CS 395/495-26: Spring 2003

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CS 395/495-26 Spring 2003

• IBMR Week 5 A Back to Chapter 2
• 3-D Projective Geometry
• Jack Tumblin
• jet_at_cs.northwestern.edu

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IBMR-Related Seminars

• 3D Scanning for Cultural Heritage Applications
  Holly Rushmeier, IBM TJ Watson
• Friday May 16 300pm, Rm 381, CS Dept.
• no title yet ...ltBRDF, BRSSDF capture? Optics of
  Hair? Inverse Rendering?gt
• Steve Marschner, Cornell University Friday May
  23 300pm, Rm 381, CS Dept.

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3D Homogeneous Coordinates

• Extend Projective space from 2D to 3D
• P2
• From 2D world space (x,y) make
• 2D homogeneous coordinates (x1,x2,x3).
• 2D projective image space (x,y) (x1/x3,
  x2/x3)
• P3
• From 3D world space (x,y,z) make
• 3D homogeneous coordinates (x1,x2,x3,x4).
• 3D projective image space (x,y,z)
  (x1/x4, x2/x4, x3/x4)

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3D Homogeneous Coordinates

• Unifies points and planes
• (but lines are messy)
• Puts perspective projection into matrix form
• No divide-by-zero, points at infinity defined

But in P3, write same point x as where
in R3, write point x as
x1 x2 x3 x4
x y z
x x1 / x4, y x2 / x4, z x3 / x4, x4
anything non-zero! (but usually defaults to 1)
y
(x,y,z)
x
z
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A Very Common Mistaek

• P2 homog. coords 2D projective map (2D point?3D
  ray is correct)
• P3 homog. coords. 3D projective map ( But 3D
  point?3D ray is WRONG!)

(x,y)
(x1, x2, x3)
(x,y,z)
(x1, x2, x3, x4)
x2
x2
x3
y
x
x1
x1
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A Very Common Mistaek

• P2 homog. coords 2D projective map (2D
  point?3D ray)
• P3 homog. coords. 3D projective map ( 3D
  point?3D ray is WRONG!)

(x,y)
(x1, x2, x3)
(x,y,z)
(x1, x2, x3, x4)
x2
NO! Dont confuse 3D z with projective x4!
x2
(x,y,z)
x3
y
x
x1
x1
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P3 Point ?? Plane duality

• Recall Plane Equations in 3D
• Normal vector (a,b,c) n
• Unaffected by scale k, with
• Min. Distance from plane to origin d
• Write in 3D homog. coordinates
• Point x and Plane ? are duals (lines are not!)

ax by cz d 0

kax kby kcz kd 0
ax by cz d 0
a b c d
0
x1 x2 x3 x4
xT.? 0
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3D Homogeneous Coordinates

• P2 homog. coords 2D projective map (2D
  point?3D ray)
• P3 homog. coords. 3D projective map 3D point?4D
  ray --an impossible-to-draw plane in R4 with
  normal x1,x2,x3 --its 3D part is a ray through
  origin

(x,y)
(x1, x2, x3)
(x,y,z)
(x1, x2, x3, x4)
x2
x2
(x,y,z)
x3
x3
y
x
x1
(Approx. as many 2D planes of constant z)
x1
(One 2D planeof constant z)
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3D Homogeneous Coordinates

• P3 homog. coords. 3D projective map 3D point?4D
  ray --Superset of P2 transformations
• --Includes translation, projection from any
  point

H
(x,y,z)
(x1, x2, x3, x4)
(x1, x2, x3, x4)
(x,y,z)
x2
x2
x3
x3
x1
x1
(Approx. as many 2D planes of constant z)
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Points?? Plane Conversions

• Find plane ? thru points P1,P2,P3?
• Easy! Stack points
• Find null space (SVD)
• (Rank 2? collinear points!)
• OR use 3D cross products

0
?
PT1 PT2 PT3
?1 ?2 ?3 ?4
p11 p12 p13 p14 p21 p22 p23 p24 p31
p32 p33 p34
0

• (p1 p3) ? (p2-p3)
• x3(x1 ? x2)

(p1 - p3) ? (p2 - p3) -p3T (p1 ? p2)
?
(scalar)
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Points?? Planes in P3

• Plane ? can define a 3D coordinate system (find
  u,v coordinates within plane, w along plane
  normal)
• Because ? is equiv. to planes normal vector
• Find 3 ? vectors in P3 (null space of 0 0 0 ?
  )
• assemble them as columns of 4x3 M vector
• To get 3D coords of any P3 point p M p x
• Finds a 2D coords in P2 plane (of ?) x

u v w
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Lines in P3 Awkward

• Geometrically, Lines are
• Intersection of 2 (or more) planes,
• An axis or pencil of planes,
• Linear combo of 2 points, p1 A(p2-p1)
• a 4 DOF entity in P3 a 4-vector wont do!
• Symbolically Three forms of P3 lines
• Span Null Space of matrix W
• Plucker Matrix
• Plucker Line Coordinates

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P3 Lines 1a (Point-Point) Span W

• Recall that a point x is on a plane ? iff
• if 2 given points A, B intersectwith a pencil
  of planes on a line,
• Define a P3 line as that intersection stack
  AT, BT to make 2x4 matrix W
• W
• If line W contains the plane ?, then W ? 0

xT.? 0
B
A
T.? 0
A B
a1 a2 a3 a4 b1 b2 b3 b4
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P3 Lines 1b (Plane-Plane) Span W

• Recall that a point x is on a plane ? if
• If 2 given planes P,Q intersect ata pencil of
  points on a line,
• Define a P3 line as that intersection stack
  PT, QT to make 2x4 matrix W
• W
• If line W contains the point x, then W x 0

xT.? 0
P
Q
xT.PQ 0
p1 p2 p3 p4 q1 q2 q3 q4
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P3 Lines 1Spans

• (Point) Span W
• (Found from points A,B)
• Used to test plane ?
• (Plane) Span W
• (Found from planes P,Q)
• Used to test point x

B
W ? 0
A
Useful property
WT W W WT 02x2 (the 2x2 null matrix)
P
Q
W x 0
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P3 Lines 1 Join reverses Spans...

• Join Line W and Point p ?? Plane ?
• W ? 0 iff plane ? holds line W, andpT ? 0
  iff plane ? holds point p stack Let M
  solve for ? in M ? 0
• Join Line W and Plane ? ?? Point p
• W p 0 iff line W holds point p? p 0
  iff plane ? holds point p stack
• Let M solve for p in M p 0

W pT
W ?
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P3 Lines 2 Plucker Matrices

• Line A 4x4 symmetric matrix, rank 2, 4DOF
• Line L through known points A, B
• L A.BT - B.AT
• Line L through known planes P,Q
• L P.QT QPT

Two forms
B
A
a1 a2 a3 a4
b1 b2 b3 b4
-
b1 b2 b3 b4
a1 a2 a3 a4
P
Q
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P3 Lines 2 Plucker Matrices

• Line A 4x4 symmetric matrix, rank 2, 4DOF
• Line L through A, B pts L A.BT - B.AT
• Line L through P,Q planes L P.QT - QPT
• L ? L convert?Note that
• Skew-symmetric
• Ls has 6 params or
• Note detL0 is written l12l34 l13l42 l14l23
  0
• Simple! Replace lij with lmn so that i,j,m,n
  1,2,3,4Examples l12 ?l34,or l42?l13 etc.

? l12 l13 l14? ? l23 l24 ? ? ?
l34? ? ? ?
? l12 l13 l14? ? l23 ? ? ? ?
l34? l42 ? ?
(to avoid minus signs)
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P3 Lines 2 Plucker Matrices

• Join Line L and Point p ? Plane ?
• L.p ? (if point is on the line,
  then L.p 0)
• Join Line L and Plane ? ? Point p
• L.? p
• (if line is in the plane, then L.? 0)

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P3 Lines 3 Plucker Line Coords

• Plucker Matrix has 6 vital elements
• Off-diagonals or
• Just make them a 6-vector
• and require that detL 0, or

? l12 l13 l14? ? l23 l24 ? ? ?
l34? ? ? ?
? l12 l13 l14? ? l23 ? ? ? ?
l34? l42 ? ?
(to avoid minus signs)
l12 l13 l14 l23 l42 l34
L
l12l34 l13l42 l14l23 0
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Projective Transformations

• Use H for transforms in P3
• Has 15 DOF (4x4 -1)
• Superset of the P2 H matrix
• Homework Hint
• ProjA show P2 objects in R3
• x1?x,x2?y,x3?z uses??

h11 h12 h13 h14 h21 h22 h23 h24 h31 h32
h33 h34h41 h42 h43 h44
H
h11 h12 0 h13 h21 h22 0 h23 0 0
0 0 h31 h32 0 h33
h11 h12 h13 h21 h22 h21 h31 h32 h33
H2
h11 h12 h13 0 h21 h22 h23 0 h31 h32
h33 0 0 0 0 1
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P3 Transformations

• Transform a point p or plane ? with H
• Lines 1 Transform a span
• Lines 2 Transform a Plucker Matrix

p H.p ? H-T. ?
W H.W W H-T.W
L H.L.HT L H-T.L.H-1
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END