The old book, geom.pptx is from intro.ppt
Description: The old book, geom.pptx is from intro.ppt Image-based Modeling, Long QUAN, Springer-Verlag, 2010. My perspective Part I What is computer vision? What is 3D reconstruction? From pixels to 3D points Structure from motion A quasi-dense
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slide1. The old book, geom.pptx is from intro.ppt Image-based Modeling,
Long QUAN,
Springer-Verlag,
2010.<br>
slide2. My perspective Part I
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<br>
slide3. 3 Overview Introduction to projective geometry
1 view geometry (calibration, …)
2-view geometry (stereo, motion, …)
3- and N-view geometry
Autocalibration (metric reconst.)
Application<br>
slide4. 4 Basic geometric concepts to understand Affine, Euclidean geometries (inhomogeneous coordinates)
projective geometry (homogeneous coordinates)
plane at infinity: affine geometry
absolute conic: Euclidean geometry<br>
slide5. 5 Introduction to projective geometry Intuitive ideas from projective geometry
(Formal definition of projective spaces)<br>
slide6. 6 Naturally everything starts from the known vector space add two vectors
multiply any vector by any scalar
zero vector – origin
finite basis Intuitive introduction<br>
slide7. 7 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 Pts at infinity<br>
slide8. 8 Algebraic extension to pts at infinity: introduction of homogeneous coordiantes Points at infinity: Rq: the homogeneous coordinates are not unique, up to a scale.<br>
slide9. 9 The direction d is a pt at infinity: On a plane, Can we see the pts at infinity?<br>
slide10. 10 a projective space is an affine space + some pts at infinity a projective space is a space of ‘homogeneous coordinates’ or Provisional summary<br>
slide11. 11 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.<br>
slide12. 12 Definition: a pt x is said to be linearly dependent on a set of pts if A projective space is nothing but a quotient space (space of equivalent classes):
A space of homogeneous coordinates
Basic structure: linear dependence of points<br>
slide13. 13 P2 and R2 Relation between Pn (homo) and Rn (in-homo): Rn --> Pn, extension, embedded in Pn --> Rn, restriction,<br>
slide14. 14 One example of construction of projective line by quotient space<br>
slide15. 15 Examples of projective spaces Projective plane P2
Projective line P1
Projective space P3<br>
slide16. 16 Pts are elements of P2 Projective plane 4 pts determine a projective basis
3 ref. Pts + 1 unit pt to fix the scales for ref. pts Relation with R2, (x,y,0), line at inf., (0,0,0) is not a pt Pts at infinity: (x,y,0), the line at infinity Space of homogeneous coordinates (x,y,t) Pts are elements of P2<br>
slide17. 17 Line equation: Lines: Linear combination of two algebraically independent pts Operator + is ‘span’ or ‘join’<br>
slide18. 18 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!<br>
slide19. 19 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.<br>
slide20. 20 Compute the intersection point of two lines, each defined by two points<br>
slide21. 21 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 Conics<br>
slide22. 22 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)<br>
slide23. 23 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<br>
slide24. 24 Line at infinity Affine classification:<br>
slide25. 25 Projective line Finite pts:
Infinite pts: how many?
Topology?
A basis by 3 pts
Fundamental inv: cross-ratio Homogeneous pair (x1,x2)<br>
slide26. 26 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))<br>
slide27. 27 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) Plane equation: ... Line equation? Projective space P3<br>
slide28. 28 planes In practice, take SVD Homework: compute plane normal vector?<br>
slide29. 29 How many d.o.f? 6 2*2 minors, Two lines intersect in space iff Plucker coordinates of lines in P3<br>
slide30. 30 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<br>
slide31. 31 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,<br>
slide32. 32 2D general Euclidean transformation: 2D general affine transformation: 2D general projective transformation: Introduction to transformation<br>
slide33. 33 Projective transformation = collineation = homography Consider all functions All linear transformations are represented by matrices A Note: linear but in homogeneous coordinates!<br>
slide34. 34 (n+1)*(n+1) -1 d.o.f.
all projective properties are left invariant by A
all transformations form a group GL(n,R) 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. Properties N+2 pts to determine a trans. = a proj. basis<br>
slide35. 35 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 Some examples of transformations<br>
slide36. 36 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:<br>
slide37. 37 How to compute canonical or standard coordinates?--- affine case 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 Given 4 pts, x1, x2, x3, x4, find the affine coord of x4 w.r.t. x1, x2 and x3: How to solve Ax=b?<br>
slide38. 38 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?<br>
slide39. 39 A transformation between 2 spaces?<br>
slide40. 40 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)<br>
slide41. 41 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 Hierarchy of geometry: All proj. Transformations nicely form a group! Each geometry is associated with a (sub)group!<br>
slide42. 42 Affine transformation is a projective one which leaves the line at inf. invariant: x3=x3’=0 Example of dim 2 From projective to affine:<br>
slide43. 43 Similarity transformation is an affine one which leaves the circular pts I and J invariant What are the circular points? From affine to euclidean<br>
slide44. 44 Intuitive introduction of circular pts The pair of circular points The line at infinity
of a usual plane Circular points<br>
slide45. 45 Affine transformation leaves the plane at inf. invariant Similarity (euclidean) leaves the absolute conic (globally, not point-wise) invariant What is the absolute conic? Example of transformation in P3<br>
slide46. 46 Absolute conic A space conic on the plane at infinity:
In point coordinates:
In plane coordinates:
rank 3 space quadric=absolute quadric Euclidean structure in projective space by the absolute conic<br>
slide47. 47<br>
slide48. 48 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!<br>
slide49. 49 Summary transformation and geometry
group of transformation
affine group: hyper-plane at inf.
euclidean group: absolute pts<br>
Long QUAN,
Springer-Verlag,
2010.<br>
slide2. My perspective Part I
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<br>
slide3. 3 Overview Introduction to projective geometry
1 view geometry (calibration, …)
2-view geometry (stereo, motion, …)
3- and N-view geometry
Autocalibration (metric reconst.)
Application<br>
slide4. 4 Basic geometric concepts to understand Affine, Euclidean geometries (inhomogeneous coordinates)
projective geometry (homogeneous coordinates)
plane at infinity: affine geometry
absolute conic: Euclidean geometry<br>
slide5. 5 Introduction to projective geometry Intuitive ideas from projective geometry
(Formal definition of projective spaces)<br>
slide6. 6 Naturally everything starts from the known vector space add two vectors
multiply any vector by any scalar
zero vector – origin
finite basis Intuitive introduction<br>
slide7. 7 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 Pts at infinity<br>
slide8. 8 Algebraic extension to pts at infinity: introduction of homogeneous coordiantes Points at infinity: Rq: the homogeneous coordinates are not unique, up to a scale.<br>
slide9. 9 The direction d is a pt at infinity: On a plane, Can we see the pts at infinity?<br>
slide10. 10 a projective space is an affine space + some pts at infinity a projective space is a space of ‘homogeneous coordinates’ or Provisional summary<br>
slide11. 11 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.<br>
slide12. 12 Definition: a pt x is said to be linearly dependent on a set of pts if A projective space is nothing but a quotient space (space of equivalent classes):
A space of homogeneous coordinates
Basic structure: linear dependence of points<br>
slide13. 13 P2 and R2 Relation between Pn (homo) and Rn (in-homo): Rn --> Pn, extension, embedded in Pn --> Rn, restriction,<br>
slide14. 14 One example of construction of projective line by quotient space<br>
slide15. 15 Examples of projective spaces Projective plane P2
Projective line P1
Projective space P3<br>
slide16. 16 Pts are elements of P2 Projective plane 4 pts determine a projective basis
3 ref. Pts + 1 unit pt to fix the scales for ref. pts Relation with R2, (x,y,0), line at inf., (0,0,0) is not a pt Pts at infinity: (x,y,0), the line at infinity Space of homogeneous coordinates (x,y,t) Pts are elements of P2<br>
slide17. 17 Line equation: Lines: Linear combination of two algebraically independent pts Operator + is ‘span’ or ‘join’<br>
slide18. 18 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!<br>
slide19. 19 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.<br>
slide20. 20 Compute the intersection point of two lines, each defined by two points<br>
slide21. 21 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 Conics<br>
slide22. 22 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)<br>
slide23. 23 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<br>
slide24. 24 Line at infinity Affine classification:<br>
slide25. 25 Projective line Finite pts:
Infinite pts: how many?
Topology?
A basis by 3 pts
Fundamental inv: cross-ratio Homogeneous pair (x1,x2)<br>
slide26. 26 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))<br>
slide27. 27 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) Plane equation: ... Line equation? Projective space P3<br>
slide28. 28 planes In practice, take SVD Homework: compute plane normal vector?<br>
slide29. 29 How many d.o.f? 6 2*2 minors, Two lines intersect in space iff Plucker coordinates of lines in P3<br>
slide30. 30 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<br>
slide31. 31 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,<br>
slide32. 32 2D general Euclidean transformation: 2D general affine transformation: 2D general projective transformation: Introduction to transformation<br>
slide33. 33 Projective transformation = collineation = homography Consider all functions All linear transformations are represented by matrices A Note: linear but in homogeneous coordinates!<br>
slide34. 34 (n+1)*(n+1) -1 d.o.f.
all projective properties are left invariant by A
all transformations form a group GL(n,R) 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. Properties N+2 pts to determine a trans. = a proj. basis<br>
slide35. 35 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 Some examples of transformations<br>
slide36. 36 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:<br>
slide37. 37 How to compute canonical or standard coordinates?--- affine case 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 Given 4 pts, x1, x2, x3, x4, find the affine coord of x4 w.r.t. x1, x2 and x3: How to solve Ax=b?<br>
slide38. 38 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?<br>
slide39. 39 A transformation between 2 spaces?<br>
slide40. 40 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)<br>
slide41. 41 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 Hierarchy of geometry: All proj. Transformations nicely form a group! Each geometry is associated with a (sub)group!<br>
slide42. 42 Affine transformation is a projective one which leaves the line at inf. invariant: x3=x3’=0 Example of dim 2 From projective to affine:<br>
slide43. 43 Similarity transformation is an affine one which leaves the circular pts I and J invariant What are the circular points? From affine to euclidean<br>
slide44. 44 Intuitive introduction of circular pts The pair of circular points The line at infinity
of a usual plane Circular points<br>
slide45. 45 Affine transformation leaves the plane at inf. invariant Similarity (euclidean) leaves the absolute conic (globally, not point-wise) invariant What is the absolute conic? Example of transformation in P3<br>
slide46. 46 Absolute conic A space conic on the plane at infinity:
In point coordinates:
In plane coordinates:
rank 3 space quadric=absolute quadric Euclidean structure in projective space by the absolute conic<br>
slide47. 47<br>
slide48. 48 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!<br>
slide49. 49 Summary transformation and geometry
group of transformation
affine group: hyper-plane at inf.
euclidean group: absolute pts<br>