Announcements. Equation of Perspective Projection. Image Formation and Cameras

Transcription

1 Announcements Image ormation and Cameras Introduction to Computer Vision CSE 52 Lecture 4 Read Trucco & Verri: pp Irfanview: is a good Windows utilit for manipulating images. Tr v for linu. Assignment : due toda 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 Equation of erspective rojection Euclidean -> Homogenous-> Euclidean Cartesian coordinates: We have, b similar triangles, that (,, ) -> (f /, f /, f ) Establishing an image plane coordinate sstem at C aligned with i and j, we get In 2-D Euclidean -> Homogenous: (, ) -> λ (,,) (can just take λ ) Homogenous -> Euclidean: (,, ) -> (/, /) In 3-D Euclidean -> Homogenous: (,, ) -> λ(,,,) (can just take λ ) Homogenous -> Euclidean: (,,, w) -> (/w, /w, /w) (,,) ( f, f )

2 Turn (,,) ( f, f ) 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 oints map to points 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 u f v f Depth () is lost X U f / V f / Y Z W Three projection models erspective all depths Affine points near point of Talor series epansion. E.g., when depth and sie variation is small, and camera is far from point relative to sie. rthographic when object is near optical ais, and depth variation of object is small compared to distance to the object. Trombone hlen&glus&clientmvgoogle&vyd9gau6y&rles&feature related Trombone Stabalied hlen&glus&clientmvgoogle&vyd9gau6y&rles&feature related What if camera coordinate sstem differs from object coordinate sstem {c} {W} 2

3 Euclidean Coordinate Sstems Coordinate Changes: ure Translations No rotation (e.g., i A i B etc) B B A + A, B A + B A A convenient notation Coordinate Changes: ure Rotations oints: A Leading superscript indicates the coordinate sstem that the coordinates are with respect to Subscript an identifier Rotation Matrices Lower left (Going from this sstem) Upper left (Going to this sstem) To add vectors, coordinate sstems (leading superscript) must agree To rotate a vector, points coordinate sstem must agree with lower left of rotation matri [ i A k A ] B B B A A A [ i B j B k B ] A A A [ i B j B k B ] [ i A k A ] B A B R A B B B [ i B j B k B ] T [ i A k A ] A A A CS252A, all 2 Computer Vision I Rotation Matri Coordinate Changes: Rigid Transformations Rotation + Translation i A.i B.i B k A i B B A R i A j B.j B k A.j B i A k B.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 ] 3

4 More about rotations matrices A rotation matri R has the following properties: Its inverse is equal to its transpose R - R T or R T R I About ais Rotation θ p' p Its determinant is equal to : det(r). r equivalentl: Rows (or columns) of R form a right-handed orthonormal coordinate sstem. Even though a rotation matri is 33 with nine numbers, it onl has three degrees of freedom can be parameteried with three numbers. There are man parameteriation. ' ' sin θ -sin θ rot(,θ) Note: coordinate doesn t change after rotation Rotation Roll-itch-Yaw About ais: ' ' sin θ -sin θ About ais: ' ' -sin θ sin θ Euler Angles Rotation Rotation b angle θ about (k, k, k), a unit vector (Rodrigues ormula) ' kk(-c)+c kk(-c)+ks kk(-c)-ks kk(-c)-ks kk(-c)+c k(-c)-ks where c & s sin θ Rotate(k, θ) θ k kk(-c)+ks kk(-c)-ks kk(-c)+c 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? 4

5 What if camera coordinate sstem differs from object coordinate sstem Intrinsic parameters {c} v u {W} R c c wt wc w T 33 homogenous matri ocal length: rincipal oint: C Units (e.g. piels) rientation and position of image coordinate sstem iel Aspect ratio Camera parameters Etrinsic arameters: Since camera ma not be at the origin, there is a rigid transformation between the world coordinates and the camera coordinates Intrinsic parameters: Since scene units (e.g., cm) differ image units (e.g., piels) and coordinate sstem ma not be centered in image, we capture that with a 33 transformation comprised of focal length, principal point, piel aspect ratio, angle between aes, etc. X U Transformation Rigid Transformation V represented b represented b Y Z W intrinsic parameters etrinsic parameters T Camera Calibration, estimate intrinsic and etrinsic camera parameters See Tet book for how to do it. Camera Calibration Toolbo for Matlab (Bouguet) Getting more light Bigger Aperture What about light? 5

6 Limits for pinhole cameras inhole Camera Images with Variable Aperture 2 mm mm.6 mm.35 mm.5 mm.7 mm The reason for lenses Thin Lens ptical ais Rotationall smmetric about optical ais. Spherical interfaces. Thin Lens: Center Thin Lens: ocus All ras that enter lens along line pointing at emerge in same direction. Incoming light ras parallel to the optical ais pass through the focus, 6

7 Thin Lens: Image of oint Thin Lens: Image of oint Z f Z All ras passing through lens and starting at converge upon Q Thin Lens: Image lane Circle of Confusion Depth of field Thin Lens: Aperture Q Image lane A price: Whereas the image of is in focus, the image of Q isn t. Image lane Smaller Aperture -> Less Blur inhole -> No Blur Light ield Camera Ltro.com ost-acquisition refocussing 7