Indian I

Ed D i

Institute

Edge Detection

of Informmation TTechnoloogy, A

llaahabad

Indian I

Wh d i i ?

Institute

What are edges in an image?

of Inform

Edges are those places i i th tm

ation T

in an image that correspond to object b d i

Technolo

boundaries.Edges are pixels where ogy, A

lla

image brightness changes abruptly. ahabad

Brightness vs. Spatial Coordinates

Indian I

M Ab Ed

Institute

More About Edges

of Inform

An edge is a property attached to an i di id l i l d i l l t d f thm

ation T

individual pixel and is calculated from the image function behavior in a neighborhood

f th i l

Technolo

of the pixel. It is a vector variable (magnitude of the ogy, A

lla

gradient, direction of an edge) .

ahabad

Indian I

I T Ed M

Institute

Image To Edge Map

of Informmation TTechnoloogy, A

llaahabad

Indian I

Ed D i

Institute

Edge Detection

of Inform

Edge information in an image is found by looking at the relationship a pixel has with itsm

ation T

at the relationship a pixel has with its neighborhoods.If a pixel’s gray-level value is similar to thoseTechnolo

If a pixel s gray level value is similar to those around it, there is probably not an edge at that point.ogy, A

lla

pIf a pixel’s has neighbors with widely varying gray levels, it may present an edge point.ahabad

Indian I

Ed D i

Institute

Edge Detection

of Inform

An edge is a boundary between an object and the background, and indicates them

ation T

and the background, and indicates the boundary between overlapping objects.Edges form the outline of an object, so if the Technolo

edges are identified in an image accurately, then all the objects can be located and basic properties such as area perimeter can beogy, A

lla

properties such as area, perimeter can be measured.Edge detection is based on the relationship a ahabad

dge de ec o s b sed o e e o s ppixel has with its neighbors.

Indian I

Ed D i

Institute

Edge Detection

of Inform

Edges, in an image are defined as locations where there is a significant variation in them

ation T

where there is a significant variation in the gray level or color of pixel in some direction.Edge detection extracts and localizes pointsTechnolo

Edge detection extracts and localizes points (pixels) around which a large change in image brightness has occurred.ogy, A

lla

age b g t ess as occu ed.This can help in the process of segmentationahabad

segmentation

Indian I

T di i l A h

Institute

Traditional Approaches

of Inform Ideal Step Edge :-mation T

(The most common definition of an edge)

Technolo

In 1-d the edge is simply a change in gray level occurring at one specific location ogy, A

lla

(fig.1a).The greater the change in the level, the ahabad

easier the edge is to detect.

Indian I

S d i

Institute

Step Edge Detection

of Informmation TTechnoloogy, A

llaahabad

Indian I

P bl i Ed D i

Institute

Problems in Edge Detection

of Inform Problems:-mation T

1)Because of digitization:-The image may be sampled in such a way so that change in gray Technolo

p y g g ylevel may extend across some number of pixels. Fig 1b-d.ogy, A

lla

p g2)Because of noise:-Due to factors such as

light intensity type of camera and lensahabad

light intensity, type of camera and lens, motion, temperature, dust and others.

Indian I NoiseInstitute

Noise

of Inform

Two types of noise are of interest in image analysis:-1) Signal –independent:-noise is a random set of gray levelsm

ation T

1) Signal independent:-noise is a random set of gray levels, statistically independent of the data. It occurs during electronically transmission of image. If A is a perfect i d N i th i th t d i

Technolo

image and N is the noise that occurs during transmission, then the final image B is

B = A + Nogy, Alla

N2) Signal-dependent noise:- In this the level of the noise

value at each point in the image is a function of the l l h

ahabad

grey level there.

Indian I

Ed D i M h d

Institute

Edge Detection Methods

of Inform

Many are implemented with convolution mask and based on discrete approximationsm

ation T

mask and based on discrete approximations to differential operators.Differential operations measure the rate ofTechnolo

Differential operations measure the rate of change in the image brightness function.Some operators return orientationogy, A

lla

Some operators return orientation information. Other only return information about the existence of an edge at each point.ahabad

bou e e s e ce o edge e c po .

Indian I A 2D grayvalue - image isInstitute

A 2D grayvalue image is a 2D -> 1D function

of Inform

v = f(x,y)

mation TTechnoloogy, A

llaahabad

Indian I Derivative OperatorsInstitute

Derivative Operators

of Inform

An operator, that is sensitive to change in gray level will operate as an edge detector- A derivative operator does this.m

ation T

Interpretation of derivative:- The rate of change of function. The rate of change of the gray levels in an image is large near an edge and small in constant areas.Technolo

In images are 2-D, so level changes are considered in many directions. For this reason partial derivatives of the image are used with respect to the principal directions x and y.ogy, A

lla

Let A (x, y) be an image then the gradient is defined as:

∆A (x ,y) = (∂A / ∂x, ∂A / ∂y)ahabad

( ,y) ( , y)

Indian I

G di

Institute

Gradient operators

of Inform

Because image is discrete, the derivative at a pixel is approximated by the difference in gray levels over some local region.m

ation T

The simplest approximation is the operator ∆1:

Technolo

Δx1A(x, y) = A(x, y) – A(x-1, y)

Δ A(x y) = A(x y) A(x y 1)

ogy, Alla

Δx2A(x, y) = A(x, y) – A(x, y-1)

Problem with this operator:- It does not compute the gradient at the point (x y) but at (x-1/2 y-1/2)ahabad

point (x, y), but at (x 1/2, y 1/2)Assumption:- the grey levels vary linearly between the pixels.

Indian I

G di

Institute

Gradient operators

of Inform

A better choice for an approximation is Δ A( ) A( +1 ) A( 1 )

mation T

Δx2 A(x, y) = A(x+1, y) – A(x-1 ,y)

Δ A(x y) = A(x y+1) – A(x y-1)Technolo

Δy2 A(x, y) = A(x, y+1) – A(x, y-1)This operator is symmetrical with respect to the pixel (x,

y). It does not consider the value of the pixel at (x, y).ogy, Alla

The edge response is given by:

√

ahabad

Gmag = √((∂A ⁄ ∂x)2 + (∂A ⁄∂y)2)

Indian I

G di

Institute

Gradient operators

of Inform

and the direction of the edge is approximately:

mation T Gdir = atan (∂A ⁄∂y) ⁄ ( ∂A ⁄ ∂x)Technolo Edge pixel- which exceeds the threshold ogy, A

lla

value

ahabad

Indian I

G di l

Institute

Gradient operators examples

of Informmation TTechnoloogy, A

llaahabad

Indian I

E l

Institute

Examples

of Informmation TTechnoloogy, A

llaahabad

Indian I

E l

Institute

Examples

of Informmation TTechnoloogy, A

llaahabad

Indian I Measures of Performance of EdgeInstitute

Measures of Performance of Edge Detection Schemes

of Inform

False PositiveFalse Negativem

ation T

gUsing Function (Pratt 1978)

∑( /( d(i)2)) / ( )

Technolo

E1 =∑(1/(1+αd(i)2)) / max(IA,II)IA = the number of edge pixels found by the edge detectorII= the number of edge pixels in the test imageogy, A

lla

II the number of edge pixels in the test imaged(i)=the distance between the actual ith pixel and the one

found by the edge detectori d f li

ahabad

α is used for scaling

Indian I

M f P f

Institute

Measures of Performance

of Inform

Evaluation scheme based on local edge coherence (Kitchen and Rosenfeld)m

ation T

)(a) It measures how well an edge pixel is continued on the

left; this function isTechnolo

L(k) =a(d,dk) a(kπ/4,d+π/2) if neighbor k is an edge pixel0 otherwise

h d i th d di ti t th i l b i t t d

ogy, Alla

where d is the edge direction at the pixel being testedd0 is the edge direction at its neighbor to the rightd1 is the direction of the upper-right neighbor and so onahabad

d1 is the direction of the upper right neighbor, and so onCounterclockwise about the pixel involveda= the measure of the angular difference between any two

Indian I

M

Institute

Measures…

of Inform

a(α,β) = π -|α-β| ⁄ π(b) A similar function measures directional continuity on them

ation T

(b) A similar function measures directional continuity on the right of the pixel being evaluated:

R(k) = a(d,dk) a(kπ/4,d+π/2) if neighbor k is an edge Technolo

pixel0 otherwise

(c) C= Overall continuity measureogy, Alla

(c) C= Overall continuity measure= average(L(k), R(k))

(d) Then measure of thinness (T) is applied.ahabad

(e) The overall evaluation of the edge detector is:(f) E2 = γC + (1-γ) T

Indian I Template-Based Edge DetectionInstitute

Template Based Edge Detection

of Inform

This uses a small, discrete template as a model of an edge instead of using a derivative operator.m

ation T

g pExampes:-

(a) Sobel Edge DetectorTechnolo

(b) Kirsch Edge Detector

ogy, Allaahabad

Indian I

P i O

Institute

Prewitt Operator

of Informmation T

⎤⎡ 111 ⎤⎡ 101

Technolo ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −−−=

111000111

y⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

=101101101

x

ogy, Alla

⎦⎣ ⎦⎣

22 yx + ⎥⎦⎤

⎢⎣⎡−

xy1tanahabad

⎦⎣

Indian I

S b l O

Institute

Sobel Operator

of Inform

Similar to the Prewitt, with different mask coefficients:m

ation T

coefficients:

⎥⎤

⎢⎡ −−− 121

⎥⎤

⎢⎡− 101Technolo

⎥⎥⎥

⎦⎢⎢⎢

⎣

=121000y

⎥⎥⎥

⎦⎢⎢⎢

⎣−−=

101202x

ogy, Alla Edge Magnitude = Edge Direction = 22 yx + ⎥⎦

⎤⎢⎣⎡−

xy1tanahabad

g g g ⎥⎦⎢⎣ x

Indian I

Ki h C M k

Institute

Kirsch Compass Masks

of Inform

Taking a single mask and rotating it to 8 major compass orientations: N NW W SWm

ation T

major compass orientations: N, NW, W, SW, S, SE, E, and NE.The edge magnitude = The maximum valueTechnolo

The edge magnitude = The maximum value found by the convolution of each mask with the image.ogy, A

lla

t e age.The edge direction is defined by the mask that produces the maximum magnitude.ahabad

p oduces e u g ude.

Indian I

Ki h C M k (C )

Institute

Kirsch Compass Masks (Cont.)

of Inform

The Kirsch masks are defined as follows:⎤⎡ 533 ⎤⎡ 553 ⎤⎡ 555 ⎤⎡ 355m

ation T ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−−=

533503533

N⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−

=333

503553

W⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−=

333303

555S

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−

=333305355

E

Technolo ⎥⎥⎤

⎢⎢⎡

−−−

= 305335

NW ⎥⎥⎤

⎢⎢⎡

−−−−

= 305333

SW ⎥⎥⎤

⎢⎢⎡

−−−−−

= 303333

SE ⎥⎥⎤

⎢⎢⎡−

−−= 503

533NEogy, A

lla EX: If NE produces the maximum value, then the

⎥⎥⎦⎢

⎢⎣ −− 335 ⎥

⎥⎦⎢

⎢⎣ −355 ⎥

⎥⎦⎢

⎢⎣ 555 ⎥

⎥⎦⎢

⎢⎣− 553

ahabad

p ,edge direction is Northeast

Indian I

Ki h Ed D

Institute

Kirsch Edge Detector

of Inform

These masks are to observe the grey level change near an edge having various orientations, rather than approximation to the gradient.

There are one mask for each of eight compass directions.mation T

There are one mask for each of eight compass directions.For example, a large response to mask K0 implies a vertical edge

(horizontal gradient) at the pixel corresponding to the center of the mask.Technolo

mask.Method:-To find the edges, an image I is convolved with all of the masks at each

pixel position The response of the operator at a pixel is the maximumogy, Alla

pixel position. The response of the operator at a pixel is the maximum of the responses of any of the eight masks. The direction of the edge pixel is quantized into eight possibilities and is π /4 *I where I is the number of the mask having the largest response.ahabad

g g pLooks for edges in both horizontal and vertical directions, then combine

the information into a single metric.

Indian I

R bi C M k

Institute

Robinson Compass Masks

of Inform

Similar to the Kirsch masks, with mask ffi i t f 0 1 d 2m

ation T

coefficients of 0, 1, and 2:

⎥⎤

⎢⎡−

202101

N ⎥⎤

⎢⎡

101210

W ⎥⎤

⎢⎡

000121

S ⎥⎤

⎢⎡

101012

E

Technolo

⎥⎥⎥

⎦⎢⎢⎢

⎣−−=

101202N

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−=

012101W

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−=

121000S

⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−=

210101E

⎤⎡ 101 ⎤⎡ 210 ⎤⎡ 121 ⎤⎡ 012

ogy, Alla ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

=101202101

NW⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−−

=012101210

SW⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −−−=

121000121

SE⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−−=

210101012

NE

ahabad

⎦⎣ ⎦⎣ ⎦⎣ ⎦⎣

Indian I •Sometimes we are interested only in edge magnitudes withoutInstitute

•Sometimes we are interested only in edge magnitudes without regard to their orientations. •The Laplacian may be used. of Inform

•The Laplacian has the same properties in all directions and is therefore invariant to rotation in the image. m

ation T

g

Technolo •The Laplace operator is a very popular operator ogy, Alla

approximating the second derivative which gives the gradient magnitude only.

ahabad

Indian I

L l i O

Institute

Laplacian Operators

of Inform

Edge magnitude is approximated in digital i b l tim

ation T

images by a convolution sum. The sign of the result (+ or -) from two Technolo

adjacent pixels provide edge orientation and tells us which side of edge brighterogy, A

llaahabad

Indian I

L l i O (C )

Institute

Laplacian Operators (Cont.)

of Inform

Masks for 4 and 8 neighborhoods ⎤⎡ 010 ⎤⎡ 111

mation T ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−

010141

010

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−−−

111181111

Technolo

⎦⎣ ⎦⎣

ogy, Alla

⎥⎥⎤

⎢⎢⎡

−−−

242121

⎥⎥⎤

⎢⎢⎡ −−

141212

ahabad

⎥⎥⎦⎢

⎢⎣ − 121

242⎥⎥⎦⎢

⎢⎣ −− 212

141

Indian I

C i

Institute

Comparison

of Inform

Sobel and Prewitt methods are very effectively providing good edge maps.Ki h d R bi th d i ti fm

ation T

Kirsch and Robinson methods require more time for calculation and their results are not better than the ones produced by Sobel and Prewitt methods.Technolo

p yRoberts and Laplacian methods are not very good as expected. ogy, A

llaahabad

Indian IInstitute

•Gradient operators can be divided into three categories

of Inform

I. Operators approximating derivatives of the image function using differences. rotationall in ariant (e g Laplacian) need one con ol tionm

ation T

•rotationally invariant (e.g., Laplacian) need one convolution mask only. Individual gradient operators that examine small local neighborhoods are in fact convolutions and can be Technolo

expressed by convolution masks.

•approximating first derivatives use several masks, theogy, Alla

approximating first derivatives use several masks, the orientation is estimated on the basis of the best matching of several simple patterns. Operators which are able to detect edge direction Each mask corresponds to a certain direction

ahabad

edge direction. Each mask corresponds to a certain direction.

Indian IInstitute of Inform

II. Operators based on the zero crossings of the image function second derivative (e.g., Marr-Hildreth or Canny edge detector). m

ation T III. Operators which attempt to match an image function to a Technolo

parametric model of edges. Parametric models describe edges more precisely than simple edge magnitude and direction and are much more computationally intensiveogy, A

lla

are much more computationally intensive.

The categories II and III will be covered in the next lecture ahabad

Indian I

A Q i k N

Institute

A Quick Note

of Inform

Matlab’s image processing toolbox provides edge function to find edges in an image:m

ation T

function to find edges in an image:I = imread('rice.tif');

BW1 = edge(I,'prewitt');Technolo

BW2 = edge(I,'canny');imshow(BW1)figure imshow(BW2)ogy, A

lla

figure, imshow(BW2)Edge function supports six different edge-finding methods: Sobel, Prewitt, Roberts, Laplacian of ahabad

pGaussian, Zero-cross, and Canny.