Lecture 12Lecture 12 Stereo Reconstruction IIStereo Reconstruction II

Mata kuliah : T0283 - Computer VisionTahun : 2010

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Learning ObjectivesLearning Objectives

After carefullyAfter carefully listening this lecture, students will listening this lecture, students will be able to do the following :be able to do the following :

demonstrate 3D stereo computation by solving demonstrate 3D stereo computation by solving point-point- correspondence problems and correspondence problems and fundamentalfundamental matrix.matrix.

Calculate object-depth information using Calculate object-depth information using disparity and disparity and triangulation techniquestriangulation techniques

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An algorithm for stereo An algorithm for stereo reconstructionreconstruction

1. For each point in the first image determine the corresponding point in the second image

(this is a search problem)

2. For each pair of matched points determine the 3D point by triangulation

(this is an estimation problem)

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Epipolar lineEpipolar line

Epipolar constraint• Reduces correspondence problem to 1D

search along an epipolar line

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Algebraic representation of epipolar Algebraic representation of epipolar geometrygeometryWe know that the epipolar geometry defines a We know that the epipolar geometry defines a mappingmapping

x l/

point in first image

epipolar line in second

image• the map only depends on the cameras P, P/ (not on structure)

• it will be shown that the map is linear and can be written as

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Stereo correspondence algorithms

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Problem statementProblem statement

GivenGiven:: two images and their associated cameras two images and their associated cameras computecompute

corresponding image points.corresponding image points.

Algorithms may be classified into two types:Algorithms may be classified into two types:

1.1. Dense:Dense: compute a correspondence at every pixel compute a correspondence at every pixel

2.2. Sparse:Sparse: compute correspondences only for features compute correspondences only for features

The methods may be top down or bottom upThe methods may be top down or bottom up

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Top down matching Top down matching

1. Group model (house, windows, etc) independently in each image

2. Match points (vertices) between images

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Bottom up matchingBottom up matching• epipolar geometry reduces the correspondence search from 2D to a 1D search on corresponding epipolar lines

• 1D correspondence problem

b/

a/

bca

CBA

c/

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Example image pair – parallel Example image pair – parallel camerascameras

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First imageFirst image

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Second imageSecond image

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Dense correspondence algorithmDense correspondence algorithm

Search problem (geometric constraint): for each point in the left image, the corresponding point in the right image lies on the epipolar line (1D ambiguity)

Disambiguating assumption (photometric constraint): the intensity neighbourhood of corresponding points are similar across images

Measure similarity of neighbourhood intensity by cross-correlation

Parallel camera example – epipolar lines are corresponding rasters

epipolar line

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Intensity profiles

• Clear correspondence between intensities, but also noise and ambiguity

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Normalized Cross CorrelationNormalized Cross Correlation

region A region B

vector a vector b

write regions as vectors

a

b

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Cross-correlation of neighbourhood Cross-correlation of neighbourhood regionsregions

epipolar line

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left image bandright image band

cross correlation

1

0

0.5

x

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left image band

right image band

cross correlation

1

0

x

0.5

target region

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Why is cross-correlation such a poor measure in the second case?

1. The neighborhood region does not have a “distinctive” spatial intensity distribution

2. Foreshortening effects

front-parallel surfaceimaged length the

same

slanting surfaceimaged lengths differ

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Limitations of similarity constraintLimitations of similarity constraint

Textureless surfaces Occlusions, repetition

Non-Lambertian surfaces, specularities

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Results with window searchResults with window search

Window-based matching Ground truth

Data

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Sketch of a dense correspondence Sketch of a dense correspondence algorithmalgorithm

For each pixel in the left imageFor each pixel in the left imageccompute the neighbourhood cross correlation ompute the neighbourhood cross correlation along the corresponding epipolar line in the right along the corresponding epipolar line in the right imageimagetthe corresponding pixel is the one with the he corresponding pixel is the one with the highest cross correlationhighest cross correlation

ParametersParameterssize (scale) of neighbourhoodsize (scale) of neighbourhoodsearch disparity search disparity

Other constraintsOther constraintsuniquenessuniquenessorderingorderingssmoothmoothness ofness of disparity field disparity field

ApplicabilityApplicabilitytextured scene, largely fronto-paralleltextured scene, largely fronto-parallel

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Example dense correspondence algorithm

left image right image

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3D Reconstruction

intensity = depthright image depth map

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range map

Pentagon exampleleft image right image

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RectificationRectification

e e /

For converging cameras epipolar lines are not paralle

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Project images onto plane parallel to baseline

epipolar plane

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Rectification continuedRectification continued

Convert converging cameras to parallel camera geometry by an image mapping

Image mapping is a 2D homography (projective transformation)

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Example original stereo pair

rectified stereo pair

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Note• image movement (disparity) is inversely proportional to depth Z • depth is inversely proportional to disparity

Example: depth and disparity for a parallel camera stereo rig

Then, y/ = y, and the disparity

Derivation

x

x /d

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TriangulationTriangulation

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Problem statementProblem statementGiven:Given: corresponding measured (i.e. noisy) points corresponding measured (i.e. noisy) points x x and and xx// , and cameras (exact) P and P , and cameras (exact) P and P//, compute the 3D point , compute the 3D point XX

Problem: in the presence of noise, back projected rays do not intersect

rays are skew in space

Measured points do not lie on corresponding epipolar lines

C C /

x x /

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1. Vector solution1. Vector solution

C C /

Compute the mid-point of the shortest line between the two rays

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2. Linear triangulation (algebraic solution)2. Linear triangulation (algebraic solution)

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3. Minimizing a geometric/statistical error3. Minimizing a geometric/statistical error