งานนำเสนอ PowerPoint - Ajay Bolar...1 20 2 30 3 10 4 15 5 15 6 10 7 0 Actual Output Histogram Histogram Matching Example (cont.) Desired histogram Transfer function

18-11-2015

1

Image Enhancement in the Spatial Domain

Spatial Domain

What is spatial domain

The space where all pixels form an image In spatial domain we can represent an image by f(x,y) where x and y are coordinates along x and y axis with

respect to an origin

There is duality between Spatial and Frequency Domains

Images in the spatial domain are pictures in the xy plane

where the word “distance” is meaningful.

Using the Fourier transform, the word

“distance” is lost but the

word “frequency” becomes alive.

Image Enhancement

Image Enhancement means improvement of images to be suitable for specific applications. Example: Note: each image enhancement technique that is suitable for one application may not be suitable for other applications.

(Images from Rafael C. Gonzalez and Richard E.

Wood, Digital Image Processing, 2nd Edition.

Image Enhancement Example

Original image Enhanced image using

Gamma correction



= Image enhancement using processes performed in the Spatial domain resulting in images in the Spatial domain.

We can written as

Image Enhancement in the Spatial Domain

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

where f(x,y) is an original image, g(x,y) is an output and T[ ] is a function defined in the area around (x,y)

Note: T[ ] may have one input as a pixel value at (x,y) only or multiple inputs as pixels in neighbors of (x,y) depending in

each function. Ex. Contrast enhancement uses a pixel value at (x,y) only

for an input while smoothing filte use several pixels around (x,y) as inputs.

Types of Image Enhancement in the Spatial Domain

- Single pixel methods

- Gray level transformations

Example

- Historgram equalization

- Contrast stretching

- Arithmetic/logic operations

Examples

- Image subtraction

- Image averaging

- Multiple pixel methods

Examples

Spatial filtering

- Smoothing filters

- Sharpening filters

18-11-2015

2

Gray Level Transformation

Transforms intensity of an original image into intensity of an

output image using a function:

( )s T r

where r = input intensity and s = output intensity

Example:

Contrast enhancement



Image Negative

White

Black

Input intensity

Ou

tpu

t in

tensi

ty

Original

digital

mammogram

1s L r

L = the number of gray levels

0 L-1

L-1

Negative

digital

mammogram



Black White

Log Transformations

Fourier

spectrum

Log Tr. of

Fourier

spectrum

log( 1)s c r

Application



Power-Law Transformations

s cr



Power-Law Transformations : Gamma Correction Application

Desired image

Image

displayed at

Monitor

After Gamma

correction



Image

displayed at

Monitor


MRI Image after Gamma Correction



18-11-2015

3


Ariel images

after Gamma

Correction



Contrast Stretching

Before contrast enhancement

After

Contrast means the difference

between the brightest and darkest intensities



How to know where the contrast is enhanced ?

Notice the slope of T(r)

- if Slope > 1 Contrast increases

- if Slope < 1 Contrast decrease

- if Slope = 1 no change

Dr

Ds

Smaller Dr yields wider Ds

= increasing Contrast



Gray Level Slicing



Bit-plane Slicing

Bit 7 Bit 6

Bit 2 Bit 1

Bit

5

Bit

3



Histogram

Histogram = Graph of population frequencies

0

2

4

6

8

10

A B+ B C+ C D+ D F

No. ofStudents

Grades of the course 178 xxx

18-11-2015

4

Histogram of an Image

( )k kh r nจ ำนว

น pix

el

จ ำนว

น pix

el

= graph of no. of pixels vs intensities



Bright image has

histogram on the right

Dark image has

histogram on the left

Histogram of an Image (cont.)

low contrast image

has narrow histogram



high contrast image

has wide histogram

Histogram Processing

= intensity transformation based on histogram

information to yield desired histogram

- Histogram equalization

- Histogram matching

To make histogram distributed uniformly

To make histogram as the desire

Monotonically Increasing Function

= Function that is only increasing or constant

)(rTs

Properties of Histogram

processing function

1. Monotonically increasing

function

2. 10for 1)(0 rrT

Probability Density Function

and relation between s and r is

Histogram is analogous to Probability Density Function (PDF) which represent density of population

Let s and r be Random variables with PDF ps(s) and pr(r ) respectively

)(rTs

We get

ds

drrpsp rs )()(

r

r dwwprTs0

)()(

Histogram Equalization

Let

We get

1)(

1)(

)(

1)(

1)()()(

0

rp

rp

dr

dwwpd

rp

dr

dsrp

ds

drrpsp

r

rr

r

r

rrs

!

18-11-2015

5


Formula in the previous slide is for a continuous PDF

For Histogram of Digital Image, we use

k

j

j

k

j

jrkk

N

n

rprTs

0

0

)()(

nj = the number of pixels with intensity = j

N = the number of total pixels

Histogram Equalization Example

Intensity # pixels

0 20

1 5

2 25

3 10

4 15

5 5

6 10

7 10

Total 100

Accumulative Sum of Pr

20/100 = 0.2

(20+5)/100 = 0.25

(20+5+25)/100 = 0.5

(20+5+25+10)/100 = 0.6

(20+5+25+10+15)/100 = 0.75

(20+5+25+10+15+5)/100 = 0.8

(20+5+25+10+15+5+10)/100 = 0.9

(20+5+25+10+15+5+10+10)/100 = 1.0

1.0

Histogram Equalization Example (cont.)

Intensity

(r)

No. of Pixels

(nj)

Acc Sum

of Pr

Output value Quantized

Output (s)

0 20 0.2 0.2x7 = 1.4 1

1 5 0.25 0.25*7 = 1.75 2

2 25 0.5 0.5*7 = 3.5 3

3 10 0.6 0.6*7 = 4.2 4

4 15 0.75 0.75*7 = 5.25 5

5 5 0.8 0.8*7 = 5.6 6

6 10 0.9 0.9*7 = 6.3 6

7 10 1.0 1.0x7 = 7 7

Total 100




Histogram Equalization (cont.)






18-11-2015

6





Original image

After histogram equalization, the image

become a low contrast image



Histogram Matching : Algorithm

r

r dwwprTs0

)()(

Concept : from Histogram equalization, we have

We get ps(s) = 1

We want an output image to have PDF pz(z) Apply histogram equalization to pz(z), we get

z

z duupzGv0

)()( We get pv(v) = 1

Since ps(s) = pv(v) = 1 therefore s and v are equivalent

Therefore, we can transform r to z by

r T( ) s G-1( ) z

To transform image histogram to be a desired histogram

Histogram Matching : Algorithm (cont.)

s = T(r) v = G(z)

z = G-1(v)

1

2

3

4



Histogram Matching Example

Intensity

( s )

# pixels

0 20

1 5

2 25

3 10

4 15

5 5

6 10

7 10

Total 100

Input image

histogram

Intensity

( z )

# pixels

0 5

1 10

2 15

3 20

4 20

5 15

6 10

7 5

Total 100

Desired Histogram Example

User define Original

data

r (nj) SPr s

0 20 0.2 1

1 5 0.25 2

2 25 0.5 3

3 10 0.6 4

4 15 0.75 5

5 5 0.8 6

6 10 0.9 6

7 10 1.0 7

Histogram Matching Example (cont.)

1. Apply Histogram Equalization to both tables

z (nj) SPz v

0 5 0.05 0

1 10 0.15 1

2 15 0.3 2

3 20 0.5 4

4 20 0.7 5

5 15 0.85 6

6 10 0.95 7

7 5 1.0 7

sk = T(rk) vk = G(zk)

18-11-2015

7

r s

0 1

1 2

2 3

3 4

4 5

5 6

6 6

7 7


2. Get a map

v z

0 0

1 1

2 2

4 3

5 4

6 5

7 6

7 7

sk = T(rk) zk = G-1(vk)

r s v z

s v

We get

r z

0 1

1 2

2 2

3 3

4 4

5 5

6 5

7 6

z # Pixels

0 0

1 20

2 30

3 10

4 15

5 15

6 10

7 0

Actual Output

Histogram


Desired histogram

Transfer function

Actual histogram




Original

image

After

histogram equalization

After

histogram matching

Local Enhancement : Local Histogram Equalization

Concept: Perform histogram equalization in a small neighborhood

Orignal image After Hist Eq. After Local Hist Eq.

In 7x7 neighborhood



Local Enhancement : Histogram Statistic for Image Enhancement

We can use statistic parameters such as Mean, Variance of Local area

for image enhancement Image of tungsten filament taken using An electron microscope

In the lower right corner, there is a

filament in the background which is

very dark and we want this to be brighter.

We cannot increase the brightness of

the whole image since the white filament will be too bright.



Local Enhancement

Example: Local enhancement for this task

otherwise ),(

and when),(),(

210

yxf

MkDkMkmyxfEyxg

GsGGs xyxy

Original image Local Variance

image

Multiplication

factor



18-11-2015

8

Local Enhancement

Output image



Logic Operations

AND

OR

Result Region

of Interest Image mask Original

image

Application:

Crop areas of interest



Arithmetic Operation: Subtraction

Error

image

Application: Error measurement



Arithmetic Operation: Subtraction (cont.)

Application: Mask mode radiography in angiography work



Arithmetic Operation: Image Averaging

Application : Noise reduction

),(),(

1yxyxg

K

Averaging results in reduction of

Noise variance

),(),(),( yxyxfyxg

Degraded image

(noise) Image averaging

K

i

i yxgK

yxg1

),(1

),(



Arithmetic Operation: Image Averaging (cont.)



18-11-2015

9

Sometime we need to manipulate values obtained from

neighboring pixels

Example: How can we compute an average value of pixels

in a 3x3 region center at a pixel z?

4

4

6 7

6

1

9

2

2

2

7

5

2

2 6

4

4

5

2 1 2

1

3

3

4

2

9

5

7

7

3 5 8 2 2 2

Pixel z

Image

Basics of Spatial Filtering

4

4

6 7

6

1

9

2

2

2

7

5

2

2 6

4

4

5

2 1 2

1

3

3

4

2

9

5

7

7

3 5 8 2 2 2

Pixel z

Step 1. Selected only needed pixels

4

6 7

6

9

1

3

3

4

… …

…

…

Basics of Spatial Filtering (cont.)

4

6 7

6

9

1

3

3

4

… …

…

…

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

19

16

9

13

9

1

69

17

9

19

9

1

49

14

9

13

9

1

y

1 1

1

1

1

1

1

1

1 9

1

X

Mask or

Window or

Template


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

4

4

6 7

6

1

9

2

2

2

7

5

2

2 6

4

4

5

2 1 2

1

3

3

4

2

9

5

7

7

Solution: Imagine that we have

a 3x3 window that can be placed

everywhere on the image

Masking Window


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.

4

4

6 7

6

1

9

2

2

2

7

5

2

2 6

4

4

5

2 1 2

1

3

3

4

2

9

5

7

7

Step 2: Compute

the average value

3

1

3

1

),(9

1

i j

jipy

Sub image p

Original image

4 1

9

2

2

3

2

9

7

Output image

Step 3: Place the

result at the pixel

in the output image Step 4: Move the

window to the next

location and go to Step 2


The 3x3 averaging method is one example of the mask

operation or Spatial filtering.

w The mask operation has the corresponding mask (sometimes

called window or template).

w The mask contains coefficients to be multiplied with pixel

values.

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

1 9

1

Example : moving averaging

The mask of the 3x3 moving average

filter has all coefficients = 1/9


18-11-2015

10

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 point

of the mask


The spatial filtering 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 coefficeints

and 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 2

until all pixel locations have been used.


Examples of Spatial Filtering Masks

Examples of the masks

Sobel operators

0 1

1

0

0

2

-1

-2

-1

-2 -1

1

0

2

0

-1

0

1

x

P

compute to

y

P

compute to

1 1

1

1

1

1

1

1

1 9

1

3x3 moving average filter

-1 -1

-1

8

-1

-1

-1

-1

-1 9

1

3x3 sharpening filter

58

Spatial filters • Purpose:

– Blur or noise reduction

• Lowpass/Smoothing spatial filtering

– Sum of the mask coefficients is 1

– Visual effect: reduced noise but blurred edge as well

• Smoothing linear filters

– Averaging filter

– Weighted average (e.g., Gaussian)

• Smoothing nonlinear filters

– Order statistics filters (e.g., median filter)

• Purpose

– Highlight fine detail or enhance detail blurred

• Highpass/Sharpening spatial filter


– Visual effect: enhanced edges on a dark background

• High-boost filtering and unsharp masking

• Derivative filters

– 1st

– 2nd

59

Smoothing filters

• Purpose:

– Blur or noise reduction

• Smoothing linear filtering (lowpass spatial filter)

– Neighborhood (weighted) averaging

– Can use different size of masks


– Drawback:

• Order-statistic nonlinear filters

– Response is determined by ordering the pixels contained in the image area covered by the mask

– Median filtering

• The gray level of each pixel is replaced by the median of its neighbor.

• Good at denoising (salt-and-pepper noise/impulse noise)

– Max filter

– Min filter

1 1 1

1 1 1

1 1 1

1/9 x

1 2 1

2 4 2

1 2 1

1/16

-60-

Smoothing: Image Averaging

Low-pass filter, leads to

softened edges

smoothing

operator

18-11-2015

11

Smoothing Linear Filter : Moving Average

Application : noise reduction

and image smoothing

Disadvantage: lose sharp details



Smoothing Linear Filter (cont.)



Order-Statistic Filters

subimage

Original image

Moving

window

Statistic parameters

Mean, Median, Mode,

Min, Max, Etc.

Output image

Consider 3x3 original image

Median filtering using the full 3×3 neighborhood: We sort the original image values on the full 3×3 window: 55 , 68 , 77 , 90 , 91 , 95 , 115 , 151 , 210 median value = 91. We replace the center pixel(i.e 68) by

91. Other pixels are unchanged

Order-Statistic Filters- Eg: Median Filter

• Median filter replaces the pixel at the center of the filter with the median value of

the pixels falling beneath the mask.

• Median filter does not blur the image but it rounds the corners.

Order-Statistic Filters: Median Filter



66

Sharpening filters

• Purpose

– Highlight fine detail or enhance detail that has been blurred

• Basic highpass spatial filter


– Visual effect: enhanced edges on a dark background

• Unsharp masking and High-boost filtering

• Derivatives

– 1st derivative

– 2nd derivative

-1 -1 -1 -1 8 -1 -1 -1 -1

1/9 x

18-11-2015

12

image derivative and sharpening Sharpening Spatial Filters

There are intensity discontinuities near object edges in an image



Laplacian Sharpening : How it works

20 40 60 80 100 120 140 160 180 200

0

0.5

1

0 50 100 150 2000

0.1

0.2

0 50 100 150 200-0.05

0

0.05

Intensity profile

1st derivative

2nd derivative

p(x)

dx

dp

2

2

dx

pd

Edge

0 50 100 150 200-0.5

0

0.5

1

1.5

0 50 100 150 200-0.5

0

0.5

1

1.5

Laplacian Sharpening : How it works (cont.)

2

2

10)(dx

pdxp

Laplacian sharpening results in larger intensity discontinuity

near the edge.

p(x)

Laplacian Sharpening : How it works (cont.)

2

2

10)(dx

pdxp

p(x) Before sharpening

After sharpening

Laplacian Masks

-1 -1

-1

8

-1

-1

-1

-1

-1

-1 0

0

4

-1

-1

0

-1

0

1 1

1

-8

1

1

1

1

1

1 0

0

-4

1

1

0

1

0

Application: Enhance edge, line, point

Disadvantage: Enhance noise

Used for estimating image Laplacian 2

2

2

22

y

P

x

PP

or

The center of the mask

is positive

The center of the mask

is negative

18-11-2015

13

Laplacian Sharpening Example

p P2

P2 PP 2



Laplacian Sharpening (cont.)

PP 2

Mask for

1 1

1

-8

1

1

1

1

1

-1 -1

-1

9

-1

-1

-1

-1

-1

-1 0

0

5

-1

-1

0

-1

0

1 0

0

-4

1

1

0

1

0

or

Mask for

P2

or



75

Unsharp masking and high-boost filters

• Unsharp masking

– To generate the mask: Subtract a blurred version of the image from itself

– Add the mask to the original

• Highboost:

– k>1

– Application: input image is very dark

gmask(x,y) = f(x,y) - f(x,y)

g(x,y) = f(x,y) + k*gmask(x,y)

Unsharp Masking and High-Boost Filtering

-1 -1

-1

k+8

-1

-1

-1

-1

-1

-1 0

0

k+4

-1

-1

0

-1

0

Equation:

),(),(

),(),(),(

2

2

yxPyxkP

yxPyxkPyxPhb

The center of the mask is negative

The center of the mask is positive

Unsharp Masking and High-Boost Filtering (cont.)



First Order Derivative

20 40 60 80 100 120 140 160 180 200

0

0.5

1

0 50 100 150 200 -0.2

0

0.2

0 50 100 150 200 0

0.1

0.2

Intensity profile

1st derivative

2nd derivative

p(x)

dx

dp

dx

dp

Edges

18-11-2015

14

First Order Partial Derivative:

Sobel operators

0 1

1

0

0

2

-1

-2

-1

-2 -1

1

0

2

0

-1

0

1 x

P

compute to

y

P

compute to

P x

P

y

P

First Order Partial Derivative: Image Gradient

22

y

P

x

PP

Gradient magnitude

A gradient image emphasizes edges (Images from Rafael C. Gonzalez and Richard E.


First Order Partial Derivative: Image Gradient

P

x

P

y

P

P

Image Enhancement in the Spatial Domain : Mix things up !

+ -

A

P2

Sharpening

P

smooth B

E C

D



E C

Multiplication

F

Image Enhancement in the Spatial Domain : Mix things up !

A

S

G

H

Power

Law Tr.



Documents

งานนำเสนอ PowerPoint - Ajay Bolar...1 20 2 30 3 10 4 15 5 15 6 10 7 0 Actual Output Histogram Histogram Matching Example (cont.) Desired histogram Transfer function