N=M=30

What is an image?

Image is a 2D rectilinear array of pixels (picture element)N=M=256

f(x,y):2

L=15(4 bits)L=255 (8 bits)

What is an image?No continuous values - Quantization

[ , ] :[1, ] [1, ] [0, ]f x y N M L

255

170

15

8

L=1 (1 bit) L=3 (2 bits)

An image is just 2D?No! – It can be in any dimensionExample 3D:

Voxel-Volume Element

An image is just 2D?No! – It can be in any dimensionAn image is a n-dimensional rectilinear array of elements

1 2[ ] :[1, ] [1, ] ...[1, ] [0, ]nf x N N N L

[ ] :[1, ] [0, ]nf x N L

f x: n

Does an image just map to scalars?

[ ] :[1, ] [1, ]n nf x N Lf x: n n

Roy van Pelt, PhD & Anna Vilanova, PhDTU/e Biomedical Image Analysis Group, 2012

Sampling and Quantization

• Sampling is digitizing the coordinate values of our function, e.g., f(x,y).

• Quantization is digitizing the amplitude values.

• In practice the sampling and quantization depend on the sensor arrangement that does the measurements.

A B

Digital vs Continuous Image

mm(11.25,11.25) [5,5]f f ( , ) [ , ]f i x j y f i j

Is the distance in mm between samples in x direction

is the distance in mm between samples in y direction

x

yy

xy

x

Spatial resolution defines the smallest spatial change that we will be able to distinguish, in spatial units!

Measures for that are dots per unit distance dpi, e.g., (dots per inch).

Contrast

• Dynamic range is lowest and highest intensity level that an image shows

• Contrast is the difference in intensity between the highest and the lowest level.

• High Dynamic range implies high contrast• Intensity resolution smallest discernible change in

intensity level. – Usually integer power of 2, measured by number of bits.– Whether you can distinguish all levels or not depends on

human perception.

False Contouring

Image Interpolation

• We use the data we know to estimate the values in unknown positions.

x

( )f x [ ] ( ) ( )f i f x f x

x?

Image Interpolation–Nearest Neighbour

• We use the data we know to estimate the values in unknown positions.

x

( )f x [ ] ( ) ( )f i f x f x

? x

0.5 [ ]( )

[ 1]

xi

x

xi f i

f x xf i

else

Example How does it work in 2D?

Image Interpolation – Linear Interpolation

• We use the data we know to estimate the values in unknown positions.

x

( )f x [ ] ( ) ( )f i f x f x

?

( ) (1 ) 1

x x i xi t

x x

f x f i t f i t

x

ExampleHow does it work in 2D?

Interpolation

• There are other methods for interpolation of higher order. Meaning more neighbors are involved and more complex curves are fitted.

Transformations

19

Motivation

How can we transform images?Apply transformation to all pixelsFirst do translation, then rotation, then scaling

tSRvv

20

Motivation

• Transformation in 2D

• Transformation using homogenous coordinatestSRvv

110012221

1211

y

x

taa

taa

y

x

y

x

21

Homogenous coordinates

• Allow to manipulate n-dim vectors in a n+1-dim space

• A point p can be written as vector • In homogenous coordinates we add a scaling factor

• To transform the homogenous coordinates in normal coordinate, divide by the n+1 coordinate.

y

xp

1

y

x

w

wy

wx

ph

22

Homogenous coordinates

• we note

• Proof:

11

:, y

x

by

x

aba

1/

/

/

1

y

x

aa

aay

aax

a

ay

ax

y

x

a

23

Translation

• Classic • Homogenous coordinates

y

x

tyy

txx

12

12

1

1

1

100

10

01

1

2

2

y

x

ty

tx

y

x

24

Rotation (clockwise)

• Classic • Homogenous coordinates

1)cos(1)sin(

1)sin(1)cos(

2

2

yaxay

yaxax

1

1

1

100

0)cos()sin(

0)sin()cos(

1

2

2

y

x

aa

aa

y

x

25

Translation and rotation

• Classic • Homogenous coordinates

tyyaxay

txyaxax

1)cos(1)sin(

1)sin(1)cos(

2

2

1

1

1

100

)cos()sin(

)sin()cos(

1

2

2

y

x

tyaa

txaa

y

x

26

Translation, rotation and scaling

• Classic • Homogenous coordinates

tyyasyxasxy

txyasyxasxx

1)cos(1)sin(

1)sin(1)cos(

2

2

1

1

1

100

)cos()sin(

)sin()cos(

1

2

2

y

x

tyasyasx

txasyasx

y

x

Affine Transformation

x2

y2

1

a11 a12 tx

a21 a22 ty

0 0 1

x1

y1

1

A transformation that preserves lines and parallelism (maps parallel lines to parallel lines) is an affine transformation.

• Demonstration

29

Rigid Transformations in 3d

• Around x-axis (counter-clockwise)

• Around y-axis• Around z-axis

• General

Rx (a)

1 0 0 0

0 cos(a) sin(a) 0

0 sin(a) cos(a) 0

0 0 0 1

Ry (b)

cos(b) 0 sin(b) 0

0 1 0 0

sin(b) 0 cos(b) 0

0 0 0 1

Rz(c)

cos(c) sin(c) 0 0

sin(c) cos(c) 0 0

0 0 1 0

0 0 0 1

T R

tx

ty

tz0 0 0 1

,R Rz(c)Ry (b)Rx (a)

30

Image transformation

For each position Pd in the destination image we searchthe pixel color I(Pd).

Source image Destination image

Tsd

31

Image transformation

First we compute a position Ps in the source image.

Source image Destination image

Tsd

1100

)cos()sin(

)sin()cos(

1d

d

yyx

xyx

s

s

y

x

tasas

tasas

y

x

32

Image transformation

• P is not integer.• How do we compute I(Pd)=I(Ps)?• Answer: by interpolation

Ps0 Ps1

Ps3Ps2

Ps

Pd

Tsd

• Demonstration