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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
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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
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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
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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
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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
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Projective TransformationUnder the point transformation X = HX, a plane transforms as: π = H−T π (IIT Kharagpur) Projective Geometry-3 Jan ’10 6 / 15
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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
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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
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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
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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
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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
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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
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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
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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
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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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