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Uncalibrated Geometry & Stratification. Stefano Soatto UCLA Yi Ma UIUC. Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial scene knowledge. Overview. calibrated coordinates.

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Uncalibrated geometry stratification

Uncalibrated Geometry & Stratification

Stefano Soatto

UCLA

Yi Ma

UIUC


Overview

Calibration with a rig

Uncalibrated epipolar geometry

Ambiguities in image formation

Stratified reconstruction

Autocalibration with partial scene knowledge

Overview

ICRA2003, Taipei


Uncalibrated camera

calibrated

coordinates

Linear transformation

pixel

coordinates

Uncalibrated Camera

ICRA2003, Taipei


Uncalibrated camera1

Calibrated camera

  • Image plane coordinates

  • Camera extrinsic parameters

  • Perspective projection

Uncalibrated camera

  • Pixel coordinates

  • Projection matrix

Uncalibrated Camera

ICRA2003, Taipei


Taxonomy on uncalibrated reconstruction

  • is known, back to calibrated case

  • is unknown

    • Calibration with complete scene knowledge (a rig) – estimate

    • Uncalibrated reconstruction despite the lack of knowledge of

    • Autocalibration (recover from uncalibrated images)

  • Use partial knowledge

    • Parallel lines, vanishing points, planar motion, constant intrinsic

  • Ambiguities, stratification (multiple views)

Taxonomy on Uncalibrated Reconstruction

ICRA2003, Taipei


Calibration with a rig
Calibration with a Rig

Use the fact that both 3-D and 2-D coordinates of feature

points on a pre-fabricated object (e.g., a cube) are known.

ICRA2003, Taipei


Calibration with a rig1

  • Eliminate unknown scales

  • Recover projection matrix

  • Factor the into and using QR decomposition

  • Solve for translation

Calibration with a Rig

ICRA2003, Taipei


Uncalibrated camera vs distorted space
Uncalibrated Camera vs. Distorted Space

  • Inner product in Euclidean space: compute

    distances and angles

  • Calibration transforming spatial coordinates

  • Transformation induced a new inner product

  • (the metric of the space) and are equivalent

ICRA2003, Taipei



Calibrated vs uncalibrated space1
Calibrated vs. Uncalibrated Space

Distances and angles are modified by

ICRA2003, Taipei


Motion in the distorted space

Calibrated space

Uncalibrated space

  • Uncalibrated coordinates are related by

  • Conjugate of the Euclidean group

Motion in the Distorted Space

ICRA2003, Taipei


Uncalibrated camera or distorted space
Uncalibrated Camera or Distorted Space

Uncalibrated camera with a calibration matrix K viewing points in Euclidean space and moving with (R,T) is equivalent to a calibrated camera viewing points in distorted space governed by S and moving with a motion conjugate to (R,T)

ICRA2003, Taipei


Uncalibrated epipolar geometry

Uncalibrated Epipolar Geometry

ICRA2003, Taipei


Properties of the fundamental matrix
Properties of the Fundamental Matrix

  • Epipolar lines

  • Epipoles

Image

correspondences

ICRA2003, Taipei


Properties of the fundamental matrix1
Properties of the Fundamental Matrix

A nonzero matrix is a fundamental matrix if has

a singular value decomposition (SVD) with

for some .

There is little structure in the matrix except that

ICRA2003, Taipei


What does f tell us

can be inferred from point matches (eight-point algorithm)

Cannot extract motion, structure and calibration from one fundamental matrix (two views)

allows reconstruction up to a projective transformation (as we will see soon)

encodes all the geometric information among two views when no additional information is available

What Does F Tell Us?

ICRA2003, Taipei


Decomposing the fundamental matrix

  • Decomposition of is not unique

  • Unknown parameters - ambiguity

  • Corresponding projection matrix

Decomposing the Fundamental Matrix

ICRA2003, Taipei


Ambiguities in image formation
Ambiguities in Image Formation algorithm)

  • Potential ambiguities

  • Ambiguity in (can be recovered uniquely – QR)

  • Structure of the motion parameters

  • Just an arbitrary choice of reference coordinate frame

ICRA2003, Taipei


Ambiguities in image formation1
Ambiguities in Image Formation algorithm)

Structure of motion parameters

ICRA2003, Taipei


Ambiguities in image formation2
Ambiguities in Image Formation algorithm)

Structure of the projection matrix

  • For any invertible 4 x 4 matrix

  • In the uncalibrated case we cannot distinguish between camera imaging word from camera imaging distorted world

  • In general, is of the following form

  • In order to preserve the choice of the first reference frame we can restrict some DOF of

ICRA2003, Taipei


Structure of the projective ambiguity

Structure of the Projective Ambiguity

  • For i-th frame

  • 1st frame as reference

  • Choose the projective reference frame

then ambiguity is

ICRA2003, Taipei


Geometric stratification cont
Geometric Stratification (cont) algorithm)

ICRA2003, Taipei


Projective reconstruction
Projective Reconstruction algorithm)

  • From points, extract , from which extract and

  • Canonical decomposition

  • Projection matrices

ICRA2003, Taipei


Projective reconstruction1
Projective Reconstruction algorithm)

  • Given projection matrices recover projective structure

  • This is a linear problem and can be solve using least-squares techniques.

  • Given 2 images and no prior information, the scene can be recovered up a 4-parameter family of solutions. This is the best one can do without knowing calibration!

ICRA2003, Taipei


Affine upgrade
Affine Upgrade algorithm)

  • Upgrade projective structure to an affine structure

  • Exploit partial scene knowledge

    • Vanishing points, no skew, known principal point

  • Special motions

    • Pure rotation, pure translation, planar motion, rectilinear motion

  • Constant camera parameters (multi-view)

ICRA2003, Taipei


Affine upgrade using vanishing points
Affine Upgrade Using Vanishing Points algorithm)

maps points on the plane

to points with affine coordinates

ICRA2003, Taipei


Vanishing point estimation from parallelism
Vanishing Point Estimation from Parallelism algorithm)

ICRA2003, Taipei


Euclidean upgrade

Exploit special motions (e.g. pure rotation) algorithm)

If Euclidean is the goal, perform Euclidean reconstruction directly (no stratification)

Direct autocalibration (Kruppa’s equations)

Multiple-view case (absolute quadric)

Euclidean Upgrade

ICRA2003, Taipei


Direct autocalibration methods
Direct Autocalibration Methods algorithm)

The fundamental matrix

satisfies the Kruppa’s equations

If the fundamental matrix is known up to scale

Under special motions, Kruppa’s equations become linear.

Solution to Kruppa’s equations is sensitive to noises.

ICRA2003, Taipei


Direct stratification from multiple views
Direct Stratification from Multiple Views algorithm)

From the recovered projective projection matrix

we obtain the absolute quadric contraints

Partial knowledge in (e.g. zero skew, square pixel) renders

the above constraints linear and easier to solve.

The projection matrices can be recovered from the multiple-view

rank method to be introduced later.

ICRA2003, Taipei


Direct methods summary
Direct Methods - Summary algorithm)

ICRA2003, Taipei


Summary of auto calibration methods
Summary of (Auto)calibration Methods algorithm)

ICRA2003, Taipei


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