1. 3D Reconstruction from Images
The second is 3d reconstruction from …
It is recently supported by google and microsoft
2009, Siggraph
A complete pipeline from pixels to finalized 3D objects through object segmentation and recognition.

2. Vision geometry Structure from motion, or 3D reconstruction, or SLAM
Example par excellence of modern computer vision integrating feature detection, object recognition and geometry computation
Camera poses
Positioning, visual GPS, for GPS denied environments such as indoors, cities …
Localization, for automation by machine vision
3D reconstruction
Obstacle avoidance, navigation, planning, and environment learning
Now the big trends in visual computing are …

3. The objects of study in geometry
Lines to points
Corners to feature points
Match correlation to descriptors of high dimension
Descriptors to recognition, beyond the geometry scope
More deterministic computational models
Robust statistics
Large-scale
Now the big trends in visual computing are …

4. New book
Image-based Modeling,
Long QUAN,
Springer-Verlag,
2010.

5. My perspective Part I Part II Conclusions What is computer vision?
What is 3D reconstruction?
From pixels to 3D points
Structure from motion
A quasi-dense approach
From 3D points to objects
Small-scale objects
Smooth surfaces, Hairs, Trees
Large-scale buildings
Façade, Buildings, Cities
Part II
Large-scale automatic 3D mapping
Conclusions

6. Overview Introduction to projective geometry
1 view geometry (calibration, …)
2-view geometry (stereo, motion, …)
3- and N-view geometry
Autocalibration (metric reconst.)
Application

7. Basic geometric concepts to understand
Affine, Euclidean geometries (inhomogeneous coordinates)
projective geometry (homogeneous coordinates)
plane at infinity: affine geometry
absolute conic: Euclidean geometry

8. Introduction to projective geometry
Intuitive ideas from projective geometry
(Formal definition of projective spaces)

9. Intuitive introduction
Naturally everything starts from the known vector space
add two vectors
multiply any vector by any scalar
zero vector – origin
finite basis

10. Vector space to affine: isomorph, one-to-one
vector to Euclidean as an enrichment: scalar prod.
affine to projective as an extension: add ideal elements
Pts, lines, parallelism
Angle, distances, circles
Affine and eucl are finite geometry in which we are handling only fnite pts
Pts at infinity

11. Points at infinity:
Algebraic extension to pts at infinity: introduction of homogeneous coordiantes
Rq: the homogeneous coordinates are not unique, up to a scale.

12. On a plane, Can we see the pts at infinity?
Yes, the vanishing pt of parallel lines in perspective drawing
Two lines intersect into a pt, so how about two parallel lines …
The direction d is a pt at infinity:

13. Provisional summary
a projective space is an affine space + some pts at infinity or
a projective space is a space of ‘homogeneous coordinates’

14. ((Formal) definition of projective geometry)
Given K=R or C, can be defined as the nonzero equivalent classes determined by the relation ~ on
If there is non-zero real number such that
Any element of the equivalent class will be called the homogeneous coordinates of the point.

15. A space of homogeneous coordinates
A projective space is nothing but a quotient space (space of equivalent classes):
A space of homogeneous coordinates
Basic structure: linear dependence of points
So proj. space is a space of homogeneous coordinates with linear dependency
In fact, the algebraic structure of proj. space is very poor! Even there is no neutral elements
Definition: a pt x is said to be linearly dependent on a set of pts if

16. Relation between Pn (homo) and Rn (in-homo):
Rn --> Pn, extension, embedded in
Pn --> Rn, restriction,
P2 and R2

17. One example of construction of projective line by quotient space

18. Examples of projective spaces
Projective plane P2
Projective line P1
Projective space P3

19. Projective plane Space of homogeneous coordinates (x,y,t)
Pts are elements of P2
Pts are elements of P2
Pts at infinity: (x,y,0), the line at infinity
4 pts determine a projective basis
3 ref. Pts + 1 unit pt to fix the scales for ref. pts
An affine plane needs 2 vectors or 3 points (including the origin!)
An proj. plane is in fact a R3, so 4 pts (in the proj. space, there is no origine, so only pts),
In proj. plane we can have at most 3 linearly independent points
Relation with R2, (x,y,0), line at inf., (0,0,0) is not a pt

20. Lines: Linear combination of two algebraically independent pts
Operator + is ‘span’ or ‘join’
Line equation:

21. Point/line duality: Point coordinate, column vector
A line is a set of linearly dependent points
Two points define a line
Line coordinate, row vector
A point is a set of linearly dependent lines
Two lines define a point
What is the line equation of two given points?
‘line’ (a,b,c) has been always ‘homogeneous’ since high school!

22. Given 2 points x1 and x2 (in homogeneous coordinates), the line connecting x1 and x2 is given by
Given 2 lines l1 and l2, the intersection point x is given by
NB: ‘cross-product’ is purely a notational device here.

23. Compute the intersection point of two lines, each defined by two points

24. Conics Conics: a curve described by a second-degree equation
3*3 symmetric matrix
5 d.o.f
5 pts determine a conic
affine classification with pts at inf
the line tangent to a conic at a pt
dual conic
pole and polar
one numerical example

25. Tangent to a conic at a pt x on C is given by l=Cx
Dual conic (in line coordinates) is given by l^T C^{-1} l = 0
Polar of a pt x is l = C x and (is also a tangent on C from x if x is on C)
Conjugacy: a pt y on l, y^T l = 0, y^T C x = 0
(in Eucl. Ortho: y^T x = 0)

26. Projective classification of (point) conics:
General rank 3: x^2+y^2+t^2=0 (imaginary)
x^2+y^2-t^2=0
Degenerate conics

27. Affine classification:
Line at infinity

28. Projective line Homogeneous pair (x1,x2) Finite pts:
Infinite pts: how many?
Topology?
A basis by 3 pts
Fundamental inv: cross-ratio

29. Euclidean coordinate:
the distance
Affine coordinate:
the ratio of the distances (x-a/a-o)
Projective coordinate:
the ratio of the ratio of the distances
(cross-ratio, double ratio)
((x-a)/(a-o)) / ((x-b)(b-o))

30. Projective space P3 Pts, elements of P3
Relation with R3, plane at inf.
lines: linear comb of 2 pts, but 3*4 matrix, complicated …back later
planes: linear comb of 3 pts
Basis by 4 (ref pts) +1 pts (unit)
quadrics: two classes---ruled and unruled
(topology of P3)
Line equation?
Plane equation: ...
3*4, two points + the variable pt

31. planes
In practice, take SVD
Homework: compute plane normal vector?

32. Plucker coordinates of lines in P3
How many d.o.f???
6 2*2 minors,
d.o.f. think of two-plane parameterisation, only 4 as two planes are given
Two lines intersect in space iff

33. (Quadric surfaces)
Ruled: hyperboloid of one sheet, 1,1,-1,-1---topo torus
Unruled: sphere, ellipsoid, hyperboloid and paraboloid: 1,1,1,-1
---- topo sphere

34. Key points Homo. Coordinates are not unique
0 represents no projective pt
finite points embedded in proj. Space (relation between R and P)
pts at inf. (x,0) missing pts, directions
hyper-plane (co-dim 1):
duality between u and x,

35. Introduction to transformation
2D general Euclidean transformation:
2D general affine transformation:
2D general projective transformation:

36. Projective transformation
= collineation = homography
Consider all functions
All linear transformations are represented by matrices A
Transformation should not be separated from the previous sections,
Only to stress the importance of the homogeneous coordinates
Note: linear but in homogeneous coordinates!

37. Properties (n+1)*(n+1) -1 d.o.f.
all projective properties are left invariant by A
all transformations form a group GL(n,R)
N+2 pts to determine a trans. = a proj. basis
Check the most important one: linear dependency,
i.e. lines into lines as line is just a span
Starting pt for new investigation: Klein’s Erlangen program
Inversely, we may also prove that any 1-1 transf. Preserving lines
is a linear trans in homogeneous coord.

38. (Some examples of transformations)
on pts, lines and conics:
Transforms contravariantly
Co-variantly to preserve incidence
Co-variantly
NB: co-,contra-variance is w.r.t. the basis trans.
Transpose is of no importance, il accommodates row/column vectors
Some numerical examples of transformation on P2

39. How to compute (canonical or standard) coordinates?--- affine case
Given 4 pts, x1, x2, x3, x4, find the affine coord of x4 w.r.t. x1, x2 and x3:
Vector(x4-x1) = a vector(x2-x1) + b (x3-x1)

40. How to compute (canonical or standard) coordinates?--- affine case
Given 4 pts, x1, x2, x3, x4, find the affine coord of x4 w.r.t. x1, x2 and x3:
by definition,
vector(x4-x1) = a vector(x2-x1) + b vector(x3-x1)
by canonical transformation,
x1->(0,0), x2->(1,0), x3->(0,1), get transfromation A, then Ax4
Vector(x4-x1) = a vector(x2-x1) + b (x3-x1)
How to solve Ax=b?

41. Canonical projective coordinates?
Given 5 pts, x1, x2, x3, x4, x5find the affine coord of x5 w.r.t. x1, x2, x3, x4:
By canonical transformation:
How to solve Ax=0?

42. A transformation between 2 spaces?

43. Exercise Compute the transformation from
(0,0,1), (1,0,1), (0,1,1) and (1,1,1) into
(0,0,1), (1,1/4,1),(0,1,1) and (1,3/4,1)

44. (Geometry as an invariant theory of transformation groups)
projective geom GL(n,R) cross-ratio
affine geom Subgroup A(n,R) ratio
Euclidean geom Subgroup E(n,R) distance
All proj. Transformations nicely form a group!
Each geometry is associated with a (sub)group!
Hierarchy of geometry:

45. Example of dim 2 From projective to affine:
Affine transformation is a projective one which leaves the line at inf. invariant: x3=x3’=0

46. From affine to euclidean
Similarity transformation is an affine one which leaves the circular pts I and J invariant
What are the circular points?

47. Circular points Intuitive introduction of circular pts
The line at infinity
of a usual plane
The pair of circular points
Intuitive introduction of circular pts

48. Example of transformation in P3
Affine transformation leaves the plane at inf. invariant
Similarity (euclidean) leaves the absolute conic (globally, not point-wise) invariant
What is the absolute conic?

49. (Absolute conic) A space conic on the plane at infinity:
Euclidean structure in projective space by the absolute conic
A space conic on the plane at infinity:
In point coordinates:
In plane coordinates:
rank 3 space quadric=absolute quadric

50. The plane at infinity
A usual plane in 3D
The absolute conic
The pair of circular points
The line at infinity
of a usual plane

51. Key message from projective geometry for vision
‘abstract camera’ is a projective transformation from P3 to P2, so 3*4 matrix
the intrinsic parameters of the camera are the image of the absolute conic!

52. Summary transformation and geometry group of transformation
affine group: hyper-plane at inf.
euclidean group: absolute pts
Affine and euclidean are particular cases of projective with particular forms of transformations