Image Sampling - Kasetsart University

Image SamplingImage SamplingImage SamplingImage Sampling

ExampleExampleExampleExample

Matrix RepresentationMatrix RepresentationMatrix RepresentationMatrix Representation

A QuantizerA QuantizerA QuantizerA Quantizer

Matrix RepresentationMatrix RepresentationMatrix RepresentationMatrix Representation

Uniform QuantizationUniform QuantizationUniform QuantizationUniform Quantization

A Uniform QuantizerA Uniform QuantizerA Uniform QuantizerA Uniform Quantizer

Example 1 of Uniform QuantizerExample 1 of Uniform Quantizer

Example 2 of Uniform Example 2 of Uniform Example 2 of Uniform Quantizer

Example 2 of Uniform Quantizer

Q i i Q i i Quantization Effect – False Quantization Effect – False Effect False

ContourEffect False

Contour

Examples of Uniform Examples of Uniform Examples of Uniform Quantizer

Examples of Uniform Quantizer

Ch t 2 Di it l I F d t lCh t 2 Di it l I F d t lChapter 2: Digital Image FundamentalsChapter 2: Digital Image Fundamentals




Chapter 2: Digital Image FundamentalsChapter 2: Digital Image Fundamentalsp g gp g g




Chapter 3Chapter 3Image Enhancement in the

Spatial DomainImage Enhancement in the

Spatial Domain

The principal objective of enhancement is to process an image so that the result image is more suitable than the original image for a specific application. There are two broad categories:categories:

•Spatial domain: These approaches are based on direct p ppmanipulation of pixels in an image.•Frequency domain: These techniques are based on

d f h f fmodifying the Fourier transform of an image.



Spatial Domain

Spatial domain refers to the aggregate of pixels composing an image. Spatial domain methods are procedures that operate directly on these pixels.p



Spatial Domain

Contrast Stretching Thresholding Function

Chapter 3Chapter 3Point Processing and mask processing

or filteringPoint Processing and mask processing

or filtering



Spatial Domain

Wh i C hi ?What is Contrast stretching?



Spatial Domain

H h h ?How to enhance the contrast?

Low contrast image values concentrated near anarrow range (mostly dark, or mostly bright, or mostly

di l )medium values)• Contrast enhancement change the image valuedistribution to cover a wide rangedistribution to cover a wide range• Contrast of an image can be revealed by its histogram

Chapter 3Chapter 3Some Basic Gray Level TransformSome Basic Gray Level Transform

- Image Negatives

- Log Transformations

- Power-Law Transformations

Chapter 3Chapter 3Image NegativesImage Negatives

Image Negatives

Chapter 3Chapter 3Log TransformationsLog Transformations

Log Transformations

Chapter 3Chapter 3Power-Law TransformationsPower-Law Transformations

Power-LawPower Law Transformations

The exponent in the l tipower-law equation

is referred to as “gamma”gamma

Chapter 3Chapter 3Gamma CorrectionGamma Correction



Chapter 3Chapter 3Log /Power-Law TransformationsLog /Power-Law Transformations

f

Chapter 3Chapter 3Piecewise Linear StretchingPiecewise Linear Stretching

Chapter 3Chapter 3Contrast StretchingContrast Stretching

Chapter 3Chapter 3Gray-level SlicingGray-level Slicing


G l lGray-level slicing




Spatial Domain



Spatial Domain



Spatial Domain

Review: Probability and Random VariablesReview: Probability and Random Variables

Random VariablesRandom Variables

Random Variable (RV)*Th S f ll ibl f i l– The set, S, of all possible outcomes of a particular

experiment is called the sample space for the iexperiment.

– An event is any collection of possible outcomes of i h i b f San experiment, that is, any subset of S.

– Random variable, x is a function from a sample S i h l bspace S into the real numbers.


Random Variables (Con’t)Random Variables (Con’t)

Examples of Random VariablesTossing two coins, X is the number of heads, and Y is the g , f ,number of tails

– X and Y take on values {0, 1, 2}– Discrete type

• X is the lifetime of a certain band of light bulbsX take on values [0 +∞)– X take on values [0, +∞)

– Continuous type• X is the intensity value (Gray level, 0-255) of pixels in an y ( y , ) f pimage

– X take on the values {0, 1, …, 255}i– Discrete type



Di t ib ti D it d M F tiDistribution, Density, and Mass Functions



S i l CSpecial Cases



In Sec. 3.3 of the book we used the notation p(rk), k = 0,1,…, L - 1, to denote the histogram of an image with L possible gray levels rkto denote the histogram of an image with L possible gray levels, rk, k = 0,1,…, L - 1, where p(rk) is the probability of the kth gray level (random event) occurring. The discrete random variables in this case are gray levels. It generally is clear from the context whether one is working with continuous or discrete random variables, and

h th th f b i ti i f l it Alwhether the use of subscripting is necessary for clarity. Also, uppercase letters (e.g., P) are frequently used to distinguish between probabilities and probability density functions (e g p)between probabilities and probability density functions (e.g., p) when they are used together in the same discussion.



If a random variable x is transformed by a monotonic transformation function T(x) to produce a new random variable y, h b bili d i f i f b b i d fthe probability density function of y can be obtained from

knowledge of T(x) and the probability density function of x, as follows:follows:

where the subscripts on the p's are used to denote the fact that they are different functions, and the vertical bars signify the b l l A f i T( ) i i ll i i ifabsolute value. A function T(x) is monotonically increasing if

T(x1) < T(x2) for x1 < x2, and monotonically decreasing if T(x1) > T(x ) for x < x The preceding equation is valid if T(x) is an> T(x2) for x1 < x2. The preceding equation is valid if T(x) is an increasing or decreasing monotonic function.

Chapter 3Chapter 3Histogram ProcessingHistogram Processing

h i hi f i ?What is a histogram of an image?

I = imread('pout tif');I = imread( pout.tif );imshow(I);figure, imhist(I,64);g , ( , );


Dark imageC ll Dark imageThe components of the histogram are concentrated on the low side

f th l

Can we tell something b h of the gray scale.

Bright imageThe components of the histogram

about the image,f j The components of the histogram

biased toward the high side of the gray scale.

from just look at its hi ? Low Contrast image

The histogram will be narrow and will be concentrated on toward the

histogram?

middle of the gray scale.High Contrast imagewhose pixels have a large variety of graylarge variety of gray tones. The histogram is not far from a uniform.


Different images But have the same histogramDifferent images But have the same histogram


More examples of histograms


Histogram CalculationHistogram Calculation

Chapter 3Chapter 3Contrast StretchingContrast Stretching

Chapter 3Chapter 3Histogram EqualizationHistogram Equalization

r is a gray level normalized to the interval [0, 1] which r = 0 representing black and r = 1 representing whitewhich r 0 representing black and r 1 representing white

S=T(r) 0 ≤ r ≤ 1The transformation function T(r) satisfies the following conditions:The transformation function T(r) satisfies the following conditions:(a) T(r) is single-valued and monotonically incresing in the interval0 ≤ r ≤ 1 and(b) 0 ≤ T( ) ≤ 1 f 0 ≤ ≤ 1(b) 0 ≤ T(r) ≤ 1 for 0 ≤ r ≤ 1

Chapter 3Chapter 3Chapter 3Histogram Equalization

Chapter 3Histogram Equalization

Transform an image with an arbitrary histogram to another image ith a flat histogramanother image with a flat histogram.

• Suppose r has pdf, pr(r) where 0 ≤ r ≤ 0

• Use the transformation

• s has a uniform pdf in (0 1)• s has a uniform pdf in (0,1)


Need and output histogram (distribution)Need and output histogram (distribution)to be uniform or flat.

A f d t l lt f b i b bilitA fundamental result from basic porbabilitytheory is that if pr(r) and T(r) are know, and T(r) is continuous and differentiable, the

h f f d i bl bThe PDF of transformed variable s can beObtained using

ps(s)ds = pr(r)dr or

Chapter 3Chapter 3

In the case of histogram:

Histogram EqualizationHistogram Equalization

e c se o s og :)()()1()(

0

wdwpLsTsr

r∫−==

Where (L-1) is the maximum intensity value of an image.

dTd )(dr

rdTdrds )(

=

⎥⎤

⎢⎡

−= ∫r

r wdwpddL )()()1( ⎥

⎦⎢⎣∫ rp

ds 0

)()()(

)()1( rpL r−=ds

Substitutes in the equationdrds

)()1(1)()()(

rpLrp

drdsrpsp rrs == 1)( =sps)()1( rpLdr r−

)1(1−

=L

10 −≤≤ Ls)1(

)(−L

ps


1)1(

1)(−

=L

sps

From the example ps(s) is a uniform PDF. Note that T(r)depends on pr(r) but as shows, the resultin ps(s) always isdepends on pr(r) but as shows, the resultin ps(s) always is uniform, independently of the form of ps(s) .



Chapter 3Chapter 3pHistogram Equalization

pHistogram Equalization

Ex: Suppose that a 3-bit image (L=8) of size 64x64 (4096 pixels) has the intensity distribution shown in table.

1 33 11.33 → 13.08 → 34.55 → 54.55 55.67 → 66.23 → 66.65 → 76.86 → 77 00 77.00 → 7




ExampleExample

rk pr(rk) sk = 7* sk ps(sk)0 0.19 0.19 [1.33]=11 0.25 0.44 [3.08]=3

0 0

1 0.191 0.25 0.44 [3.08] 32 0.21 0.65 [4.55]=53 0 16 0 81 [5 67] 6

12 0

3 0 253 0.16 0.81 [5.67]=64 0.08 0.89 [6.03]=6

3 0.25

4 0

5 0.06 0.95 [6.65]=76 0.03 0.98 [6.86]=7

5 0.21

6 0.16+0. = 0.0.246 0.03 0.98 [6.86] 77 0.02 1.00 [7]=7

67 0.06+0.03+0.02=0.11


ExampleExampleIntensity (rk)

No. of Pixels(nj)

sk = 7* sk

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





Hi li i i l f i h i i l h hHistogram equalization involves transforming the intensity values so that the histogram of the output image approximately matches a specified histogram. (By default, histeq tries to match a uniform histogram with 64 bins, but you can specify a different histogram instead.)

I = imread('pout.tif');i h (I)imshow(I);[J,T] = histeq(I);figure;imshow(J);figure;imshow(J);figure, imhist(J,64);figure,plot((0:255)/255,T);


Chapter 3Chapter 3Histogram EqualizationSample Matlab Code

Histogram EqualizationSample Matlab Code

Chapter 3Chapter 3Chapter 3Project 1Chapter 3Project 1

N t ll i b tl t d i t f d d b k d i i lNot all images can be neatly segmented into foreground and background using simple thresholding. Whether or not an image can be correctly segmented this way can be determined by looking at an intensity histogram of the image.

If it is possible to separate out the foreground of an image on the basis of pixel intensity, then the intensity of pixels within foreground objects must be distinctly different from the intensity of pixels within the background. In this case, we expect to see a distinct peak in the histogram corresponding to foreground objects such that thresholds can be chosen to isolate this peak accordingly. If such a peak does not exist, then it is unlikely that simple thresholding will produce a good segmentation. In this case, adaptive thresholding may be a better answer.

Chapter 3Chapter 3Chapter 3Project 1Chapter 3Project 1

P j t 1Project 1

Chapter 3Chapter 3Histogram SpecificationHistogram Specification

zk=G-1[T(rk)]zk G [T(rk)]zk = G-1(sk)

k = 0, 1, 2,,…,L-1

Chapter 3Chapter 3Specific pdfHistogram SpecificationHistogram Specification

’

Calculate from a given imageSpecific pdf

rk pr(rk) pz(zk) zk p’z(zk)

0 0.19 0.19 0.0 0.0 0 0.01 0 25 0 44 0 0 0 0 1 0 01 0.25 0.44 0.0 0.0 1 0.02 0.21 0.65 0.0 0.0 2 0.03 0 16 0 81 0 15 0 15 3 0 193 0.16 0.81 0.15 0.15 3 0.194 0.08 0.89 0.35 0.20 4 0.255 0.06 0.95 0.65 0.30 5 0.216 0.03 0.98 0.85 0.20 6 0.247 0.02 1.00 1.00 0.15 7 0.11

rk 0 1 2 3 4 5 6 7

zk 3 4 5 6 6 7 7 7


Histogram ของ input image เปนดังนี้ตองการให Histogramของ output image เปนดังนี้

Intensity( s )

# pixels

Histogram ของ input image เปนดงนIntensity ( z )

# pixels

ของ output image เปนดงน

โจทยตัวอยาง ( s )0 20

1 5

( )0 5

1 101 5

2 25

3 10

2 15

3 20

User เปนผูกําหนดขอมูลตั้งตน

3 10

4 15

5 5

4 20

5 155 5

6 10

7 10

6 10

7 57 10

Total 100 Total 100


1. ทํา Histogram Equalization ทัง้สองตาราง

rk (nj) Σpr sk

0 20 0 2 1

zk (nj) Σpz vk

0 5 0 05 00 20 0.2 1

1 5 0.25 2

2 25 0 5 3

0 5 0.05 0

1 10 0.15 1

2 15 0 3 22 25 0.5 3

3 10 0.6 4

4 15 0 75 5

2 15 0.3 2

3 20 0.5 4

4 20 0 7 54 15 0.75 5

5 5 0.8 6

4 20 0.7 5

5 15 0.85 6

6 10 0.9 6

7 10 1.0 7

6 10 0.95 7

7 5 1.0 7

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


2. ไดตาราง Mapไดเปน

Actual Output Hi t

rk sk vk zk

rk sk vk zk

sk vk

ไดเปน

rk zk zk # Pixels

Histogram

0 1

1 2

0 0

1 1

0 1

1 2

0 0

1 20

2 3

3 4

2 2

4 3

2 2

3 3

2 30

3 10

4 5

5 6

5 4

6 5

4 4

5 5

4 15

5 15

6 6

7 7

7 6

7 7

6 5

7 6

6 10

7 07 7 7 7

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



Spatial Domain



Spatial Domain



Spatial Domain



Spatial Domain



Spatial Domain

Chapter 3Chapter 3Logic OperationLogic Operation

Chapter 3Chapter 3Image SubtractionImage Subtraction

Chapter 3Chapter 3Image SubtractionImage Subtraction

Chapter 3Chapter 3Image AveragingImage Averaging

Then



Spatial Domain

Chapter 3Chapter 3

Sometime we need to manipulate values obtained from i hb i i l

Basics of Spatial FilteringBasics of Spatial Filtering

neighboring pixels

Example: How can we compute an average value of pixelsin a 3x3 region center at a pixel z?

4 1 2 262Pixel z

44 12

2 2643 4

29

676

92

725135

7

54 212735 8222

Image

Chapter 3Chapter 3Basic of Spatial FilterBasic of Spatial Filter

Chapter 3Chapter 3

S 1 S l d l d d i l


Step 1. Selected only needed pixels

4 1 2 262Pixel z

…

4679

27243 49

74

6793 4

……

6254

5212

1357

6 13

…

35 8222

Chapter 3Chapter 3

S 2 M l i l i l b 1/9 d h h l


…

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

4679

3 4

…

676

913

……

…

4914

913

91

⋅+⋅+⋅=y

…

6917

919

91 ⋅+⋅+⋅+

X1

916

913

91 ⋅+⋅+⋅+1 1

1 1111 Mask or

Window or1

11

111

9 Window orTemplate

Chapter 3Chapter 3

Q ti H t t th 3 3 l t i l ?


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

4 1 2 262Solution: Imagine that we havea 3x3 window that can be placed

h h i4679

27243 49

7

everywhere on the image

6254

5212

1357

Masking Window

Chapter 3Chapter 3

Step 1: Move the window to the first location where we want to


compute the average value and then select only pixelsinside the window.

44 12

2 2643 4

29

Step 2: Computethe average value

3 3 1

4 12 3

29

676

92

725135

7 ∑∑= =

⋅=3

1

3

1

),(91

i j

jipy

Sub image p927

6254

5212

1357

Sub image p

Original imageStep 3: Place theresult at the pixel

4.3Original image result at the pixel

in the output imageStep 4: Move the

i d hOutput image

window to the next location and go to Step 2

Chapter 3Chapter 3

The 3x3 averaging method is one example of the mask


operation or Spatial filtering.

The mask operation has the corresponding mask (sometimes p p g (called window or template).

The mask contains coefficients to be multiplied with pixelp pvalues.

Example : moving averagingw(2,1) w(3,1)

w(2,2) w(3,2)

w(1,1)

w(1,2)

1 11 1

111

w(3,3)w(3,2)w(3,1)

k ffi i

1119

Th k f h 3 3 iMask coefficients The mask of the 3x3 moving averagefilter has all coefficients = 1/9

Chapter 3Chapter 3

The mask operation at each point is performed by:1 Move the reference point (center) of mask to the


1. Move the reference point (center) of mask to thelocation to be computed

2. Compute sum of products between mask coefficients and pixels in subimage under the mask.

… Mask frame

p(2,1)p(1,1) p(3,1)

…

w(2,1) w(3,1)w(1,1)

p(3,2)p(2,2)

p(2,3)

p(2,1)

p(3,3)p(1,3)

……

w(3,3)

w(2,2)

w(3,2)

w(3,2)w(1,2)

w(3,1)

…Subimage Mask coefficientsN M

∑∑= =

⋅=N

i

M

j

jipjiwy1 1

),(),(The reference pointof the mask

Chapter 3Chapter 3Basics of Spatial FilteringBasics of Spatial Filtering

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 coefficeintsand 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 2until all pixel locations have been used.

Chapter 3Chapter 3Chapter 3Basics of Spatial Filtering

Chapter 3Basics of Spatial Filtering

• Consider when the mask is moving near the edge of image.g g– Mask would not go out the image, so what is the

size of a result image?size of a result image?



• Consider when the mask is moving near the edge of image. How can we g g gkeep the size of original image? If the size of the mask is nxn.– Pad the image by adding (n-1)/2 rows or columns with 0’s– Pad with Replicating adding (n-1)/2 rows or columns.

• What are the effects?• How to do these with matlab or OpenCV?• How to do these with matlab or OpenCV?

0 0 0 00000

00

00

00

00

00

0 0 0 00000

00



• Is this Spatial Filter is linear operation?– What are Linear Spatial Filters?p

– What are Nonlinear Spatial Filters?

Chapter 3Chapter 3

Examples of the masks


Examples of the masksSobel operators

0 11 2 111 11

3x3 moving average filter

0 1

100

2-1-21

-2 -1

102

0-101

1 1

111

1111

91

10-1 121P∂∂ compute to P

∂∂ compute to

111

x∂ y∂

3x3 sharpening filter (laplacian)-1 -18 -1

-1-11

-1-1-19



• Smoothing Spatial Filters– Smoothing Linear Filtersg

– Order-Statistics Filter (Nonlinear Filters)

Chapter 3Chapter 3Smoothing Linear FiltersSmoothing Linear Filters

• The first mask is a standard average of pixels under the mask sometimes called a box filter. An m x n mask would have a so e es ca ed a box filte . m x n as wou d ave anormalizing constant equal to 1/nm.

• The second mask called weighted average. Pixels are inversely weighted to as a function of their distance from the center of the mask.

• In practice it is difficult in general to see differences between• In practice, it is difficult in general to see differences between images smoothed by either of the masks.

Chapter 3Smoothing Linear Filter : Moving Average


Application : noise reductionand image smoothing

Di d t l h d t ilDisadvantage: lose sharp details

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.g g g


Chapter 3Smoothing Linear Filter : Moving AverageSmoothing Linear Filter : Moving AverageSmoothing Linear Filter : Moving Average

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Chapter 3O d St ti ti Filt

Chapter 3O d St ti ti FiltOrder-Statistic FiltersOrder-Statistic Filters

subimageOriginal image

Statistic parametersM M di M dMean, Median, Mode, Min, Max, Etc.

Moving window

Output imageNonlinear Filters

Chapter 3Chapter 3pOrder-Statistic Filters

pOrder-Statistic Filters


Chapter 3Chapter 3Chapter 3Order-Statistic Filters

Chapter 3Order-Statistic Filters

• Median Filters are particularly effective in presence of impulse noise, also called salt-p p ,pepper noise. Why?

• Salt-pepper noise appears as white and black dots superimposed on an image.

Chapter 3Chapter 3Chapter 3Sharpening Spatial Filters

Chapter 3Sharpening Spatial Filters

• Foundation– A basic definition of the first –order derivative of a one dimensional

function f(x) is the difference:function f(x) is the difference:–

( ) ( )xfxfxf

−+=∂∂ 1

An image function has two variables f(x,y), we use partial derivative in order to keep the notation the same.

x∂

– Similarly, we define a second-order derivative as the difference:

( ) ( ))1()()()1(2

xfxfxfxff−−−−+=

∂ ( ) ( )

)(2)1()1(

)1()()()1(

2

2

2

xfxfxff

xfxfxfxfx

−−++=∂

+∂

)(2)1()1(2 xfxfxfx

++∂

Chapter 3Sh i S ti l Filt

Chapter 3Sh i S ti l Filt

There are intensity discontinuities near object edges in an imageSharpening Spatial FiltersSharpening Spatial Filters


Chapter 3L l i Sh i H it k


1Intensity profileEd

Laplacian Sharpening : How it worksLaplacian Sharpening : How it works

0

0.5f(x)Edge

20 40 60 80 100 120 140 160 180 2000.2

1st derivative

0 50 100 150 2000

0.1

dxdf

0 50 100 150 200

0

0.052nd derivative

0 50 100 150 200-0.05

0

2

2

dxfd

0 50 100 150 200

1 5

Chapter 3Laplacian Sharpening : How it works

Chapter 3Laplacian Sharpening : How it works

1

1.5

f(x)

0

0.5f(x)

0 50 100 150 200-0.5

1

1.5

2

)( fdf

0

0.5210)(dx

fdxf −

0 50 100 150 200-0.5

Laplacian sharpening results in larger intensity discontinuityLaplacian sharpening results in larger intensity discontinuity near the edge.


Chapter 3L l i Sh i H it kLaplacian Sharpening : How it worksLaplacian Sharpening : How it works

B f h if(x)

Before sharpening

2 fdAfter sharpening

2

2

10)(dx

fdxf −

Chapter 3Chapter 3Chapter 3Summary First and Second order derivatives

Chapter 3Summary First and Second order derivatives

1. First-order derivatives generally produce thicker edges in an image.2. Second-order derivatives have a stronger response to fine detail, such as

thin lines and isolated pointsthin lines and isolated points.3. First-order derivatives generally have a stronger response to a gray-level

step.4. Second-order derivatives produce a double response at step changes in

gray level.

• The second derivative is better suited than the first derivative for image enhancement because of the ability of the former to enhance fine detail.

• The first derivative is for edge extraction, they do have important uses in conjunction with second derivative to obtain some impressive enhancement resultenhancement result

Chapter 3S d D i ti f E h t Th L l i

Chapter 3S d D i ti f E h t Th L l iSecond Derivatives for Enhancement The LaplacianSecond Derivatives for Enhancement The Laplacian

Chapter 3Laplacian Masks

Chapter 3Laplacian Masks

Used for estimating image Laplacian 2

2

2

22

yf

xff

∂∂

+∂∂

=∇Laplacian MasksLaplacian Masks

g g p

1 18 1

11

1 04 1

01

The center of the mask i i

1-81

1110

-41

110

is negative90o Isotropic Filters are 45o Isotropic

or-1 -1-1-1 00

Th f h k

Filters are rotation invariant

45 Isotropic Filters

-18-1

-1-1-10

4-1

-1-10

The center of the mask is positive

Application: Enhance edge, line, pointpp g pDisadvantage: Enhance noise

Chapter 3Laplacian Sharpening Example


f f2∇

f2∇ ff 2∇f∇ ff 2∇−


Chapter 3Chapter 3Chapter 3Laplacian Sharpening

Chapter 3Laplacian Sharpening

• Laplacian is a derivative operation, its use highlights gray-level discontinuities in an image and deemphasizes regions with slowly varying gray levels This will trend to producewith slowly varying gray levels. This will trend to produce images that have grayish edge lines and other discontinuities, all superimpose on a dark, featureless background.p p , g

• To recover background features while still preserve the sharpening effect of Laplacian simply by adding the original and Laplacian images.

Chapter 3Chapter 3Chapter 3Laplacian Sharpening

Chapter 3Laplacian Sharpening

SimplificationWe implemented the Laplacian sharpening by first computing the L l i filt d i d th bt ti it f th i i lLaplacian-filtered image and then subtracting it from the original image. We can computer a one pass mask:

Mask forMask forf2∇ ff 2∇−

1 18 1

11

-1 -19 1

-11

-1 05 1

01

1 0-4 1

01 oror

1-81

111 -1

9-1

-1-1-10

5-1

-1-100

-41

110

o



ff 2∇−Mask for

-1 -19 1

-11

-1 05 1

01 or

-19-1

-1-1-10

5-1

-1-10

orMask for

f2∇

1 11

f∇

1-81

111

or1 0-4 1

01

or

041

110


Chapter 3Chapter 3Chapter 3Unsharp Masking and High-Boost Filtering

Chapter 3Unsharp Masking and High-Boost Filtering

• A process to sharpen images (in the publishing industry) consists of subtracting a blurred version of image from the image itself called unsharp masking expressed as:image itself, called unsharp masking, expressed as:

• A slight further generalization of unsharp masking is called• A slight further generalization of unsharp masking is called high-boost filtering. fhb is defined at any point (x,y) as:

• As before is a blurred version of f, so fhb can rewrite as:f

• Then

Chapter 3Chapter 3Unsharp Masking and High-Boost FilteringUnsharp Masking and High-Boost Filtering

-1 -1-1 -1 00

1

A+8

1

-1-1

1 0

A+4

1

-1-1

0-1-1-1 0-10

Equation:Equation:

⎨⎧ ∇−

=),(),(

),(2

2 yxfyxAfyxfhb

The center of the mask is negative

⎩⎨

∇+ ),(),(),(

2 yxfyxAfyfhb The center of the mask is positive




Chapter 3First Order Derivative


0.5

1

f(x) Edges

First Order DerivativeFirst Order Derivative

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Intensity profile( ) g

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Sobel operators


M h i ll h di f i bl f i (h h iMathematically, the gradient of a two-variable function (here the image intensity function) is at each image point a 2D vector with the components given by the derivatives in the horizontal and vertical directions. At each image point, the gradient vector points in the direction of largest possible intensity increase, and the length of the gradient vector corresponds to the rate of change in that direction. g



0 1-1 -2 -1-1f∂ f∂tt


Sobel operators1

0

0

2-2

-1 1

0

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00

1xf∂∂computeto y

f∂

computeto

Sobel operators

fxf∂∂

yf∂∂

Chapter 3First Order Partial Derivative: Image Gradient


f∂f x

f∂∂

f∂ f∇

y∂

imfilterimfilter imfilter O ti D i tiimfilter Multidimensional image filtering Syntax:B = imfilter(A,H)B = imfilter(A H option1 option2 )

Option Description

X Input array values outside the bounds of the array are implicitly assumed toB = imfilter(A,H,option1,option2,...)

DescriptionB = imfilter(A,H) filters the multidimensional array A

with the multidimensional filter H. The array A can be a nonsparse numeric array of any class

the array are implicitly assumed to have the value X. When no boundary option is specified, imfilter uses X = 0.

' t i I t l t id th b d fp y y

and dimension. The result B has the same size and class as A.

Each element of the output B is computed using double precision floating point If A is an integer

'symmetric'

Input array values outside the bounds of the array are computed by mirror-reflecting the array across the array border.

double-precision floating point. If A is an integer array, then output elements that exceed the range of the integer type are truncated, and fractional values are rounded.

'replicate' Input array values outside the bounds of the array are assumed to equal the nearest array border value.

B = imfilter(A,H,option1,option2,...) performs multidimensional filtering according to the specified options. Option arguments can have the following values.

Boundary Options

'circular' Input array values outside the bounds of the array are computed by implicitly assuming the input array is periodic

ou da y Op o s

fspecialp

f i l V l D i ifspecial Create special 2-D filtersSyntax

Value Description

'average' Averaging filter

h = fspecial(type)h = fspecial(type,parameters)Description

'disk' Circular averaging filter (pillbox)

'gaussian' Gaussian lowpass filter

h = fspecial(type) creates a two-dimensional filter h of the specified type. fspecial returns h as a correlation kernel hich is

'laplacian' Approximates the two-dimensional Laplacian operator

'log' Laplacian of Gaussian filteras a correlation kernel, which is the appropriate form to use with imfilter. type is a string having one of these values.

'motion' Approximates the linear motion of a camera

' itt' P itt h i t l d'prewitt' Prewitt horizontal edge-emphasizing filter

'sobel' Sobel horizontal edge-emphasizing filterfilter

'unsharp' Unsharp contrast enhancement filter



22

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=∇fffGradient magnitude

First Order Partial Derivative: Image GradientFirst Order Partial Derivative: Image Gradient

⎟⎠

⎜⎝ ∂⎠⎝ ∂ yx

fGradient magnitude

A gradient image emphasizes edges(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.


Chapter 3First Order Partial Derivative: Image GradientFirst Order Partial Derivative: Image GradientFirst Order Partial Derivative: Image Gradient

xf∂∂

yf∂∂

I=imread('fig345.jpg');I=double(I);( )h1=fspecial('sobel');h2=h1';I1=imfilter(I,h1);figure; imshow(I1);g ( )I2=imfilter(I,h2);figure; imshow(I2);I3=sqrt(imfilter(I,h1).^2+imfilter(I,h2).^2);figure; imshow(uint8(I3));


Image Enhancement in the Spatial Domain : Image Enhancement in the Spatial Domain : Mix things up !

f2∇ f∇

f∇DA smoothB D

+ -+

Sharpening EC (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Image Enhancement in the Spatial Domain : Image Enhancement in the Spatial Domain : Mix things up !

G PowerL TEC

G Law Tr.

Σ

F

Σ

MultiplicationF A

H(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Filtering Using imfilter in MatlabFiltering Using imfilter in MatlabFiltering Using imfilter in MatlabFiltering Using imfilter in Matlab

Filtering of images either by correlation or convolution can be performed using theFiltering of images, either by correlation or convolution, can be performed using the toolbox function imfilter. This example filters an image with a 5-by-5 filter containing equal weights. Such a filter is often called an averaging filter.

I = imread('coins png');I = imread( coins.png );h = ones(5,5) / 25;I2 = imfilter(I,h);imshow(I), title('Original Image');figure, imshow(I2), title('Filtered Image')

Filtering Using imfilter in MatlabFiltering Using imfilter in MatlabFiltering Using imfilter in MatlabFiltering Using imfilter in Matlab

B = imfilter(A H option1 option2 )B = imfilter(A, H, option1, option2,...)

Imfilter pad with 0Imfilter pad with 0Imfilter pad with 0Imfilter pad with 0

I = imread('eight.tif');h = ones(5,5) / 25;I2 = imfilter(I,h);imshow(I), title('Original Image');figure, imshow(I2), title('Filtered Image with Black Border')

Effect of zeros padding

Imfilter pad with replicationImfilter pad with replicationImfilter pad with replicationImfilter pad with replication

I = imread('eight tif');I = imread( eight.tif );h = ones(5,5) / 25;I3 = imfilter(I h 'replicate');I3 = imfilter(I,h, replicate );figure, imshow(I3); title('Filtered Image with Border Replication')title('Filtered Image with Border Replication')

No Effect

Imfilter paddingImfilter paddingImfilter paddingImfilter padding

• What are results from 'circular' and 'symmetric‘ options?y p

Create predefined 2-D filterCreate predefined 2-D filterpp

h = fspecial(type)h = fspecial(type parameters)h = fspecial(type, parameters)

>> h = fspecial('sobel')h =

1 2 11 2 10 0 0

-1 -2 -1>> h'>> hans =

1 0 -12 0 22 0 -21 0 -1

>> h=fspecial('disk',5)>> h=fspecial('laplacian',)

Filtering Using imfilterFiltering Using imfilter

• Correlation and Convolution Options

Option Description'corr' imfilter performs multidimensional filtering usingcorr imfilter performs multidimensional filtering using

correlation, which is the same way that filter2 performs filtering. When no correlation or convolution option is specified, imfilter uses correlation.

'conv' imfilter performs multidimensional filtering using convolution.

CorrelationCorrelationCorrelationCorrelation

The correlation operation to compute a weighted sum of neighboring pixels.p p g g g pThe following figure shows how to compute the (2,4) output pixel of the correlation of A,

assuming h is a correlation kernel instead of a convolution kernel, using these steps: 1. Slide the center element of the correlation kernel so that lies on top of the (2,4) element of p ( , )

A. 2. Multiply each weight in the correlation kernel by the pixel of A underneath. 3. Sum the individual products from step 2.

The (2,4) output pixel from the correlation isthe correlation is

L ti I F tL ti I F tLocating Image FeaturesLocating Image Features

C l i b d l f i hi i i hiCorrelation can be used to locate features within an image; in this context correlation is often called template matching.

bw =double( imread('text.png'));a = bw(32:45,88:98); ( , );C = imfilter(bw,a);figure, imshow(C,[]);

(C( ))max(C(:))ans = 68.0000thresh = 60;thresh 60;figure, imshow(C > thresh)

ConvolutionConvolutionConvolutionConvolution

Convolution is a neighborhood operation in which each output pixel is the weighted sum ofConvolution is a neighborhood operation in which each output pixel is the weighted sum of neighboring input pixels. The matrix of weights is called the convolution kernel, also known as the filter. A convolution kernel is a correlation kernel that has been rotated 180 degrees

The following figure shows how to compute the (2,4) output pixel using these steps: h l i k l d b i l1. Rotate the convolution kernel 180 degrees about its center element.

2. Slide the center element of the convolution kernel so that it lies on top of the (2,4) element of A.

3 Multiply each weight in the rotated convolution kernel by the pixel of A underneath3. Multiply each weight in the rotated convolution kernel by the pixel of A underneath. 4. Sum the individual products from step 3.

Statistics Filter (nonlinear filter)Statistics Filter (nonlinear filter)Statistics Filter (nonlinear filter)Statistics Filter (nonlinear filter)

B = ordfilt2(A, order, domain)domain is equivalent to the structuring element used for binary i ti It i t i t i i l 1' d 0' th 1'image operations. It is a matrix containing only 1's and 0's; the 1's define the neighborhood for the filtering operation.

For example, B dfilt2(A 5 (3 3)) i l t 3 b 3 di filtB = ordfilt2(A,5,ones(3,3)) implements a 3-by-3 median filter; B = ordfilt2(A,1,ones(3,3)) implements a 3-by-3 minimum filter; B dfilt2(A 9 (3 3)) i l t 3 b 3 i filtB = ordfilt2(A,9,ones(3,3)) implements a 3-by-3 maximum filter. B = ordfilt2(A,1,[0 1 0; 1 0 1; 0 1 0]) replaces each element in A

by the minimum of its north east south and west neighborsby the minimum of its north, east, south, and west neighbors.

Statistics Filter (nonlinear filter)Statistics Filter (nonlinear filter)Statistics Filter (nonlinear filter)Statistics Filter (nonlinear filter)

B = medfilt2(A, [m n]) performs median filtering of the matrix A in two dimensions.

E h t t i l t i th di l i th bEach output pixel contains the median value in the m-by-n neighborhood around the corresponding pixel in the input image medfilt2 pads the image with 0's on the edges so theimage. medfilt2 pads the image with 0 s on the edges, so the median values for the points within [m n]/2 of the edges might appear distorted.

B = medfilt2(A) f ( )performs median filtering of the matrix A using the default 3-by-

3 neighborhood.

Median FilterMedian FilterMedian FilterMedian Filter

I = imread('eight.tif');figure, imshow(I);J = imnoise(I 'salt & pepper' 0 02);J imnoise(I, salt & pepper ,0.02);K = medfilt2(J);figure, imshow(J), figure, imshow(K)

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