Remember: The equation of projection. Imaging Geometry 1. Basic Geometric Coordinate Transforms. C306 Martin Jagersand

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1 Imaging Geometr 1. Basic Geometric Coordinate Transorms emember: The equation o rojection Cartesian coordinates: (,, z) ( z, z ) C36 Martin Jagersand How do we develo a consistent mathematical ramework or rojection calculations? Size illusions Distant objects are smaller 1

Parallel lines meet Vanishing oints common to draw ilm lane in ront o the ocal oint each set o arallel lines (=direction) meets at a dierent oint The vanishing oint or this direction How would ou

2 Parallel lines meet Vanishing oints common to draw ilm lane in ront o the ocal oint each set o arallel lines (=direction) meets at a dierent oint The vanishing oint or this direction How would ou show this? Sets o arallel lines on the same lane lead to collinear vanishing oints. The line is called the horizon or that lane Geometric roerties o rojection Polhedra roject to olgons Points go to oints Lines go to lines Planes go to whole image Polgons go to olgons Degenerate cases line through ocal oint to oint lane through ocal oint to line (because lines roject to lines) 2

Junctions are constrained This leads to a rocess called line labelling one looks or consistent sets o labels, bounding olhedra disadv - can t get the lines and junctions to label rom real images Back

3 Junctions are constrained This leads to a rocess called line labelling one looks or consistent sets o labels, bounding olhedra disadv - can t get the lines and junctions to label rom real images Back to rojection Cartesian coordinates: (,, z) ( z, z ) We will develo a ramework to eress rojection as =PX, where is 2D image rojection, P a rojection matri and X is 3D world oint. Basic geometric transormations: Translation A translation is a straight line movement o an object rom one ostion to another. A oint (,) is transormed to the oint (, ) b adding the translation distances T and T : = + T = + T z = z + T z Coordinate rotation Eamle: Around -ais Z = " # = z P 2 4 X 3 " cos sin 1 5 # à sin cos z = X 3

4 = Euler angles Note: Successive rotations. Order matters. " # cos àsin sin cos 1 = z cos sin 1 5 àsin cos cos àsin sin cos 3 5 otation and translation Translation t in new o coordinates 2 3 cos sin = t Z à sin cos P X X Basic transormations Scaling Basic transormations Scaling A scaling transormation alters the scale o an object. Suose a oint (,) is transormed to the oint (,) b a scaling with scaling actors S and S, then: = S = S z = z S A uniorm scaling is roduced i S = S = S z. The revious scaling transormation leaves the origin unaltered. I the oint (, ) is to be the ied oint, the transormation is: = + ( - ) S = + ( - ) S This can be rearranged to give: = S + (1 - S ) = S + (1 - S ) 4

5 Aine Geometric Transorms A Simle 2-D Eamle In general, a oint in n-d sace transorms b P = rotate(oint) + translate(oint) In 2-D sace, this can be written as a matri equation: Cos( θ ) Sin( θ ) t = + Sin( θ ) Cos( θ ) t In 3-D sace (or n-d), this can generalized as a matri equation: = + T or = t ( T) = (,1) = (1,) Suose we rotate the coordinate sstem through 45 degrees (note that this is measured relative to the rotated sstem! Cos( π / 4) Sin( π / 4) 1 = Sin( π / 4) Cos( π / 4) Cos( π / 4) = Sin( π / 4) Cos( π / 4) Sin( π / 4) = Sin( π / 4) Cos( π / 4) 1 Sin( π / 4) = Cos( π / 4) Matri reresentation and Homogeneous coordinates Matri reresentation and Homogeneous coordinates Oten need to combine transormations to build the total transormation. I all transormations could be reresented as matri oerations then the combination o transormations siml involves the multilication o the resective matrices As translations do not have a 2 2 matri reresentation, we introduce homogeneous coordinates to allow a 3 3 matri reresentation. The Homogeneous coordinate corresonding to the oint (,,z) is the trile ( h, h, z h, w) where: h = w h = w z h = wz We can (initiall) set w = 1. Suose a oint P = (,,z,1) in the homogeneous coordinate sstem is maed to a oint P = (,,z,1) b a transormations, then the transormation can be eressed in matri orm. 5

6 Matri reresentation and Homogeneous coordinates For the basic transormations we have: Translation T P = T 4 z 5 = T z z w 1 w Scaling s P 6 = 7 s 4 z 5 = s z z w 1 w Geometric Transorms Using the idea o homogeneous transorms, we can write: = T 1 and T both require 3 arameters. " # 2 32 cos àsin cos sin 1 = sin cos cos àsin 1 àsin cos sin cos 3 5 Geometric Transorms otation about a Seciied Ais I we comute the matri inverse, we ind that T = 1 It is useul to be able to rotate about an ais in 3D sace This is achieved b comosing 7 elementar transormations (net slide) and T both require 3 arameters. These corresond to the 6 etrinsic arameters needed or camera calibration 6

7 otation through θ about Seciied Ais Comarison: z P2 P1 z initial osition rotate through requ d angle, θ z z translate P1 to origin rotate ais z to orig orientation z rotate so that P2 lies on z-ais (2 rotations) P2 P1 translate back Homogeneous coordinates otations and translations are reresented in a uniorm wa Successive transorms are comosed using matri roducts: = Pn*..*P2*P1* Aine coordinates Non-uniorm reresentations: = A + b Diicult to kee track o searate elements 2. Camera models and rojections Using geometr and homogeneous transorms to describe: Persective rojection Weak ersective rojection Orthograhic rojection The equation o rojection Cartesian coordinates: We have, b similar triangles, that (,, z) -> ( /z, /z, -) Ignore the third coordinate, and get (,, z) ( z, z ) 7

The camera matri Homogenous coordinates or 3D our coordinates or 3D oint equivalence relation (X,Y,Z,T) is the same as (k X, k Y, k Z,k T) Turn revious eression into HC s HC s or 3D oint are

8 The camera matri Homogenous coordinates or 3D our coordinates or 3D oint equivalence relation (X,Y,Z,T) is the same as (k X, k Y, k Z,k T) Turn revious eression into HC s HC s or 3D oint are (X,Y,Z,T) HC s or oint in image are (U,V,W) U 1 X V = 1 Y W 1 Z T U V ( U, V, W ) (, ) = ( u, v) W W Issue Camera arameters camera ma not be at the origin, looking down the z- ais etrinsic arameters one unit in camera coordinates ma not be the same as one unit in world coordinates intrinsic arameters - ocal length, rincial oint, asect ratio, angle between aes, etc. X U Transormation Transormation Transormation Y V = reresenting reresenting reresenting Z W intrinsic arameters rojection modeletrinsic arameters T Intrinsic Parameters Intrinsic Parameters describe the conversion rom metric to iel coordinates (and the reverse) w i / s = mm = - ( i o ) s mm = - ( i o ) s or / s o o 1 w mm = M Note: Focal length is a roert o the camera and can be incororated as above int Eamle: A real camera Laser range inder Camera 8

elative location Camera-Laser In homogeneous coordinates Camera =1deg T=(16,6,-9) Laser otation: 2 3 cosà1 sinà1 = 4 1 5 àsinà1 cosà1 Translation 1 1 16 1 6 T = B C @ A 1 à 9 1 Full rojection model

9 elative location Camera-Laser In homogeneous coordinates Camera =1deg T=(16,6,-9) Laser otation: 2 3 cosà1 sinà1 = àsinà1 cosà1 Translation T = B A 1 à 9 1 Full rojection model esult camera = Camera internal Camera arameters rojection!! 1278: : Camera image Laser measured 3D structure :985 à : :6612 B 1 CB 1 6 CBà A@ A@ A = :174 :985 1 à 9 18: 1 1 1! :47 9

10 Orthograhic rojection The undamental model or orthograhic rojection u = v = X U 1 V = Y 1 Z W 1 T Persective and Orthograhic Projection ersective Orthograhic (arallel) Issue Weak ersective ersective eects, but not over the scale o individual objects collect oints into a grou at about the same deth, then divide each oint b the deth o its grou Adv: eas Disadv: wrong u = T v = T T = / Z 1

11 The undamental model or weak ersective rojection = T Z Y X Z W V U * / 1 1 Camera Model Structure Assume and T eress camera in world coordinates, then T w c = 1 Combining with a ersective model

11 11 The undamental model or weak ersective rojection = T Z Y X Z W V U * / 1 1 Camera Model Structure Assume and T eress camera in world coordinates, then T w c = 1 Combining with a ersective model (and neglecting internal arameters) ields T T T M u w z z w c = = Note the M is deined onl u to a scale actor at this oint! I M is viewed as a 34 matri deined u to scale, it is called the rojection matri. Camera Model Structure Assume and T eress camera in world coordinates, then T w c = 1 Combining with a weak ersective model (and neglecting internal arameters) ields T P T T M u w z w c = = ) ( Where is the nominal distance to the viewed object P Histor o Persective oman Prehistoric:

enaissance Leonardo da Vinci: The last suer Other Models The aine camera is a generalization o weak ersective. The rojective camera is a generalization o the ersective camera.

12 enaissance Leonardo da Vinci: The last suer Other Models The aine camera is a generalization o weak ersective. The rojective camera is a generalization o the ersective camera. Both have the advantage o being linear models on real and rojective saces, resectivel. But in general will recover structure u to an aine or rojective transorm onl. (ie distorted structure) Learn about in cmut Dimensional Comuter Vision ecall: Intrinsic Parameters Intrinsic Parameters describe the conversion rom metric to iel coordinates (and the reverse) w i 1/ s = mm = - ( i o ) s mm = - ( i o ) s or 1/ s o o 1 w mm = M int CAMEA CALIBATION: A WAMUP Known distance d known regular oset r A simle wa to get scale arameters; we can comute the otical center as the numerical center and thereore have the intrinsic arameters rki = ( i o) s d r = ( i+ 1 i ) s d 12

Camera calibration Stereo Vision Issues: what are intrinsic arameters o the camera? what is the camera matri?

obtain intrinsic arameters rom camera matri Error minimization: Linear least squares eas roblem numericall solution can be

with linear least squares Numerical scaling is an issue GOAL: Passive 2- camera sstem or triangulating 3D osition o oints in

Humans use stereo vision to obtain deth Stereo deth calculation: Simle case, aligned cameras Eiolar constraint DISPAITY= (XL

13 Camera calibration Stereo Vision Issues: what are intrinsic arameters o the camera? what is the camera matri? (intrinsic+etrinsic) General strateg: view calibration object identi image oints obtain camera matri b minimizing error obtain intrinsic arameters rom camera matri Error minimization: Linear least squares eas roblem numericall solution can be rather bad Minimize image distance more diicult numerical roblem solution usuall rather good, but can be hard to ind start with linear least squares Numerical scaling is an issue GOAL: Passive 2- camera sstem or triangulating 3D osition o oints in sace to generate a deth ma o a world scene. Humans use stereo vision to obtain deth Stereo deth calculation: Simle case, aligned cameras Eiolar constraint DISPAITY= (XL - X) Similar triangles: Z = (/XL) X Z= (/X) (X-d) Z Solve or X: (/XL) X = (/X) (X-d) X = (XL d) / (XL - X) Solve or Z: Z = d* (XL - X) XL X (,) (d,) X Secial case: arallel cameras eiolar lines are arallel and aligned with rows 13

Stereo measurement eamle: How wide is the hallwa? General strateg Similar triangles: W v = Z Need deth Z Then solve or W W Z Let image esolution = 128 124 iels = 136 iels ight image Baseline d = 1.

14 Stereo measurement eamle: How wide is the hallwa? General strateg Similar triangles: W v = Z Need deth Z Then solve or W W Z Let image esolution = iels = 136 iels ight image Baseline d = 1.2m Q: How wide is the hallwa v How wide is the hallwa? Stes in solution: Focal length: 1. Comute ocal length in meters rom iels 2. Comute deth Z using stereo ormula (aligned camera lanes) Z = 3. Comute width: d* (XL - X) = 136 iels 136 = *.224 =.238m 128 v W = Z.224m is 128 iels 14

But in the case o sa 1.2*.238 X-coordinate calculation it should be w.r.t. the center Z = = 4.

15 How wide Deth calculation How wide? Answer: Similar triangles: XL =.144m X =.74m v W = Z The width o the hallwa is:.135 W = 4.1* = 2. 3m.238 W Z Disarit: XL X =.7m Deth (Note in the disarit calculation the choice o reerence (here the edge) doesn t matter. But in the case o sa 1.2*.238 X-coordinate calculation it should be w.r.t. the center Z = = 4. 1m o the image as in the stereo ormula derivation.7 V =.135m v Visual ambiguit Ambiguit Will the scissors cut the aer in the middle? Will the scissors cut the aer in the middle? NO! 15

16 Visual ambiguit Ambiguit Is the robe contacting the wire? Is the robe contacting the wire? NO! Visual ambiguit Ambiguit Is the robe contacting the wire? Is the robe contacting the wire? NO! 16

17 Camera Models Visual Invariance Internal calibration: Weak calibration: Aine calibration: Stratiication o stereo vision: - characterizes the reconstructive certaint o weakl, ainel, and internall calibrated stereo rigs sim a roj a roj roj inj inj inj inj C sim u to a similarit (scaled Euclidean transormation) C a u to an aine transormation o task sace C roj u to a rojective transormation o task sace C inj reconstruction u to a bijection o task sace 17

Computer Vision cmput 428/615

Computer Vision cmput 428/615 Basic 2D and 3D geometry and Camera models Martin Jagersand The equation of projection Intuitively: How do we develop a consistent mathematical framework for projection calculations?