what is an image? - gunadarma

1

Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 1

What is an image?

Definition: An image is a 2-dimensional light intensity function, f(x,y), where x and y are spatial coordinates, and fat (x,y) is related to the brightness of the image at that point.Definition: A digital image is the representation of a continuous image f(x,y) by a 2-d array of discrete samples. The amplitude of each sample is quantized to be represented by a finite number of bits. Definition: Each element of the 2-d array of samples is called a pixel or pel (from „picture element“)


A digital image can be written as a matrix

Note: For a color image, f(x,y) might be one of the components.

f x, y( ) =

f (0,0) f (0,1) L f (0, N −1)f (1,0) f (1,1) L f (1, N −1)

M M M

f (L −1,0) f (L −1,1) L f (L −1, N −1)

2


Image Size and Resolution

200x200 100x100 50x50 25x25

• These images were produced by simply picking every n-th sample horizontally and vertically and replicating that value nxn times.


Color Components

Red R(x,y) Green G(x,y) Blue B(x,y)

Monochrome image

R(x,y) = G(x,y) = B(x,y)

3


Different numbers of gray levels

256 32 16

8 4 2

„Contouring“


How many gray levels are required?

Contouring is most visible for a ramp

Digital images typically are quantized to 256 gray levels.

32 levels

64 levels

128 levels

256 levels

4


Storage requirements for digital images

Image LxN pixels, 2B gray levels, c color components

Size = LxNxBxc

– Example: L=N=512, B=8, c=1 (i.e., monochrome)Size = 2,097,152 bits (or 256 kByte)

– Example: LxN=1024x1280, B=8, c=3 (24 bit RGB image)Size = 31,457,280 bits (or 3.75 MByte)

Much less with (lossy) compression!


130129128127126125

Brightness discrimination experiment

Can you see the circle?

Visibility threshold

I

I + ∆I

∆I I ≈ const. ≈ 1K2% „Weber fraction“„Weber‘s Law“

Note: I is luminance,

measured in cd m2

5


Contrast with 8 Bits According to Weber‘s Law

Assume that the luminance difference between two successive representative levels is just at visibility threshold

For

Typical display contrastCathode ray tube 100:1Print on paper 10:1

Suggests uniform quantization in the log(I) domain

Imax

Imin

= 1+ const.( )255

const. = 0.01L0.02

Imax

Imin

= 13L156


Cathode ray tubes (CRT) are nonlinear

Cameras contain γ -predistortion circuit

Gamma characteristic

γ = 2.0 . . . 2.3

Luminance

I

Voltage U, rep. level f

I ~ U γ

U ~ I1 γ

6


log vs. γ-predistortion

Similar enough for most practical applications

U

I

U ~ I1 γ

U ~ log(I)

Imax

Imin

= 100


Image Scaling

Original image Scaled image

f x, y( ) a ⋅ f x,y( )

Scaling in the γ-domain is equivalent to scaling the linear luminance domain

. . . same effect as adjusting camera exposure time.

I ~ a ⋅ f x, y( )( )γ= aγ ⋅ f x, y( )( )γ

7


Adjusting γ

Original image γ increased by 50%

f x, y( ) a ⋅ f x,y( )( )γ with γ = 1.5

. . . same effect as using a different photographic film . . .


Photographic film

γ measures film contrastGeneral purpose films: γ = -0.7 . . . -1.0High-contrast films: γ = -1.5 . . . -10

Lower speed films tend to have higher absolute γ

slope -γ

log E

E is exposure

dens

ity d

shoulder

toe

„linear“ region

I = I0 ⋅10−d

= I0 ⋅10− −γ log E+ d0( )

= I0 ⋅10−d0 ⋅ Eγ

Luminance

Hurter & Driffield curve (H&D curve)for photographic negative

d0

0

2.0

1.0

8


Changing gradation by γ-adjustment

Original ramp γ0

Scaled ramp 2γ0

Scaled ramp 0.5γ0

Scaling chosen toapproximately preservebrightness of mid-gray


Histograms

Distribution of gray-levels can be judged by measuring a histogram:

For B-bit image, initialize 2B counters with 0

Loop over all pixels x,yWhen encountering gray level f(x,y)=i, increment counter #ι

Histogram can be interpreted as an estimate of the probability density function (pdf) of the underlying random process.You can also use fewer, larger bins to trade off amplitude resolution against sample size.

9


Example histogram

gray level

#pix

els

Cameramanimage


Example histogram

gray level

#pix

els

Poutimage

10


Histogram equalization

Idea: find a non-linear transformation

to be applied to each pixel of the input image f(x,y), such that a uniform distribution of gray levels in the entire range results for the output image g(x,y).Analyse ideal, continuous case first, assuming

T(f) is strictly monotonically increasing, i.e., there exists

Goal: pdf pg(g) = const. over the range

0 ≤ f ≤1

f = T −1 g( )

g = T f( )

0 ≤ g ≤1

0 ≤ g ≤1


Histogram equalization for continuous case

From basic probability theory

Consider the transformation function

Then . . .

pg g( ) = pf f( )dfdg

f =T −1 g( )

g = T f( ) = pf α( )0

f

∫ dα 0 ≤ f ≤ 1

dgdf

= pf f( )

pg g( ) = pf f( )dfdg

f =T −1 g( )

= pf f( ) 1pf f( )

f =T −1 g( )

= 1 0 ≤ g ≤1

11


Histogram equalization for discrete case

Now, f only assumes discrete amplitude values with probabilities

Discrete approximation of

The resulting values gk are in the range [0,1] and need to be scaled and rounded appropriately.

f0, f1,L, fL−1

P0 =

n0

n P1 =

n1

n L PL −1 =

nL −1

n

g = T f( ) = pf α( )0

f

∫ dα

gk = T fk( )= Pii= 0

k

∑


Histogram equalization example

Original image Pout Pout after histogram equalization

12



Original image Pout . . . after histogram equalization

gray level

#pix

els

gray level

#pix

els



Original image Cameraman

Cameramanafter histogram equalization

13



Original image Cameraman . . . after histogram equalization

gray level

#pix

els

gray level

#pix

els



Original image Moon Moonafter histogram equalization

14



Original image Moon . . . after histogram equalization

gray level

#pix

els

gray level

#pix

els


Luminance-based segmentation

Original imagePeter f(x,y)

Thresholded Peter m(x,y)

const. ⋅ f (x,y) ⋅m(x,y)

Holes could befilled by morphologicalimage processing algorithms

15


Chroma key

Color is more powerful for pixel-wise segmentation: 3-d vs. 1-d spaceTake picture in front of a blue screen (or green, or orange)


Soft chroma key

α

1 −α∑

Extract„blueness“

for each pixelα

16


Landsat image processing

Original Landsat imagefalse color picture out of bands 4,5,6

Water area segmented and enhancedto show sediments

Source: US Geological Survey USGS, http://sfbay.wr.usgs.gov/

what is an image? - gunadarma

Documents