Mosaics Plenoptic Function and Light Field Rendering Topics in Image Based Modeling and Rendering CSE291 J00 Lecture 3 David Kriegman 2001 CS348 Fall 2001 Last Lecture Camera Models Pinhole perspective Affine Orthographic models Homogeneous coordinates Coordinate transforms Lenses Radiometry Irradiance Radiance BRDF CS348 Fall 2001 David Kriegman 2001 1 Pinhole cameras Abstract camera model box with a small hole in it Pinhole cameras work in practice David Kriegman 2001 CS348 Fall 2001 The equation of projection Mapping from 33 D world coordinates to 2 D image coordinates Cartesian coordinates We have by similar triangles that x y z f x z f y z f Ignore the third coordinate and get CS348 Fall 2001 x y x y z f f z z David Kriegman 2001 2 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 David Kriegman 2001 CS348 Fall 2001 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 CS348 Fall 2001 David Kriegman 2001 3 Orthographic projection Take Taylor series about 0 0 z0 a point on optical axis David Kriegman 2001 CS348 Fall 2001 Coordinate Changes Rigid Transformations B CS348 Fall 2001 P ABR AP BOA David Kriegman 2001 4 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 David Kriegman 2001 CS348 Fall 2001 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 CS348 Fall 2001 4x4 David Kriegman 2001 5 The reason for lenses David Kriegman 2001 CS348 Fall 2001 Thin Lens Image of Point P f F P Z O Z 1 1 1 z z f CS348 Fall 2001 David Kriegman 2001 6 Radiometry Solid Angle Irradiance Radiance BRDF Lambertian Phong BRDF David Kriegman 2001 CS348 Fall 2001 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 CS348 Fall 2001 dA cos r2 David Kriegman 2001 7 Radiance Power is energy per unit time d Radiance Power traveling at some point in a specified direction per unit area perpendicular to the direction of travel per unit solid angle x dA Symbol L x L Units watts per square meter per steradian w m2 sr1 P dA cos d David Kriegman 2001 CS348 Fall 2001 Irradiance How much light is arriving at a surface E x Units w m2 Incident power per unit area not foreshortened This is a function of incoming angle A surface experiencing radiance L x coming in from solid angle d experiences irradiance L x cos d L x Crucial property Total power arriving at the surface is given by adding irradiance over all incoming angles Total power is L x cos sin d d N x CS348 Fall 2001 x David Kriegman 2001 8 Radiance transfer What is the power received by a small area dA2 at distance r from a small emitting area dA1 From definition of radiance L P dA cos d From definition of solid angle d P CS348 Fall 2001 dA cos r2 L dA1dA2 cos 1 cos 2 r2 David Kriegman 2001 BRDF With assumptions in previous slide Bi directional Reflectance Distribution Function in in out out Ratio of incident irradiance to emitted radiance in in n Function of Incoming light direction in in out out Outgoing light direction out out x in in out out CS348 Fall 2001 Lo x out out Li x in in cos in d David Kriegman 2001 9 Surface Reflectance Models Common Models Arbitrary Reflectance Lambertian Phong Physics based Specular Blinn 1977 Cook Torrance 1982 Ward 1992 Diffuse Hanrahan Kreuger 1993 Generalized Lambertian Oren Nayar 1995 Thoroughly Pitted Surfaces Koenderink et al 1999 Phenomenological Non parametric model Anisotropic Non uniform over surface BRDF Measurement Dana et al 1999 Marschner Koenderink Van Doorn 1996 David Kriegman 2001 CS348 Fall 2001 Rough Specular Surface Phong Lobe CS348 Fall 2001 David Kriegman 2001 10 Announcements Mailing list cse291 j cs ucsd edu Has been setup with class list as of Sunday night If you re not on it and want to be added e g auditing and not on course list send the email msg to majordomo cs ucsd edu with body saying subscribe cse291 j my email something something Class presentations requests from Sameer Agarwal Jin Su Kim Satya Mallick Peter Schwer Diem Vu Cindy Wang Yang Yu David Kriegman 2001 CS348 Fall 2001 This lecture S Chen Quicktime VR an image based approach to virtual environment navigation SIGGRAPH pages 29 38 Los Angeles California August 1995 E H Adelson and J R Bergen The plenoptic function and the elements of early vision In M Landy and J A Movshon editors Computational Models of Visual Processing pages 3 20 MIT Press Cambridge MA 1991 S J Gortler R Grzeszczuk R Szeliski M F Cohen The Lumigraph SIGGRAPH pp 43 54 1996 M Levoy P Hanrahan Light Field Rendering SIGGRAPH 1996 CS348 Fall 2001 David Kriegman 2001 11 CS348 Fall 2001 Next few slides courtesy Paul Debevec SIGGRAPH 99 course notes David Kriegman 2001 CS348 Fall 2001 David Kriegman 2001 12 CS348 Fall 2001 David Kriegman 2001 CS348 Fall 2001 David Kriegman 2001 13 CS348 Fall 2001 David Kriegman 2001 Mosaics Quicktime VR S Chen Quicktime VR An image based approach to virtual environment navigation SIGGRAPH pages 29 38 Los Angeles California August 1995 CS348 Fall 2001 David Kriegman 2001 14 View Synthesis Without Motion Analysis Peri and Nayar 1997 Shum and Szeliski 1998 Quicktime VR Chen 1995 David Kriegman 2001 CS348 Fall 2001 Constructing Mosaic Cylindrical images Easy to acquire with camera tripod Any two planar perspective projections of a scene which share a common viewpoint are related by a two dimensional projective transform u 2 a11 v a 2 21 w2 a31 a12 a22 a32 a13 u1 a23 v1 a33 w1 Can be estimated from a minimum of four points correspondences in two images CS348 Fall 2001 David Kriegman 2001 15 Stitched panoramic image from photos CS348 Fall 2001 David Kriegman 2001
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