Uncalibrated geometry stratification
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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

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Uncalibrated geometry stratification

Uncalibrated Geometry & Stratification

Stefano Soatto


Yi Ma



Calibration with a rig

Uncalibrated epipolar geometry

Ambiguities in image formation

Stratified reconstruction

Autocalibration with partial scene knowledge


ICRA2003, Taipei

Uncalibrated camera



Linear transformation



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

  • Given 3-D coordinates on known object

  • 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 space

Calibrated vs. Uncalibrated Space

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

  • Epipolar constraint

  • Fundamental matrix

  • Equivalent forms of

Uncalibrated Epipolar Geometry

ICRA2003, Taipei

Properties of the fundamental matrix

Properties of the Fundamental Matrix

  • Epipolar lines

  • Epipoles



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 the fundamental matrix into a skew

  • symmetric matrix and a nonsingular 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

  • 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

Structure of motion parameters

ICRA2003, Taipei

Ambiguities in image formation2

Ambiguities in Image Formation

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

  • can be further decomposed as

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)

ICRA2003, Taipei

Projective reconstruction

Projective Reconstruction

  • From points, extract , from which extract and

  • Canonical decomposition

  • Projection matrices

ICRA2003, Taipei

Projective reconstruction1

Projective Reconstruction

  • 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

  • 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

maps points on the plane

to points with affine coordinates

ICRA2003, Taipei

Vanishing point estimation from parallelism

Vanishing Point Estimation from Parallelism

ICRA2003, Taipei

Euclidean upgrade

Exploit special motions (e.g. pure rotation)

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

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

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

ICRA2003, Taipei

Summary of auto calibration methods

Summary of (Auto)calibration Methods

ICRA2003, Taipei

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