C OMPUTER V ISION : P ROJECTIVE G EOMETRY 3D                               IIT Kharagpur                   Computer Scienc...
The projective geometry of 3D                                                   P3A point X in 3-space is represented in h...
Planes A plane in 3-space may be written as                         π1 X + π2 Y + π3 Z + π4 = 0 This equation is unaffecte...
Join and incidence relations  A plane is defined uniquely by the join of 3 points, or the join of a  line and a point, (in ...
Three points define a plane               A point Xi incident                     T                   on a plane π would ...
Projective TransformationUnder the point transformation X = HX, a plane transforms as:                             π = H−T...
Lines  A line is defined by the join of two points or the intersection of two  planes.  A line has 4 degrees of freedom in ...
The hierarchy of transforms                       A    t   Projective:                  with 15 dof.                      ...
Invariants                                                    P3  Projective:       Intersection and tangency of surfaces ...
ComparisonIn planar P2 projective           In 3-space P3 projective geometrygeometry    Identifying the line at          ...
The plane at infinity  π∞ has the canonical position π∞ = (0, 0, 0, 1)T in affine 3-space.  Two planes are parallel, if and ...
Affine properties of areconstruction  Identify π∞ in the projective coordinate frame.  Move π∞ to its canonical position at...
The absolute conic                                                              Ω∞  The absolute conic Ω∞ is a (point) con...
The absolute conic                                                 Ω∞  The absolute conic Ω∞ is only fixed as a set by a ge...
Metric Properties  Once Ω∞ and its support plane π∞ have been identified in  projective 3-space then metric properties, suc...
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Lecture 3

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Lecture 3

  1. 1. C OMPUTER V ISION : P ROJECTIVE G EOMETRY 3D IIT Kharagpur Computer Science and Engineering, Indian Institute of Technology Kharagpur. (IIT Kharagpur) Projective Geometry-3 Jan ’10 1 / 15
  2. 2. The projective geometry of 3D P3A point X in 3-space is represented in homogeneous coordinates as:    X1    T    X2   X=   X3  = X1 X2 X3 X4            X4 A projective transformation acting on P3 is a non-singular 4 × 4 matrix. X = HX The matrix H has 15 degrees of freedom. The map is a collineation (lines are mapped to lines) which preserves incidence relations such as intersection point of a line with a plane, order of contact. (IIT Kharagpur) Projective Geometry-3 Jan ’10 2 / 15
  3. 3. Planes A plane in 3-space may be written as π1 X + π2 Y + π3 Z + π4 = 0 This equation is unaffected by scalar multiplication. The homogeneous representation of the plane is the 4-vector π = (π1 , π2 , π3 , π4 )T Homogenizing by replacements: X → X1 /X4 , Y → X2 /Y4 , Z → X3 /X4 π1 X1 + π2 X2 + π3 X3 + π4 X4 = 0 πT X = 0 The normal to the plane is given by: n = (π1 , π2 , π3 )T (IIT Kharagpur) Projective Geometry-3 Jan ’10 3 / 15
  4. 4. Join and incidence relations A plane is defined uniquely by the join of 3 points, or the join of a line and a point, (in general position). Two distinct planes intersect in a unique line. Three distinct planes intersect in a unique point. (IIT Kharagpur) Projective Geometry-3 Jan ’10 4 / 15
  5. 5. Three points define a plane A point Xi incident  T  on a plane π would  X1    satisfy πT Xi = 0         π = 0  T  X     2            T   X3   This is a 3 × 4 matrix with rank 3. The intersection  π1  T  point X of 3 planes    πi is obtained using:          π  T    2 X = 0             π3  T    (IIT Kharagpur) Projective Geometry-3 Jan ’10 5 / 15
  6. 6. Projective TransformationUnder the point transformation X = HX, a plane transforms as: π = H−T π (IIT Kharagpur) Projective Geometry-3 Jan ’10 6 / 15
  7. 7. Lines A line is defined by the join of two points or the intersection of two planes. A line has 4 degrees of freedom in 3-space. A line can be defined by its intersection with two orthogonal planes. (IIT Kharagpur) Projective Geometry-3 Jan ’10 7 / 15
  8. 8. The hierarchy of transforms A t Projective: with 15 dof. vT v A t Affine: with 12 dof. 0T 1 sR t Similarity: with 7 dof. 0T 1 R t Euclidean: with 6 dof. 0T 1 (IIT Kharagpur) Projective Geometry-3 Jan ’10 8 / 15
  9. 9. Invariants P3 Projective: Intersection and tangency of surfaces in contact Affine: Parallelism of planes, volume ratios, centroids, The plane at infinity π∞ Similarity: The absolute conic Euclidean: Volume (IIT Kharagpur) Projective Geometry-3 Jan ’10 9 / 15
  10. 10. ComparisonIn planar P2 projective In 3-space P3 projective geometrygeometry Identifying the line at Plane at infinity π∞ infinity l∞ allowed affine properties of the plane to be measured. Identifying the circular points on l∞ allows the Absolute conic Ω∞ measurement of metric properties. (IIT Kharagpur) Projective Geometry-3 Jan ’10 10 / 15
  11. 11. The plane at infinity π∞ has the canonical position π∞ = (0, 0, 0, 1)T in affine 3-space. Two planes are parallel, if and only if, their line of intersection is on π∞ . A line is parallel to another line, or to a plane, if the point of intersection is on π∞ . The plane π∞ is a geometric representation of the 3 degrees of freedom required to specify affine properties in a projective coordinate frame. The plane at infinity is a fixed plane under the projective transformation H if, and only if, H is an affinity. (IIT Kharagpur) Projective Geometry-3 Jan ’10 11 / 15
  12. 12. Affine properties of areconstruction Identify π∞ in the projective coordinate frame. Move π∞ to its canonical position at π∞ = (0, 0, 0, 1)T . The scene and the reconstruction are now related by an affine transformation. Thus affine properties can now be measured directly from the coordinates of the entity. (IIT Kharagpur) Projective Geometry-3 Jan ’10 12 / 15
  13. 13. The absolute conic Ω∞ The absolute conic Ω∞ is a (point) conic on π∞ . In the metric frame π∞ = (0, 0, 0, 1)T and points on Ω∞ satisfy X2 + X2 + X2   1 2 3  ( X 1 , X 2 , X 3 ) I ( X 1 , X 2 , X 3 )T = 0  =0    X  4 Ω∞ corresponds to a conic C with matrix C = I. It is a conic of purely imaginary points on π∞ . The conic Ω∞ is a geometric representation of the 5 additional degrees of freedom that are required to specify metric properties in an affine coordinate frame. The absolute conic Ω∞ is a fixed conic under the projective transformation H if and only if H is a similarity transformation. (IIT Kharagpur) Projective Geometry-3 Jan ’10 13 / 15
  14. 14. The absolute conic Ω∞ The absolute conic Ω∞ is only fixed as a set by a general similarity; it is not fixed pointwise. This means that under a similarity transformation, a point on Ω∞ may travel to another point on Ω∞ , but it is not mapped to a point off the conic. All circles intersect Ω∞ in two points. These points are the circular points of the support plane of the circle. All spheres intersect π∞ in Ω∞ . (IIT Kharagpur) Projective Geometry-3 Jan ’10 14 / 15
  15. 15. Metric Properties Once Ω∞ and its support plane π∞ have been identified in projective 3-space then metric properties, such as angles and relative lengths, can be measured. Consider two lines with directions (3-vectors) d1 and d2 . The angle between these directions: In Euclidean frame In a projective frame dT d2 1 dT Ω∞ d2 cos θ = cos θ = 1 (dT d1 )(dT d2 ) 1 2 (dT Ω∞ d1 )(dT Ω∞ d2 ) 1 2 These expressions are equivalent since in the Euclidean world frame Ω∞ = I (IIT Kharagpur) Projective Geometry-3 Jan ’10 15 / 15

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