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Lecture 8.2

Structure from Motion

Thomas Opsahl

More-than-two-view geometry

2

Correspondences (matching)

•More views enables us to reveal and remove more mismatches than we can do in the two view case •More views also enables us to predict correspondences that can be tested with or without the use of descriptors

Scene geometry (structure)

•Effect of more views on determining the 3D structure of the scene?

Camera geometry (motion)

•Effect of more views on determining camera poses?

Structure from Motion

Problem

Given ݉ images of ݊ fixed 3D points, estimate the

݉ projection matrices ܲ

and the ݊ points ࢄ from the ݉ή݊ correspondences ࢛ 3

Structure from Motion

Problem

Given ݉ images of ݊ fixed 3D points, estimate the

݉ projection matrices ܲ

and the ݊ points ࢄ from the ݉ή݊ correspondences ࢛ •We can solve for structure and motion when •In the general/uncalibrated case, cameras and points can only be recovered up to a projective ambiguity (࢛ •In the calibrated case, they can be recovered up to a similarity (scale) -Known as Euclidean/metric reconstruction 4

Structure from motion

Problem

Given ݉ images of ݊ fixed 3D points, estimate the

݉ projection matrices ܲ

and the ݊ points ࢄ from the ݉ή݊ correspondences ࢛ •We can solve for structure and motion when •In the general/uncalibrated case, cameras and points can only be recovered up to a projective ambiguity (࢛ •In the calibrated case, they can be recovered up to a similarity (scale) -Known as Euclidean/metric reconstruction 5 Images courtesy of Hartley & Zisserman http://www.robots.ox.ac.uk/~vgg/hzbook/

Projective reconstruction

Metric reconstruction

Structure from motion

Problem

Given ݉ images of ݊ fixed 3D points, estimate the

݉ projection matrices ܲ

and the ݊ points ࢄ from the ݉ή݊ correspondences ࢛ •We can solve for structure and motion when •In the general/uncalibrated case, cameras and points can only be recovered up to a projective ambiguity (࢛ •In the calibrated case, they can be recovered up to a similarity (scale) -Known as Euclidean/metric reconstruction 6 •This problem has been studied extensively and several different approaches have been suggested •We will take a look at a couple of these -Sequential structure from motion -Bundle adjustment

Sequential structure from motion

7

Structure

Cameras

݆ •Initialize motion from two images

•Initialize the 3D structure by triangulation

Sequential structure from motion

8

Structure

Cameras

݆ •Initialize motion from two images

•Initialize the 3D structure by triangulation •For each additional view -Determine the projection matrix ܲ , e.g. from 2D

3D correspondences ࢛

-Refine and extend the 3D structure by triangulation

Sequential structure from motion

9

Structure

Cameras

New points

observed

New observations of

old points

݆ •Initialize motion from two images

•Initialize the 3D structure by triangulation •For each additional view -Determine the projection matrix ܲ , e.g. from 2D

3D correspondences ࢛

-Refine and extend the 3D structure by triangulation

Sequential structure from motion

10

Structure

Cameras

Extended structure

Refine old structure by

triangulation over 3 views

݆ •Initialize motion from two images

•Initialize the 3D structure by triangulation •For each additional view -Determine the projection matrix ܲ , e.g. from 2D

3D correspondences ࢛

-Refine and extend the 3D structure by triangulation

Sequential structure from motion

11

Structure

Cameras

݆ •Initialize motion from two images

•Initialize the 3D structure by triangulation •For each additional view -Determine the projection matrix ܲ , e.g. from 2D

3D correspondences ࢛

-Refine and extend the 3D structure by triangulation

Sequential structure from motion

12

Structure

Cameras

Extend

structure

Refine structure

•Initialize motion from two images •Initialize the 3D structure by triangulation •For each additional view -Determine the projection matrix ܲ , e.g. from 2D

3D correspondences ࢛

-Refine and extend the 3D structure by triangulation

Sequential structure from motion

13

Structure

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