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Structure from Motion
Course web page:vision.cis.udel.edu/~cv
April 25, 2003 Lecture 26
Announcements
• Read Hartley & Zisserman Chapter 17.2 (skip 17.2.1) and Forsyth & Ponce Chapter 12.3 on affine structure from motion for Monday
• Homework 4 due on Monday
Outline
• Triangulation• Stratified reconstruction
Computing Structure
• Recall that canonical camera matrices P, P’ can be computed from fundamental matrix F– E.g. P = [Id j 0] and P’ = [[e’]£F j e’],
• Triangulation: Back-projection of rays from image points x, x’ to 3-D point of intersection X such that x = PX and x’ = P’X
from Hartley & Zisserman
Triangulation: Issues
• Errors in points x, x’ ) @ F such that x’T
F x = 0 or X such that x = PX and x’ = P’X
• This means that rays are skew — they don’t intersect
from Hartley & Zisserman
Triangulation with Non-Intersecting Rays
• Define some heuristic for best estimate of X– Idea: Find midpoint of common perpendicular to the two rays
• But this is not invariant to projective transformations – Recall that without calibration the camera matrices are only known
up to projection—i.e., PH, P’H are the “true” camera matrices for some non-singular H—so we will get different answers for X
X
from Hartley & Zisserman
Definition of Projectively Invariant Triangulation
• Suppose we compute a 3-D point X from the image points x, x’ and camera matrices P, P’ by some triangulation method ¿
• We say that ¿ is projective-invariant if for any projective transformation H:
X = ¿ (x, x’, P, P’) = H-1
¿ (x, x’, PH-1, P’H-
1)
Optimal Projective-Invariant Triangulation: Reprojection Error
• Pick that exactly satisfies camera geometry so that and , and which minimizes
• Can use as error
function for nonlinear minimization on two views– Polynomial solution exists
from Hartley & Zisserman
DLT Triangulation
• There is a Direct Linear Transformation method for triangulation (see Hartley & Zisserman Chapter 11.2)– Not projectively invariant– Easily extends to > 2 views (whereas
nonlinear method does not)
Covariance of Structure Recovery
• Bigger angle between rays ) Less uncertainty
• Can’t triangulate points on baseline (epipoles) because rays intersect along entire length
from Hartley & Zisserman
Projective Reconstruction Theorem
• With uncalibrated cameras alone, we can reconstruct a scene (e.g., via triangulation) up to a projective ambiguity
• Calibrated cameras give metric reconstruction
Example: Projective Reconstruction Ambiguity
from Hartley & Zisserman
Reconstructions related by a 4 x 4 projection H
Two views from which F and
hence P, P’ are computed
Hierarchy of Transformations
Less ambiguity
Properties of transformations (2-D)from Hartley & Zisserman
Stratified Reconstruction• Idea: Try to upgrade reconstruction to differ from the truth
by a less ambiguous transformation • Use additional constraints imposed by:
– Scene – Motion– Camera calibration
• “Cheats”– Again: Cameras with known K, K’ ! Metric reconstruction– ¸ 5 known 3-D points (no 4 coplanar) ! Euclidean reconstruction
from Hartley & Zisserman
Projective ! Affine Upgrade
• Identify plane at infinity ¼1 (in the “true”
coord-inate frame, ¼1 = (0, 0, 0, 1)T)
– E.g., intersection points of three sets of parallel lines define a plane
– E.g., if one camera is known to be affine
from Hartley & Zisserman
Projective ! Affine Upgrade• Then apply 4 x 4 transformation:
• This is the 3-D analog of affine image rectification via the line at infinity l1
• Things that can be computed/constructed with only affine ambiguity:– Midpoint of two points– Centroid of group of points– Lines parallel to other lines, planes
Example: Affine Reconstruction
Ambiguity
Affine reconstructionsfrom Hartley & Zisserman
Affine ! Metric Upgrade
• Identify absolute conic 1 on ¼1 via image of absolute conic (IAC) ! – From scene
• E.g., orthogonal lines– From known camera calibration
• Completely constrained: ! = K-T K-1
• Partially constrained:– Zero skew– Square pixels
– Same camera took all images
• 1 and ! are beyond scope of this class—they won’t be on the final
Example: Metric Reconstruction with Texture
Mapping
Only overall scale ambiguity remains—i.e., what are units of length?
from Hartley & Zisserman
Original views Synthesized views of reconstruction