GEOMETRIC OPERATIONS - pub.roimag.pub.ro › ro › cursuri › archive › IPA_03_Geom_op.pdf ·...

LABORATORUL DE ANALIZA ŞI PRELUCRAREA IMAGINILOR - LAPI

GEOMETRIC OPERATIONS


Introduction

• Permit elimination of the geometric distortions.

• Geometric distortion may arise:

– Because of the lens


Lens Geometric Distortions


Image taken from http://nofilmschool.com/2011/11/lens-choice-affects-subjects-appearance

Introduction




– Because of perspective


Perspective Distortions


Image taken from www.techsupportalert.com

Introduction





– Because of human error



Human Error Example: Tilt Horizon

Image taken from http://thehdrimage.com/5-quick-steps-to-better-hdrs-step-1/

Introduction





– Because of human error

– In situations where there are distortions inherent in the

imaging process such as remote sensing:

• For example is an attempt to match remotely sensed images of

the same area not taken from precisely the same position. In

this case it is necessary first to execute a geometric

transformation, and then compare the images.


C. VERTAN

Panorama Stitching


Image taken from http://matthewalunbrown.com/autostitch/autostitch.html


Definition ?

Class of operations modifying the neighboring structure of image

pixels (the structure of spatial organization of the image).

Geometric transform allow pixels to move within the image,

to new positions.

Intuitive model: image is printed onto a thin sheet of rubber, that

can be deformed at will.


C. VERTAN


Geometric operations without constraints

If the movement of each pixel is independent with respect to the

movements of the rest of image pixels, the visual content could be

mixed or cut-off.

image quarters

are interchanged

“random” pixel

displacement

Constraint: the law describing pixel movement is NOT random and

is spatially invariant (all image pixels move according to the same

equations).


C. VERTAN


Defining geometric transforms

Transform = equation describing the modification of pixel coordinates

),('

),('

yxYy

yxXx

x’ , y’ :

pixel coordinates

after transform

x, y :

pixel coordinates

before the transform

X, Y:

equations of the

coordinate modification

Particular forms of the functions of the coordinate transform,

X(x,y) and Y(x,y) particularize the geometric transforms.


C. VERTAN


Image coordinates system

x

y(0,0) Origin is upper-left

corner, coordinates increase

from left to right and from top

to bottom.

First coordinate is the line coordinate.

Coordinates can be continuous (x, y)

or discrete (m, n)

Transforming image coordinates to plane Cartesian coordinates :

xplan

yplan

xy

yx

plan

plan


C. VERTAN


Affine elementary geometric transforms


An affine transformation = function between affine spaces which

preserves points, straight lines and planes.

• from the Latin affinis: "connected with“

• parallel lines remain parallel;

• does not necessarily preserve angles between lines;

• does not necessarily preserve distances between points;

• preserve ratios of distances between points lying on a straight

line.

Examples: translation, scaling, reflection, rotation, shear mapping,

and compositions of them in any combination and sequence.



Translation

Displacement of the image content within the plane;

is equivalent to change of the origin of the image coordinate system.

0

0

),('

),('

yyyxYy

xxyxXx

Does it preserve inter-pixel distances?

Parameters : x0, y0 – vertical/ horizontal displacement.


C. VERTAN

Yes


Translation

x

y(0,0)

x’

y’(0,0)

(x0, y0)

x

y(0,0)

x’

y’(0,0)



C. VERTAN


Scaling

0,

),('

),('

yyxYy

xyxXx Parameters : , - horizontal/ vertical scaling factors

Stretching/compressing image content according to one or

both coordinate axes

= : homogeneous scaling



C. VERTAN

Does it preserve inter-pixel distances? No

, > 1:

, < 1:

stretching

compressing


Scaling

x

y(0,0)

x’

y’(0,0)

> 1, < 1

< 1, > 1



C. VERTAN

Scaling

• Compressing => image

reduction or subsampling

•Stretching => image

magnification or zooming

•Performed by replacement (of a

group of pixel values by one

arbitrarily chosen pixel value

from within this group) or by

interpolating between pixel

values in a local neighborhoods



C. VERTAN


Important issue when downscaling: aliasing

Aliasing: comes from the inappropriate sampling of a signal.

When reconstructed, an undersampled sine wave will create an

“alias”, i.e., a parasite low-frequency component, that might alter the

signal


C. VERTAN


Image downscaling = projecting the same information onto a

coarser grid = downsampling.

Some of the high-frequency image components might create

visible aliases:


C. VERTAN


Solution: low-pass filtering (blurring) the image prior to

downsampling it.

The high-frequency components susceptible of creating aliases

are removed from the image.

without filtering with filtering


C. VERTAN


Skewing

Skewing is the displacement of image pixels according to a single

coordinate, dependent by the global location within the image.

The second coordinate remains unchanged.

0

),('

),('

t

ytxyxYy

xyxXx

0

),('

),('

s

yyxYy

syxyxXx

Horizontal skewing Vertical skewing

Parameters : t, s – skewing coefficients.



C. VERTAN

Does it preserve inter-pixel distances? No


Horizontal skewing

x

y(0,0)

x’

y’(0,0)

x=0 points are fixed points

0

),('

),('

t

ytxyxYy

xyxXxAffine elementary geometric transforms


C. VERTAN


Vertical skewing

x

y(0,0)

x’

y’(0,0)

y=0 points are fixed points

0

),('

),('

s

yyxYy

syxyxXxAffine elementary geometric transforms


C. VERTAN


Rotation

cossin),('

sincos),('

yxyxYy

yxyxXx

Circular displacement of pixels around the rotation center

(the origin of the coordinate system).

Does it preserve inter-pixel distances?

Parameter : - rotation angle



C. VERTAN

Yes


Rotation


• Rotates points by an angle θ about origin

(θ >0: counterclockwise rotation)

• From OBP triangle:

• From OCP’ triangle:

=>


C. VERTAN

O C B


Rotation

x

y(0,0)

x

y(0,0)



C. VERTAN


Reflection with respect to a reflection center (x0,y0)

The initial and final positions of any pixel form a

line segment; the reflection center is the middle of the segment.

yyyxYy

xxyxXx

0

0

2),('

2),('

x

y(0,0)

(x, y)

(x’, y’)

(x0, y0)

Composed affine geometric transforms


C. VERTAN


Reflection with respect to a rotation axis

The initial and final positions of any pixel form a line segment;

the reflection axis is the median axis of the line segment.

x

y(0,0)

(x, y)

(x’, y’)

reflection

axis

y=mx+n

1

''

1

1

'

'

22

m

nmxy

m

nmxy

mxx

yy

22

2

2

222

2

1

2

1

1

1

2),('

1

2

1

2

1

1),('

m

ny

m

mx

m

myxYy

m

mny

m

mx

m

myxXx



C. VERTAN


horizontal axis: x = k = ct (1/m = 0, -n/m = k)

vertical axis: y = k = ct (m=0, n=k)

ykyxYy

xyxXx

2),('

),('

22

2

2

222

2

1

2

1

1

1

2),('

1

2

1

2

1

1),('

m

ny

m

mx

m

myxYy

m

mny

m

mx

m

myxXx

22

2

2

222

2

1

2

1

1

1

2),('

1

2

1

2

1

1),('

m

ny

m

mx

m

myxYy

m

mny

m

mx

m

myxXx

yyxYy

xkyxXx

),('

2),('


Reflection with respect to a rotation axis: particular cases


C. VERTAN


Matrix form of affine geometric transforms

By

xA

y

x

'

'1. Translation

0

0

'

'

y

x

y

x

y

x2. Scaling

y

x

y

x

0

0

'

'

3a. Horizontal skewing

y

x

ty

x

1

01

'

'

3b. Vertical skewing

y

xs

y

x

10

1

'

'

4. Rotation

y

x

y

x

cossin

sincos

'

'


By

xA

y

x

'

'

1. Reflection with respect to a point

2. Reflection with respect to an axis

0

0

2

2

10

01

'

'

y

x

y

x

y

x

2

2

2

2

2

22

2

1

21

2

1

1

1

21

2

1

1

'

'

m

mm

mn

y

x

m

m

m

mm

m

m

m

y

x

transforms

composed by

translation and

rotation

Matrix form of affine geometric transforms


Composed geometric transforms

Composing = iteration of elementary transforms

Composing geometric transforms is simple in matrix form ...

but translation has not the same matrix form and cannot be

expressed as a matrix product.

Solution : homogeneous coordinates

1

y

x

y

x(plane of z=1 from the 3D Cartesian space)


Homogeneous matrix form of geometric transforms

11

'

'

y

x

Ay

x

1. Translation

1100

10

01

1

'

'

0

0

y

x

y

x

y

x

1100

0cossin

0sincos

1

'

'

y

x

y

x

4. Rotation

2. Scaling

1100

00

00

1

'

'

y

x

y

x

3a. Horizontal skewing

1100

01

001

1

'

'

y

x

ty

x

3b. Vertical skewing

1100

010

01

1

'

'

y

xs

y

x


Composed geometric transforms

• Important: preserve the order of transformations!

translation + rotation rotation + translation

Images taken from: http://www.cse.unr.edu/~bebis/CS485/Lectures/GeometricTransformations.ppt


Decomposing complex geometric operations

In the most general form, expressed in homogeneous coordinates:

11001

'

'

02221

01211

y

x

yaa

xaa

y

x

aij coefficients are due to rotation, skewing and scaling.

cossin

sincos

10

1

0

0

2221

1211 s

aa

aa

given a series of elementary

transforms, find an unique

equivalent transform.

given an affine transform,

find the composing

elementary geometric

transforms


given a series of elementary

transforms, find an unique

equivalent transform.Trivial problem.

Ex. The equivalent transform for a rotation by 45º (=45º), vertical

skewing by coefficient 0.1 (s=0.1), horizontal scaling 2 (=2)

and vertical scaling 0.5 (=0.5).

cossin

sincos

10

1

0

0

2221

1211 s

aa

aa

2/22/2

2/22/2

10

1.01

20

05.0

2221

1211

aa

aa

2728.14142.1

3889.03535.0

2221

1211

aa

aa



cossin

cossinsincos

2221

1211 s

aa

aa

2

22

2

21 aa 22

21arctana

a

221221

2

22

2

21aa

a

aa

221221

2

22

2

21

22

2

22

2

21

21

11 aaa

aaaaa

a

as

Decomposing complex geometric operationsgiven an affine transform,

find the composing

elementary geometric

transforms


Why using a single skewing ?

horizontal skewing

vertical skewing

rotation /2

reflection with respect to

vertical



Nonlinear geometric transforms

The “pillow” effect

displacement controlled by

sinusoidal law

yyxYy

W

yA

H

Hy

HyxXx x

),('

sin2

122

),('

Horizontal pillow effect

H

xA

W

Wy

WyxYy

xyxXx

y sin2

122

),('

),('

Vertical pillow effect


General affine transform


W

xA

H

Hy

Hy

H

yA

W

Wx

Wx

y

x

sin2

122

'

sin2

122

'


Lens distortion: pincushion and barrel effects


Regular grid Distortion introduced by

typical lens




2

2

1'

1'

PAPP

PAPP

yyy

xxx

Distortion equation:

Px and Py: normalized coordinates.

Ax and Ay: constants between 0 (no action) and 0.1 (wide angle

lens).




22

22

1/1

'

1/1

'

PAPA

PP

PAPA

PP

yy

y

y

xx

xx

Images taken from: http://local.wasp.uwa.edu.au/~pbourke/miscellaneous/lenscorrection/

http://local.wasp.uwa.edu.au/~pbourke/miscellaneous/lenscorrection/


A geometric transform consists of two basis steps:

1. Pixel co-ordinate transformation, which maps the co-

ordinates of the input image pixel to the point in the output image.

The output point co-ordinates should be computed as continuous

values (real numbers) as the position does not necessarily match

the digital grid after the transform.

2. The second step is to find the point in the digital raster which

matches the transformed point and determine its brightness value.

Its brightness is usually computed as an interpolation of the

brightness of several points in the neighborhood .

Implementing geometric transforms



Two components:

1. where the pixel is moved

2. what value is placed in the new location ?

geometric transform

interpolation

Images are discrete: pixel coordinates are integer numbers.

In most cases the coordinates obtained after the geometric

transform are real numbers.

There is thus the need of producing integer coordintes.


Solution 1: Pixel carry-over

Solution 2: Pixel filling

m

n

x’

y’T

(x’, y’) non-integer coordinates

after the transform(m, n) integer coordinates




Pixel carry-over

m

nT

(x’, y’) non-integer coordinates

after the transform(m, n) integer coordinates


the value must be distributed

to neighboring integer locations

x’

y’



Problems related to the direct implementation

2/),('

2),('

yyxYy

xyxXx

non-integer coordinates

non-appearing coordinates

0

1

2

3

4

5

0 1 2 3 4 5 6

initial coordinates

0

1

2

3

4

5

0 1 2 3 4 5 6

transformed coordinates


Implementing geometric operations

Pixel filling

T-1

(m’, n’) integer coordinates

after the transform(x, y) non-integer coordinates


m’

n’

x

y

the non-existing value from the initial image

must be obtained by the interpolation of the values

from the neighboring pixels.


Interpolation methods

m

m+1

n n+1

integer coordinate positions;

image values are given

f(m, n), f(m+1, n), ...

x

y

non-integer coordinate position

where image value must be computed;

image function is supposed to be

continuous on [m, m+1] x [n, n+1]

Simplest: 0th-order interpolation

The value in (x,y) is the value of the closest point

having integer coordinates.


Linear (order 1 interpolation) according to each dimension.

[x] [x]+1

f1=f ([x])

f2=f ([x]+1)

x

f (x)

])([][][)1]([

])([)1]([)( xfxx

xx

xfxfxf

112 ])[)(()( fxxffxf

Bilinear interpolation


(m+1,n) (m+1,n+1)

(m,n)

(m,n+1)

m = [x]

n = [y]

f1

f2

f3

f1 = f(m, n)

f2 = f(m+1, n)

f3 = f(m, n+1)

f4 = f(m+1, n+1)

f4x

g1

g2

f (x, y)

g1 is linearly interpolated from f1 and f2g2 is linearly interpolated from f3 and f4

f is linearly interpolated from g1 and g2



(m+1,n) (m+1,n+1)

(m,n)

(m,n+1)

f1

f2

f3

f4x

g1

g2

f (x, y)m = [x]

n = [y]

f1 = f(m, n)

f2 = f(m+1, n)

f3 = f(m, n+1)

f4 = f(m+1, n+1)

g1 is linearly interpolated from f1 and f2 1121 ))(( fmxffg

g2 is linearly interpolated from f3 and f4 3342 ))(( fmxffg

f is linearly interpolated from g1 and g2 112 ))(( gnyggf




1121 ))(( fmxffg

3342 ))(( fmxffg

112 ))(( gnyggf

112

112334

))((

)())(())((

fmxff

nyfmxfffmxfff

113

121234

))((

))(()(

fnyff

mxffnymxfffff

)()()( nymxnymxf


Interpolation methods

Higher-order interpolation: 2, 3, ...

(spline functions – order 3 interpolation)

GEOMETRIC OPERATIONS - pub.roimag.pub.ro › ro › cursuri › archive › IPA_03_Geom_op.pdf ·...

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