3D from Pictures

Jiajun ZhuSept.29 2006

University of Virginia

What can we compute from a collection of

pictures?

- 3D structure- camera poses and

parameters

One of the most important / exciting results in computer vision from 90s’

It is difficult, largely due to numerical computation in practice.

But this is SO powerful!!!

2 SIGGRAPH papers with several sketches this year!

show a few demo videos

Now let’s see how this works!

Input: (1) A collection of pictures.

Output:(1) camera parameters(2) sparse 3D scene

structure

Consider 1 camera first

What’s the relation between pixels and rays in space?

10100

ZYX

ff

ZfYfX

1010101

1ZYX

ff

ZfYfX PXx

0|I)1,,(diagP ff

C~-X~RX~ cam

X10RCR

110C~RRXcam

ZYX

camX0|IKx XC~|IKRx

~

C~|IKRP P is a 3x4 Matrix7 degree of freedom:1 from focal length3 from rotation3 from translation

t|RKP

Simplified projective camera model P

x = P X = K [ R | t ] X

x = P X

Consider 1 cameraP3x4 has 7 degrees of freedom

Given one image, we observe xCan we recover X or P?

If P is known, what do we know about X?If X is known, can we recover P?

# unknown = 7Each X gives 2 equations

2n >= 7 i.e. n >= 4

This is a Camera Calibration Problem

Input: n>4 world to image point correspondences {Xi xi}

Output:camera parameters P = K[R|T]

Direct Linear Transform (DLT)ii PXx

ii PXx where [Xi]x =

0 -w yw 0 –x-y x 0

Direct Linear Transform (DLT)

n 4 pointsApminimize subject to constraint 1p

use SVD TVUΣA

p is the last column vector of V: p = Vn

ObjectiveGiven n≥4, 3D to 2D point correspondences {Xi↔xi’}, determine P

Algorithm(i) Linear solution:

(a) Normalization: (b) DLT

(ii) Minimization of geometric error: Iteratively optimization (Levenberg-Marquardt):

(iii)Denormalization:

ii UXX~ ii Txx~

UP~TP -1

~~~

Implementation in Practice

Camera centre C is the point for which PC = 0i.e. the right null vector of P

~ ~

ObjectiveGiven camera projection matrix P, decompose P = K[R|t]

Algorithm

Perform RQ decomposition of M, so that K is the upper-triangular matrix and R is orthonormal matrix.

write M = KR, then P = M[I|- C]~

How to recover K, R and t from P?

~P = K[R|t] = K[R|-RC] = KR[I|-C]

~

This is what we learn from 1 Camera

Let’s consider 2 cameras

(i) Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the corresponding point x’ in the second image?

(ii) Camera geometry (motion): Given a set of corresponding image points {xi ↔x’i}, i=1,…,n, what are the cameras P and P’ for the two views?

(i)Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the corresponding point x’ in the second image?

The Fundamental Matrix F

x’T Fx = 0

What does Fundamental Matrix F tell us?

x’T Fx = 0

Fundamental matrix F relates corresponding pixels

If the intrinsic parameter (i.e. focal length in our camera model) of both cameras are known, as K and K’.Then we can derive (not here) that: K’TFK = t cross product

R

t and R are translation and rotation for the 2nd camera

i.e. P = [I|0] and P’ = [R|t]

Good thing is that …

x’T Fx = 0Fundamental matrix F can be computed:

from a set of pixel correspondences: {x’ x}

Compute F from correspondence:

0Fxx'T

separate known from unknown

0'''''' 333231232221131211 fyfxffyyfyxfyfxyfxxfx

0,,,,,,,,1,,,',',',',',' T333231232221131211 fffffffffyxyyyxyxyxxx

(data) (unknowns)(linear)

0Af

0f1''''''

1'''''' 111111111111

nnnnnnnnnnnn yxyyyxyxyxxx

yxyyyxyxyxxx

How many correspondences do we need?

What can we do now?(1) Given F, K and K’, we can estimate the relative translationand rotation for two cameras:

(2) Given 8 correspondences: {x’ x}, we can compute F

P = [I | 0] and P’ = [R | t]

Given K and K’, and 8 correspondences {x’ x}, we can compute: P = [I | 0] and P’ = [R | t]

This answers the 2nd question

(i) Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the corresponding point x’ in the second image?

(ii) Camera geometry (motion): Given a set of corresponding image points {xi ↔x’i}, i=1,…,n, what are the cameras P and P’ for the two views?

But how to make this automatic?

Given K and K’, and 8 correspondences {x’ x}, we can compute: P = [I | 0] and P’ = [R | t] (1) Estimating intrinsic K and K’ (auto-calibration)

will not be discussed here. (involve much projective geometry knowledge)(2) Let’s see how to find correspondences automatically. (i.e. Feature detection and matching)

Lowe’s SIFT features invariant to with position, orientation and

scale

Scale• Look for strong responses of DOG filter

(Difference-Of-Gaussian) over scale space

• Only consider local maxima in both position and scale

Orientation• Create histogram of local

gradient directions computed at selected scale

• Assign canonical orientation at peak of smoothed histogram

• Each key specifies stable 2D coordinates (x, y, scale, orientation)

0 2

Simple matchingFor each feature in image 1 find the feature in

image 2 that is most similar (compute correlation of two vectors) and vice-versa

Keep mutual best matchesCan design a very robust RANSAC type

algorithm

What have we learnt so far?

What have we learnt so far?

Consider more then 2 cameras

KK’

PP’

X

P’’

Objective

Given N images { Q1, …, QN } with reasonable overlaps

Compute N camera projection matrices { P1, …,

PN }, where each Pi = Ki[Ri |ti], Ki is the intrinsic parameter, Ri and ti are rotation and translation matrix respectively

Algorithm(1) Find M tracks T = {T1, T2, …, TN }

(i ) for every pair of image {Qi , Qj}: detect SIFT feature points in Qi and Qj

match feature points robustly (RANSAC)(ii) match features across multiple images, construct tracks.

(2) Estimate { P1… PN } and 3D position for each track { X1… XN }

(i ) select one pair of image {Q1’ , Q2’} (well-conditioned). Let T1’2’ = {their associate overlapping track};(ii) Estimate K1’ and K2’, compute {P1’ , P2’} and 3D position of

T1’2’ from fundamental matrix.(iii) incrementally add new camera Pk into the system, estimate

its camera matrix by DLT (calibration) (iv) repeat (iii) until all the cameras are estimated.

Algorithm(1) Find M tracks T = {T1, T2, …, TN }

(i ) for every pair of image {Qi , Qj}: detect SIFT feature points in Qi and Qj

match feature points robustly (RANSAC)(ii) match features across multiple images, construct tracks.

(2) Estimate { P1… PN } and 3D position for each track { X1… XN }

(i ) select one pair of image {Q1’ , Q2’} (well-conditioned). Let T1’2’ = {their associate overlapping track};(ii) Estimate K1’ and K2’, compute {P1’ , P2’} and 3D position of

T1’2’ from fundamental matrix.(iii) incrementally add new camera Pk into the system, estimate

its camera matrix by DLT (calibration) (iv) repeat (iii) until all the cameras are estimated.

However, this won’t work!

Algorithm(1) Find M tracks T = {T1, T2, …, TN }

(i ) for every pair of image {Qi , Qj}: detect SIFT feature points in Qi and Qj

match feature points robustly (RANSAC)(ii) match features across multiple images, construct tracks.

(2) Estimate { P1… PN } and 3D position for each track { X1… XN }

(i ) select one pair of image {Q1’ , Q2’} (well-conditioned). Let T1’2’ = {their associate overlapping track};(ii) Estimate K1’ and K2’, compute {P1’ , P2’} and 3D position of

T1’2’ from fundamental matrix. Then non-linearly minimize reprojection errors (LM).

(iii) incrementally add new camera Pk into the system, estimate initial value by DLT, then non-linearly optimize the system.

(iv) repeat (iii) until all the cameras are estimated.

Replaces with more robust non-linear optimization

Tired?

Recall the camera calibration algorithm

ObjectiveGiven n≥4, 3D to 2D point correspondences {Xi↔xi’}, determine P

Algorithm(i) Linear solution:

(a) Normalization: (b) DLT

(ii) Minimization of geometric error: Iteratively optimization (Levenberg-Marquardt):

(iii)Denormalization:

ii UXX~ ii Txx~

UP~TP -1

~~~

We are lucky! 1st time huge amount of visual data is easily accessible. High-level description of these data also become available. How do we explore them? Analysis them? Wisely use them?

What’s the contribution of this paper?

How to extract high-level information?- Computer Vision, Machine Learning Tools. Structure from motion, and more computer vision tools reach a certain robust point for graphics application.- InternetImage search

- Human Labelgame with purpose

What is the space of all the pictures?

in the past

present

the future?

What’s the space of all the videos?

in the past

present

the future?

What else?

Using Search Engine?

Using human computation power?

Using human computation power?

Using human computation power?

What else?

What else?

Book:“Multiple View Geometry in Computer Vision” Hartley and Zisserman

Online Tutorial:http://www.cs.unc.edu/~marc/tutorial.pdfhttp://www.cs.unc.edu/~marc/tutorial/

Matlab Toolbox:http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/TORR1/index.html