ifsr transformation functions

8/3/2019 Ifsr Transformation Functions

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Transformation Functions

Arthur Goshtasb

Wright State University


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corresponding points in the images:

xi i Xi Yi : i=1 N

we want to determine function f(x,y)

with com onents x(x, ) and y(x, )(x,y)

such that

Xi = fx(xi,yi),

Yi = fy(xi,yi), i = 1,,N. (X,Y)

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Rearrange the coordinates of corresponding

points into two sets of 3-D points:{(xi,yi,Xi): i=1,,N}, y

, , ,, ,

then, fx andfy can be considered two single-

valued surfaces fitting to two sets of 3-D

x

X

po nts.

We will consider the problem of findingfunctionf(x,y) that approximates/interpolates

{(xi,yi,fi): i=1,,N}. y

Y

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Rigid

-

Multiquadric

Similarity

Affine

Weighted mean

Piecewise methods

Projective Weighted linear

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Corres ondin ima e

points are related by

= x +

Y = y + k

One pair of corresponding

points is sufficient to

eterm ne t e reg strat onparameters.

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X = x cos sin + h

Y = x sin(+y cos() + k and (h,k) are the rotational and translational

differences between the images.

Knowing minimum of two corresponding points Reference

determined.

This transformation is useful when registering

images as rigid bodies.

Under rigid transformation shape and size are Sensed

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.


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X = s x cos( s sin() + h

Y = s x sin(+ s y cos() + k s, , and (h,k) are the scaling, rotational, and

.

Minimum two corresponding points in the

images are required to determine s, h, and k. Reference

Angles are preserved under the similaritytransformation.

This transformation is useful when re isterindistant orthographic images of flat scenes.

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Sensed


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8

.


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X = ax + by + c

Y = dx + ey + f

This transformation is a combination of shearing andsimilarity transformations.

each other.

Under the affine transformation parallel lines remain

parallel.Reference

When the components are made independent, thetransformation becomes a linear transformation.

Knowing the coordinates of three non-colinearcorres ondin oints in the ima es the six arameters of

the transformation can be determined. This transformation is useful when registering images

taken from a distant platform of a flat scene.

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Lunarimage 1.

Lunarimage 2.

Registered

lunar

Subtracted

registered

images. images.

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X = (ax+by+c)/(dx+ey+1)

Y = (fx+gy+h)/(dx+ey+1)

Knowing the coordinates of four non-colinearcorresponding points in the images, parametersa can e e erm ne .

This transformation is useful when registering

images obtained from different views of a flatscene.Reference

Under the projective transformation straight linesremain straight.

If ima es are from camera ver far from a flat

scene, projective transformation can be replaced bythe affine transformation when registering theimages.

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-N

where

i

iii rryxyx1

321 n,

2222 )()( dyyxxr iii

Parameters A1, A2, A3, and Fi: i=1,Nare

determined using {(xi,yi,fi): i=1,,N} and theReference

o ow ng ree cons ra n s:

N

N

i iF

10

N

i ii

i ii

Fyx

1

1

0

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the control point are not uniformly spaced, largeerrors may be obtained away from the control points.

TPS is useful when

the local geometric difference between images is not large

the control points are rather uniformly spaced

the density of the control points does not change

the number of corresponding control points is not veryhigh.

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Reference

image. Sensedimage.

Registered

using all

Registered

using

gr po n s. grid points.

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: . ; : . : . ; : .


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N

i

ii yxyx1

,,

2/1222

a a as s unct ons:

)yy()xx(

]d)yy()xx[()y,x(R

,

2i2i

2/122i

2ii

iii

When inverse multiquadrics:

Note that whenGaussian:

Multiquadric basis functions are also radially symmetric, therefore, their

2,

2i

proper es are s m ar o ose o .

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MAX: 0.0; RMS: 0.0 MAX: 23.1; RMS: 2.9

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eg s ra on us ng as s unc ons.


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N

i ii yxbfyxf 1 ),(),(

N

i iii yxRyxRyxb 1 ),(),(),(where

andRi(x,y) is a monotonically decreasing radialfunction.

This is an approximation method; therefore, there isno need to solve a system of equations.

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When the wei ht functions are narrow the functions roduce

flat spots at and in the neighborhood of the data points. Flat spots imply that many points in the sensed image map to

t e same po nt n t e re erence mage, resu t ng n arge

registration errors.

Consider oints: 0 0 0 0 1 0 1 1 0 1 0 0 0.5 0.5 0.5

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Narrow widthWider width

Density of points

(narrow width)

Density of points

(wider width)


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To produce a uniform density, use parametricsurfaces as components of thetransformation.

=y

f

,

x = f2(u,v) (2)

y = f3(u,v) (3)

x

For a given (x,y), find corresponding (u,v)from (2) and (3). Then, use (u,v) in (1) tofindX.

f

u

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Resampled

usingsingle-valued

.

Resampled

using

parametricsurfaces.

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MAX: 1.4; RMS: 0.6 MAX: 9.0; RMS: 1.3

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.


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Piecewise linear and

piecewise cubic functions

1. Triangulate the points in the referenceimage.

2. From the point correspondences, find

corresponding triangles in the sensed

.

3. Map triangular regions in the sensed

image one by one to the corresponding

triangular regions in the sensed imageusing linear or cubic functions.

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MAX: 0.0; RMS: 0.0 MAX: 16.1; RMS: 4.0

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Registered using piecewise linear functions.


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Note that wei hted mean is a wei hted sum of constants.

Instead of using a weighted sum of constants, use a weightedsum of linear functions, each representing the geometric

erence etween t e mages oca y.

N

i

ii

1

,,,

iiii cybxayxf ),(where

The local functions at (xi,yi) is determined by fitting a plane

to (xi,yi,fi) and at least two other points in vicinity of(xi,yi).

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Properties of weighted linear

functions The weighted mean method produces horizontal

spots because a surface is obtained from a

weighted sum of constants (horizontal planes). In weighted linear, since the planes can have any Weighted mean

, .This implies parametric surfaces are not neededto find the components of a transformation.

If rational wei hts are used because the wei htsstretch toward the gaps, the functions can adjustthemselves to the irregular spacing of the points.

If the width of the weight functions are setWeighted linear

proportiona to the oca density of points, thefunctions can adjust themselves to the localdensity of the points also.

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MAX: 3.6; RMS: 0.7 MAX: 16.1; RMS: 4.0

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Registered using weighted linear functions with rational Gaussian weights.


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What do transformation functions

tell us about the images?

geometric difference betweenimages.

Plot offx(x,y) x.

fx(x,y) = x 8sin(y/16)

fy(x,y) = y +4cos(x/32)

They predict the geometry of

the scene.

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Plot offy(x,y) y.


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Transformation functions contain information about the

mismatches.

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o o x x,y x, us ng

incorrect correspondences

o o y x,y y, us ng

incorrect correspondences


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Transformation functions can be used to generate flow diagrams,

from one image to the next.

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Volumetric registration Image flow


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1. A. Goshtasby, Registration of image with geometric distortions,IEEE Trans. Geoscience

, .

2. F. L. Bookstein, Principal warps: Thin-plate splines and the decomposition ofdeformations, IEEE Trans. Pattern Analysis and Machine Intelligence, 11(6):567585(1989).

. . , ., . . , . , . . , . , . . ,Landmark-based elastic registration using approximating thin-plate splines,IEEETransactions on Medical Imaging, 20(6):526534 (2001).

4. M. Fornefett, K. Rohr, and H. S. Stiehl, Radial basis functions with compact support for, , .

5. A. Goshtasby, Piecewise linear mapping functions for image registration, PatternRecognition, 19(6):459466 (1986).

6. A. Goshtasby, Piecewise cubic mapping functions for image registration, Pattern, .

7. A. Goshtasby, Image registration by local approximation methods,Image and VisionComputing, 6(4):255261 (1988).

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ifsr transformation functions

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