lecture 3 - stenography

Steganography & Pixel neighbors

Dr Khurram Khurshid

Digital Image Processing

2

Steganography

The science of writing hidden messages in such a way that no one, apart from the

sender and intended recipient, suspects the existence of the message

3

Steganography

Before Moving On …. Recall – Logical Shift Operators

Logical left shift one bit Logical right shift one bit

SteganographySteganography

8-bit Image 6-bit Image

Two

lea

st s

igni

fica

nt b

its a

re 0

If an image is quantized, say from 8 bits to 6 bits and redisplayed it can be all but impossible to tell the difference between the two.

5

Steganography

That other information could be a message, perhaps encrypted, or even another image.

If the 6-bit version is displayed as an 8-bit image then the 8-bit pixels all have zeros in the lower 2 bits:

00bb bb b b

b = 0 or 1 always 0

This introduces the possibility of encoding other information in the low-order bits.

Image 2

R-Shift 6

Image Out

Image 1

R-Shift 2

L-Shift 2

X-Shift n = logical left or right shift by n bits.

logical OR

0 0 1 0 1 1 0 1

1 0 1 1 0 1 0 0

180

1 1 0 1 1 1 0 0

220

0 0 0 0 0 0 1 103

1 0 1 1 0 1 1 1

183

1 0 1 1 0 1 1 0

182

R-Shift 2

L-Shift 2

SteganographySteganography

R-Shift 6

1 0 1 1 0 1 1 1

183

SteganographySteganography

L-Shift 6

1 1 0 0 0 0 0 0

192

Extracted Second Image

How

to

get

the

seco

nd im

age

How

to

get

the

seco

nd im

age

0 0 1 0 1 1 0 1

1 0 1 1 0 1 0 0

180

0 0 0 0 0 0 1 1

03

0 0 0 0 0 0 1 103

1 0 1 1 0 1 1 1

183

1 0 1 1 0 1 1 0

182

R-Shift 2

L-Shift 2

SteganographySteganography

R-Shift 6

If we have only 4 colors (2-bits) and we put them in the lower order bits

If we have only 4 colors (2-bits) and we put them in the lower order bits

No need to shift rightNo need to shift right

What would you do to get back original data?

What would you do to get back original data?

9

Steganography

8-bit-per-band, 3-band, “original” image

8-bit-per-band, 3-band, “original” image

10

Steganography

6-bit-per-band, 3-band, quantized image

6-bit-per-band, 3-band, quantized image

11

Steganography

The histograms of the two versions indicate which is which. If the 6-bit version is displayed as an 8-bit image it has only pixels with values 0, 4, 8, … , 252.

green-band histogram of 8-bit image green-band histogram of 6-bit image

12

Steganography

The second image is invisible because the value of each pixel is between 0 and 3. For any given pixel, its value is added to the to the collocated pixel in the first image that has a value from the set {0, 4, 8, … , 252}. The 2nd image is noise on the 1st.

13

Steganography

To recover the second image (which is 2 bits per pixel per band) simply left shift the combined image by 6 bits.

L-S

hift

6

?

14

Steganography

To recover the second image (which is 2 bits per pixel per band) simply left shift the combined image by 6 bits.

L-S

hift

6

image

15

Steganography

Images 1 and 2 each have 4-bits per pixel when combined.

Images 1 and 2 each have 4-bits per pixel when combined.

This is so effective that two 4-bit-per-pixel images can be superimposed with only the image in the high-order bits visible. Both images contain the same amount of information but the image in the low-order bits is effectively invisible

Image 2

R-Shift 4

Image Out

Image 1

R-Shift 4

L-Shift 4

16

Steganography

Original ImageOriginal Image

17

Steganography

Image quantized to 4-bits per pixel.

Image quantized to 4-bits per pixel.

18

Steganography

Image 1 in upper 4-bits. Image 2 in lower 4-bits.

Image 1 in upper 4-bits. Image 2 in lower 4-bits.

19

Steganography

Extracted Image

20

Steganography

http://mozaiq.org/encrypt/

21

Relationships between pixels

Neighbors of pixel are the pixels that are adjacent pixels of an identified pixel

x-1 x+1x

y-1

y+1

y

22

4- Neighbors of a Pixel –N4(p)

x-1 x+1x

y-1

y-1

y

(x-1,y), (x+1,y), (x, y-1), (x, y+1)

Wh

at a

re th

e

coo

rdin

ate

s of

ea

ch o

f th

e b

lue

pix

els

Wh

at a

re th

e

coo

rdin

ate

s of

ea

ch o

f th

e b

lue

pix

els

23

Diagonal Neighbors of a Pixel –ND(p)

x-1 x+1x

y-1

y-1

y

(x-1,y-1), (x+1,y-1), (x-1, y+1), (x+1, y+1)

24

8- Neighbors of a Pixel –N8(p)

x-1 x+1x

y-1

y-1

y

(x-1,y), (x+1,y), (x, y-1), (x, y+1)

(x-1,y-1), (x+1,y-1), (x-1, y+1), (x+1, y+1)

8 4( ) ( ) ( )DN p N p N p

25

Determine different regions in the image

26

Connectivity

Establishing boundaries of objects and components in an image

Group the same region by assumption that the pixels being the same color or

equal intensity

Two pixels p & q are connected if

They are adjacent in some sense If their gray levels satisfy a specified criterion of

similarity

27

Connectivity

V: Set of gray levels used to define the criterion of similarity

4-connectivity4-connectivity

If gray level4( , ) , ( )p q V and q N p

Set of gray levels V = {1}Set of gray levels V = {1}

28

Connectivity

V: Set of gray levels used to define the criterion of similarity

8-connectivity8-connectivity

If gray level8( , ) , ( )p q V and q N p

Set of gray levels V = {1}Set of gray levels V = {1}

29

Connectivity

V: Set of gray levels used to define the criterion of similarity

m-connectivity (Mixed Connectivity)m-connectivity (Mixed Connectivity)

If gray level

( , ) , satisfies one of the following:p q V and q4 ( )q N p or

4 4( ) ( ) ( ) has no pixels

whose values are from VDq N p And N p N q

a.

b.

30

Example: m – Connectivity

Set of gray levels V = {1}

Note: Mixed connectivity can eliminate the multiple path connections that often occurs in 8-connectivity

31

Paths

Path: Let coordinates of pixel p: (x, y), and of pixel q: (s, t)

A path from p to q is a sequence of distinct pixels with

coordinates: (x0, y0), (x1, y1), ......, (xn,yn)

where (x0, y0) = (x, y) & (xn,yn) = (s, t), and (xi,yi) is adjacent

to (xi-1,yi-1) 1≤i ≤n

32

CC labeling – 4 Connectivity

Process the image from left toright, top to bottom:

1.) If the next pixel to process is 1 i.) If only one of its neighbors

(top or left) is 1, copy its label.

ii.) If both are 1 and have the same label, copy it.

iii.) If they have different labels Copy the label from the left. Update the equivalence table.

iv.) Otherwise, assign a new label.

Re-label with the smallest of equivalentlabels

Pass 1

Pass 2

33

CC labeling – 4 Connectivity

34

CC labeling – 4 Connectivity

35

CC labeling – 8 Connectivity

Same algorithm but examine also the upper diagonal neighbors of pSame algorithm but examine also the upper diagonal neighbors of p

36

CC labeling – 8 Connectivity

Background pixel

Unlabeled Pixel

Background pixel

Unlabeled Pixel

Label 1

37

CC labeling – 8 Connectivity

Background pixel

Unlabeled Pixel

Label 1

Label 2

Background pixel

Unlabeled Pixel

Label 1

Label 2

Label 3

38

CC labeling – 8 Connectivity

Background pixel

Unlabeled Pixel

Label 1

Label 2

Label 3

Background pixel

Unlabeled Pixel

Label 1

Label 2

Label 3

39

CC labeling – 8 Connectivity

Background pixel

Label 1

Label 2

Label 3

Unlabeled pixel

Background pixel

Label 1

Label 2

Label 3

Unlabeled pixel

40

CC labeling – 8 Connectivity

Background pixel

Label 1

Label 2

Label 3

Unlabeled pixel

Label 4

Background pixel

Label 1

Label 2

Label 3

Unlabeled pixel

Label 4

41

Distance Metrics

Let pixels p, q and z have coordinates (x,y), (s,t) and (u,v) respectively.

D is a distance function or metric if D(p,q) ≥ 0 and D(p,q) = 0 iff p = q and D(p,q) = D(q,p) and D(p,z) ≤ D(p,q) + D(q,z)

42

City block distance (D4 distance)

4 ( , )D p q x s y t

Diamond with center at (x,y)

D4 = 1 are the 4 neighbors of pixel p(x,y)

43

Chessboard distance (D8 distance)

8 ( , ) max( , )D p q x s y t

Square centered at p(x,y) D8 = 1 are the 8 neighbors

of pixel p(x,y)

44

Euclidean Distance

2 2( , ) ( ) ( )eD p q x s y t

A circle with radius r centered at (x,y)

p(x,y)

q(s,t)

r

45

Arithmetic Operations

Carried out between corresponding pixel pairs ( , ) ( , ) ( , )s x y f x y g x y

( , ) ( , ) ( , )d x y f x y g x y ( , ) ( , ) ( , )p x y f x y g x y

( , ) ( , ) ( , )d x y f x y g x y

46

Arithmetic Operations

Conversion to range 0 – 255 Difference of two 8-bit images: -255 to 255 Sum of two 8-bit images: 0 to 510 Solution?

Set all values < 0 to 0

Set all values > 255 to 255

Full range of arithmetic operation not captured

47

Arithmetic Operations

First perform the operation

Then perform

min( )mf f f

Creates an image whose minimum value

is 0

max( )s m mf K f f

Creates a scaled image fs with values in

the range [0 K]

48

Logical Operations (Binary Images)

49

Acknowledgements

Digital Image Processing”, Rafael C. Gonzalez & Richard E. Woods, Addison-Wesley, 2002

Peters, Richard Alan, II, Lectures on Image Processing, Vanderbilt University, Nashville, TN, April 2008

Brian Mac Namee, Digitial Image Processing, School of Computing, Dublin Institute of Technology

Computer Vision for Computer Graphics, Mark Borg

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teria

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Acknowledgements

Statistical Pattern Recognition: A Review – A.K Jain et al., PAMI (22) 2000 Pattern Recognition and Analysis Course – A.K. Jain, MSU Pattern Classification” by Duda et al., John Wiley & Sons. Digital Image Processing”, Rafael C. Gonzalez & Richard E. Woods, Addison-Wesley,

2002 Machine Vision: Automated Visual Inspection and Robot Vision”, David Vernon, Prentice

Hall, 1991 www.eu.aibo.com/ Advances in Human Computer Interaction, Shane Pinder, InTech, Austria, October 2008

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teria

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