1 Formation et Analyse d’Images Session 2 Daniela Hall 26 September 2005

2 Course Overview Session 1: –Overview –Human vision –Homogenous coordinates –Camera models Session 2: –Tensor notation –Image transformations –Homography computation Session 3: –Reflection models –Color spaces Session 4: –Pixel based image analysis Session 5: –Gaussian filter operators –Scale Space

3 Course overview Session 6: –Contrast description –Hough transform Session 7: –Kalman filter Session 8: –Tracking of regions, pixels, and lines Session 9: –Stereo vision Session 10: –Epipolar geometry Session 11: exercises and questions

4 Trifocal Tensor A tensor is used in 3d position estimation with multiple cameras. The (trifocal) tensor encapsulates all the (projective) geometric relations between 3 camera views independent of the scene. Reference –book: R. Hartley, A.Zisserman: Multiple view geometry in computer vision, Appendix 1, Cambridge University Press, 2000 –Exists on-line

5 Tensor notation In tensor notation a superscript stands for a column vector a subscript for a row vector (useful to specify lines) A matrix is written as

6 Tensor notation Tensor summation convention: –an index repeated as sub and superscript in a product represents summation over the range of the index. Example:

7 Tensor notation Scalar product can be written as where the subscript has the same index as the superscript. This implicitly computes the sum. This is commutative Multiplication of a matrix and a vector This means a change of P from the coordinate system i to the coordinate system j (transformation).

8 Line equation In R 2 a line is defined by the equation In homogenous coordinates we can write this as In tensor notation we can write this as

9 The tensor operator E ijk and E ijk The tensor E ijk is defined for i,j,k=1,...,3 as even odd

10 Determinant in tensor notation

11 Cross product in tensor notation

12 Example Line equation in tensor notation

13 Example Intersection of two lines L: l 1 x+l 2 y+l 3 =0, M: m 1 x+m 2 y+m 3 =0 Intersection: Tensor: Result:

14 Translation Classic Tensor notation T is a transformation from the system A to B Homogenous coordinates

15 Rotation Homogenous coordinates Classic Tensor notation

16 Image transformation For each position P d in the destination image we search the pixel color I(P d ). Source image Destination image TsdTsd

17 Image transformation First we compute a position P s in the source image. Source image Destination image TsdTsd

18 Image transformation P is not integer. How do we compute I(P d )=I(P s )? Answer: by a linear combination of the neighboring pixels I(P si ) (interpolation). P s0 P s1 P s3 P s2 PsPs PdPd TsdTsd

19 Interpolation methods 0 th order: take value of closest neighbor –fast, applied for binary images 1 st order: linear interpolation and bi-linear interpolation 3 rd order: cubic spline interpolation

20 1D linear interpolation P s0 P s1 P s3 P s2 PsPs position P intensity I(P) P s0 PsPs P s1 Gradient Pixel color

21 2D linear interpolation P s0 P s1 P s3 P s2 PsPs x intensity I(P) P s0 PsPs P s1 Gradient Pixel color I(P s ) y P sx P sy

22 Bi-linear interpolation P s0 P s1 P s3 P s2 PsPs The bilinear approach computes the weighted average of the four neighboring pixels. The pixels are weighted according to the area. A B C D

23 Higher order interpolation Cubic spline interpolation takes into account more than only the closest pixels. Result: more expensive to compute, but image has less artefacts, image is smoother.

24 Homographie: projection from one plane to another Homographie H B A is bijective Q B = H B A P A

25 Homography computation H can be computed from 4 point correspondences. Ps1 Ps2 Ps3 Ps4 Rd1 Rd2 Rd3 Rd4 Source image (observed) Destination image (rectified)

26 Homography computation H is 3x3 matrix and has 8 degrees of freedom (homogenous coordinates) gives 8 equations and one solution for H.

27 Application: Rectifying images

28 Applications