1. 1 Formation et Analyse d’Images Daniela Hall 19 Septembre 2005 Daniela.Hall@inrialpes.fr

2. 2 Course material course slides (.pdf) –http://www- prima.inrialpes.fr/perso/Hall/Courses/FAI05/http://www- prima.inrialpes.fr/perso/Hall/Courses/FAI05/ References on the slides last years documents –http://www-www- prima.inrialpes.fr/perso/Hall/Courses/FAI04/http://www-www- prima.inrialpes.fr/perso/Hall/Courses/FAI04/ CVonline: –http://homepages.inf.ed.ac.uk/rbf/CVonline/ Contact: email Daniela.Hall@inrialpes.frDaniela.Hall@inrialpes.fr

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

4. 4 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

5. 5 Session Overview 1.Course overview 2.Image formation in the eye and the camera 1.The eye 2.The retina 3.Capacities of the eye 3.Homogenous coordinates 4.Camera models 1.Pinhole camera 2.Perspective projection 3.Mathematical formulation

6. 6 Image formation in the eye Biological vision: process of using light reflected from surrounding world for modifying behaviour. Humans require conscious understanding of the 3d world from 2d projections on the retina. Surrounding environment is interpreted by visual input. Ref: CVonline/LOCAL_COPIES/OWENS/LECT1/

7. 7 The eye aqueous humor

8. 8 The eye Light enters the eye through the transparent cornea, passes through the aqueous humor, the lens and the vitreous humor, where it forms the image on the retina. Accomodation: muscular adjustment of the lens that focuses the image directly on the retina.

9. 9 The retina Complex tiling of photoreceptors (rods and cones). Photoreceptors stimulated by light transmit electrical signals to the brain via the optic nerve. Location of the optic nerve on the retina has no photoreceptors. No light is perceived within this region (blind spot).

10. 10 The retina Rods and cones communicate through several layers of cells to the ganglion cells Synapses: junctions between layers Rods and cones are situated at the back of the retina. Light passes through the different layers and the signal is transmitted back via synaptic junctions to the optic nerve.

11. 11 The retina Ganglion cell responds to the photostimulus according to a receptive field. The spatial organization of the receptive field on the retinal ganglion cell is circular symmetric (either excitatory center and inhibitory suround or inverse). Such cells are known as mexican hat operators Other spatial organization are –simple cells (orientation sensitive receptive fields), –complex cells (non-linear combination of even and odd responses) and –end-stopped cells (simple differentiation operators).

12. 12 Capacities of the eye Sensitivity and performance approach the limits set by quantum physics. Eye can detect a single photon Eye adapts to ranges in light of many orders of magnitude No camera can even partially match this performance Inputs from left and right are processed in the optic chiasma. The slight difference of viewpoint of the eyes is used to deduce depth. Nerve fibers lead from the optic chiasma to the striate cortex (the seat of visual processing in the brain).

13. 13 Session Overview 1.Course overview 2.Image formation in the eye and the camera 1.The eye 2.The retina 3.Capacities of the eye 3.Camera models 1.Pinhole camera 2.Homogenous coordinates 3.Perspective projection 4.Mathematical formulation

14. 14 Camera model Physical geometry The pinhole camera is the simplest. It has a infinitesimally small hole through which light enters and forming an inverted image on the camera surface (retina). To simplify things, we model a pinhole camera by placing the retina between the camera center and the object (projective model)

15. 15 Camera model Projective model Scene coordinates Camera coordinates Image coordinates

16. 16 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 This mapping from 3 dimensions to 2 is called perspective projection

17. 17 Camera parameters 4 intrinsic parameters: –2 for the origin of the image coordinate frame –2 for the scale of the axes –Focal length F 6 extrinsic parameters: –3 for the 3D position of the center of projection –3 for the orientation of the image plane

18. 18 Intrinsic camera parameters F: focal distance C i, C j : Optical image center (in pixels) D i, D j : Physical size of the pixel on the retina (in pixel/mm) i, j : image coordinates (in pixels) Transformation Retina-Image

19. 19 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 w.

20. 20 Homogenous coordinates we note Proof:

21. 21 Translation ClassicHomogenous coordinates

22. 22 Scaling ClassicHomogenous coordinates x y p plpl

23. 23 Rotation (clockwise) ClassicHomogenous coordinates x y p plpl

24. 24 Translation and rotation ClassicHomogenous coordinates

25. 25 Translation, rotation and scaling ClassicHomogenous coordinates

26. 26 3d rotation Around x-axis (counter-clockwise) Around y-axis Around z-axis General

27. 27 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).

28. 28 Transformation Camera-Retina Imagine a 1D camera in a 2D space. The transformation M R c can be found by considering similar triangles z (x c,z c ) F x xrxr O

29. 29 Transformation Camera-Retina

30. 30 Transformation Retina-Image A frame: the image is composed of pixels (picture elements) Pixels are in general not squared. There physical sizes depends on the used material. i columns j rows (0,0) (i-1,j-1)

31. 31 Exercise Soit une camera avec un axe optique a la position (Ci,Cj) [pixels] et une taille de pixel de Di [mm/ligne], Dj [mm/colonne]. L'horloge du numerisateur est mal regle: chaque ligne d l'image est decalee vers la droite par k pixels par ligne. Ecrivez la matrice de projection, C i r de l'image vers la retine en coordonees homogenes.