Announcements. The equation of projection. Image Formation and Cameras

Transcription

1 Announcements Image ormation and Cameras Introduction to Computer Vision CSE 52 Lecture 4 Read Trucco & Verri: pp. 5-4 HW will be on web site tomorrow or Saturda. Irfanview: is a good windows utilit for manipulating images. Tr v for linu. inhole Camera: erspective projection Abstract camera model - bo with a small hole in it Geometric Aspects of erspective rojection oints project to points Lines project to lines Angles & distances (or ratios) are NT preserved under perspective Vanishing point Image plane orsth&once The equation of projection Euclidean -> Homogenous-> Euclidean In 2-D Euclidean -> Homogenous: (, ) -> λ (,,) (can just take λ =) Homogenous -> Euclidean: (,, ) -> (/, /) Cartesian coordinates: We have, b similar triangles, that (,, ) -> (f /, f /, -f) Ignore the third coordinate, and get In 3-D Euclidean -> Homogenous: (,, ) -> λ(,,,) (can just take λ =) Homogenous -> Euclidean: (,,, w) -> (/w, /w, /w)

2 Turn The camera matri into homogenous coordinates HC s for 3D point are (X,Y,Z,) HC s for point in image are (U,V,W) Affine Camera Model Take erspective projection equation, and perform Talor Series Epansion about (some point (,, ). Drop terms of higher order than linear. Resulting epression is called affine camera model. roperties ts. map to pts, lines map to lines arallel lines map to parallel lines (no vanishing point at infinit) Ratios of distance/angles preserved rthographic projection Start with affine camera model, and take Talor series about (,, o )= (,, ) a point on optical ais What if camera coordinate sstem differs from object coordinate sstem {c} {W} Depth () is lost Euclidean Coordinate Sstems Coordinate Changes: ure Translations No rotation (e.g., i A =i B etc) B = B A + A, B = A + B A 2

3 Rotation Matri Coordinate Changes: ure Rotations i A.i B j A.i B k A i B B A R = i A j B j A.j B k A.j B = i A k B j A.k B k A k B A i B T A j B T A k B T = [ B i B A j B A k A ] A rotation matri R has the following properties: Coordinate Changes: Rigid Transformations Its inverse is equal to its transpose R - = R T Its determinant is equal to : det(r)=. r equivalentl: Rows (or columns) of R form a right-handed orthonormal coordinate sstem. Rotation: Homogenous Coordinates About ais ' ' ' = sin θ -sin θ rot(,θ) Note: coordinate doesn t change after rotation θ p' p About ais: About ais: ' ' ' ' ' ' = = Rotation -sin θ sin θ -sin θ sin θ 3

4 Roll-itch-Yaw Rotation Rotate(k, θ) Rotation b angle θ about (k, k, k), a unit vector (Rodrigues ormula) θ k Euler Angles ' ' ' = kk(-c)+c kk(-c)+ks kk(-c)-ks kk(-c)-ks kk(-c)+c kk(-c)-ks kk(-c)+ks kk(-c)-ks kk(-c)+c where c = & s = sin θ Homogeneous Representation of Rigid Transformations Transformation represented b 4 b 4 Matri Block Matri Multiplication Given A = A A 2 B = B B 2 A 2 A 22 B 2 B 22 What is AB? Camera parameters Issue camera ma not be at the origin, looking down the -ais etrinsic parameters (Rigid Transformation) one unit in camera coordinates ma not be the same as one unit in world coordinates intrinsic parameters - focal length, principal point, aspect ratio, angle between aes, etc. X U Transformation Transformation V = representing representing Y Z W intrinsic parameters etrinsic parameters T Camera Calibration What about light?, estimate intrinsic and etrinsic camera parameters See tetbook for how to do it. 4

5 Getting more light Bigger Aperture Limits for pinhole cameras inhole Camera Images with Variable Aperture The reason for lenses 2 mm mm.6 mm.35 mm.5 mm.7 mm Thin Lens Thin Lens: Center ptical ais Rotationall smmetric about optical ais. Spherical interfaces. All ras that enter lens along line pointing at emerge in same direction. 5

6 Thin Lens: ocus Thin Lens: Image of oint Incoming light ras parallel to the optical ais pass through the focus, All ras passing through lens and starting at converge upon Thin Lens: Image of oint Thin Lens: Image lane Z f Z Q Q Image lane A price: Whereas the image of is in focus, the image of Q isn t. Thin Lens: Aperture ield of View Image lane Smaller Aperture -> Less Blur inhole -> No Blur Image lane f ield of View 6

7 Deviations from the lens model Deviations from this ideal are aberrations Two tpes. geometrical 2. chromatic spherical aberration astigmatism distortion coma Aberrations are reduced b combining lenses Spherical aberration Ras parallel to the ais do not converge uter portions of the lens ield smaller focal lenghts Compound lenses Distortion magnification/focal length different for different angles of inclination Chromatic aberration Inde of refraction of lens depends on wavelength of light pincushion (tele-photo) barrel (wide-angle) Can be corrected! (if parameters are know) Chromatic aberration Spatial Non-Uniformit Ras of different wavelengths focused in different planes Cannot be removed completel Sometimes achromatiation is achieved for more than 2 wavelengths camera Iris Litvinov & Schechner, radiometric nonidealities 7

8 Vignetting nl part of the light reaches the sensor eripher of the image is dimmer 8