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Motion from normal flow . The aperture problem. Depth discontinuities. Optical flow difficulties. Translational Normal Flow. In the case of translation each normal flow vector constrains the location of the FOE to a half-plane. Intersection of half-planes provides FOE.

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Optical flow difficulties l.jpg

The aperture problem

Depth discontinuities

Optical flow difficulties


Translational normal flow l.jpg
Translational Normal Flow

  • In the case of translation each normal flow vector constrains the location of the FOE to ahalf-plane.

  • Intersection of half-planes provides FOE.


Egoestimation from normal flow l.jpg
Egoestimation from normal flow

  • Idea: choose particular directions: patterns defined on the sign of normal flow along particular orientation fields

  • positive depth constraint

  • 2 classes of orientation fields: copoint vectors and coaxis vectors




Coaxis vectors l.jpg
Coaxis vectors

with respect to axis (A,B,C)




Translational coaxis vectors10 l.jpg
Translational coaxis vectors

h passes through FOE and (Af/C, Bf/C), defined by 2 parameters




Rotational coaxis vectors13 l.jpg
Rotational coaxis vectors

g passes through AOR and (Af/C, Bf/C), defined by 1 parameter


Combine translation and rotation l.jpg
Combine translation and rotation

Positive + positive  positive

Negative + negative negative

Positive + negative don’t know (depends on structure)


Coaxis pattern l.jpg
Coaxis pattern

translational

rotational

combined


G vectors translation l.jpg
g-vectors: Translation


G vectors rotation l.jpg
g-vectors: Rotation


G vectors translation and rotation l.jpg
g-vectors: Translation and Rotation


Slide19 l.jpg

Three coaxis vector fields

alpha beta gamma




Copoint vectors22 l.jpg
Copoint vectors

defined by point (r,s)



Translational copoint vectors25 l.jpg
Translational copoint vectors

k passes through FOE and (r,s) defined by 1 parameter



Rotational copoint vectors27 l.jpg
Rotational copoint vectors

l passes through AOR and (r,s), is defined by 2 parameters


Slide28 l.jpg

translational component

rotational component



Slide30 l.jpg

a,b,c : positive and negative

a,b,g vectors

c,d,e: Fitting of a,b,g patterns

g: Separation of (a,b,g) coaxis pattern

h: Separation of (x0, y0) copoint pattern


Slide33 l.jpg

Optical

illusion


What is the problem l.jpg
What is the Problem?

  • Flow can be accurately estimated in an image patch corresponding to a smooth scene patch,

  • But erroneous flow estimates are obtained for image patches corresponding to scene patches containing discontinuities

Image Flow

Scene structure

Discontinuities

3D Motion


Depth variability constraint l.jpg
Depth variability constraint

  • Errors in motion estimates lead to distortion of the scene estimates.

  • The distortion is such that the correct motion gives the “smoothest” (least varying) scene structure.


Depth estimation l.jpg
Depth estimation

  • Scene depth can be estimated from normal flow measurements:


Visual space distortion l.jpg
Visual Space Distortion

  • Wrong 3D motion gives rise to a rugged (unsmooth) depth function (surface).

  • The correct 3D motion leads to the “smoothest” estimated depth.


Inverse depth estimates l.jpg

correct motion

incorrect motion

Inverse depth estimates


The error function l.jpg
The error function

  • A normal flow measurement:

    For an estimate

  • The error function to be minimized:


The error function41 l.jpg
The error function

  • Estimated normal flow

  • The error function to be minimized:

  • Global parameters:

  • Local parameter:

    locally planar patches:


Error function evaluation l.jpg
Error function evaluation

  • Given a translation candidate , each local depth can be computed as a linear function of the rotation .

  • We obtain a second order function of the rotation; its minimization provides both the rotation and the value of the error function.


Is derived from image gradients only l.jpg
Is derived from image gradients only

Brightness consistency:

Flow:

Planar patch:


Handling depth discontinuities l.jpg
Handling depth discontinuities

  • Given a candidate motion, the scene depth can be estimated and further processed to find depth discontinuities.

  • Split a region if it corresponds to two depth values separated in space.


The algorithm l.jpg
The algorithm

  • Compute spatio-temporal image derivatives and normal flow.

  • Find the direction of translation that minimizes the depth-variability criterion.

    • Hierarchical search of the 2D space.

    • Iterative minimization.

    • Utilize continuity of the solution in time; usually the motion changes slowly over time.


The algorithm46 l.jpg

Depth variation small?

Divide image into small patches

YES

Search in the 2D space of translations

Use the error

NO

Distinguish between two cases

For each candidate 3D motion, using normal flow measurements in each patch, compute depth of the scene.

existence of a discontinuity at the patch

wrong 3D motion

For each image patch

Use the error

Split the patch and repeat the process

The Algorithm


3d reconstruction l.jpg
3D reconstruction

  • Comparison of the original sequence and the re-projection of the 3D reconstruction.



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