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Affine Structure-from-Motion: Understanding Two Frames and Transformations - Prof. David J, Exams of Computer Science

University of MarylandComputer Science
Prof. David Jacobs

The concept of affine structure-from-motion, focusing on two frames and the transformations involved. It covers topics like weak perspective, perspective-n-point, and affine transformation, as well as the limitations of determining the exact 3d structure of a scene. It also discusses the use of matrices and their role in solving for motion and structure.

Typology: Exams

Pre 2010

Uploaded on 07/29/2009

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Structure-from-Motion
• Determining the 3-D structure of the world, and/or the
motion of a camera using a sequence of images
taken by a moving camera.
– Equivalently, we can think of the world as moving and the
camera as fixed.
• Like stereo, but the position of the camera isn’t
known (and it’s more natural to use many images
with little motion between them, not just two with a lot
of motion).
– We may or may not assume we know the parameters of the
camera, such as its focal length.
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Download Affine Structure-from-Motion: Understanding Two Frames and Transformations - Prof. David J and more Exams Computer Science in PDF only on Docsity!

Structure-from-Motion

-^

Determining the 3-D structure of the world, and/or themotion of a camera using a sequence of imagestaken by a moving camera.^ – Equivalently, we can think of the world as moving and the

camera as fixed.

-^

Like stereo, but the position of the camera isn’tknown (and it’s more natural to use many imageswith little motion between them, not just two with a lotof motion).^ – We may or may not assume we know the parameters of the

camera, such as its focal length.

Structure-from-Motion

  • As with stereo, we can divide problem:
    • Correspondence.– Reconstruction.
      • Again, we’ll talk about reconstruction

first.^ – So for the next few classes we assume that

each image contains some points, and weknow which points match which.

Movie

Reconstruction

  • A lot harder than with stereo.• Start with simpler case: scaled

orthographic projection (weakperspective).^ – Recall, in this we remove the z coordinate

and scale all x and y coordinates the sameamount.

Perspective -> Scaled

Orthographic

  • Recall: (

xi

,y

,zi

) -> (xi

/zi

, yi

/zi

)i

  • Let

Z = (z

+z 1

2

+ …+z

)/nn

  • Then, (

xi

,y

,zi

) approx-> (xi

/Z, yi

/Z)i

The Equation of Weak

Perspective

) , ( ) , ,

(^

y x s z y x

→

-^ s

is constant for all points.

  • Parallel lines no longer converge, they remainparallel.

           

  

     

1 0 0

0

0 0 0

X Y^ Z

s s

x y

First: Represent motion

-^

We’ll talk about a fixed camera, and moving object.

-^

Key point:

     

    

1

1 1

. . . 2 1

2 1

2 1

n n z n

z z

y

y y

x

x x

P

Points

=^
x ty
s
s
s
t
s
s
s
S^

(^3) , 2

(^2) , 2

(^1) , 2

(^3) , 1

(^2) , 1

(^1) , 1 Some matrix

  

  

n v n

v v

u

u u I

2 1

2 1

. . .

The image

SP

I

Then:

Remember what this means.• We are representing moving a set of points,projecting them into the image, and scaling them.• Matrix multiplication: take inner product betweeneach row of S and each point. First row of Sproduces X coordinates, while second row producesY. • Projection occurs because S has no third row.• Translation occurs with tx and ty.• Scaling can be encoded with a scale factor in S.• The rest of S must be allowing the object to rotate.

Structure-from-Motion

  • S encodes:
    • Projection: only two lines– Scaling, since S can have a scale factor.– Translation, by tx/s and ty/s.– Rotation:

SP

I

Rotation

P

r

r

r

r

r

r

r

r

r

    

    

(^3) , 3

(^2) , 3

(^1) , 3

(^3) , 2

(^2) , 2

(^1) , 2

(^3) , 1

(^2) , 1

(^1) , 1

Represents a3D rotation ofthe points in P.

Why does multiplying points by R rotate them?• Think of the rows of R as a new coordinate system.Taking inner products of each points with these expressesthat point in that coordinate system.

  • This means rows of R must be orthonormal vectors(orthogonal unit vectors).
    • Think of what happens to the points (1,0) and (0,1). Theygo to (cos theta, -sin theta), and (sin theta, cos theta). Theyremain orthonormal, and rotate clockwise by theta.
      • Any other point, (a,b) can be thought of as a(1,0) +b(0,1). R(a(1,0)+b(0,1) = Ra(1,0) + Ra(0,1) = aR(1,0) +bR(0,1).

So it’s in the same position relative to the

rotated coordinates that it was in before rotation relativeto the x, y coordinates. That is, it’s rotated.

Simple 3D Rotation

    

      

    

−

n n z n

z

z

y

y

y

x

x

x

2

1

2

1

2

1

... 1 0 0

0

cos

sin

0

sin

cos

θ

θ

θ

θ Rotation about z axis.Rotates x,y coordinates. Leaves z coordinates fixed.

  • Intuitively, it makes sense that 3D rotations can beexpressed as 3 separate rotations about fixed axes.Rotations have 3 degrees of freedom; two describe anaxis of rotation, and one the amount.• Rotations preserve the length of a vector, and the anglebetween two vectors. Therefore, (1,0,0), (0,1,0), (0,0,1)must be orthonormal after rotation. After rotation, theyare the three columns of R. So these columns must beorthonormal vectors for R to be a rotation. Similarly, ifthey are orthonormal vectors (with determinant 1) R willhave the effect of rotating (1,0,0), (0,1,0), (0,0,1). Samereasoning as 2D tells us all other points rotate too.
    • Note if R has determinant -1, then R is a rotationplus a reflection.

Questions?