digital image fundamentals

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Digital Image Fundamentals

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What is Digital Image Processing ?

Processing of a multidimensional pictures by a digital computer

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Why we need Digital Image Processing ?

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Image “After snow storm”

Fundamentals of Digital Images

f(x,y)

x

y

An image: a multidimensional function of spatial coordinates. Spatial coordinate: (x,y) for 2D case such as photograph,

(x,y,z) for 3D case such as CT scan images (x,y,t) for movies

The function f may represent intensity (for monochrome images) or color (for color images) or other associated values.

Origin

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Digital Images

Digital image: an image that has been discretized both inSpatial coordinates and associated value.

Consist of 2 sets:(1) a point set and (2) a value set

Can be represented in the formI = {(x,a(x)): x X, a(x) F}

where X and F are a point set and value set, respectively.

An element of the image, (x,a(x)) is called a pixel where- x is called the pixel location and- a(x) is the pixel value at the location x

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CCD KAF-3200E from Kodak.(2184 x 1472 pixels,

Pixel size 6.8 microns2)

Image Sensor:Charge-Coupled Device (CCD)

Used for convert a continuous image into a digital image

Contains an array of light sensors

Converts photon into electric chargesaccumulated in each sensor unit

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Horizontal Transportation Register

Out

put G

ate

Image Sensor: Inside Charge-Coupled Device

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Photosites Output

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Image Sensor: How CCD works

abc

ghi

def

abc

ghi

def

abc

ghi

def

Vertical shift

Horizontal shift

Image pixel

Horizontal transportregister

Output

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Image Types

Intensity image or monochrome image each pixel corresponds to light intensitynormally represented in gray scale (gray level).

39871532

22132515

372669

28161010

Gray scale values

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39871532

22132515

372669

28161010

39656554

42475421

67965432

43567065

99876532

92438585

67969060

78567099

Image Types

Color image or RGB image:each pixel contains a vectorrepresenting red, green andblue components.

RGB components

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Image TypesBinary image or black and white imageEach pixel contains one bit :

1 represent white0 represents black

1111

1111

0000

0000

Binary data

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Image TypesIndex imageEach pixel contains index numberpointing to a color in a color table

256

746

941

Index value

Index No.

Redcomponent

Greencomponent

Bluecomponent

1 0.1 0.5 0.3

2 1.0 0.0 0.0

3 0.0 1.0 0.0

4 0.5 0.5 0.5

5 0.2 0.8 0.9

… … … …

Color Table

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Image Sampling

Image sampling: discretize an image in the spatial domain

Spatial resolution / image resolution: pixel size or number of pixels

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How to choose the spatial resolution

= Sampling locationsO

rigi

nal i

mag

eS

ampl

ed im

age

Under sampling, we lost some image details!

Spatial resolution

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How to choose the spatial resolution : Nyquist RateO

rigi

nal i

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e

= Sampling locations

MinimumPeriod

Spatial resolution(sampling rate)

Sampled image

No detail is lost!

Nyquist Rate: Spatial resolution must be less or equalhalf of the minimum period of the imageor sampling frequency must be greater orEqual twice of the maximum frequency.

2mm

1mm

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Effect of Spatial Resolution

256x256 pixels

64x64 pixels

128x128 pixels

32x32 pixels

Down samplingis an irreversibleprocess.

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Image Quantization

Image quantization: discretize continuous pixel values into discrete numbers

Color resolution/ color depth/ levels: - No. of colors or gray levels or- No. of bits representing each pixel value- No. of colors or gray levels Nc is given by

bcN 2

where b = no. of bits

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Image Quantization : Quantization function

Light intensity

Qua

ntiz

atio

n le

vel

0

1

2

Nc-1

Nc-2

Darkest Brightest

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Effect of Quantization Levels

256 levels 128 levels

32 levels64 levels

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Effect of Quantization Levels (cont.)

16 levels 8 levels

2 levels4 levels

In this image,it is easy to seefalse contour.

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Basic Relationship of Pixels

x

y

(0,0)

Conventional indexing method

(x,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)

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Neighbors of a Pixel

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

(x,y-1)

(x,y+1)

4-neighbors of p:

N4(p) =

(x1,y)(x+1,y)(x,y1)(x,y+1)

Neighborhood relation is used to tell adjacent pixels. It is useful for analyzing regions.

Note: q N4(p) implies p N4(q)

4-neighborhood relation considers only vertical and horizontal neighbors.

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p (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)

Neighbors of a Pixel (cont.)

8-neighbors of p:

(x1,y1)(x,y1)

(x+1,y1)(x1,y)(x+1,y)

(x1,y+1)(x,y+1)

(x+1,y+1)

N8(p) =

8-neighborhood relation considers all neighbor pixels.

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p

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

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

Diagonal neighbors of p:

ND(p) =

(x1,y1)(x+1,y1)(x1,y1)(x+1,y+1)

Neighbors of a Pixel (cont.)

Diagonal -neighborhood relation considers only diagonalneighbor pixels.

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Template, Window, and Mask Operation

Sometime we need to manipulate values obtained from neighboring pixels

Example: How can we compute an average value of pixelsin a 3x3 region center at a pixel z?

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35 8222

Pixel z

Image

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Template, Window, and Mask Operation (cont.)

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Pixel z

Step 1. Selected only needed pixels

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9

1

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3

4……

……

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4

67

6

9

1

3

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4……

……


Step 2. Multiply every pixel by 1/9 and then sum up the values

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1 1

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X

Mask orWindow orTemplate

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Question: How to compute the 3x3 average values at every pixels?

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Solution: Imagine that we havea 3x3 window that can be placedeverywhere on the image

Masking Window

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4.3


Step 1: Move the window to the first location where we want to compute the average value and then select only pixels inside the window.

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Step 2: Computethe average value

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Sub image p

Original image

4 1

9

2

2

3

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9

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Output image

Step 3: Place theresult at the pixelin the output image

Step 4: Move the window to the next location and go to Step 2

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The 3x3 averaging method is one example of the mask operation or Spatial filtering.

The mask operation has the corresponding mask (sometimes called window or template).

The mask contains coefficients to be multiplied with pixelvalues.

w(2,1) w(3,1)

w(3,3)

w(2,2)

w(3,2)

w(3,2)

w(1,1)

w(1,2)

w(3,1)

Mask coefficients

1 1

1

1

1

1

1

1

19

1

Example : moving averaging

The mask of the 3x3 moving average filter has all coefficients = 1/9

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Template, Window, and Mask Operation (cont.)The mask operation at each point is performed by:1. Move the reference point (center) of mask to the location to be computed 2. Compute sum of products between mask coefficients and pixels in subimage under the mask.

p(2,1)

p(3,2)p(2,2)

p(2,3)

p(2,1)

p(3,3)

p(1,1)

p(1,3)

p(3,1)

……

……

Subimage

w(2,1) w(3,1)

w(3,3)

w(2,2)

w(3,2)

w(3,2)

w(1,1)

w(1,2)

w(3,1)

Mask coefficients

N

i

M

j

jipjiwy1 1

),(),(

Mask frame

The reference pointof the mask

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The mask operation on the whole image is given by:

1. Move the mask over the image at each location.

2. Compute sum of products between the mask coefficeintsand pixels inside subimage under the mask.

3. Store the results at the corresponding pixels of the output image.

4. Move the mask to the next location and go to step 2until all pixel locations have been used.

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Examples of the masks

Sobel operators

0 1

1

0

0

2

-1

-2

-1

-2 -1

1

0

2

0

-1

0

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dx

dP compute to

dy

dP compute to

1 1

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1

3x3 moving average filter

-1 -1

-1

8

-1

-1

-1

-1

-19

1

3x3 sharpening filter

digital image fundamentals

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