Image Registration Mapping of Evolution

Registration Goals

• Assume the correspondences are known

• Find such f() and g() such that the images are best matched

I2(x,y)=g(I1(f(x,y))

f() – 2D spatial transformationg() – 1D intensity transformation

Spatial Transformations

• Rigid

• Affine

• Projective

• Perspective

• Global Polynomial

• Spline

Rigid Transformation

• Rotation(R)• Translation(t)• Similarity(scale)

2

22 y

xp

1

11 y

xp

12 pRstp

)cos()sin(

)sin()cos(

R

2

11 s

ss

2

11 t

tt

Affine Transformation

• Rotation• Translation• Scale• Shear

1

1

2221

1211

23

13

2

2

y

x

aa

aa

a

a

y

x

No more preservation of lengths and angles

Parallel lines are preserved

Perspective Transformation (Planar Homography)

Perspective Transformation(2)

• (xo,yo,zo) world coordinates

• (xi,yi) image coordinates

• Flat plane tilted with respect to the camera requires Projective Transformation

fz

fyy

o

oi

fz

fxx

o

oi

Projective Transformation

• (xp,yp) Plane Coordinates• (xi,yi) Image Coordinates

• amn coefficients from the equations of the scene and the image planes

333231

131211

ayaxa

ayaxax

pp

ppi

333231

232221

ayaxa

ayaxay

pp

ppi

Complex Transformations

Global Polynomial Transformation(splines)

Methods of Registration

• Correlation

• Fourier

• Point Mapping

Correlation Based TechniquesGiven a two images T & I, 2D normalized

correlation function measures the similarity for each translation in an image patch

x y

x y

vyuxI

vyuxIyxTvuC

),(

),(),(),(

2

Correlation must be normalized to avoid contributions from local image intensities.

Correlation Theorem

• Fourier transform of the correlation of two images is the product of the Fourier transform of one image and the complex conjugate of the Fourier transform of the other.

Fourier Transform Based Methods

• Phase-Correlation• Cross power spectrum• Power cepstrum

All Fourier based methods are very efficient, only only work in cases of rigid transformation

Point Mapping Registration

• Control Points

• Point Mapping with Feedback

• Global Polynomial

Control Points

Intrinsic

Markers within

the Image

Extrinsic

Manually or Automatically selected

After the control points have been

determined, cross correlation, convex hull

edges and other common methods are used

to register the sets of control points

Point mapping with Feedback

• Clustering example: determine the optimal spatial transformation between images by an evaluation of all possible pairs of feature matches.

• Initialize a point in cluster space for each transformation

• Use the transformation that is closest to the best cluster

• Too many points, thus use a subset

Global Polynomial Transformation(1)

• Use a set of patched points to generate a single optimal transformation

• Bi-Variate transformation:

m

l

i

j

jllj

m

l

i

j

jllj

yxbv

yxau

0 0

1

0 0

1

(x,y) – reference image

(u,v) – working image

Global Polynomial Transformation(2)

• When is polynomial transformation bad?

• Splines approximate polynomial transformations(B-spline, TP-spline)

Characteristics of Registration Methods

• Feature Space

• Similarity Metrics

• Search Strategy

Feature Spaces

Similarity Metrics

Search Strategies

Robust Multi-Sensor Image Alignment

Irani & Anandan

Direct Method(vs. Feature Based)

Multi Sensor Images

EO IR

Original Image (Intensity Map)

Find features

Assume global statistical correlation

Loss of important information

Often violated

Multi Sensor Image Representation(1)

• Same Modality Camera Sensors enough correlated structure at all resolution levels

• Different Modality Camera Sensors primary correlation only in high resolution levels

Multi Sensor Image Representation(2)

Goal: Suppress non-common information & capture the common scene details

Solution: High pass energy images

Laplacian Energy Images

• Apply the Laplacian high pass filter to the original images

• Square the results NO contrast reversal

BUT

The Laplacian is directionally invariant

Directional Derivative Energy Images

• Filter with Gaussian • Apply directional

derivative filter to the original image in 4 directions

• Square the resultant images

Alignment Algorithm

• Do not assume global correlation, use only local correlation information

• Use Normalized Correlation as a similarity measure

• Thus, no assumptions about the original data

Behavior of Normalized Correlation with Energy Images

• NC=1

Two images are linearly related

• NC<1(high)

Two images are not linearly related, yet local fluctuations are low

• NC<1(low)≈0

Incorrect displacements

Global Alignment with Local Correlation(1)

a, b – original images

{ai,bi} – directional derivatives (i=1..4)

p=(p1…p6)T affine(u,v) – shift from one image to another

Si(x,y)(u,v) – correlation surface at a pixel(x,y)

),(),(),(),( vyuxNbyxavuS iiyx

i

Global Alignment with Local Correlation(2)

));,(),;,(()(.

),( yx i

yxi pyxvpyxuSpM

Goal: Find the parametric transformation p, which maximizes the sum of all normalized correlation values. global similarity M(p)

yx i

yxi pyxuS

,

),( ));,((

Solving for M(p)

Newton’s method is used to solve for M(p)

pMTpp

Tp pHpMpMpM

)())(()()( 000

)())(( 01

0* pMpH pMp

iyx iu

T

iyx ipp uSXuSpM,,,,

))(()()(

iyx S

TM XuHXpH

i,,))(()(

The quadratic approximation of M around p is obtained by combining the quadratic approximations of each of the local correlation surfaces S around local displacement.

Steps of the Algorithm

• Construct a Laplacian resolution pyramid

• Compute a local normalized-correlation surface around a given displacement

• Compute the parametric refinement

• Update p

• Start over(process terminates at the highest resolution level of the last image)

Outlier Rejection

Due to the different modalities of thesensors, the number of outliers may be very large

1. Accept pixels based on concavity of the correlation surface

2. Weigh the contribution of a pixel by det|H(u)|

Results

EO image

IR imageComposite before Alignment

IR image

Composite after Alignment

Acknowledgements

• I would like to thank Compaq for making awful computers

• Professor Belongie for reviewing the slides