Background Camera models Transforms Radiometry Topics in Image Based Modeling and Rendering CSE291 J00 Lecture 2 Kriegman 2003 CSE 291 J00 Winter 03 Outline Camera Models Pinhole perspective Affine Orthographic models Homogeneous coordinates Coordinate transforms Lenses Radiometry Irradiance Radiance BRDF CSE 291 J00 Winter 03 Kriegman 2003 1 Effect of Lighting Monet Kriegman 2003 CSE 291 J00 Winter 03 Change of Viewpoint Monet Haystack at Chailly at sunrise 1865 CSE 291 J00 Winter 03 Kriegman 2003 2 Pinhole cameras Abstract camera model box with a small hole in it Pinhole cameras work in practice Kriegman 2003 CSE 291 J00 Winter 03 The equation of projection Cartesian coordinates We have by similar triangles that x y z f x z f y z f Ignore the third coordinate and get CSE 291 J00 Winter 03 x y x y z f f z z Kriegman 2003 3 Homogenous coordinates Add an extra coordinate and Basic notion use an equivalence relation Possible to represent for 2D points at infinity equivalence relation Where parallel lines k X Y Z is the same as intersect X Y Z Where parallel planes for 3D intersect equivalence relation Possible to write the k X Y Z T is the same action of a perspective as X Y Z T camera as a matrix CSE 291 J00 Winter 03 Kriegman 2003 Euclidean Homogenous Euclidean In 2 D Euclidean Homogenous x y k x y 1 Homogenous Euclidean x y z x z y z In 3 D Euclidean Homogenous x y z k x y z 1 Homogenous Euclidean x y z w x w y w z w CSE 291 J00 Winter 03 Kriegman 2003 4 The camera matrix Turn previous expression into Homogenous Coordinates HC s for 3D point are X Y Z T HC s for point in image are U V W U 1 0 V 0 1 W 0 0 0 0 1 f X 0 Y 0 Z 0 T Kriegman 2003 CSE 291 J00 Winter 03 Affine Camera Model Take Perspective projection equation and perform Taylor Series Expansion about some point x0 y0 z0 Drop terms of higher order than linear Resulting expression is affine camera model CSE 291 J00 Winter 03 Kriegman 2003 5 Orthographic projection Take Taylor series about 0 0 z0 a point on optical axis Kriegman 2003 CSE 291 J00 Winter 03 The projection matrix for orthographic projection X U 1 0 0 0 Y V 0 1 0 0 Z W 0 0 0 1 T CSE 291 J00 Winter 03 Kriegman 2003 6 Euclidean Coordinate Systems x OP i y OP j z OP k x OP xi yj zk P y z CSE 291 J00 Winter 03 Kriegman 2003 Coordinate Changes Pure Translations OBP O BOA OAP BP AP BOA CSE 291 J00 Winter 03 Kriegman 2003 7 Rotation Matrix i A i B B A R i A j B i A k B j A i B jA jB j A k B CSE 291 J00 Winter 03 k A i B k A j B k A k B A iTB A jTB A k TB B iA B jA B kA Kriegman 2003 A rotation matrix is characterized by the following properties Its inverse is equal to its transpose and its determinant is equal to 1 Or equivalently Its rows or columns form a right handed orthonormal coordinate system CSE 291 J00 Winter 03 Kriegman 2003 8 Coordinate Changes Pure Rotations OP i A B jA A x k A A y i B Az jB B x k B B y B z P ABR AP Kriegman 2003 CSE 291 J00 Winter 03 Coordinate Changes Rigid Transformations B CSE 291 J00 Winter 03 P ABR AP BOA Kriegman 2003 9 Block Matrix Multiplication A A 11 A21 A12 A22 B B 11 B21 B12 B22 What is AB A B A12 B21 AB 11 11 A21 B11 A22 B21 A11 B12 A12 B22 A21 B12 A22 B22 Homogeneous Representation of Rigid Transformations B P AB R T 1 0 B OA AP AB R AP BOA B A P AT 1 1 1 1 Kriegman 2003 CSE 291 J00 Winter 03 Camera parameters Issue camera may not be at the origin looking down the z axis extrinsic parameters one unit in camera coordinates may not be the same as one unit in world coordinates intrinsic parameters focal length principal point aspect ratio angle between axes etc U V W X Transformation 1 0 0 0 Transformation Y representing 0 1 0 0 representing Z intrinsic parameters 0 0 1 0 extrinsic parameters T 3x3 CSE 291 J00 Winter 03 4x4 Kriegman 2003 10 The reason for lenses CSE 291 J00 Winter 03 Kriegman 2003 Pinhole too big many directions are averaged blurring the image Pinhole too smalldiffraction effects blur the image Generally pinhole cameras are dark because a very small set of rays from a particular point hits the screen CSE 291 J00 Winter 03 Kriegman 2003 11 Thin Lens O Optical axis Rotationally symmetric about optical axis Spherical interfaces Kriegman 2003 CSE 291 J00 Winter 03 Thin Lens Center F O All rays that enter lens along line pointing at O emerge in same direction CSE 291 J00 Winter 03 Kriegman 2003 12 Thin Lens Focus F O Parallel lines pass through the focus F Kriegman 2003 CSE 291 J00 Winter 03 Thin Lens Image of Point P F O P All rays passing through lens and starting at P converge upon P CSE 291 J00 Winter 03 Kriegman 2003 13 Thin Lens Image of Point P f F P O Z Z 1 1 1 z z f Kriegman 2003 CSE 291 J00 Winter 03 Thin Lens Image Plane Q P F O P Image Plane Q A price Whereas the image of P is in focus the image of Q isn t CSE 291 J00 Winter 03 Kriegman 2003 14 Thin Lens Aperture P O P Image Plane Smaller Aperture Less Blur Pinhole No Blur Kriegman 2003 CSE 291 J00 Winter 03 Field of View O Field of View Image Plane CSE 291 J00 Winter 03 Kriegman 2003 15 Earliest Surviving Photograph First photograph on record la table service by Nicephore Niepce in 1822 Note First photograph by Niepce was in 1816 CSE 291 J00 Winter 03 Kriegman 2003 Radiometry Solid Angle Irradiance Radiance BRDF Lambertian Phong BRDF CSE 291 J00 Winter 03 Kriegman 2003 16 Solid Angle By analogy with angle in radians the solid angle subtended by a region at a point is the area projected on a unit sphere centered at that point The solid angle subtended by a patch area dA is given by d dAcos r2 Kriegman 2003 CSE 291 J00 Winter 03 Radiance Power is energy per unit time Radiance Power traveling at some point in a specified direction per unit area perpendicular to the direction of travel per unit solid angle Symbol L x x Units watts per square meter per steradian wm 2 sr 1 CSE 291 J00 Winter 03 Kriegman 2003 17 Irradiance How much light is arriving at a surface Sensible unit is Irradiance Incident …
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