Robotics - Projective Geometry and Camera model. Marcello Restelli

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1 Robotics - Projective Geometr and Camera model Marcello Restelli marcello.restelli@polimi.it Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano Ma 2013 Inspired from Matteo Pirotta s slides Como 2013)

2 2/26 Outline 1 Camera Geometr 2

3 3/26 Outline 1 Camera Geometr 2

4 4/26 What is an image Image Two-dimensional brightness arra: I 3 two-dimensional arra: I R, I G, I B RGB: Red, Green, Blue others: YUV, HSV, HSL, Ideal: I : Ω R 2 R + Discrete: I : Ω N 2 R + e.g., Ω = [0, 639] [0, 479] N 2 e.g., Ω = [1, 1024] [1, 768] N 2 e.g., R + = [0, 255] N e.g., R + = [0, 1] R I(x, ) is the intensit I result of 3D 2D projection: flat we lose: angles, distances (lengths)

5 5/26 Camera Optical sstem Set of lenses to direct light change in the direction of propagation CCD sensor integrate energ both in time (exposure time) in space (pixel size)

6 6/26 Thin lenses model Thin lenses Mathematical model Optical axis (z) Focal plane π f ( z) Optical center o π f f Parameters o z Propert f distance o, π f Parallel ras converge π f Ras through o undeflected

7 7/26 Ras from scene Image from a scene point P P = (Z, Y ) Ra through o undeflected Ra parallel to z cross in ( f, 0) f Z Similarities P Blue triangles: h Y = r f f Green triangles: h Y = r Z Fresnel s Law 1 Z + 1 r = 1 f Note: Z r f h p r o Y z

8 8/26 The image plane Image plane π I Plane z at distance d Blur Circle If d r π I π f f a Z P image of P is a circle C o z Diameter of C: a(d r) φ(c) = r a is the aperture C p d r Focused image φ(c) < pixel size Depth of field : range [Z 1, Z 2] : φ(c) < pixel size set of values that satisfies Fresnel s law

9 Camera Geometr Depth of field - Example 1 9/26

10 Camera Geometr Depth of field - Example 2 The same scene - different aperture 10/26

11 11/26 Aperture controls depth of field Wh not make the aperture as small as possible? Less light gets through; Diffraction effects...

12 12/26 Aperture controls depth of field Changing the aperture size affects depth of field A smaller aperture increases the range in which the object is approximatel in focus; But small aperture reduces amount of light need to increase exposure.

13 13/26 Outline 1 Camera Geometr 2

14 14/26 Pinhole model - Definition Hpothesis Z a Z f r f Image of P O (I) p x z Pin-Hole Model -f image plane O X camera centre Y P Z principal axis l Po : line that join P and o Frontal Pin-Hole Model X p = π I l Po Notes p is the image of P i l Po l Po : interpretation line of p O f camera centre Y x p O (I) image plane P Z principal axis

15 15/26 Pinhole model - Geometr camera centre: centre of projection or optical centre (O); principal axis: line from the camera centre perpendicular to the image plane (z); principal point: intersection between image plane an principal axis (O (I ) ); principal plane: plane through the camera centre, parallel to the image plane. X O f camera centre Y p x O (I) image plane P Z principal axis

16 16/26 Pinhole model - Geometr (Cont d) Given [ ] T P (O) = X, Y, Z, 1 [ P (O) = X, Y, Z, 1 [ ] T p (I) = x,, 1 ] T Z = f, X = f X Z, Y = f Y Z x = X, = Y Projection (O) = f Y Z Z x = f X Z look at the triangles P (O) Y Note I O z (O) λp (O) projects on p (I) [ sx, sy, sz, 1] T projects on p (I), p (I) (I) f s 0

17 17/26 Pinhole model - Frontal Camera Model The pinhole camera model is inconvenience that the focal length f is negative. The coordinate sstem: We will use the pinhole model as an approximation; Put the camera centre (O) at the origin; Put the image plane in front of the camera centre; The camera looks up the positive z axis. Projection equations Compute intersection with IP of ra from P to O; Derived using similar triangles x = f X Z, = f Y Z Ignoring the final image coordinate: (X, Y, Z) T (f X Z, f Y Z )T

18 18/26 Pinhole model - Matrix Projection equations = f Y Z x = f X Z In matrix form x X f = 0 f Y w Z W = X f f 0 0 Y Z W Define f 0 0 K = 0 f 0 : intrinsic parameters p (I) = π P (O) π = [ K 0 ] : projection matrix

19 19/26 Distortions from Phsical Lenses Pinhole camera model assumes: Perfect pinhole camera lenses; Image x = I (p) R 2 be measured in infinite accurac; Principal point is at the center of the image. Phsical camera lenses give us: Distorted imaging projections; Finite resolutions dened b the sensing devices in digital cameras; Offset between image center and optical center. p I x I p x O z x

20 20/26 Pinhole model - Image coordinates - 1 Reference sstem on image Metric I: origin centered on z (O) π I I : origin centered top-left image [ ] T c (I ) = c x, c : position of I in I I metric I in pixel c (I ) in pixel Definition [ ] T (I) [ ] T (I ) 0, 0 c x, c : principal point Image of the optical center (o) or z (O) π I I x I x

21 21/26 Pinhole model - Image coordinates - 2 The origin of coordinates in image planes is not [ ] T (I ) at the principal point c x, c : I x (X, Y, Z) T (f X Z + cx, f Y Z + c )T c (I ) x X f 0 c x 0 Y = 0 f c 0 w Z W p (I ) = π P (O) I x

22 22/26 Pinhole model - Image coordinates - 3 (CCD Camera) The pinhole camera model assumes that image coordinate Euclidean (equal scales in both axial direction); In CCD there is the possibilit of non-square pixels; Image coordinates measured in pixel introduces unequal scale factors in each direction. If the number of pixels per unit distance coordinates are m x and m in the x and direction: m x 0 0 f 0 c x 0 π = 0 m 0 0 f c where (c x, c ) is the principal point in terms of pixel dimensions.

23 23/26 Pinhole model - Image coordinates - 4 Meters to pixels Consider I : origin on I, in pixel Scale meters to pixels p (I ) x p (I ) = s x p (I) x = s p (I) s x = 1 d x, d x: width of a pixel [m] s = 1 d, d : height of a pixel [m] s x = s : square pixel s x 0 0 p (I ) = 0 s 0 p (I) Translation 1 0 c x p (I ) = 0 1 c p (I ) I x c (I ) I I x x

24 24/26 Pinhole model - Intrinsic camera matrix Consider f p (I) = 0 f 0 0 P (O) s x 0 0 p (I ) = 0 s 0 p (I) c x p (I ) = 0 1 c p (I ) In one step s xf 0 c x 0 p (I ) = 0 s f c 0 P (O) p (I ) = πp (O) = K [ I 0 ] P (O) The intrinsic camera matrix or calibration matrix f x s c x K = 0 f c f x, f : focal lenght (in pixels) f /f x = s /s x = a: aspect ratio s: skew factor pixel not orthogonal usuall 0 in modern cameras c x, c : principal point (in pixel) usuall half image size due to misalignment of CCD

25 25/26 Camera rotation and translation Consider p (I ) = πp (O) Y point is expressed in terms of world coordinate frame: P (W ). W and O are related via a rotation and a translation. ) P (O) = T (O) OW P(W ) = R (O) OW (P (W ) O (W ) extrinsic camera matrix O X Z R, t X W Z Y One step p (I ) = π = [ ] [ ] R RC K 0 [ KR 0 1 ] KRC complete projection matrix P (W ) p (W ) p (O) O Note R is R (O) OW, C is O(O) OW (camera centre) i.e., the position and orientation of W in O W T (O) OW

26 26/26 Camera rotation and translation - 2 General pinhole camera [ ] p (I ) = KR KRC [ ] π = KR I C P (W ) C is the camera centre in world reference 9 (11) dof: 3 (5) for K, 3 for R and 3 for C parameters in K are called internal parameters in R and C are called external Other formulation do not make the camera center explicit [ ] [ ] p (I ) R t = K 0 P (W ) 0 1 [ ] π = KR Kt t = RC The intrinsic parameters f x s c x K = 0 f c The extrinsic parameters R = R (φ, θ, ρ) C = [c 1, c 2, c 3] T

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