Camera Projection Model
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1 amera Projection Model
2 3D Point Projection (Pixel Space) O ( u, v ) ccd ccd f Projection plane (, Y, Z ) ( u, v ) img Z img (0,0) w img ( u, v ) img img uimg f Z p Y v img f Z p x y Zu f p Z img img x Zv fy p Z uimg f px v img fk py Y Z y Pixel space Metric space h ccd + w ccd (0,0) h img ( p, p ) x y D sensor (mm) Image (pixel) amera intrinsic parameter : metric space to pixel space
3 2D Inverse Projection O f Projection plane ( u, v ) img img (, Y, Z ) 2D point == 3D ray uimg - K v img K Pixel space Metric space uimg f px v img fk py Y Z - uimg vimg Y Z 3D ray The 3D point must lie in the 3D ray passing through the origin and 2D image point.
4 oordinate Transform (Rotation) R W amera 3D world Origin at world coordinate Ground plane = ( x, y, z ) = ( x, y, z ) oordinate transformation from world to camera: RW r r r = r r r = rz rz2 rz3 x x2 x3 y y2 y3 RW Degree of freedom? SO(3) Orthogonal matrix T R W RW I3 RW, det World amera
5 oordinate Transform (Rotation) amera oordinate transformation from world to camera: R W 3D world r r r = r r r = r r r r x x2 x3 y ry2 Y y3 RW r z z Z 2 z3 Origin at world coordinate Ground plane = ( x, y, z ) = ( x, y, z ) RW SO(3) Orthogonal matrix Right hand rule r T R W RW I3 RW, det r Y amera World rz = r ry
6 oordinate Transform (Rotation) amera oordinate transformation from world to camera: R W 3D world r r r = r r r = r r r r x x2 x3 y ry2 Y y3 RW r z z Z 2 z3 Origin at world coordinate Ground plane RW SO(3) Orthogonal matrix Right hand rule T R W RW I3 RW, det r : camera x axis seen from the world coord. amera r Y r World rz = r ry
7 oordinate Transform (Rotation) amera oordinate transformation from world to camera: R W 3D world r r r = r r r = r r r r x x2 x3 y ry2 Y y3 RW r z z Z 2 z3 Origin at world coordinate r Z Ground plane r Y : camera y axis seen from the world coord. r : camera x axis seen from the world coord. amera RW SO(3) Orthogonal matrix Right hand rule r Y r T R W RW I3 RW, det World rz = r ry
8 oordinate Transform (Rotation) amera oordinate transformation from world to camera: R W 3D world r r r = r r r = rz rz2 rz3 x x2 x3 r y r r y2 2 y3 3 RW Origin at world coordinate Ground plane RW SO(3) Orthogonal matrix Right hand rule T R W RW I3 RW, det r r 2 amera World r : world x axis seen from the camera coord. r3 = r r2
9 oordinate Transform (Rotation) amera oordinate transformation from world to camera: R W 3D world r r r = r r r = rz rz2 rz3 x x2 x3 r y r r y2 2 y3 3 RW Origin at world coordinate r 3 Ground plane RW SO(3) Orthogonal matrix Right hand rule T R W RW I3 RW, det World r : world y axis seen from 2 the camera coord. amera r : world x axis seen from the camera coord. r 2 r3 = r r2 r
10 amera Projection (Pure Rotation) amera oordinate transformation from world to camera: R W 3D world r r r = r r r = rz rz2 rz3 x x2 x3 r y r r y2 2 y3 3 RW Origin at world coordinate amera projection of world point: r 3 Ground plane u f p x v f K py Y Z World r : world y axis seen from 2 the camera coord. amera r : world x axis seen from the camera coord. Y r z rz2 rz3 Z f p x rx rx2 rx3 f py ry ry2 ry3
11 amera Projection (Pure Rotation) amera oordinate transformation from world to camera: R W 3D world r r r = r r r = rz rz2 rz3 x x2 x3 r y r r y2 2 y3 3 RW Origin at world coordinate amera projection of world point: r 3 Ground plane u f p x v f K py Y Z World r : world y axis seen from 2 the camera coord. amera r : world x axis seen from the camera coord. Y r z rz2 rz3 Z f p x rx rx2 rx3 f K py ry ry2 ry3
12 Euclidean Transform=Rotation+Translation R W, t amera 3D world Origin at world coordinate Ground plane = ( x, y, z ) World
13 R W, Euclidean Transform=Rotation+Translation amera t 3D world Origin at world coordinate Ground plane = ( x, y, z ) = ( x, y, z ) amera World
14 R W, Euclidean Transform=Rotation+Translation amera t 3D world Origin at world coordinate Ground plane = ( x, y, z ) = ( x, y, z ) amera World
15 R W, Euclidean Transform=Rotation+Translation amera oordinate transformation from world to camera: t rx rx2 rx3 t x = = RW t ry ry2 ry3 t y z z2 z3 z 3D world r r r t Origin at world coordinate where t is the world orgin seen from camera. Ground plane = ( x, y, z ) = ( x, y, z ) t amera World
16 R W, Euclidean Transform=Rotation+Translation amera oordinate transformation from world to camera: t rx rx2 rx3 t x = = RW t ry ry2 ry3 t y z z2 z3 z 3D world r r r t Origin at world coordinate where t is the world orgin seen from camera. Ground plane = ( x, y, z ) = ( x, y, z ) t amera World
17 Geometric Interpretation R W, t amera 3D world oordinate transformation from world to camera: rx rx2 rx3 t x = = RW t ry ry2 ry3 t y rz rz2 rz3 t z Origin at world coordinate where t is the world orgin seen from camera. Ground plane = ( x, y, z ) = ( x, y, z ) Rotate and then, translate. t amera World
18 Geometric Interpretation R W, t amera 3D world oordinate transformation from world to camera: rx rx2 rx3 t x = = RW t ry ry2 ry3 t y rz rz2 rz3 t z Origin at world coordinate where t is the world orgin seen from camera. Ground plane = ( x, y, z ) = ( x, y, z ) Rotate and then, translate. cf) Translate and then, rotate. amera World rx rx2 r x3 - x = RW = ry ry2 ry3 - y where r r r - z z2 z3 z is the camera location seen from world.
19 Projective Line
20 Point-Point in Image A 2D line passing through two 2D points: au bv c 0 au2 bv 2 c 0 T T xl 0 xl 2 0 x 2 ( uv, ) x where x u v x u 2 v 2 2 l a b c ( u2, v2) x x T T 2 A 2x3 l null A or l = x x 2
21 Line-Line in Image Two 2D lines in an image intersect at a 2D point: au bv c 0 a2u b2v c2 0 lx T 0 lx T 2 0 l a b c x ( uv, ) (804,934) l a b c where u x v T l x 0 T l2 l a b c l a b c A 2x3 0 0 null A or x = ll2
22 Vanishing Point Parallel lines: l u u l2 u u2 4 3 Ground plane l l 2 u 3 u 4 u 2 u
23 Vanishing Point Parallel lines: l u u l2 u u2 4 3 l u u l22 u3 u4 2 4 Ground plane l 22 l2 u 4 u 3 l u 2 l 2 u
24 Vanishing Point Parallel lines: l u u l2 u u2 4 3 l u u l22 u3 u4 2 4 x 2 Ground plane Vanishing points: x l l 2 x l l x l 22 l2 u 4 u 3 l u 2 l 2 u
25 Vanishing Point Parallel lines: l u u l2 u u2 4 3 l u u l22 u3 u4 2 4 Vanishing points: v l l 2 v l l v2 Ground plane Vanishing line: l v l l v v 2 l 22 l2 u 3 l 2 u 4 u 2 u
26 Where was I (how high)? Taken from my hotel room (6 th floor) Taken from beach
27 Vanishing point Vanishing point Multiple vanishing point
28 Vanishing point Vanishing line for horizon Vanishing point Vanishing line: Horizon
29 Vanishing point Vanishing line for horizon Vanishing point What can vanishing line tell us about me? Horizon amera pitch angle (looking down) amera roll angle (tilted toward right) Keller Hall
30 Single Vanishing Point amera K - v r 3 Z r r 3D world r 2 Ground plane 0 0 v K r r2 r3 t W 0 v Kr 3 v
31 Single Vanishing Point amera K - v r 3 Z r r 3D world r 2 Ground plane 0 0 v K r r2 r3 t W 0 v Kr 3 r Kv because r3 is a unit vector. Kv Z vanishing point tells us about the surface normal of the ground plane v
32 Single Vanishing Point amera K - v r 3 Z r r 3D world r 2 Ground plane 0 0 v K r r2 r3 t W 0 v Kr 3 r Kv because r3 is a unit vector. Kv Z vanishing point tells us about the surface normal of the ground plane Rotation ambiguity v Ground plane
33 Two Vanishing Points amera r 3 r 3D world r v K r r2 r3 t W Kr v Y Ground plane v Y D world
34 v Y amera Two Vanishing Points Ground plane r 3 v r 3D world r 2 Y v K r r2 r3 t W Kr v Y K r r2 r3 t W Y Kr 2 3D world
35 v Y amera Two Vanishing Points Ground plane 0 0 v K r r2 r3 t W r 0 3 Kr r r 2 v 3D world Y v Y K r r2 r3 t W Y K v K v - -, r - 2 K v - r Kr K v Y Y r r r : Orthogonality constraint 3D world
36 Single View Metrology
37 ross Ratio a' b' c' d' d A a B b c Reference points D Target points A BD ac bd ac bd B AD bc ad bc ad ross ratio (perspective transformation invariant) Ground plane
38 v Z h p2 p3 p sh_f jp_h jp_f v h_prime = norm(p-p2); h_r = norm(p-p3); h_prime_inf = norm(p4-p2); h_inf = norm(p4-p3); H = H_R * h_prime * h_prime_inf / h_r / h_inf h R h h H =.6779 Ground truth:.7m H H R hh h h R p4 omputeheightfromrossratio.m
39 Homography
40 Perspective Transform (Homography) I I 2 v x h h2 h3 u x u x v y h2 h22 h23 u y Hu y 3 32 h h : General form of plane to plane linear mapping
41 Perspective Transform (Homography) u amera plane = Ground plane u Y 0 u K R t amera plane Ground plane 2 3 Y u Kr r r t 0 u K r r2 t Y u x uy H Y
42 Perspective Transform (Homography) I I 2 Invariant properties ross ratio oncurrency olinearity v x h h2 h3 u x u x v y h2 h22 h23 u y Hu y 3 32 h h Degree of freedom 8 (9 variables scale)
43 Hierarchy of Transformations Euclidean (3 dof) Similarity (4 dof) Affine (6 dof) Projective (8 dof) Length Length ratio Parallelism ross ratio Angle Angle Ratio of area oncurrency Area Ratio of length olinearity cosθ sinθ sinθ cosθ t x t y cosθ sinθ sinθ cosθ t x t y a a2 a3 a2 a22 a h h h h h h h h32
44 Image Transform via Plane Keller entrance left Keller entrance right
45 Image Transform via 3D Plane v Plane u Y = 0 u K R t u = K r r2 t Y v = Kr r2 t Y How are two image coordinates (u,v) related? = = r r t K u Y r r2 t K v v = Kr r tr r t K u v = Hu
46 Image Transform by Pure 3D Rotation amera
47 Image Transform by Pure 3D Rotation amera u = K v KR 2 = = K u = R K v v = - T KRK u H = KRK R = - - K HK
48 Homography omputation v x h h2 h3 u x y = v h2 h22 h23 u y h h h
49 Homography omputation v v x y h u + h u h x 2 y 3 h u + h u + h h h 3 x 32 y 33 u + h u h 2 x 22 y 23 u + h u + h 3 x 32 y 33 v x h h2 h3 u x y = v h2 h22 h23 u y h h h
50 Homography omputation v v x y h u + h u h x 2 y 3 h u + h u + h h h 3 x 32 y 33 u + h u h 2 x 22 y 23 u + h u + h 3 x 32 y 33 h u + h u h h u v + h u v + h v 0 h x 2 y 3 3 x x 32 y x 33 x u + h u h + h u v + h u v + h v 0 2 x 22 y 23 3 x y 32 y y 33 y v x h h2 h3 u x y = v h2 h22 h23 u y h h h Unknowns: h,, h33 Equations: 2 per correspondence
51 Homography omputation v v x y h u + h u h x 2 y 3 h u + h u + h h h 3 x 32 y 33 u + h u h 2 x 22 y 23 u + h u + h 3 x 32 y 33 v x h h2 h3 u x y = v h2 h22 h23 u y h h h h u + h u h h u v h u v h v 0 h x 2 y 3 3 x x 32 y x 33 x u + h u h h u v h u v h v 0 2 x 22 y 23 3 x y 32 y y 33 y h h h h 2 u x u y -u xv x -u yv x -v x 0 h 22 x y x y y y y 0 u u -u v -u v -v h23 h 3 h32 h33 2 3
52 Homography omputation v v x y h u + h u h x 2 y 3 h u + h u + h h h 3 x 32 y 33 u + h u h 2 x 22 y 23 u + h u + h 3 x 32 y 33 v x h h2 h3 u x y = v h2 h22 h23 u y h h h h u + h u h h u v h u v h v 0 h x 2 y 3 3 x x 32 y x 33 x u + h u h h u v h u v h v 0 2 x 22 y 23 3 x y 32 y y 33 y h h h h A x 0 2 u x u y -u xv x -u yv x -v x 0 h 22 x y x y y y y 0 u u -u v -u v -v h23 2x9 h 3 h32 h33 2 3
53 Homography omputation How many correspondences are needed? v x h h2 h3 u x y = v h2 h22 h23 u y h h h u u -u v -u v -v h h h h A x 0 2 x y x x y x x 0 h 22 x y x y y y y 0 u u -u v -u v -v h23 2x9 h 3 h32 h33 2 3
54 Homography omputation How many correspondences are needed? 4 v x h h2 h3 u x y = v h2 h22 h23 u y h h h u u -u v -u v -v h h h h A x 0 2 x y x x y x x 0 h 22 x y x y y y y 0 u u -u v -u v -v h23 8x9 h 3 h32 h33 x = V null A :end 2 3
55 Epipolar Geometry
56 Fundamental Matrix u v e bob e alice u v Properties of Fundamental Matrix Transpose: if F is for Pbob, Palice, then F T is for Palice, Pbob. Epipolar line: l Fu l u v F T v e bob e alice Epipole: Fe = bob 0 F T e 0 alice = rank(f)=2: degree of freedom 9 (3x3 matrix)- (scale)- (rank)=7 Bob Pbob K I P K R t alice Alice F K t RK -T - rank 2 matrix
57 Essential Matrix Essential Matrix: F = F(R,t) = K t RK K EK -T - -T - u v E = T K FK where alibrated fundamental matrix E t R = Bob Alice Property of essential matrix: 0 T T E = UDV = U V
58 HW #4 RANSA Fundamental matrix
59 Triangulation General camera pose P mike u v u = bob P Two 3D vectors are parallel. u bob P 0 u bob P 0 : Knowns : Unknowns Bob Pbob Baseline Palice Alice v w P P alice mike
60 PnP 3D-2D correspondence: u u P = t where R SO (3) u p + p Y + p Z p p + p Y + p Z p x x x x x 4 Y Z -u -uy -uz -u y y y y p2 Y Z -u -uy -uz -u p22 x x x x p23 m Ym Z m -um -umy -umz -um y y y y 24 p m Ym Z m -um -umy -umz -um p3 2mx2 u p + p Y + p Z p p + p Y + p Z p y p p p p p p p
61 Bundle Adjustment
62 Nonlinear System E Find x such that the following equation is satisfied: f x T T ( x) f( x) f ( x) x b How? Strategy: Given x, move x such that E( x x) E( x) x x x x Taylor expansion: f ( x) f( x x) f( x) x H.O.T. x T T f ( x) f ( x) f ( x) f ( x) x b x x x f.) x = T A A - A T b T T f ( x ) f ( x ) f ( ) x x b f ( x) x x x
63 Point Jacobian P bob u u E geom f ( ) u w v w 2 2 u v x y w w - T T ( x) ( x) ( x) where f f f x b f ( x) x x x Black: given variables Red: unknowns u v KR w - u v KR w u w w u u 2 f ( ) w w v u w v v w 2 w
64 u w w u u 2 f ( ) w w v u w v v w 2 w - T T ( x) ( x) ( x) f f f x b f ( x) x x x Damping factor (Levenberg-Marquardt algorithm) - T T ( x) ( x) ( x) f f f x I b f ( x) x x x
65 amera Jacobian p P bob u u = r p r 2 r parameters E geom 2 2 u v x y w w u v -KR w - T T ( p) ( p) ( p) where f f f p b f ( p) p p p Black: given variables Red: unknowns u v KR - w K22 - K u K 0 3 K3 v R w 3 3
66 amera Jacobian p P bob u u = p q 3+4 parameters E geom 2 2 u v x y w w u v -KR w where Black: given variables Red: unknowns u v KR - w K22 - K u K 0 3 K3 v R w 3 3 u u R v v q R q w w : hain rule Quaternion jacobian
67 amera & Point Jacobian Black: given variables Red: unknowns f( p ) ( ) j,i f p j,i J ij Jpij Jij pj i J Jp,bob 0 J 27,bob 02 7 J p,alice J,alice u v p p Bob Pbob Palice Alice
68 amera & Point Jacobian 2 f( p ) ( ) j,i f p j,i J ij Jpij Jij pj i Black: given variables Red: unknowns J Jp,bob 0 J 27,bob 02 7 J p,alice J,alice u v Jp2,bob J 23 2,bob 02 7 J p2,alice 02 3 J 2,alice p p Bob Pbob Palice Alice
69 Pt 2 J= am am 2 am 3 Pt Pt 2 Pt 3 Pt 4 Pt Pt 4 Pt 3 amera amera 2 amera 3 # of unknowns: 3x7+4x3 # of projections: 3x4
70 Pt 2 J= am am 2 am 3 Pt Pt 2 Pt 3 Pt 4 Pt Pt 4 Pt 3 amera amera 2 amera 3 # of unknowns: 3x7+4x3 # of projections: 9 (not all points are visible from cameras)
71 Pt 2 J= am am 2 am 3 Pt Pt 2 Pt 3 Pt 4 Pt Pt 4 Pt 3 amera amera 2 amera 3 # of unknowns: 3x7+4x3 # of projections: 9 (not all points are visible from cameras) T - T x J J J b f(x) size of JJ T : 24x24
72 Pt 2 J= am am 2 am 3 Pt Pt 2 Pt 3 Pt 4 Pt Pt 4 Pt 3 amera amera 2 amera 3 # of unknowns: 3x7+4x3 # of projections: 9 (not all points are visible from cameras) size of T - T x J J J b f(x) Main computational bottle neck JJ T : 24x24
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