Combining Photometric and Geometric Constraints

Y. Moses 1

Combining Photometric and Geometric

Constraints

Yael Moses

IDC, Herzliya

Joint work with Ilan Shimshoni and Michael Lindenbaum, the Technion

Y. Moses 2

Recover the 3D shape of a general smooth surface from a set of calibrated images

Problem 1:

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Problem 2:

Recover the 3D shape of a smooth bilaterally symmetric object from a single image.

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Shape Recovery

Geometry: Stereo Photometry:

Shape from shading Photometric stereo

Main problems: Calibrations and Correspondence

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3D Shape Recovery

Photometry: Shape from

shading Photometric

stereo

Geometry:Stereo

Structure from motion

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Geometric Stereo

2 different images Known camera parameters Known correspondence

+ +

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Photometric Stereo

3D shape recovery: surface normals from two or more

images taken from the same viewpoint

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Photometric Stereo

Solution:

)ˆ(

3

2

1

3

2

1

n

l

l

l

I

I

I

IT

T

T

LnI

IL

ILn

1

1

nlyxI

nlyxI

nlyxI

ˆ),(

ˆ),(

ˆ),(

33

22

11

Three images Matrix notation

IL

ILn

1

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Photometric Stereo

3D shape recovery (surface normals) Two or more images taken from the

same viewpoint

Main Limitation:

Correspondence is obtained by a fixed viewpoint

IL

ILn

1

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Overview

Combining photometric and geometric stereo: Symmetric surface, single image Non symmetric: 3 images

Mono-Geometric stereo Mono-Photometric stereo Experimental results.

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The input Smooth featureless surface Taken under different viewpoints Illuminated by different light sources

The Problem: Recover the 3D shape from a set of calibrated images

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Assumptions

Given correspondence the normals can be computed (e.g., Lambertian, distant point light source …)

*

*

ln

nlI ˆ

n

Three or more images

Perspective projection

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Our method

Combines photometric and geometric stereo

We make use of:

Given Correspondence:

Can compute a normal

Can compute the 3D point

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IL

ILn

1

1

ˆ

Basic Method

GivenCorrespondence

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First Order Surface Approximation

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P() = (1 - )O1 + P,

N)O - (P

N)O - (PT

1

T1

N (P() - P) = 0

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PMp ii

New Correspondence

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IL

ILn

1

1

ˆ

New Surface Approximation

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Dense Correspondence

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Basic Propagation

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Basic Propagation

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Basic method: First Order

Given correspondence pi and L

P and n Given P and n T Given P, T and Mi

a new correspondence qi

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Extensions

Using more than three images Propagation:

Using multi-neighbours Smart propagation

Second error approximation Error correction:

Based on local continuity Other assumptions on the surface

IL

ILn

1

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Multi-neighbors Propagation

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Smart Propagation

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Second Order: a Sphere

P()

N+N

N

N

P

(P-P())(N+N)=0

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Second Order Approximation

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Second Order Approximation

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Using more than three images

Reduce noise of the photometric stereo

Avoid shadowed pixels Detect “bad pixels”

Noise Shadows Violation of assumptions on the surface

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Smart Propagation

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Error correction

The compatibility of the local 3D shape can be used to correct errors of:

Correspondence Camera parameters Illumination parameters

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Score

Continuity: Shape Normals Albedo

The consistency of 3D points locations and the computed normals: General case: full triangulation Local constraints

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Extensions

Using more than three images Propagation:

Using multi-neighbours Smart propagation

Second error approximation Error correction:

Based on local continuity Other assumptions on the surface

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Real Images

Camera calibration Light calibration

Direction Intensity Ambient

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Error correction + multi-neighbor5 Images

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5pp

3pp 3nn

5nn5pn

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Detected Correspondence

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Error correction + multi-neighbordMulti-neighborsBasic scheme (3 images)Error correction no multi-neighbors

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New ImagesSynthetic Images

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Sec a

Ground truthBasic schemeMulti-neighbors Error correction

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Sec b


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Sec c


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Ground truthBasic schemeMulti-neighbors

approx.Error correction

Sec d


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Combining Photometry and

Geometry

Yields a dense correspondence and dense shape recovery of the object

in a single path

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Assumptions

Bilaterally Symmetric object Lambertian surface with constant

albedo Orthographic projection Neither occlusions nor

shadows Known “epipolar geometry”

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Geometric Stereo

2 different images Known camera parameters Known viewpoints Known correspondence

3D shape recovery

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Computing the Depth from Disparity

pl

pr

P

ql

qrZ

Z

Orthographic

Projection

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Symmetry and Geometric Stereo

Non frontal view of a symmetric object

Two different images of the same object

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Symmetry and Geometric Stereo

Non frontal view of a symmetric object


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Geometry

Weak perspective projection:

)(~ tsRpprojp

Around X Around Z Around Y

zyx RRRR

xR zR yRI I

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Geometry

Projection of Ry:

010

)sin(0)cos()(' yRprojR

Around Y

is the only pose parameter

PRp '~

Image point

Object point

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objectx

z

image

),,( zyxP l Tr zyxP ),,(

)~,~(~ rrr yxp )~,~(~ lll yxp

Correspondence

Assume YxZ is the symmetry plane.

x~

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Mono-Geometric Stereo

3D reconstruction: given correspondence and ,

unknown

known

PRp '~

x~

z

image

x object

)sin()cos(~)sin()cos(~

~~

zxx

zxx

yyy

l

r

lr

xx

rx~lx~

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Viewpoint Invariant

Given the correspondence and unknown

2

1

22

11

~~

~~

x

x

pp

pplr

lr

*1

~p

*2~p

*3~p 3*

~p2*

~p1*

~p

lr pp 11~~


~~

zxx

zxx

yyy

l

r

lr

Invariant

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Photometric Stereo

2 images Lambertian reflectance Known illuminations Known correspondence

(same viewpoint)

3D shape recovery

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Symmetry and Photometric Stereo

Non-frontal illumination of a symmetric object


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Notation: Photometry

Corresponding object points:

Illumination:

Tyx

r zzn )1,,( Tyx

l zzn )1,,( Tzyx

r nnnn ),,(ˆ Tzyx

l nnnn ),,(ˆ

Tzyx eeeE ),,(

Tzyx

r nnnn ),,(ˆ Tzyx

l nnnn ),,(ˆ

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Mono-Photometric Stereo

3D reconstruction given correspondence and E (up to a twofold ambiguity):

unknown

known

zzyyxxr neneneI

zzyyxxl neneneI

)(2

2

zzyylr

xxlr

neneIII

neIII

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Invariance to Illumination

Given correspondence and E unknown

Invariant:x

x

n

n

I

I

2

1

2

1

)(2

2

zzyylr

xxlr

neneIII

neIII

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Mono-Photometric Stereo

3D reconstruction E unknown but correspondence is given

Frontal viewpoint with non-frontal illumination. Use image first derivatives.

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Mono-Photometric Stereo Using image derivatives

3 global unknowns: E For each pair:

5 unknowns zx zy zxx zxy zyy

6 equations 3 pairs are sufficient

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Mono-Photometric StereoUnknown Illumination

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Correspondence

No correspondence => no stereo. Hard to define correspondence in

images of smooth surfaces. Almost any correspondence is legal

when: Only geometric constraints are

considered. Only photometric constraints are

considered.

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Combining Photometry and

Geometry Yields a dense correspondence (dense shape recovery of the

object).

Enables recovering of the global parameters.

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Self-Correspondence A self-correspondence function:

lr ppC ~)~(

*~ lp

rp~*lrr xyxC ~)~,~(

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Dense Correspondence using Propagation

Assume correspondence between a pair of points, p0

l and p0r.

rr yxCydCxdpCpdpC ~~

~~)~()~~(

lp2~

rp0*~

rp1*~

rp2~

*1~ lp

*0~ lp

)~,~(~ ydxdpd

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Dense Correspondence using Propagation

* ***

**

lp2~

rp0*~

rp1*~

rp2~

*1~ lp

*0~ lp

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x

x~

z

image

objectn̂

dxdx

dz

)sin()cos(~ dzdxxd )cos(~ dxxd

xdxcxdxc ~)~()~~(

dx

dz

xCxdxCxdxC ~~)~()~~(

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Tyx zzn )1,,(

First derivatives of the Correspondence

Assume known Assume known E

)sin()cos(

)2sin(

;)sin()cos(

)sin()cos(

~

~

x

y

y

x

xx

z

zC

z

zC

r

r

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~~

zxxzxx

yyy

l

r

lr

Computing and

ryy

ryxy

rxy

rxxx

yCxCC

yCxCC

rr

rr

~~

~~

~~

~~

rr yxCC ~~

Object coordinates:Given computing and is trivial

Moving from object to image coordinates depends on the viewing parameter

rr yxCCn̂

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Derivatives with respect to the object coordinates:

Derivatives with respect to the image coordinates:

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x

x~

z

image

object

xd~

yx cc ~~

E

)~,~(~ rrr yxp

)~,~(~ lll yxp

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Given a corresponding pair and E n=(zx,zy,-1)T Given and n

cx and cy

Given cx and cy

a new corresponding pair

General Idea

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Results on Real Images: Given global parameters

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Finding Global Parameters

Assume E and are unknown. Assume a pair of corresponding

points is given. Two possibilities:

Search for E and directly.

Compute E and from the image second derivatives.

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All roads lead to Rome …

Find and verify correct correspondence

Recover global parameters, E and

Integration Constraint:Circular Tour

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Finding Global Parameters

Consider image second derivatives Due to foreshortening effect:

and

We can relate image and object derivatives by

xll

xxrr

xIIII rr ~~

ry

r

y

ry

r

x

ry

rx

r

y

rx

r

x

rx

yIxII

yIxII

rr

rr

~~

~~

~~

~~

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For each corresponding pair:

and

Plus 4 linear equations in 3 unknown.

Where

Testing E and : Image second derivatives

),,,(34

EzzfA yxx

yy

xy

xx

l

y

r

y

l

x

r

x

zzz

A

I

III

r

r

r

r

~

~

~

~

rrll nEInEI ˆˆ

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Counting

5 unknowns for each pair: zx zy,zxx zxy zyy

4 global unknowns: E, For each pair: 6 equations. For n pairs: 5n+4 unknowns

6n equations. 4 pairs are sufficient

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Results on Simulated Data

Ground Truth Recovered Shape

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Recovering the Global Parameters

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Degenerate Case

Close to frontal view: problems with geometric-stereo.

reconstruction problem Close to frontal illumination:

problems with photometric-stereo. correspondence problem

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Future work

Perspective photometric stereo Use as a first approximation to global

optimization methods Test on other reflection models Recovering of the global parameters:

Light Cameras

Detect the first pair of correspondence

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Future Work

Extend to general 3 images under 3 viewpoints and 3 illuminations.

Extend to non-lambertian surfaces.

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Assumptions

Bilaterally Symmetric object Lambertian surface with constant

albedo Orthographic projection Neither occlusions nor shadows Known “epipolar geometry”

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Thanks

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x

x~

z

image

object

)cos(~

dx

xd

n̂

dx

dz

dx

dz

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Integration Constraint

ry

r

y

ry

r

x

ry

rx

r

y

rx

r

x

rx

yIxII

yIxII

rr

rr

~~

~~

~~

~~

l

xylx

l

y

r

xyrx

r

y

x

lxl

x

x

rxr

x

lr

rr

r

r

IznEI

IznEI

z

nEI

z

nEI

~~

~~

~

~

)sin(ˆ

)sin(ˆ

)sin()cos(

ˆ

)sin()cos(

ˆ

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Integration Constraint

)(2

2

zzyy

xx

neneI

neI

l

xylx

l

y

r

xyrx

r

y

x

lxl

x

x

rxr

x

lr

rr

r

r

IznEI

IznEI

z

nEI

z

nEI

~~

~~

~

~

)sin(ˆ

)sin(ˆ

)sin()cos(

ˆ

)sin()cos(

ˆ

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Searching for E

Illumination must satisfy:

E is further constrained by the image second derivatives.

),max( lr IIE

)(2 zzyy neneI

xxneI 2

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Image second derivatives:

xll

x

xrr

x

yrl

y

yrr

y

II

II

II

II

r

r

r

r

~

~

~

~

lx

l

x

rx

r

x

ly

l

y

ry

r

y

nEI

nEI

nEI

nEI

r

r

r

r

ˆ

ˆ

ˆ

ˆ

~

~

~

~

),,,,(ˆ yyxyxxyxrx zzzzzn

yy

xy

xx

l

y

r

y

l

x

r

x

zzz

A

I

III

r

r

r

r

~

~

~

~ll nEI ˆrr nEI ˆ

Where ),,(34

EzzfA yxx

4 linear equations in 3 unknown

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For each corresponding pair and E: 4 linear equations in 3 unknown.

Where

Image second derivatives

),,(34

EzzfA yxx

yy

xy

xx

l

y

r

y

l

x

r

x

zzz

A

I

III

r

r

r

r

~

~

~

~

zzyyxxr neneneI

zzyyxxl neneneI

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Counting

5 unknowns for each pair: zx,zy,zxx,zxy,zyy

3 global unknowns: E For each pair: 6 equations. For n pairs: 5n+3 unknowns

6n equations.3 pairs are sufficient

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Correspondence

Tr zyxP ),,( Tl zyxP ),,(

)~,~(~ rrr yxp )~,~(~ lll yxp

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Variations

Known/unknown distant light source Known/unknown viewpoint Symmetric/non-symmetric image

Frontal/non-frontal viewpoint Frontal/non-frontal illumination

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Correspondence

Epipolar geometry is the only geometric constraint on the correspondence.

Weak photometric constraint on the correspondence.

)(2

2

zzyy

xx

neneI

neI

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Lambertian Surface

Basic radiometric

)cos(ˆ),(Re EnEqpf I =

E

*

*P

EE

n2

n1

n̂

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E

Photometric Stereo

First proposed by Woodham, 1980. Assume that we have two images ..

nEInEI ˆˆ 2211

Combining Photometric and Geometric Constraints

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