Uncalibrated reconstruction

Uncalibrated reconstruction

Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial scene knowledge

S. Soatto, J. Kosecka

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Uncalibrated Camera

• Image Plane Coordinates

• Perspective projection

Calibrated camera

• Pixel coordinates

• Camera intrinsic parameters

• Projection Matrix

Uncalibrated camera

xyc

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Taxonomy of uncalibrated reconstruction

K is known, back to calibrated case K is unknown

Calibration (with a rig) – estimate K Uncalibrated reconstruction despite the lack

of knowledge of K Autocalibration (recover K from images)

Use complete scene knowledge (a rig) Use partial knowledge

Parallel lines, vanishing points, planar motion, constant intrinsic

Ambiguities, stratification (multiple views)

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Calibration with a rig

• Given 3-D coordinates on known object

• Eliminate unknown scales

• Recover Projection Matrix•

• Factor the into and using QR decomposition• Solve for translation

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Uncalibrated camera vs. distorted space

Inner product in Euclidean space : compute distances and angles

Calibration K transforming spatial coordinates

Transformation induced on the inner product

S (the metric of the space) and K are equivalent

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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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Uncalibrated Camera or Distorted Space

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

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Uncalibrated Epipolar Geometry

• Fundamental matrix

• equivalent forms of F

• Epipolar Constraint

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Imagecorrespondences

• Epipolar lines• Epipoles

Properties of the fundamental matrix

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Characterization 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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Decomposing the fundamental matrix• Decomposition of the Fundamental matrix into skewsymmetric matrix and nonsingular matrix

• Decomposition of F is not unique

• Unknown parameters - ambiguity

• Corresponding projection matrix

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What does F tell us?

F can be inferred from point matches (8-point algorithm)

Cannot extract motion, structure and calibration from one fundamental matrix (2 views)

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

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

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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 the image formationStructure of the Motion Parameters

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Ambiguities in the image formation

For any invertible 4 x 4 matrix H In the uncalibrated case we cannot

distinguish between camera imaging word

from camera imaging distorted world In general, H is of the following form

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

Structure of the projection matrix

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Structure of the projective ambiguity

• H can be further decomposed as

• For i-th frame

• 1st frame as reference

• Choose the projective reference frame

then ambiguity is

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Geometric stratification (contd.)

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Projective reconstruction

From points, extract F; from F extract X

Canonical decomposition

Projection matrices

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Projective reconstruction• Given projection matrices recover projective structure

• Given 2 images and no prior information, the scene canbe recovered up a a 4-parameter family of solutions. This is the best one can do!

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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

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Vanishing point estimation

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Euclidean upgrade

Exploit special motions (e.g. pure rotation)

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

Multi-view reconstruction (tomorrow)

Autocalibration (Kruppa)

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Overview of the methods

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Direct stratification for multiple views

Absolute Quadric Constraint

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Direct methods - Summary

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Summary