ECU 337 Digital Image Processing - National Institute of ... · PDF fileECU 337 Digital Image Processing ... Department of ECE NIT Calicut December 26, 2012 Dr. Praveen Sankrana DIP

Review and furtherSummary

ECU 337 Digital Image Processing

Dr. Praveen Sankaran

Department of ECE

NIT Calicut

December 26, 2012

Dr. Praveen Sankaran DIP Winter 2013


Outline

1 Review and further

Review

Discrete structure and mathematical analysis

Pixel Relationships

2 Summary



ReviewDiscrete structure and mathematical analysisPixel Relationships

Outline


Review


Pixel Relationships

2 Summary




Review Summary

A pixel is a small image area indexed by [m,n];

g [m,n] is the associated pixel value;

the possible values of g [m,n] are the gray levels l = 0,1 · · ·L;a digital image is an M×N array of gray levels.




Scene Sampling

Discerete structure of dgital image

f (x ,y)(scene brightness) at scene element is di�erent from

estimate g (m,n) at digitized image.Dr. Praveen Sankaran DIP Winter 2013



Sampling and Quantization - further example




Spatial Sampling Reduction E�ect




Bit Change E�ect




A Compromise?

If M and N are large and b is small, then high spatial resolution

has been achieved at the expense of low brightness resolution.

If b is too small (too few gray levels) the digital image will

exhibit objectionable �gray level contouring�.

Conversely, if M and N are too small the image will have low

spatial resolution (but high brightness resolution) and exhibit

an objectionable �pixelation�.

In terms of visual quality, it is generally felt that minimum

requirements are: M and N should be at least 256 and b

should be at least 5 (i.e., 32 gray levels).




Outline


Review


Pixel Relationships

2 Summary




Why is Discrete Structure a Problem?

We will look at two cases,

1 Where the conversion of operation to discrete domain is

obvious enough.

2 Where there are ambiguities, and our selection seriously a�ects

our end results.

Obvious case: Average brightness of an image

1MN

∫M−1/2−1/2

∫ N−1/2−1/2 f (x ,y)dxdy

- would give a true average.

- but how do we integrate an image?

Digital Image Equivalent

1MN ∑

M−1m=0 ∑

N−1n=0 g (m,n)




Why is Discrete Structure a Problem?

We will look at two cases,

1 Where the conversion of operation to discrete domain is

obvious enough.

2 Where there are ambiguities, and our selection seriously a�ects

our end results.

Obvious case: Average brightness of an image

1MN

∫M−1/2−1/2

∫ N−1/2−1/2 f (x ,y)dxdy

- would give a true average.

- but how do we integrate an image?

Digital Image Equivalent

1MN ∑

M−1m=0 ∑

N−1n=0 g (m,n)




Why is this a problem?

Not so obvious case: derivatives

∂ f (x ,y)∂x

?

This is a very realistic case. e.g. if we want to �nd the edge image

from a given image:

Laplacian ∇2f (x ,y) = ∂2f (x ,y)∂x2

+ ∂2f (x ,y)∂y2

(Why would we do this anyway?)

From Taylor's theorem,

f (x±ξ ,y) = f (x ,y)+ ∂ f (x ,y)∂x

(±ξ )+ 12!

∂2f (x ,y)∂x2

ξ 2+ · · ·Ignoring higher order terms we end up with three di�erent scenarios.




Di�ering Approximations

1∂ f (x ,y)

∂xw f (x+ξ ,y)−f (x ,y)

ξ

2∂ f (x ,y)

∂xw f (x ,y)−f (x−ξ ,y)

ξ

3∂ f (x ,y)

∂xw f (x+ξ ,y)−f (x−ξ ,y)

2ξ

Substituting ξ = 1, three separate, corresponding operators can be

formed for a digital image.

Which one do we choose?

g [m+1,n]−g [m,n]g [m,n]−g [m−1,n]g [m+1,n]−g [m−1,n]

2




Outline


Review


Pixel Relationships

2 Summary




Neighbors, Adjacency, Regions, Connectivity, Boundaries,

Edges

We took ξ = 1 before. We are essentially de�ning our neighbors in

this step.




Distance Measures Within an Image

Consider p (x ,y) and q (s, t)

Euclidean distance =[(x− s)2+(y − t)2

] 12

D4 distance?

D8 distance?



Summary

Looked at how bits and samples allocation can a�ect an image.

Mathematical models developed for continuous systems can

only be approximated in digital imagery.

There can be variation in �nal output based on how you

approximate your equations.

Simple pixel relationships.

Merry Christmas!


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ECU 337 Digital Image Processing - National Institute of ... · PDF fileECU 337 Digital Image Processing ... Department of ECE NIT Calicut December 26, 2012 Dr. Praveen Sankrana DIP