1 Digital Images WorldCameraDigitizer Digital Image (i) What determines where the image of a 3D point appears on the 2D image? (ii) What determines how bright that image point is? Reflectance, radiometry geometry

2 change of coordinate system: the same pt in two different systems oxy and ox’y’ point transfomation: a point (x,y) is transformed (translated) into (x’,y’) within the same coordinate frame Two different interpretations: Review of some basic geometry (compulsary for vision, graphics and robotics)

3 R is a rotation matrix=orthonormal = orthogonal and unit vectors, 2*2 matrix (only 1 d.o.f.) such that 2D general Euclidean transformation:

4 One example of R might be: 3D Euclidean transformation: Different sign

5 Naturally everything starts from the known vector space add two vectors multiply any vector by any scalar zero vector – origin finite basis One step further … vector, affine, and Euclidean spaces

6 Affine geometry and affine coordinates: E1 and e2 are any noncolinear vectors, not necessarily orthogonal unit ones No more rotation as no perpendicularity (as no dot prod.)

7 Distances -- eucl. Coord Angles, ortho Ratios – affine coord. parallelism Dot product Linear dependency

8 Given 3 points (2 vectors) on the plane, we can define an affine coordinate frame (affine basis), Any 4 th point can be expressed in terms of affine coordinates …

9 Geometric modeling of a camera u v X u O X’ u’ P3 P2 How to relate a 3D point X (in oxyz) to a 2D point in pixels (u,v)?

10 Pinhole cameras Abstract camera model - box with a small hole in it Pinhole cameras work in practice

11 Distant objects are smaller:

12 Parallel lines meet:

13 each set of parallel lines (=direction) meets at a different point –The vanishing point for this direction Sets of parallel lines on the same plane lead to collinear vanishing points. –The line is called the horizon for that plane Good ways to spot faked images –scale and perspective don’t work –vanishing points behave badly –supermarket tabloids are a great source. Vanishing points:

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15 Vector space to affine: isomorph, one-to-one (pt=vector) vector to Euclidean as an enrichment: scalar prod. Pts, lines, parallelism Angle, distances, circles

16 X Y Z x y u v X x O f Camera coordinate frame

17 In more familiar matrix form:

18 x o y uv X Y Z x y u v X x O f Image coordinate frame

19 Image coordiante frame: intrinsic parameters If u not perpendicular to v, but an angle alpha:

20 Camera calibration matrix

21 Focal length in horizontal/vertical pixels (2) (or focal length in pixels + aspect ratio) the principal point (2) the skew (1) 5 intrinsic parameters one rough example: 135 film In practice, for most of CCD cameras: alpha u = alpha v i.e. aspect ratio=1 alpha = 90 i.e. skew s=0 (u0,v0) the middle of the image only focal length in pixels?

22 Xw Yw Zw XwXw X Y Z x y u v X x O f World (object) coordinate frame

23 World coordinate frame: extrinsic parameters Relation between the abstract algebraic and geometric models is in the intrinsic/extrinsic parameters! 6 extrinsic parameters

24 Finally, we should count properly... Finally, we have a map from a space pt (X,Y,Z) to a pixel (u,v) by

25 Summary of camera modelling 3 coordinate frame projection matrix decomposition intrinsic/extrinsic param