image enhancement spatial

8/3/2019 Image Enhancement Spatial

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CIS 601

Image ENHANCEMENTin the

SPATIAL DOMAIN

Dr. Rolf Lakaemper


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Most of these slides base on thebook

Digital Image Processingby Gonzales/Woods

Chapter 3


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Introduction

Image Enhancement ?

enhance otherwise hidden information Filter important image features Discard unimportant image features

Spatial Domain ? Refers to the image plane (the natural

image)

Direct image manipulation


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Remember ?

A 2D grayvalue - image is a 2D -> 1Dfunction,

v = f(x,y)


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Remember ?

As we have a function, we can applyoperators to this function, e.g.

T(f(x,y)) = f(x,y) / 2

Operator Image (= function !)


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Remember ?

T transforms the given image f(x,y)into another image g(x,y)

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


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Spatial Domain

The operator T can be defined over

The set of pixels (x,y) of the image The set of neighborhoods N(x,y) of

each pixel

A set of images f1,f2,f3,


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Operation on the set of image-pixels

6 8 2 0

12 200 20 10

3 4 1 0

6 100 10 5

Spatial Domain

(Operator: Div. by 2)


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Operation on the set of neighborhoodsN(x,y) of each pixel

6 8 2 0

12 200 20 10

226

Spatial Domain

6 8

12 200

(Operator: sum)


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Operation on a set of images f1,f2,

6 8 2 0

12 200 20 10

Spatial Domain

5 5 1 0

2 20 3 4

11 13 3 0

14 220 23 14

(Operator: sum)


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Operation on the set of image-pixelsRemark: these operations can also be seen as operations on the

neighborhood of a pixel (x,y), by defining the neighborhood as the

pixel itself.

The simplest case of operators g(x,y) = T(f(x,y)) depends only on the value

of f at (x,y)

T is called a

gray-level or intensity transformation

function

Spatial Domain


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Basic Gray Level Transformations

Image Negatives Log Transformations Power Law Transformations Piecewise-Linear Transformation

Functions

For the following slides L denotes the max. possible gray value of the

image, i.e. f(x,y) [0,L]

Transformations


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Image Negatives: T(f)= L-f

Transformations

Input gray level

O

utputgray

level

T(f)=L-f


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Log Transformations:T(f) = c * log (1+ f)

Transformations


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Log Transformations

Transformations

InvLog Log


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Log Transformations

Transformations


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Power Law TransformationsT(f) = c*fK

Transformations


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varying gamma (K) obtains familyof possible transformation curves K > 0

Compresses dark values Expands bright values

K t,

0 else

t = threshold level

Piecewise Linear Transformations

Input gray level

Outputgr

ay

level


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Gray Level Slicing

Purpose: Highlight a specific range of grayvalues

Two approaches:

1. Display high value for range of interest, low valueelse (discard background)

2. Display high value for range of interest, originalvalue else (preserve background)



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Gray Level Slicing



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Bitplane Slicing

Extracts the information of asingle bitplane



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BP 7

BP5

BP0


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Exercise:

How does thetransformation function lookfor bitplanes 0,1, ?

What is the easiest way to filter a single bitplane(e.g. in MATLAB) ?



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Histograms

Histogram Processing

gray level

Numbero

fPixels

1 4 5 0

3 1 5 1


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Histograms

Histogram Equalization:

Preprocessing technique to

enhance contrast in naturalimages

Target: find gray level

transformation function T totransform image f such that the

histogram of T(f) is equalized


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Histogram Equalization

Equalized Histogram:

The image consists of an equal

number of pixels for every gray-value, the histogram is constant !


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Example:

We are looking for

this transformation !

T


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Target:Find a transformation T to transform the

grayvalues g1[0..1] of an image I tograyvalues g2 = T(g1) such that the

histogram is equalized, i.e. theres an equalamount of pixels for each grayvalue.

Observation (continous model !):Assumption: Total image area = 1 (normalized). Then:

The area(!) of pixels of the transformed

image in the gray-value range 0..g2 equals

the gray-value g2.


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The area(!) of pixels of the transformed image in the gray-

value range 0..g2 equals the gray-value g2.

Every g1 is transformed to a grayvaluethat equals the area (discrete: number of

pixels) in the image covered by pixelshaving gray-values from 0 to g1.

The transformation T function t is the

area- integral: T: g2 = 0..g1 I da


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Discrete:

g1 is mapped to the (normalized)number of pixels having

grayvalues 0..g1 .


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Mathematically the transformation

is deducted by theorems in

continous (not discrete) spaces.

The results achieved do NOT hold

for discrete spaces !

(Why ?)

However, its visually close.


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Conclusion:

The transformation function that yields an imagehaving an equalized histogram is the integral of

the histogram of the source-image The discrete integral is given by the cumulative sum,

MATLAB function: cumsum() The function transforms an image into an image,

NOT a histogram into a histogram ! The

histogram is just a control tool !

In general the transformation does not create an

image with an equalized histogram in the

discrete case !


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Operations on a set of images

Operation on a set of images f1,f2,

6 8 2 0

12 200 20 10

5 5 1 0

2 20 3 4

11 13 3 0

14 220 23 14

(Operator: sum)


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Logic (Bitwise) Operations

AND

OR

NOT


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The operators AND,OR,NOT arefunctionally complete:

Any logic operator can be implementedusing only these 3 operators


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Any logic operator can be implementedusing only these 3 operators:

A B Op

0 0 1

0 1 1

1 0 0

1 1 0

Op=

NOT(A)AND NOT(B)

OR

NOT(A)AND B


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Image 1 AND Image 2

1 2 3 9

7 3 6 4

1 1 1 1

2 2 2 2

1 0 1 1

2 2 2 0

(Operator: AND)


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Image 1 AND Image 2:Used forBitplane-Slicing and

Masking


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Exercise: Define the mask-image, thattransforms image1 into image2 using

the OR operand

1 2 3 9

7 3 6 4

255 2 7 255

255 3 7 255

(Operator: OR)


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Operations

Arithmetic Operations on a set of images

1 2 3 9

7 3 6 4

1 1 1 1

2 2 2 2

2 3 4 10

9 5 8 6

(Operator: +)


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Operations

Exercise:

What could the operators +and be used for ?


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Operations

(MATLAB) Example: Operator Foreground-Extraction


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Operations

(MATLAB)Example: Operator +

Image Averaging


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Part 2

CIS 601

Image ENHANCEMENTin the

SPATIAL DOMAIN


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Histograms

So far (part 1) :

Histogram definition Histogram equalization

Now:

Histogram statistics


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Histograms

Remember:The histogram shows the number of

pixels having a certain gray-value

numberofpixels

grayvalue (0..1)


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Histograms

The NORMALIZED histogram is thehistogram divided by the total number

of pixels in the source image.

The sum of all values in the normalizedhistogram is 1.

The value given by the normalizedhistogram for a certain gray value can

be read as the probability of randomly

picking a pixel having that gray value


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Histograms

What can the (normalized)

histogram tell about the

image ?


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Histograms

1.The MEAN VALUE (or average gray level)

M =7g g h(g)

1*0.3+2*0.1+3*0.2+4*0.1+5*0.2+6*0.1=

2.60.3

0.2

0.1

0.0

1 2 3 4 5 6


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Histograms

The MEAN value is the average gray

value of the image, the overallbrightness appearance.


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Histograms

2. The VARIANCE

V = 7g (g-M)2 h(g)

(with M = mean)

or similar:

The STANDARD DEVIATION

D = sqrt(V)


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Histograms

VARIANCE gives a measure about the

distribution of the histogram values

around the mean.

0.3

0.2

0.1

0.0

0.3

0.2

0.1

0.0

V1 > V2


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Histograms

The STANDARD DEVIATION is a value

on the gray level axis, showing the

average distance of all pixels to the

mean

0.3

0.2

0.1

0.0

0.3

0.2

0.1

0.0

D1 > D2


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Histograms

VARIANCE and STANDARD DEVIATION

of the histogram tell us about the

average contrast of the image !

The higher the VARIANCE (=the higherthe STANDARD DEVIATION), the

higher the images contrast !


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Histograms

Example:

Image and blurred version


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Histograms

Histograms with MEAN andSTANDARD DEVIATION

M=

0.7

3D=

0.32 M=

0.7

1D=

0.27


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Histograms

Exercise:

Design an autofocus system for a digital

camera !

The system should analyse an area in the middle of the picture and

automatically adjust the lens such that this area is sharp.


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Histograms

In between the basics

histograms can give us a first hinthow to create image databases:


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Feature Based Coding


Determine a feature-vector for a given image

Compare images by their feature-vectors

Two operations need to be defined: a mapping of shape

into the feature space and a similarity of feature vectors.

Where are the histograms ?

Representation Feature Extraction Vector Comparison


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Feature Based Coding Determine a feature-vector for a given image

Compare images by their feature-vectors

Two operations need to be defined: a mapping of shape

into the feature space and a similarity of feature vectors.

HERE !

Question: how can we compare histograms (vectors) ?

Representation HISTOGRAM Histogram Comp.


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Vector Comparison

Vector Comparison

,

V t C i


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Vector Comparison

V t C i


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Vector Comparison

V t C i


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Vector Comparison

Whats the meaning of the Cosine Distance with

respect to histograms ?

i.e.: whats the consequence of eliminating the vectors length information ?

V t C i


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Vector Comparison

More Vector Distances:

Quadratic Form Distance Earth Movers Distance

Proportional Transportation Distance

V t C i


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Vector Comparison

Histogram Intersection

(non symmetric):

d(h1,h2) = 1 - min(h1,h2 ) / h1

Ex.: What could be a huge drawback of image

comparison using histogram intersection ?

i i i ii

Hi t


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Histograms

Exercise:

Outline an image database system,using statistical (histogram)

information

Histograms


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Histograms

Discussion:

Which problems could occur if thedatabase consists of the following

images ?

Histograms


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Histograms

Spatial Filtering


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Spatial Filtering

End of histograms.

And now to something completelydifferent

image enhancement spatial

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