MASKS © 2004 Invitation to 3D vision

Uncalibrated Camera

Uncalibrated Geometry & Stratification

Sastry and Yang

MASKS © 2004 Invitation to 3D vision

Uncalibrated Camera

pixelcoordinates

calibratedcoordinates

Linear transformation

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Overview

• Calibration with a rig

• Uncalibrated epipolar geometry

• Ambiguities in image formation

• Stratified reconstruction

• Autocalibration with partial scene knowledge

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Uncalibrated Camera

xyc

• Pixel coordinates

• Projection matrix

Uncalibrated camera

• Image plane coordinates • Camera extrinsic parameters

• Perspective projection

Calibrated camera

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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)

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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.

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Calibration with a Rig

• Eliminate unknown scales

• Factor the into and using QR decomposition

• Solve for translation

• Recover projection matrix

• Given 3-D coordinates on known object

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Uncalibrated Epipolar Geometry

• Epipolar constraint

• Fundamental matrix

• Equivalent forms of

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Properties of the Fundamental Matrix

Imagecorrespondences

• Epipolar lines

• Epipoles

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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

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• Fundamental matrix can be estimated up to scale

• Find such F that the epipolar error is minimized

Estimating Fundamental Matrix

• Denote

• Rewrite

• Collect constraints from all points

Pixel coordinates

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• Project onto the essential manifold:

• cannot be unambiguously decomposed into pose and calibration

Two view linear algorithm – 8-point algorithm

• Solve the LLSE problem:

• Compute SVD of F recovered from data

• Solution eigenvector associated with smallest eigenvalue of

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Calibrated vs. Uncalibrated Space

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Calibrated vs. Uncalibrated Space

Distances and angles are modified by S

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Motion in the distorted space

• Uncalibrated coordinates are related by

• Conjugate of the Euclidean group

Calibrated space Uncalibrated space

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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

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Decomposing the Fundamental Matrix

• Decomposition of is not unique

• Unknown parameters - ambiguity

• Corresponding projection matrix

• Decomposition of the fundamental matrix into a skew symmetric matrix and a nonsingular matrix

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Projective Reconstruction

• From points, extract , followed by computation of projection matrices and structure • Canonical decomposition

• Given projection matrices – recover structure

• Projective ambiguity – non-singular 4x4 matrix

Both and are consistent with the epipolar geometry – give the same fundamental matrix

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Projective Reconstruction

• Given projection matrices recover projective structure

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

• Projective reconstruction – projective camera matrices and projective structure

Euclidean Structure Projective Structure

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Euclidean vs Projective reconstruction

• Euclidean reconstruction – true metric properties of objects lenghts (distances), angles, parallelism are preserved • Unchanged under rigid body transformations• => Euclidean Geometry – properties of rigid bodies under rigid body transformations, similarity transformation

• Projective reconstruction – lengths, angles, parallelism are NOT

preserved – we get distorted images of objects – their distorted 3D counterparts --> 3D projective reconstruction

• => Projective Geometry

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Homogeneous Coordinates (RBM)

3-D coordinates are related by:

Homogeneous coordinates:

Homogeneous coordinates are related by:

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Homogenous and Projective Coordinates

• Homogenous coordinates in 3D before – attach 1 as the last

coordinate – render the transformation as linear transformation

• Projective coordinates – all points are equivalent up to a scale

• Each point on the plane is represented by a ray in projective space

2D- projective plane 3D- projective space

• Ideal points – last coordinate is 0 – ray parallel to the image plane • points at infinity – never intersects the image plane

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Vanishing points – points at infinity

Representation of a 3-D line – in homogeneous coordinates

When -> 1 - vanishing points – last coordinate -> 0

Similarly in the image plane

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Ambiguities in the image formation

• Potential Ambiguities

• Ambiguity in K (K can be recovered uniquely – Cholesky or QR)

• Structure of the motion parameters

• Just an arbitrary choice of reference frame

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Ambiguities in Image Formation

• 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

Structure of the (uncalibrated) projection matrix

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Structure of the Projective Ambiguity

• 1st frame as reference

• Choose the projective reference frame

then ambiguity is

• can be further decomposed as

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Stratified (Euclidean) Reconstruction

• General ambiguity – while preserving choice of first reference

frame

• Decomposing the ambiguity into affine and projective one

• Note the different effect of the 4-th homogeneous coordinate

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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)

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Affine upgrade using vanishing points

Maps the points

To points with affine coordinates

How to compute

Vanishing points – last homogeneous affine coordinate is 0

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Affine Upgrade

Need at least three vanishing points

3 equations, 4 unknowns (-1 scale) Solve for

Set up and update the projectivestructure

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Euclidean upgrade

• We need to estimate remaining affine ambiguity

• In the case of special motions (e.g. pure rotation) – no projective ambiguity – cannot do projective reconstruction

• Estimate directly (special case of rotating camera – follows)

• Multi-view case – estimate projective and affine ambiguity together

• Use additional constraints of the scene structure (next)• Autocalibration (Kruppa equations)

Alternatives:

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Euclidean Upgrade using vanishing points

• Vanishing points – intersections of the parallel lines

• Vanishing points of three orthogonal directions

• Orthogonal directions – inner product is zero

• Provide directly constraints on matrix

• S – has 5 degrees of freedom, 3 vanishing points – 3 constraints (need additional assumption about K)

• Assume zero skew and aspect ratio = 1

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Geometric Stratification

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Special Motions – Pure Rotation

• Calibrated Two views related by rotation only

• Mapping to a reference view – rotation can be estimated

• Mapping to a cylindrical surface

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Pure Rotation - Uncalibrated Case

• Calibrated Two views related by rotation only

• Conjugate rotation can be estimated

• Given C, how much does it tell us about K (or S) ?

• Given three rotations around linearly independent axes – S, K can be estimated using linear techniques• Applications – image mosaics

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Overview of the methods

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Summary

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Direct Autocalibration Methods

satisfies the Kruppa’s equations

The fundamental matrix

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.

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Direct Stratification from Multiple Views

we obtain the absolute quadric contraints

From the recovered projective projection matrix

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.

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Summary of (Auto)calibration Methods