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
Review and furtherSummary
Outline
1 Review and further
Review
Discrete structure and mathematical analysis
Pixel Relationships
2 Summary
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
Outline
1 Review and further
Review
Discrete structure and mathematical analysis
Pixel Relationships
2 Summary
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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.
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
Sampling and Quantization - further example
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
Spatial Sampling Reduction E�ect
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
Bit Change E�ect
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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).
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
Outline
1 Review and further
Review
Discrete structure and mathematical analysis
Pixel Relationships
2 Summary
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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)
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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)
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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.
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
Outline
1 Review and further
Review
Discrete structure and mathematical analysis
Pixel Relationships
2 Summary
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
Neighbors, Adjacency, Regions, Connectivity, Boundaries,
Edges
We took ξ = 1 before. We are essentially de�ning our neighbors in
this step.
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
ReviewDiscrete structure and mathematical analysisPixel Relationships
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?
Dr. Praveen Sankaran DIP Winter 2013
Review and furtherSummary
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!
Dr. Praveen Sankaran DIP Winter 2013
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