Jorge Salvador Marques, geometric camera model

Transcription

1 geometric camera model

2 image acquisition 6 years ago today curc of te Holy Spirit, Brunellesci, 436 te camera projects 3D points into an image plane

3 geometric primitives and transformations

4 How do we see te world? Let s design a camera Idea : put a piece of film in front of an object Do we get a reasonable image? Slide by Steve Seitz

5 Pinole camera Add a barrier to block off most of te rays is reduces blurring e opening known as te aperture How does tis transform te image? Slide by Steve Seitz

6 Pinole camera model Pinole model: Captures pencil of rays all rays troug a single point e point is called Center of Projection (COP) e image is formed on te Image Plane Effective focal lengt f is distance from COP to Image Plane Slide by Steve Seitz

7 Dimensionality Reduction Macine (3D to 2D) 3D world 2D image Point of observation Wat ave we lost? Angles Distances (lengts) Figures Stepen E. Palmer, 22

8 Funny tings appen

9 Parallel lines aren t Figure by David Forsyt

10 Distances can t be trusted... Figure by David Forsyt

11 but umans adopt! Müller-Lyer Illusion We don t make measurements in te image plane ttp://

12 How to Build a Camera? Camera Obscura Camera Obscura, Gemma Frisius, 558 e first camera Known to Aristotle Dept of te room is te effective focal lengt

13 Home-made pinole camera Wy so blurry? ttp://

14 Srinking te aperture Less ligt gets troug Wy not make te aperture as small as possible? Less ligt gets troug Diffraction effects Slide by Steve Seitz

15 Srinking te aperture

16 e reason for lenses Slide by Steve Seitz

17 Image Formation using Lenses Ideal Lens: Same projection as pinole but gaters more ligt! i o P P f Lens Formula: i + o f f is te focal lengt of te lens determines te lens s ability to bend (refract) ligt f different from te effective focal lengt f discussed before! Slide by Sree Nayar

18 Modeling Projections

19 Modeling gprojection e coordinate system We will use te pin-ole model as an approimation Put te optical center (Center Of Projection) at te origin Put te image plane (Projection Plane) in front of te COP Wy? e camera looks down te negative z ais we need tis if we want rigt-anded-coordinates d di t Slide by Steve Seitz

20 Modeling gprojection Projection equations Compute intersection wit PP of ray from (,y,z) to COP Derived using similar triangles (on board) We get te projection by trowing out te last coordinate: Slide by Steve Seitz

21 Homogeneous coordinates Is tis a linear transformation? no division by z is nonlinear rick: add one more coordinate: omogeneous image coordinates omogeneous scene coordinates Converting from omogeneous coordinates Slide by Steve Seitz

22 Perspective Projection Projection is a matri multiply using omogeneous coordinates: divide by tird coordinate is is known as perspective projection e matri is te projection matri Can also formulate as a 44 divide by fourt coordinate Slide by Steve Seitz

23 Ortograpic Projection Special case of perspective projection Distance from te COP to te PP is infinite Image World Also called parallel projection Wat s te projection matri? Slide by Steve Seitz

24 geometric primitives: 2D points y Cartesian coordinates (, y ) IR y 2 omogeneous coordinates λ (, y,) λ y IR 3 \ {} λ te same 2D point can be represented by several (infinite) omogeneous vectors. Points of te form (a,b,) are called points at infinity since tey do not correspond to (finite) 2D points.

25 geometric primitives: 2D lines 2D line l a + by + c y d n normalization: l ( nˆ, d) nˆ - normal vector ( nˆ ) d - distance to te origin line at infinity l (,, ) (includes all points at infinity) intersection of two lines line joining two points l l 2 l 2

26 geometric primitives: 3D points Cartesian coordinates (, y, z) y IR z 3 z omogeneous coordinates y λ (, y, z,) λ IR 4 \ {} λ z y te same 2D point can be represented by several (infinite) omogeneous vectors. Points of te form (a,b,c,) are called points at infinity since tey do not correspond to (finite) 2D points.

27 geometric primitives: 3D lines and planes plane l a + by + cy + d z nˆ l ( n ˆ, d ) ( nˆ - normal vector ( nˆ ) d - distance to te origin y z line αp + ( α ) q p, q - two points p q α p + β q y

28 geometric transformations in te plane translation rotation +translation affine projective invariance lines paralelism lengt orientation Y Y Y Y +R Y Y Y N A Y Y N N P Y N N N

29 transformation of points in te plane translation ' + t ' I t rotation +translation ' R + t ' R t R cosθ sinθ - sinθ cosθ affine ' A + t ' A t a a A a a projective ' y + y + + y + y' y + 22 ' H

30 transformation of lines Previous transformations can be written as ' H can we use tis equation to transform lines? l I I H ' l ' ' tis is te equation of a line in te transformed space wit l ' H I

31 transformation of points in 3d space translation ' + t ' I t rotation +translation R t ' R + t ' R R RR I affine ' A + t ' A t a a A a a projective ' y' y' y + y + y + y + y + y ' H

32 camera model

33 camera model arbitrary position internal parameters: focal lengt, scaa factors, principal point O

34 pin-ole model P f y p Y O Z Inverted image point in space: [ Y Z] [ y] point in te image: perspective projection f Z y f Y Z

35 Perspective projection wit frontal plane non inverted image point in space: [ Y Z] [ y] point in te image: perspective projection f y Z f Y Z

36 omogeneous coordinates Y α y, arbitrary (, β α β α β Z ) All tese vectors represent te same point in space

37 Perspective projection (ideal case) p p j ( ) Y Z Y y Z [ ] I Π Π λ Π λ

38 etrinsic parameters aamera a in arbitrary ab aypositiono Z R + t Cartesian coordinates Y Z world Y g omogeneous coordinates g R t λ Π g λ λ [ R t ] λ R, são parâmetros etrínsecos da câmara

39 Intrinsic parameters o y ' 2 ' 2 o internal model: conversion from metric coordinates to piels ' y' fs fs y + o y + o y (,y ) in piels ' fs o y ' fsy oy y ' K K upper triangular matri

40 comment Matri K is usually considered as a upper triangular matriz witout additional constraints is is equivalent to assuming tat te angle θ between te two coordinate ais can be sligtly different from 9º. K α α cotθ β sinθ c c2 ' 2 θ '

41 Full perspective p model Camera model: Π Π K[ R ] Π is a 3 4 matri denoted as camera matri. te camera model is linear in omogeneous coordinates!

42 Camera model in Cartesian equations q Cartesian coordinates Π 2 Y π π π Π Z, 2, π π π π graus de liberdade Matri P é specified apart from a scale factor!

43 Projection of straigt lines general case special cases

44 vanising point gp α + y Z Y Z Y z Z Z Π + Π α Wen a goes to infinity Π π π 2 Π y π π e vanising point does not depend on. Eac set of parallel lines as its on valising point. gp p p gp

45 Optical center plane 3 C plane plane 2 optical center ΠC

46 optical ray How to obtain te optical ray projected on? C is line is defined by 2 pontos e.g., te optical center C and Π ( ΠΠ )

47 Projective transformation from te plane Z C + p Y + p 2 4 p3 + p32y + p34 Z p2 + p22y + p y 24 p3 + p32y + p34 p em coordenadas omogéneas H ' ' ' [ Y ] 2 3 H ij p ij, i, 2 3 j p 4 j e converse i also true ' H'

48 projective transformation from a plane C in omogeneous coordinates ' H H

49 Camera calibration ( i, i ) Calibration involves te estimation of te intrinsic and etrinsic parameters K, R, t from eperimental data. data: we assume tat we know several pairs of corresponding points in 3D and in te image plane 3 i R, i {(, ), i,..., n} R i i 2

50 linear metod P j ti dl Projective model 3 π π ( 3 ) π π y π π π ) ( ) ( yπ π Há um par de equações lineares que relacionam e. Conecendo n pares ( i, i ) obtém-se π M y M M M M 2 π π π Mπ n n n n n n y M M M M 3 2 π π π

51 Linear metod II Minimize te norm of te residual r Mπ π 2 Solução: π é o vector próprio (unitário) associado ao menor valor próprio de M M.