Camera Models. Acknowledgements Used slides/content with permission from

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1 Camera Models Acknowledgements Used slides/content with ermission rom Marc Polleeys or the slides Hartley and isserman: book igures rom the web Matthew Turk: or the slides Single view geometry Camera model Camera calibration Single view geom. Aril 24 Camera Models 2

2 2 Aril 24 Camera Models 3 Pinhole camera geometry A general rojective camera P mas world oints to image oints according to P. Aril 24 Camera Models 4 T T ) / / ( ) ( a a Central rojection in homogeneous coordinates

3 3 Aril 24 Camera Models 5 P [ ] I ) diag( P Camera rojection matri P P: rincial oint Princial lane Aril 24 Camera Models 6 T y T ) / / ( ) ( + + a rincial oint T y ) ( + + y a Pinhole oint oset Image (y) and camera (_cam y_cam) coordinate systems.

4 Camera calibration matri K + + y calibration matri K y K[ I ] cam camera is assumed to be located at the center o a Euclidean coordinate system with the rincial ais o the camera oint in the direction o z-ais. Aril 24 Camera Models 7 Camera rotation and translation Euclidean transormation between world and camera coordinate rames Inhomogeneous 3-vector o coordinates o a oint in the world coordinate rame. ~ ( - C ~ cam R ~ ) cam R R C R R C K I [ C ] [ ] cam KR I - Same oint in the camera coordinate rame Coordinates o camera center in world coordinates P P K R t RC ~ [ t] Aril 24 Camera Models 8 4

The arameters o R and which relate the camera orientation and osition to a world coordinate system are called the eternal arameters or eterior

5 Internal and eterior orientation has 9 do 3 or K ( y) 3 or R 3 or Parameters contained in K are called the internal camera arameters or the internal orientation o the camera. The arameters o R and which relate the camera orientation and osition to a world coordinate system are called the eternal arameters or eterior orientation. Oten convenient not to make the camera center elicit and instead to reresent the world->image transormation as where Aril 24 Camera Models 9 CCD Cameras K m m y y CCD Cameras: may have non-square iels! α K α y y CCD camera: do Aril 24 Camera Models 5

6 Finite rojective camera S: skew arameter; or most normal cameras A camera with K as above is called a a inite rojective camera. A inite rojective camera has degrees o reedom. This is the same number o degrees o reedom as a 3 4 matri deined u to an arbitrary scale. Note that the let hand 3 3 submatri o P equal to KR is nonsingular. any 3 4 matri P or which the let hand 3 3 submatri is non-singular is the camera matri or some inite rojective camera. Aril 24 Camera Models Camera anatomy Camera center Column oints Princial lane Ais lane Princial oint Princial ray Aril 24 Camera Models 2 6

7 Camera Center null-sace camera rojection matri Consider: PC Consider the line containing C and any other oint A in 3-sace. For all A all oints on ray AC roject on image o A thereore C is camera center Image o camera center is () T i.e. undeined Aril 24 Camera Models 3 Column Vectors The columns o the rojective camera are 3-vectors that have a geometric meaning as articular image oints. [ ] [ ] P: vanishing oint o the world coordinate -ais P2: vanishing oint o y-ais P3: vanishing oint o z ais : image o the world origin. Aril 24 Camera Models 4 7

8 Row Vectors and the Princial Plane The rincial lane is the lane through the camera center arallel to the image lane. It consists o the set o oints which are imaged on the line at ininity o the image. i.e. A oint lies on the image lane i In articular the camera center C lies on the rincial lane. P3 is the vector reresenting the rincial lane o the camera Aril 24 Camera Models 5 Princial Plane Aril 24 Camera Models 6 8

Ais lanes Consider the set o oints on lane P. This set satisies: These are imaged at P (yw)^t these are oints on the image y-ais. Plane P is deined by the camera center and the line in the image.

9 Ais lanes Consider the set o oints on lane P. This set satisies: These are imaged at P (yw)^t these are oints on the image y-ais. Plane P is deined by the camera center and the line in the image. Similarly P2 is given by P2. note: 2 deendent on image and y ais (choice o image coordinage system). Aril 24 Camera Models 7 The rincial oint rincial oint ( ) 3 ˆ Princial ais: is the line assing through the camera center C with direction erendicular to the rincial lane P3. The ais intersects the image lane at the rincial oint. Aril 24 Camera Models 8 9

10 Resectioning Estimating the camera rojection matri rom corresonding 3-sace and image measurements -> resectioning. i i P? Similar to the 2D rojective transormation H. H was 33 whereas P is 34. Aril 24 Camera Models 9 Basic equations i P i : is a 4-vector the i-th row o P. A Each oint corresondence gives 2 indeendent equations. A 2n 2 matri : 2 column vector. Aril 24 Camera Models 2

11 Camera matri P A minimal solution P has do 2 indeendent eq./oints 5.5 corresondences needed (say 6) Over-determined solution n 6 oints minimize ˆ 3 A P subject to constraint ˆ 3 Aril 24 Camera Models 2 HW #3: Comuting P Will be osted soon. Will be due net week. Aril 24 Camera Models 22

Calibration Issues. Linear Models. Interest Point Detection + Description Algorithms

Calibration Issues. Linear Models. Interest Point Detection + Description Algorithms Calibration Issues Linear Models Homograhy estimation H Eiolar geometry F, E Interior camera arameters K Eterior camera arameters R,t Camera ose R,t h 2 H h 3 h 2 3 4 y ~ 2 22 23 24 3 32 33 34 Interest