) u -1X K CR W v C 1 3D ray direction 3D ray origin
CheiralityCameraCu u v KX C 1 K R W u CC-1XGround plane3D world K C R W X C t K CR W ( X C ) u -1X K CR W v C 1 3D ray direction 3D ray originwhere 0
Perspective CameraCameraPerspective camera model:WorldStrong perspectiveness u p11 p12X v P p21 p22 1 p 1 31 p 32p13p23p 33p14 X p24 1 p 34
Affine CameraCameraPerspective camera model:World u p11 p12X v P p21 p22 1 p 1 31 p 32p13p23p 33p14 X p24 1 p 34 Affine camera model:Strong perspectiveness u p11 p12 v P X pp A 1 21 22 1 00p13p230p14 X p23 1 1 u p11 p12 v p 21 p22p13p23p14 X p23 1
Affine CameraCameraCameraPerspective camera model:World u p11 p12X v P p21 p22 1 p 1 31 p 32p13p23p 33p14 X p24 1 p 34 Affine camera model:Weak perspectivenessStrong perspectiveness u p11 p12 v P X pp A 1 21 22 1 00p13p230p14 X p23 1 1 u p11 p12 v p 21 p22p13p23p14 X p23 1
Orthographic CameraAffine camera: u p11 p12 v p 21 p22p13p23p14 X p23 1 Orthographic camera:f 1 pX pY 0 u r11 r12 r13 t x X v r r 21 22 r23 t y 1
Camera Anatomy
Lens Radial Distortion
Radial Distortion ModelAssumption: Lens distortion is a function of distance from the principal point.udistortedp x ,p y
Radial Distortion ModelAssumption: Lens distortion is a function of distance from the principal point.uundistortedudistortedp x ,p yudistorted L( ρ ) uundistortedwhereρ udistortedL( ρ ) 1 k 1ρ2 k 2 ρ4
Radial Distortion Modeludistorted L( ρ ) uundistortedL( ρ ) 1 k 1ρ2 k 2 ρ4 udistorted L( ρ ) uundistortedwhereρ udistortedL( ρ ) 1 k 1ρ2 k 2 ρ4 k1 0k1 0
Radial Distortion ModelAssumption: Lens distortion is a function of distance from the principal point.[X, Y] meshgrid(1:(size(im,2)), 1:(size(im,1)));h size(X, 1); w size(X,2);X X(:);Y Y(:);pt [X'; Y'];pt bsxfun(@minus, pt, [px;py]);pt bsxfun(@rdivide, pt, [f;f]);r u sqrt(sum(pt. 2, 1));pt bsxfun(@times, pt, 1 k * r u. 2);pt bsxfun(@times, pt, [f;f]);pt bsxfun(@plus, pt, [px;py]);imUndistortion(:,:,1) reshape(interp2(im(:,:,1), pt(1,:), pt(2,:)), [h, w]);imUndistortion(:,:,2) reshape(interp2(im(:,:,2), pt(1,:), pt(2,:)), [h, w]);imUndistortion(:,:,3) reshape(interp2(im(:,:,3), pt(1,:), pt(2,:)), [h, w]);
f p r r r f p r r r r X Y rrZ X Y Z Ground plane Camera 3D world z Origin at world coordinate Camera Projection (Pure Rotation) X C 1 R W Coordinate transformation from world to camera: Camera World 3 C C W 3 ªº «» «» «» ¼ X X R X r r r r 1: world x axis seen from the camera coord. r1r2 r 3 r 2: world y axis seen from the camera .
CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1 2 The square root of a matrix (if unique), not elementwise
CONTENTS CONTENTS Notation and Nomenclature A Matrix Aij Matrix indexed for some purpose Ai Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not elementwise
A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not .
2-9V in unit & 2 AA in camera. Match polarities ( ) and ( ). Set camera date back, close camera lens and connect plug to camera port. 2 3 Secure camera, open camera shutter, and slide unit power switch to (ON) and back to (OFF), then push camera test button. Close camera Shutter, remove camera & load film, connect plug to camera, close cover. 4
CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1 2 The sq
Orthographic Projection. TOPICS Object representation Glass box concept Line convention Orthographic projection of point, line, plane, surfaceand object. Multiview projection. OBJECT REPRESENTATION Axonometric projection Multiview projection. MULTIVIEW PROJECTION Three principle dimensions
User Manual Replace a Pro 3 Camera battery You can leave the camera housing in place so the camera position stays the same. 1. Release the camera from the camera housing. Press the button on the charging port underneath the camera. The camera clicks as it disengages from the camera housing. 2. Pull the camera all the way out of the camera .
Korean and Chinese belong to different language families. In terms of their linguistic structures, they are extremely dissimilar. Beginning Korean: A Grammar Guide 2 Autumn 2004 Finally, hangeul is uniquely associated with the language, literature, and people of the Korean peninsula. No other community uses the hangeul system for graphically representing the sounds of their language. Given the .
