is a range imaging technique; it refers to the process of estimating three-dimensional structures from two-dimensional image sequences which may be coupled with local motion signals

1. Structure From motion
Computer Vision
By Fatima Radi
Kufa University
College of Computer Science and mathematics
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2. Outline.
Introduction
Review to pose estimation model
How to find Essential Matrix
How it can predict the locations of epipolar lines
Show how to recover rotation and translation
Show how to recover point positions
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3. Introduction
There are several ways to inferring 3D information from 2D
to method so far one is
• Model based pose Estimation
• Stereo vision
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4. Structure From motion
Structure from motion (SFM) (might be better termed “structure-and-
motion from a moving camera”) is a range imaging technique; it
refers to the process of estimating three-dimensional structures from
two-dimensional image sequences which may be coupled with local
motion signals.(i.e Estimate the 3D structure and 3D (camera) motion.)
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5. Review to pose estimation model
Create a Scene
• Create some points on the face of cube
• Render image from two views
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6. The Essential Matrix
• The Essential Matrix is The Matrix E , That relates the Image of a point
in one Camera to its image in other camera , given a translation and
rotation
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7. Cont.
The essential matrix is E = [t]x R
where
• [t]x is the skew symmetric matrix corresponding to t
• t is the translation of camera 2 with respect to camera 1
• R is rotation of camera 2 with respect to camera 1
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8. Calculating E.
E=[tx]R
where
[tx] is cross prodect :- [tx]=
0 − t 3 t(2)
t 3 0 − t 1
−t 2 t 1 0
and general Rotation is:- R=Rx * Ry * Rz
Rx=[1 0 0 ; 0 cos(x) -sin(x) ; 0 sin(x) cos(x)]
Rx=[1 0 0 ; 0 cos(x) -sin(x) ; 0 sin(x) cos(x)]
Rx=[1 0 0 ; 0 cos(x) -sin(x) ; 0 sin(x) cos(x)]
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9. Epipolar lines
• Representation of a line:
• Equation of a line in the (x,y) plane is ax + by + c =
0
• line may be represented by the homogeneous
coordinates l = (a,b,c)T
• The point p lies on the line l if and only if pT l = 0
• l = (a,b,c)T = E p1 are the parameters of the line
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10. Visualization of Epipolar lines .
• Draw epipolar lines to verify that corresponding points lie on these
lines
• Epipolar lines
• E𝑃1 is the epipolar line in the first view corresponding to P1 in the second
view
• 𝐸 𝑇 𝑃0 is the epipolar line in the second view corresponding to P1 in the first
view
• Draw the line on the first image
• –Find two points (xa,ya) and (xb,yb) on the line, and draw a line between them
• –Let xa = -1, solve for ya
• –Let xb = +1, solve for yb
• y = (– c – ax)/b
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11. Recovering Motion Parameters .
• once you have Essential matrix E , you can recover the relative motion
between cameras
• Recall that the Essential matrix is made up of the translation and
rotation matrices ;ie.E = [t]x R
• we can extract the translation and rotation by taking SVD of E again E
= U D VT
• than from the following combinations :
• t is either u3 or –u3 , where u3 is the third (last) column of U
• R is either U W VT or U WT VT
• where W=[0 -1 0;1 0 0;0 0 1];
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12. Reconstruction:
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13. Find Reconstructing rest of points
• for each point pair of corresponding image points p1 and p2
• Form the skew symmetric matrices [𝑃1𝑥] And [𝑃2𝑥]
• Form the matrix equation
𝑃1𝑥 𝑀1
𝑃2𝑥 𝑀2
= 0
• M1=[1 0 0 0 ;0 1 0 0;0 0 1 0] And M2=[ R t]
• solve for P using SVD
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