Stereo Imaging - Aegean

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Stereo ImagingStereo Imaging

Introduction

Example of Human VisionPerception of Depth from Left and right eye imagesDifference in relative position of object in left and right eyes.Depth information in the 2 views??Depth information in the 2 views??

Saurav Basu

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Stereopsis

Retinal disparityRetinal disparityp yp y– The horizontal distance between the corresponding left and right image points of the superimposed retinal images.

– The disparity is zero if the eyes are converged.

Stereopsis– The sense of depth combined from two different perspective views by the mind.

The Stereo Problem

– The stereo problem is usually broken in to two subproblemstwo subproblems•Extraction of Depth information from Stereo Pairs

•Use of depth data to visualize the world scene in 3‐dimensions by a suitable projection techniqueprojection technique.

Saurav Basu

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Stereo Images

Depth Estimation

Visualization

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What are Stereo Images?

Images of the same world scene taken from slightly displaced view points are called stereoslightly displaced view points are called stereo images.To illustrate how a typical stereo imaging system let us take a look at the camera model for obtaining stereo images

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Camera Model Of A Stereo System

Image 1

Image 2

Image 1

W (X,Y,Z)

BaseLine distance

y

x

x

y

(x2,y2)

(x1,y1)

Optical Axis

BaseLine distance

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Important Points about the Model

The cameras are identicalTh di t t f b thThe coordinate systems of both cameras are perfectly alligned.Once camera and world co‐ordinate systems are alligned the xy plane of the image is allignedwith the XY plane of the world co‐ordinate system then Z coordinate of W is same for bothsystem,then Z coordinate of W is same for both camera coordinate systems.

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Relating Depth with Image Coordinates

X

(x1,y1) Z - λZ

W (X,Y,Z)Origin Of

World Coordinate

B

λ

Image 1

Z λ

System λ

Image 2(x2,y2)

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By Similar Triangles:

)( 11

1 −= λλ

ZxX

)(

)()(

)(

21

22

111

1

21

12

22

2

−=−⇒

−=+⇒−=⇒

==+=

−=

λλ

λλ

λλ

λλ

λ

xxBZ

ZxBXZxX

ZZZBXX

ZxX

Disparity x2- x1where

Disparity

1

KB Depth, - Z

21

21

=

⇒−

=⇒

==

α

λλ

Depthxx

KDepth

Putxx

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Thus Depth is inversely proportional to (x1

Important Result

Thus Depth is inversely proportional to (x1‐x2) where x1 and x2 are pixel coordinates of the same world point when projected on the stereo image planes.(x1‐ x2) is called the DISPARITY The problem of finding x1 and x2 in theThe problem of finding x1 and x2 in the stereo pairs is done by a stereo matching technique.

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Stereo Matching

– The goal of stereo matching algorithms is to find matching locations in the left and rightfind matching locations in the left and right images .

– Specifically find the coordinates of the pixel on the left and right images which correspond to the same world point. It is also called the correspondence problem– It is also called the correspondence problem.

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Correlation based approaches

– A common approach to finding correspondences is to search for local regionscorrespondences is to search for local regions that appear similar

– Try to match a window of pixels on the left image with a corresponding sized window on the right image.

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Matching PixelsMatching Window

Left Image Right Image

Diagram to illustrate the Stereo Matching

Assumption :Matching Pixels lie on same horizontal Raster Line(Rectified stereo)

Disparity of this pixel is 1 since x1=0 and x2=1,x2-x1=1

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The SSD and SAD are commonly used correlation functions

SSD:Sum of Squared Deviations

2)),(),(()),(),,(( ydxIyxIydxIyxI rlrl +−=+ψ

),(),()),(),,(( ydxIyxIydxIyxI rlrl +−=+ψ

q

SAD:Sum of Absolute Differences

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The Multi Window Algorithm

In this algorithm technique 9 different i d d f l l tiwindows are used for calculating

disparity of a single pixel.The window which gives the maximum correlation is used for disparity calculations.a u a io

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Left Image Right Imageg g g

Demonstration of the 9 different windows used for the Correlation

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Disparity Map

Based on the calculated disparities a di it i bt i ddisparity map is obtainedThe disparity map is a gray scale map where the intensity represents depth.The lighter shades (greater disparities) represent regions with less depth asrepresent regions with less depth as opposed to the darker regions which are further away from us.

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Visualization

Visualization is the process by which we th d th ti t f th tuse the depth estimates from the stereo

matching to build projections .3‐D information can be represented in many ways :

‐Orthographic projectionsO og ap i p oje io‐Perspective Projections

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Perspective Projections

Perspective projections allow a more li ti i li ti f ldrealistic visualization of a world scene

The visual effect of perspective projections is similar to the human visual system and photographic systems.Hence perspective projection of the 3‐dHence perspective projection of the 3‐d data was implemented for the stereo pairs.

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A

B

•In Perspective projections the projectors are of finite length and converge at a pointA’

B’ Projection Plane

converge at a point called the center of projection.

•In perspective projection size of an object is

Center Of Projection

inversely proportional to its distance from ooint of projection

Projectors

V

(Umax,Vmax)VUP

Window

CW

U

(Umin,Vmin)

VPN

VRP

VIEW PLANE

N

CW

Center of Projection

THE 3-D VIEWING REFERENCE COORDINATE SYSTEM

DOP

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Specifying a 3‐D View

To specify a 3‐d view we need to specify a projection plane and a center of Projection:projection plane and a center of Projection: The Projection plane specified by 1. A point on the plane called the

View Reference Point (VRP)2. The normal to the view plane,i.e.

Vi Pl N l (VPN)View Plane Normal (VPN)

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We define a VRC (View Reference Coordinate system) on the projection plane with u v and n being


system) on the projection plane with u,v,and n being its 3 axes forming a right handed coordinate system The origin of the VRC system is the VRPThe VPN defines the ‘n’ axis of the VRC systemA View Up Vector (VUP) determines the ‘v’ axis of the VRC system. The projection of the VUP parallel t th i l i i id t ith th ‘ ’ ito the view plane is coincident with the ‘v’ axis.The ‘u’ axis direction is defined such that the ‘u’,’v’ and ‘n’ form a right handed coordinate system.

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A view Window on the view plane is defined ,projections


p ,p jlying outside the view window are not displayed .The coordinates (Umin,Vmin) and (Umax,Vmax) define this window. The center of projection Projection Reference Point(PRP).

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The semi infinite pyramid formed by the PRP d th j t i th h


PRP and the projectors passing through the corners of the view window form a view volume.A Canonical view volume is one where the VRC system is alligned with the World e y e i a ig e i e oCoordinate system.

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X or YBack Plane

1

Front Plane

-1

-Z

The 6 bounding planes of the canonical view volume have equations:

PRP

-1

Canonical view volume for Perspective Projections

equations:

x=z ,x=-z ,y=z, y=-z z=zmin, z=-1

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Perspective projection when VRC alligned with World Coordinate system

V

P(X,Y,Z)Y

X

U

N

P(Xp,Yp,d)

P(X,Y,Z)

Zd

PRPCW

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d Zd

ZYY

dZXX

TrianglesSimilar From

pp ===

matrix a as drepresente becan tion Transforma The

dd

0100010000100001

Mper

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

1] /

/

[] z [1

.

0d100

ddz

ydz

xdzyxz

yx

Mper ==⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎢⎣

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Only true when view volume is canonicalFor arbitrary view volume ‐For arbitrary view volume First transform the view volume into

canonical form and then apply the above formula to take projections

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For transforming a view volume we do the following:1 Translate VRP to origin1. Translate VRP to origin2. Rotate VRC to allign u,v and n axes with the X,Y and Z

axes.3. Translate the PRP to origin4. Shear to make center line of view volume the z‐axis.5. Scale such that the view volume becomes the canonical

perspective view volume

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The translation matrix is

100010001

),,(dzdydx

dzdydxT⎥⎥⎥⎤

⎢⎢⎢⎡

=Z

)(1000

100

VRPT

dz

−

⎥⎥

⎦⎢⎢

⎣

2. The Rotation matrix is

⎥⎤

⎢⎡ 0111 zyx rrr

N

V

U

X

YVRP

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

100000

333

222

zyx

zyx

rrrrrr

R

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VPNwhere

][

][

][

222

1 1 1

3 3 3

RRrrrRRVUPRVUPrrrR

VPNVPNrrrR

z

zzyxx

zyxz

×==

××

==

==

)( 3.][ 2 2 2

PRPTRRrrrR xzzyxy

−×==

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The Shear Matrix

SH shyshx

⎥⎥⎤

⎢⎢⎡

010001

Y

dopx

Let

SH shyper

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

WindowofCenter :CWProjection ofDirection :DOP

PRP-CWDOP 10000100010 Y

CW

dopzdopyshy

dopzdopxshx

−=

−=

PRP -Z

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The scale transformation

⎥⎥⎤

⎢⎢⎡

00s0000s

y

x

Y

BVRPuuVRPs

PRPTSHVRPLet

S

z

zx

per

per

+=

−=

⎥⎥⎥

⎦⎢⎢⎢

⎣

=

− )')(('2

1] 0 0 0).[(.' 10000s0000s0

minmax

z

y Y

CW

Y=-Z

BVRPs

BVRPvvVRPs

zz

z

zy

+−

=

+=

−

'1

)')(('2

minmax

PRP -Z

Y=-Z

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Once all the projected points have been l l t d l th di t t fit thcalculated, scale the coordinates to fit the

display screen.A wire frame display of the image is obtained by joining the projections of all points lying on the same row or column.poi yi g o e a e o o o uMap the pixel colors of the image on to the projected points to create a realistic effect.

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Limitations

Can work well only for stereo images h i t d t il t i dwhere minute details are not required.

More suited for depth estimation of landscape through images taken from top.No accurate metric calculations done.

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Stereo ImagingThere are two methods for creating stereo pairs:

1. Side-by-side display of both images

2. Use different colors for each image eg. Anaglyphs of red-green which produce a monochrome 3D image

Slide 38 t:/powerpnt/course/ /524lect6.ppt© 1994-2004 J.Paul Robinson - Purdue University Cytometry Laboratories

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Creating Stereo pairs

Pixel shifting -ive pixel shift for left+ive pixel shift for right

z

Slide 39 t:/powerpnt/course/ /524lect6.ppt© 1994-2004 J.Paul Robinson - Purdue University Cytometry Laboratories

xy

Method 1: Separated Images

Actually present two different imagesActually present two different images to the eyes

Left eye

Right eye

Left eye view Right eye view

AJ MinichFinal Project

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Method 1: Separated Images



Method 2: Anaglyphs

Use colored glasses to receive different visualsUse colored glasses to receive different visuals

Left eye

Right eye



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Method 2: Anaglyphs

Left eye view

Right eye view

Method 2: Anaglyphs


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Method 2: Anaglyphs


Method 2: Anaglyphs

Background Image


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Method 2: Anaglyphs

Background Image


Method 2: Anaglyphs


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Stereo Imaging - Aegean