1 Formation et Analyse d’Images Daniela Hall 30 Septembre 2004

2 Course material course slides (.pdf) – prima.inrialpes.fr/perso/Hall/Courses/FAI04/ prima.inrialpes.fr/perso/Hall/Courses/FAI04/ last years documents – prima.inrialpes.fr/Prima/Homepages/jlc/Courses/2003/ ENSI3/FAI/ENSI3.FAI.htmlhttp://www- prima.inrialpes.fr/Prima/Homepages/jlc/Courses/2003/ ENSI3/FAI/ENSI3.FAI.html Tensor notation –book: R. Hartley, A.Zisserman: Multiple view geometry in computer vision, Appendix 1, Cambridge University Press, 2000

3 Course Overview Session 1: –Homogenous coordinates and tensor notation –Image transformations –Camera models Session 2: –Camera models –Reflection models –Color spaces Session 3: –Review color spaces –Pixel based image analysis –Gaussian filter operators Session 4: –Scale Space

4 Course overview Session 5: –Contrast description –Hough transform Session 6: –Kalman filter –Tracking of regions, pixels, and lines Session 7: –Stereo vision –Epipolar geometry Session 8: exam

5 Session Overview 1.Homogeneous coordinates 2.Tensor notation 3.Image transformations 4.Camera models

6 Motivation How can we transform images? Apply transformation to all pixels First do translation, then rotation, then scaling

7 Motivation Transformation in 2D Transformation using homogenous coordinates

8 Session Overview 1.Homogeneous coordinates 2.Tensor notation 3.Image transformations 4.Camera models

9 Homogenous coordinates Allow to manipulate n-dim vectors in a n+1-dim space A point p can be written as vector In homogenous coordinates we add a scaling factor To transform the homogenous coordinates in normal coordinate, divide by the n+1 coordinate.

10 Homogenous coordinates we note Proof:

11 Translation ClassicHomogenous coordinates

12 Rotation (clockwise) ClassicHomogenous coordinates x y p plpl

13 Translation and rotation ClassicHomogenous coordinates

14 Translation, rotation and scaling ClassicHomogenous coordinates

15 Session Overview 1.Homogeneous coordinates 2.Tensor notation 3.Image transformations 4.Camera models

16 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

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

18 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).

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

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

21 Determinant in tensor notation

22 Cross product in tensor notation

23 Example Line equation in tensor notation

24 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:

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

26 Rotation Homogenous coordinates Classic Tensor notation

27 Session Overview 1.Homogeneous coordinates 2.Tensor notation 3.Image transformations 4.Camera models

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

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

30 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

31 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

32 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

33 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

34 Bi-linear interpolation P s0 P s1 P s3 P s2 PsPs x intensity I(P) P s0 PsPs P s1 I(P s ) y P s3 The bilinear approach computes the weighted average of the four neighboring pixels. P sx P sy Bilinear formula:

35 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.

36 Session Overview 1.Homogeneous coordinates 2.Tensor notation 3.Image transformations 4.Camera model

37 Camera model Physical geometry

38 Camera model Projective model Scene coordinates Camera coordinates Image coordinates

39 Change of coordinate systems Transformation from scene to camera coordinates Projection of camera coordinates to retina coordinates Transformation from retina coordinates to image coordinates Composition

40 Transformation Scene - Camera (xs,ys,zs) is position of the origin of the camera system with respect to the scene coordinates (translation). R is the orientation of the camera system with respect to the scene system (3d rotation).

41 3d rotation Around x-axis (counter-clockwise) Around y-axis Around z-axis General