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1/12/2012 Computer Graphics & ImageProcessing

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Computer GraphicsComputer Graphics&&Image ProcessingImage Processing

Anuja Dharmaratne,Anuja Dharmaratne,PhDPhD

UCSCUCSC

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Lecture 4Lecture 4

Point Detection

Line Detection Edge Detection

Edge Thinning

Edge Linking

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Point DetectionPoint Detection

Smallest feature in an image.

The simplest point detection operatoris a high value surrounded by lowones.

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4

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1/12/2012 Computer Graphics & ImageProcessing 5

Line Detection

Line Detection Filters

-1 -1 -1

2 2 2-1 -1 -1

-1 2 -1

-1 2 -1-1 2 -1

2 -1 -1

-1 2 -1-1 -1 2

-1 -1 2

-1 2 -12 -1 -1

The above line detector filters should be appliedto detect lines in an image with the relevantdirections.

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DetectingDetectingLinesLines

orientedoriented

inin --4545oo

directiondirection

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Detecting linesDetecting lines ininhorizontalhorizontal directiondirection

Gray scale image binary output image afterthresholding

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Edge/Boundary DetectionEdge/Boundary Detection

What is Edge/Boundary detection ?

- Image processing technique in whichedge pixels are identified by examiningtheir neighboring pixels.

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What is an Edge?

Edges of objects in a digital image arerepresented by the discontinuities ofthe intensity or the grey-level.

High grey-level value

Low grey-level value

Edge

If a pixel haswidelyvarying greylevels, thenit might bean edge.

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An edge is a set of connected pixels thatlies on a boundary between two regions

A fast rate of change of intensity at somedirection

What is an edge?What is an edge?

A ramp edgeAn ideal edge

Intensitylevel

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Real EdgesReal Edges

Noisy and Discrete!

We want an Edge Operator that produces:

Edge Magnitude

Edge Orientation

High Detection Rate and Good Localization

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The gradient direction is given by:

The edge strength is given by the gradient

magnitude

The gradient of an image: = [ Gx , Gy ]

Image GradientImage Gradient

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First Derivative Based Techniques

The overall gradient at (x,y ) is given by

G(x,y) = ((df/dx)2+(df/dy)

2)

This is approximated by

G(x,y) } |df/dx| + |df/dy|

If G(x,y) >T then (x,y) is in an edge point. WhereT is a threshold. (10% of the highest G(x,y) maybe a good estimate).

Derivative of the image function has twocomponents

f(x,y) = ( d f(x,y) , d f(x,y) )dx dy

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Edge DetectionEdge DetectionOperatorsOperators

All the operators used forimage sharpening can be usedfor edge detection, too.

Successive Difference Operator

Roberts Cross operator

Prewitt Operator Sobel Operator

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Successive DifferenceSuccessive DifferenceOperatorOperator

1 -1

0 0

1 0

-1 0

Gx Gy

Roberts Cross operatorRoberts Cross operator

1, jiI 1,1 jiI

jiI , jiI ,1I

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-1 -1 -1

0 0 0

1 1 1

-1 0 1

-1 0 1

-1 0 1

Prewitt operators

-1 -2 -1

0 0 0

1 2 1

-1 0 1

-2 0 2

-1 0 1

Sobel operators

0 1 2

-1 0 1

-2 -1 0

-2 -1 0

-1 0 1

0 1 2

Weighted value 2 is given to achieve some smoothingby giving more importance to center point

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Sobel OperatorSobel Operator

More sensitive to X and Y directional edges When compared with Successive DifferenceOperator, Roberts Cross Operator and PrewittOperator

Calculation is more complexLess sensitive to noise

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Other operatorsOther operators

Kirsh Compass Mask

Robinson Compass Mask

Laplacian Operator Frei-Chen Mask

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ResultsResults

Original image Sobel Operator

PrewittOperator Frei-ChenOperator RobertsOperator

LaplacianOperator

RobinsonOperator

Kirsh Operator

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Comparing Edge OperatorsComparing Edge Operators

-1 0 1

-1 0 1

-1 0 1

1 1 1

0 0 0

-1 -1 1

Gradient:

Roberts (2 x 2):

Sobel (3 x 3):

Sobel (5 x 5): -1 -2 0 2 1

-2 -3 0 3 2

-3 -5 0 5 3

-2 -3 0 3 2

-1 -2 0 2 1

1 2 3 2 1

2 3 5 3 2

0 0 0 0 0

-2 -3 -5 -3 -2

-1 -2 -3 -2 -1

0 1

-1 0

1 0

0 -1

Good Localization

Noise SensitivePoor Detection

Poor LocalizationLess Noise SensitiveGood Detection

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22ndnd Derivative techniquesDerivative techniques

Sign of the second derivative is used tofind the edge pixels.

Zero crossing property of the secondderivative is used to find thick edges ofan image.

i.e. Line joining the extreme positive andnegative values of the second derivative

would cross zero near the midpoint of theedge.

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22ndnd derivative techniquesderivative techniques

Laplacian Operator

Lalpacian of Gaussian Marr-Hildreth Operator

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Laplacian FiltersLaplacian FiltersIdea:

Smooth the image,

compute the second derivative ofthe (2D) image,

Find the pixels where the brightnessfunction crosses 0 and mark them.

We can actually devise convolution

filters that carry out the smoothing andthe computation of the secondderivative.

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Laplacian Operator

Digital Laplacian Discrete variants (applied after smoothing):

2 f(x,y) = d2 f(x,y) + d2 f(x,y)

dx2 dy2

0 1 0

1 -4 1

0 1 0

1 1 1

1 -8 1

1 1 1

2 -1 2

-1 -4 -1

2 -1 2

Continuous variant:

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Laplacian FiltersLaplacian Filters

3v3 Laplacian 5v5 Laplacian

zerodetection

zerodetection

zerodetection

7v7 Laplacian

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Finding Zero-crossings

(x-1, y-1) (x,y-1) (x+1,y-1)

(x-1, y) (x,y) (x+1,y)

(x-1, y+1) (x,y+1) (x+1,y+1)

If (G(x,y)=0 Then

I

f (G(x-1,y-1)* G(x+1,y+1))

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Sensitive to noise: This sensitivity comesnot just from the sensitivity of the zero-crossings, but also of the differentiation. Ingeneral, the higher the derivative, the more

sensitive the operator.

Need to reduce the sensitivity.Therefore, usually convolution is applied ona smoothed image. Thus, smooth the imagewith a Gaussian filter, first. Then, apply theLaplacian filter.

Properties ofLaplacianProperties ofLaplacianOperatorOperator

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Gaussian Distribution

The 2D Gaussian distribution function intwovariables, G(x,y), is illustrated and defined by,

Where isthe standard deviation.

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Simply, first Convolving the Laplacian tothe Gaussian and then applying theresulting filter for edge detection is called

Laplacian of Gaussian (LoG). Therefore the total operation of edge

detection after smoothing on the originalimage is

s(x,y) = (f(x,y)*G(x,y)) *

Since Convolution is Associatives(x,y) = f(x,y) * (G(x,y) * )

Laplacian of GaussianLaplacian of Gaussian

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It issimple to show thatthis operation can be

reduced to convolve the original image f(x,y) with a``Laplacian of a Gaussian''(LOG) operator

,

2nd Derivative ofthe 2D Gaussian

Laplacian of GaussianLaplacian of Gaussian

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Laplacian and GradientLaplacian and Gradient Zero-crossings of Laplacian operators only

tell you where the edge is, not how strongthe edge is.

To get the strength of the edge, we canlook at the gradient magnitude at theposition of the Laplacian zero crossings.

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MarrMarrHildreth OperatorHildreth Operator This is a

derivation ofLOG.

First, filter

the imagewith aGaussianfilter

Then findzerocrossings

Variance

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MarrMarrHildreth OperatorHildreth Operator

sigma = 2original

sigma = 0.9sigma = 4

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DisadvantagesDisadvantages

Marr-Hildreth operator is symmetric, so itfinds edges in all orientations.

However, it generates responses that do

not correspond to edges, so-called "falseedges", and the localization error may besevere at curved edges.

Since it uses a second derivative operator,the influence of noise is considerable.

It produces unrealistic closed contours.

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Canny OperatorCanny Operator

Properties

Low error rate

Edge points be well localized

Have only one response to a single edge

This is the First derivative of the GaussianFunction

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Canny AlgorithmCanny Algorithm

1. Smooth the image

2. Find the image gradient in both X and Y directions

3. Apply Non-maximum suppression

4. Reduce gradient array by hysteresis Using two

thresholds

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Canny Algorithm (contd.)Canny Algorithm (contd.)1. Filter outnoise using Gaussian filter

2. Find the edge strength bytaking the gradient

ofthe image uses Sobel operator

Uses a pair of3x3 convolution masks

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Canny Algorithm (contd.)Canny Algorithm (contd.)After calculating Gx=P[i,j] and Gy=Q[i,j],

compute the magnitude and orientationof the gradient vector at position [i, j]:

22

],[],[],[ jiQjiPjiM !]),[],,[arctan(],[ jiPjiQji !U

This is exactly the information that wewant it tells us where edges are, howsignificant they are, and what theirorientation is.

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Canny Algorithm (contd.)Canny Algorithm (contd.)

After finding the direction ofthe gradient vector

U = tan-1(Gy / Gx),

relate the calculated directionto a directionthat can betraced in an image

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Canny Algorithm (contd.)Canny Algorithm (contd.)We first assign a sector ^[i, j] to each pixel according

to its associated edge orientation U[i, j]. There are foursectors with numbers 0, , 3:

180r 0r

135r

90r

45r

315r

270r

225r

0

0

0

0

11

1

1

2 2

2 2

33

3

3

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Canny Algorithm (contd.)Canny Algorithm (contd.) Then we iterate through the array M[i, j]

and compare each element to two of its8-neighbors.

Which two neighbors we choose dependson the value of^[i, j].

In the following diagram, the numbers inthe neighboring squares of [i, j] indicatethe value of^[i, j] for which theseneighbors are chosen:

3 2 1

0 [i,j] 0

1 2 3

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Canny Algorithm (contd.)Canny Algorithm (contd.)3. Applynon maximum suppression

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Predicting the next edge point

Assume the marked point is an

edge point. Then we constructthe tangent to the edge curve

(which is normal to the gradient at

that point) and use this to predict

the next points (here either r or s).

At q, we have a

maximum ifthe value islargerthanthose at both

p and at r.


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4. Eliminating streaking by using hysteresis

- Usestwo thresholds high and low


If the magnitude is below the fir st threshold (T1), it

is set to zero (made a non-edge). If the magnitude

is above the high threshold (T2), it is made an edge.

And if the magnitude is between the 2 thresholds,

then it is set to zero unless there is a path from this

pixel to a p ixel with a gradient above T2.

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First derivative based techniques alwaysproduce more than one pixel as edge pointsin a boundary.

Selecting the highest gradient pixel andelimination (suppression) of other non-maximums is performed by Non-MaximumSuppression operation.

Thus, this non-maxima suppression techniquethins the detected edges to a width of onepixel.

dd

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1/12/2012 Computer Graphics & Image

Processing

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Canny Edge DetectorCanny Edge Detector --ExampleExample

Let us look at the following sample imageand perform Canny edge detection on it:

grayscale image

385456978579

69618490105100

82100101121165176

7187119182202211

101131180200215209

78120170183213220

intensity values of

same image

C EdC Ed

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Processing

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Canny Edge DetectorCanny Edge Detector --ExampleExample

Applying the horizontal and vertical gradient filtersgives us the following results:

P[i,j]

000000

0-4-12.5-23.5-1.55.5

0-5-12-13-29.5-3

0-17-16.5-41.5-32-10

0-23-40.5-41.5-17.5-1.5

0-36-49.5-16.5-22.5-0.5

Q[i,j]

000000

01917.510.56.520.5

026282445.568

0-122.539.54936

03752.539.515.5-5.5

0-17-10.5-13.5-9.54.5

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Processing

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Canny Edge DetectorCanny Edge Detector -- ExampleExample

Based on P[i, j] and Q[i, j], we can now compute themagnitude and orientation of the gradient:

M[i, j]

000000

019.421.525.76.721.2

026.530.527.354.268.1

020.816.757.358.537.4

043.666.357.323.45.7

039.850.621.324.44.5

U[i, j]

180r180r180r180r180r180r

180r281r305r335r282r254r

180r280r293r298r302r272r

180r35r351r316r303r285r

180r301r307r316r318r285r

180r25r11r39r22r276r

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Processing

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This allows us to compute the sector ^[i, j]for each angle U[i, j]:

^[i, j]

000000

023322

023332

010332

033332

010112 Remember thedirections:

321

0[i,j]0

123

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Processing

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The last steps are nonmaxima suppression andthresholding

(here we use a single threshold T = 30):

E[i, j]

000000

000000

026.50054.268.1

00057.358.50

0066.357.300

0050.6024.40

final result

000000

000000

000011

000110

001100

001000

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Processing

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Canny Edge DetectorCanny Edge Detector -- ExampleExampleFinally we visualize the detected edge pixels (0 =white, 1 = black) and indicate the precise edgelocations (the pixels lower right corners) in theoriginal image:

edge locations inoriginal image

edge pixels

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Processing

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Important ParametersImportant Parameters

Width ofthe Gaussian kernel

Upperthreshold

Lowerthreshold

Weak point of Canny OperatorWeak point of Canny Operator

Doesnt trace Y junctions properly

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Processing

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ExampleExample

W!8!8! W!8!8!

W 8 8 W 8 8

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