Image Enhancement in the Spatial Domainraul/ImageAnalysis/Enhancement.pdf · 8/21/2019 Enhancement in the Spatial Domain 22 Local Enhancement The parameters of most functions presented

Image Enhancement in the

Spatial Domain

Raul Queiroz Feitosa

8/21/2019 Enhancement in the Spatial Domain 2

Enhancement in the Spatial

Domain Contents

Introduction

Background

Point Processing

Histogram Processing

Local Enhancement

Spatial Filters

Smoothing Filters

Sharpening Filters


Introduction

Two Groups of Enhancement Techniques :

In the Spatial Domain

Based on the direct manipulation of the pixels in an image.

In the Frequency Domain

Based on modifying the Fourier Transform of an image.

pre-

processing

segmen-

tation

feature

extraction

feature

selection

classifi-

cation

post-

processing data label


Background

Notation

g(x,y) =T [ f(x,y) ] where

f(x,y) is the input image,

g(x,y) is the output image, and

T is an operator defined over a neighborhood of (x,y).

x

y

(x,y)


Background

Two groups of approaches:

Point Processing: when the neighborhood is of size

1×1, it becomes a gray level (or intensity or mapping)

transformation function of the form:

s =T (r)

where r and s denote the gray level at f(x,y) and g(x,y)

at any point g(x,y) .

Larger neighborhoods: it bases on the use of small

arrays called mask, whose coefficients determine the

nature of the process.


Point Processing

Image Negatives

Ou

tpu

t g

ray

leve

l -

s

Input gray level - r

T(r)


Point Processing

(r1,s1)

(r2,s2)

T(r)

Ou

tpu

t g

ray

leve

l -

s


Contrast Stretching


Point Processing

Power Law-Transformation

Correção Gama


maxmax/where, rsccrs

Ou

tpu

t g

ray

leve

l -

s

γ=0.04

γ=0.1

γ=0.2

γ=0.4

γ=0.67

γ=1

γ=1.5

γ=2.5

γ=5.0

γ=10

γ=25


Point Processing

Power Law-Transformation

γ=0.04

γ=0.1

γ=0.2

γ=0.4

γ=0.67

γ=1

γ=1.5

γ=2.5

γ=5.0

γ=10

γ=25

Correção Gama


maxmax/where, rsccrs

Ou

tpu

t g

ray

leve

l -

s

γ=0.04

γ=0.1

γ=0.2

γ=0.4

γ=0.67

γ=1

γ=1.5

γ=2.5

γ=5.0

γ=10

γ=25


Point Processing

Brightness Adjustment

10 tosubjected, sbrs


Ou

tpu

t g

ray

leve

l -

s



Definition

The histogram of a digital image with gray levels

in the range [0,L-1] is a discrete function

h (rk) = nk/n

where

rk is the kth gray level,

nk is the number of pixels having gray level rk, and

n is the number of pixels in the image.



Example: dark image



Example: bright image



Example: image with low contrast



Example: image with high contrast



Equalization

Problem formulation:

Design a function s =T (r) that transforms a given input

image so that the output image has uniformly

distributed gray levels.



Equalization

Solution:

Let pr(r) and ps(s) be the histogram of the input and of the output

images, respectively.

Assume that r and s represent continuous gray levels in the

interval [0,1].

Search for the transformation function T (r) satisfying the

following conditions:

a) T(r) is a single-valued and monotonically increasing in [0,1].

b) 0 ≤ T(r) ≤ 1, for 0 ≤ r≤ 1,

c) the inverse function T-1(s) must also meet the above conditions.



Equalization

In the continuous case

In the discrete case

for k = 0,1,…,L-1

r

rrdwwprTs

0

10)()(

j

k

j

r

k

j

j

kkrp

n

nrTs

00

sk

rk



Equalization: example

T(r)

r

Equalization


Histogram Processing Histogram Matching

Problem formulation:

Design a transformation function s =T (r) that transforms a given

input image so that the output image has a specified histogram.

Solution:

Let pr(r) and ps(s) be the histogram respectively of the input image and

the desired histogram. The function s =T (r) will be computed by:

r

rdwwprH

0

.)()( s

sdwwpsG

0

.)()(

.1rHGz

)(1

zG

pr(r) uniform ps(s)


Histogram Matching : example


Source: Matthew Perry

https://www.mapbox.com/about/team/matthew-perry/


Local Enhancement

The parameters of most functions presented so far are

computed based on the whole image.

To enhance details in small image areas (example):

1. Select a neighborhood, whose center moves from pixel to pixel

over the image.

2. At each position find the transformation function based on the

pixel neighborhood.

3. Apply the function to the pixel at the center of the neighborhood.

4. Move the center to the next pixel and repeat steps 2 and 3 till all

image is covered.


Local Enhancement

Example: local histogram equalization

Original Image After Global Equalization After Local Equalization


Local Enhancement

Histogram Statistic for Enhancement

Global statistics: the nth moment about the mean

where

Local statistics: Let (x,y) be the pixel coordinates and Sxy a neighborhood with

a given size around (x,y). We define

the average gray level around Sxy

the contrast around Sxy

i

L

i

n

inrpmrr

1

0

1

0

L

i

iirprm

xy

xyxy

Sts

tsStsSrpmr

),(

,

2

,

2

xy

xy

Sts

tstsSrprm

,

,,


Local Enhancement Histogram Statistic for Enhancement (example)

f (x,y) / g (x,y) the input/output images,

MG and DG are the global mean and standard deviation respectively,

E, k0, k1 e k2, are specified parameters.

only dark, non uniform areas

with low contrast are modified

where, otherwise,,

AND if,,

210

yxf

DkDkMkmyxfEyxg

GSGGSxyxy

{E=4;k0=0.4;k1=0.02;k2=0.4}


Local Enhancement Histogram Statistic for Enhancement (example)

f (x,y) / g (x,y) the input/output images,

MG and DG are the global mean and standard deviation respectively,

E, k0, k1 e k2, are specified parameters;

only dark, non uniform areas with low contrast are modified

where , otherwise , ,

AND if , ,

2 1 0

y x f

D k D k M k m y x f E y x g

G S G G S xy xy

0 M k m G S xy

1 D k S G xy

2 D k G S xy


Region of Interest Processing

(ROI) • logical matrices called masks can be used to define regions in

an image (roi) where an operator is to be applied.

• Example: blurring uninteresting regions

input image mask (roi) output image


Spatial Filters

Operation: g(x,y)=T [ f(x,y)],

w1 w3 w2

w7 w9 w8

w4 w6 w5

Response of a linear spatial filter

R=w1z1+ w2z2+ . . .+ w9z9

w1

w1

w1

w1 w3 w2

w7 w9 w8

w4 w6 w5

w1

w1

w1

w1 w3 w2

w7 w9 w8

w4 w6 w5

w1

w1

w1

w1

w1

w1

w1 w3 w2

w7 w9 w8

w4 w6 w5

w1

w1

w1

w1

w1

w1

w1

w1

w1

w1 w3 w2

w7 w9 w8

w4 w6 w5

w1

w1

w1 . . .

w1 w1 w1

w1 w3 w2

w7 w9 w8

w4 w6 w5

w1 w1 w1 w1 w1 w1 w1

w1

w1

w1

w1 w3 w2

w7 w9 w8

w4 w6 w5

Image

a

as

b

bt

tysxftswyxg ,,,


Smoothing Filters

Aplications:

Removal of small details prior to large object extraction.

Noise reduction.

Side Effect

It blurs the image.

Smoothing masks

All elements are non-negative, and

sum up to 1.

Masks

1 1 1

1 1 1

1 1 1 1/9

averaging filter 3×3

1 1 7

1 1 7

7 7 54 1/88

gaussian filter 3×3


Smoothing Filters

Examples of applying an averaging filter :

original 3×3 5×5

7×7 15×15 25×25


Smoothing Filters

Order Statistic Filters

are non-linear spatial filters whose response is

based on ordering (ranking) the pixels contained

in the image area encompassed by the mask, and

then replacing the value of the center pixel with

the value determined by the ranking result.


Smoothing Filters

Order Statistic Filters:

Median Filter

replaces the value of a pixel by the median of the gray

levels in its neighborhood

A median ξ of a set of values is such that half of the

values in the set is lower than ξ and half of the values

in the set is greater than or equal to ξ .

Percentile Filter

replaces the value of a pixel by n-th percentile of the

gray levels in its neighborhood.


Smoothing Filters

Examples :

Original image Image with salt-and-pepper noise

After average 5x5 filtering After median filtering


Adaptive Filter

The output g(x,y) of a adaptive filter depends on

the statistical characteristics of the input image

f(x,y) inside a m×n rectangular window Sxy

centered at (x,y).

L

L

myxfyxfyxg ),(),(),(2

2

variance in Sxy mean in Sxy

variance of the noise (?)


Adaptive Filter

Example

image corrupted by

additive Gaussian

noise of zero mean

and variance 1000

result of an adaptive

noise reduction filter

result of an arithmetic

mean filter


Sharpening Filters

Applications

Highlight of fine details.

Enhance details that have been blurred due to the

acquisition method.

Side Effect

Emphasizes the noise.


Sharpening Filters

Accomplished by spatial differentiation:

Approximation for the 1ª derivative

Approximation for the 2ª derivative

xfxf

x

xfxxf

x

xf

ox

1lim

121

22

2

lim

xfxfxf

x

xxfxfxfxxf

x

xf

ox


Sharpening Filters

Illustration


Sharpening Filters

From the previous slide we may conclude:

1. First-order derivatives generally produce thicker

edges in the image,

2. Second-order derivatives have a stronger response at

the fine details,

3. First-order derivatives generally have a stronger

response to a gray-level step, and

4. Second-order derivatives have a double-response at

step changes in gray level.


Sharpening Filters

Laplacian Filter

Starting from

we obtain

2

2

2

2

2

y

f

x

ff

yxfyxfyxfyxfyxff ,41,1,,1,12

yxfyxfyxfx

f,1,2,1

2

2

1,,21,2

2

yxfyxfyxf

y

f


Sharpening Filters

Laplacian Filter: example of masks

0 1 0

0 1 0

1 -4 1

1 4 1

1 4 1

4 -20 4

1 1 1

1 1 1

1 -8 1


Sharpening Filters

)(2

xf

)( xf

)()(2

xfxf

shaper

transition

Laplacian based filter : an example

if center coefficient <0

if center coefficient >0

The sharpening effect in 1-D

yxfCyxf

yxfCyxfyxf

s

,,

,,,

2

2


Sharpening Filters

Laplacian based filter: a mask example

= 1 0 0

0 0 0

0 0 0

-4 1 1

1 0 0

1 0 0

4C+1 - C - C

-C 0 0

- C 0 0

-C


Sharpening Filters

Laplacian Filter: application example

Imagem Original C=4

C=10 C=6

C=2

C=8


Sharpening Filters

Unsharp masking

Subtracts a blurred version of an image from the image

itself

Example of a mask

yxfyxfyxfum

,,,

= 1 0 0

0 0 0

0 0 0

8/9 -1/9 -1/9

-1/9 -1/9 -1/9

-1/9 -1/9 -1/9

- 1 1 1

1 1 1

1 1 1

1/9

output of a smoothing filter


Sharpening Filters

Filter High-Boost

It is a generalization of unsharp masking, given by

If we elect to use the Laplacian filter for sharpening

if center coefficient <0

if center coefficient >0

1for ,,, AyxfyxAfyxfhb

output of a sharpening filter

yxfyxAf

yxfyxAfyxf

hb

,,

,,,

2

2

1for ,,1, AyxfyxfAyxfshb


Sharpening Filters

Filter High-Boost: example

original

image

high-boost

with A=1

laplacian

high-boost

with A=1.7


Sharpening Filters

The Gradient

y

f

x

f

f

2/122

y

f

x

fmagf f


Sharpening Filters

Masks for the Gradient

-1 0

0 1

0 -1

1 0

Roberts

0 0 0

1 1 1

-1 -1 -1

0 1 -1

0 -1

1

0 -1 1

Prewitt

0 0 0

2 1 1

-2 -1 -1

0 2 -2

0 -1

1

0 -1 1

Sobel


Sharpening Filters Gradient: example

original image

|Gx|+|Gy| (sobel)

|Gy| (sobel) |Gx| (sobel)


Next Topic

Image Enhancement

in the Frequency

Domain

Documents

Image Enhancement in the Spatial Domainraul/ImageAnalysis/Enhancement.pdf · 8/21/2019 Enhancement in the Spatial Domain 22 Local Enhancement The parameters of most functions presented