Edge Detection2

EDGE DETECTION

Presentation by Sumit TandonDepartment of Electrical Engineering University of Texas at Arlington

Course # EE6358 Computer Vision

September 13, 2005 EE - 6358 Computer Vision 2

CONTENTS Introduction Types of Edges Steps in Edge Detection Methods of Edge Detection

First Order Derivative Methods First Order Derivative Methods - Summary

Second Order Derivative Methods Second Order Derivative Methods - Summary

Optimal Edge Detectors Canny Edge Detection

Edge Detector Performance Line Detection

Convolution based technique Hough transform

Application areas


INTRODUCTION Edge - Area of significant

change in the image intensity / contrast

Edge Detection – Locating areas with strong intensity contrasts

Use of Edge Detection – Extracting information about the image. E.g. location of objects present in the image, their shape, size, image sharpening and enhancement


TYPES OF EDGES Variation of

Intensity / Gray Level Step Edge Ramp Edge Line Edge Roof Edge


Steps in Edge Detection Filtering – Filter image to improve

performance of the Edge Detector wrt noise Enhancement – Emphasize pixels having

significant change in local intensity Detection – Identify edges - thresholding Localization – Locate the edge accurately,

estimate edge orientation


Noisy Image Example of Noisy Image


METHODS OF EDGE DETECTION First Order Derivative / Gradient Methods

Roberts Operator Sobel Operator Prewitt Operator

Second Order Derivative Laplacian Laplacian of Gaussian Difference of Gaussian

Optimal Edge Detection Canny Edge Detection


First Derivative At the point of

greatest slope, the first derivative has maximum value E.g. For a

Continuous 1-dimensional function f(t)


Gradient For a continuous two dimensional

function Gradient is defined as

yfxf

GyGx

yxfG )],([

GyGxGyGxG 22

GxGy1tan


Gradient Approximation of Gradient for a

discrete two dimensional function Convolution Mask

Gx=

Gy =

Differences are computed at the interpolated points [i, j+1/2] and [i+1/2, j]

Demo

-1 1-1 1

],1[],[],[]1,[jifjifGyjifjifGx

1 1-1 -1

-1 11-1

http://homepages.inf.ed.ac.uk/rbf/HIPR2/convolutiondemo.htm


Gradient Methods – Roberts Operator Provides an approximation to the gradient

Convolution Mask Gx=

Gy =

Differences are computed at the interpolated points [i+1/2, j+1/2] and not [i, j]

Demo

1 00 -10 -11 0

)1,(),1()1,1(),()],([ jifjifjifjifGyGxjifG

http://homepages.inf.ed.ac.uk/rbf/HIPR2/robertsdemo.htm


Roberts Operator - Example The output image

has been scaled by a factor of 5

Spurious dots indicate that the operator is susceptible to noise


Gradient Methods – Sobel Operator The 3X3 convolution mask smoothes the image by

some amount , hence it is less susceptible to noise. But it produces thicker edges. So edge localization is poor

Convolution Mask

Gx = Gy=

The differences are calculated at the center pixel of the mask.

Demo

-1 0 1-2 0 2-1 0 1

1 2 10 0 0-1 -2 -1

http://homepages.inf.ed.ac.uk/rbf/HIPR2/sobeldemo.htm


Sobel Operator - Example Compare the output of

the Sobel Operator with that of the Roberts Operator: The spurious edges are

still present but they are relatively less intense compared to genuine lines

Roberts operator has missed a few edges

Sobel operator detects thicker edges

Will become more clear with the final demo

Outputs of Sobel (top) and Roberts operator


Gradient Methods – Prewitt Operator It is similar to the Sobel operator but uses slightly

different masks Convolution Mask

Px =

Py =

Demo

-1 0 1-1 0 1-1 0 11 1 10 0 0-1 -1 -1

http://homepages.inf.ed.ac.uk/rbf/HIPR2/convolutiondemo.htm


First Order Derivative Methods - Summary Noise – simple edge detectors are affected

by noise – filters can be used to reduce noise

Edge Thickness – Edge is several pixels wide for Sobel operator– edge is not localized properly

Roberts operator is very sensitive to noise Sobel operator goes for averaging and

emphasizes on the pixel closer to the center of the mask. It is less affected by noise and is one of the most popular Edge Detectors.


Second Order Derivative Methods Zero crossing of the second derivative

of a function indicates the presence of a maxima


Second Order Derivative Methods - Laplacian Defined as

Mask

Demo Very susceptible to noise, filtering

required, use Laplacian of Gaussian

0 1 01 -4 10 1 0

http://homepages.inf.ed.ac.uk/rbf/HIPR2/laplaciandemo.htm


Second Order Derivative Methods - Laplacian of Gaussian Also called Marr-Hildreth Edge

Detector Steps

Smooth the image using Gaussian filter Enhance the edges using Laplacian

operator Zero crossings denote the edge location Use linear interpolation to determine the

sub-pixel location of the edge


Laplacian of Gaussian – contd. Defined as

Greater the value of , broader is the Gaussian filter, more is the smoothing

Too much smoothing may make the detection of edges difficult


Laplacian of Gaussian - contd. Also called the Mexican Hat operator


Laplacian of Gaussian – contd. Mask

DemoDiscrete approximation to LoG function with Gaussian = 1.4

http://homepages.inf.ed.ac.uk/rbf/HIPR2/logdemo.htm


Second Order Derivative Methods - Difference of Gaussian - DoG LoG requires large computation time

for a large edge detector mask To reduce computational

requirements, approximate the LoG by the difference of two LoG – the DoG

22

)2

22(

21

)2

22(

22),(

22

21

yxyx

eeyxDoG


Difference of Gaussian – contd. Advantage of DoG

Close approximation of LoG Less computation effort Width of edge can be adjusted by

changing 1 and 2


Second Order Derivative Methods - Summary Second Order Derivative methods

especially Laplacian, are very sensitive to noise

Probability of false and missing edges remain

Localization is better than Gradient Operators


Optimal Edge Detector Optimal edge detector depending on

Low error rate – edges should not be missed and there must not be spurious responses

Localization – distance between points marked by the detector and the actual center of the edge should be minimum

Response – Only one response to a single edge One dimensional formulation

Assume that 2D images have constant cross section in some direction


Mathematical Formulation of Performance Criteria Notation

Edge : G(x) Noise : n(x) Impulse response of Filter : f(x)

Assumptions The edge is located at x=0 Filter has finite response bounded by [-W, W] Noise is of Gaussian nature n0 : Root mean squared noise amplitude per unit

length


First Criterion: Edge Detection Response of filter to the edge:

RMS response of filter to noise :

First criterion: output Signal to Noise Ratio


Second Criterion: Edge Localization A measure that increases as the

localization increases is needed Reciprocal of RMS distance of the

marked edge from center of true edge is taken as the measure of localization

Localization is defined as:


Maximizing the product The design problem is reduced to

maximization of the product:


Third Criterion – Elimination of multiple responses In presence of noise several maxima are

detected – it is difficult to separate noise from edge

We try to obtain an expression for the distance between adjacent noise peaks

The mean distance between the adjacent maxima in the output is twice the distance between the adjacent zero crossings in the derivative of output operator


Elimination of Multiple Responses – contd. Expected number of noise maxima in

the region of width 2W is given by

Fixing k will restrict the number of noise maxima that could lead to false response – this is a constraint imposed on the design

kxWNn

22

max


Finding Optimal Detectors by Numerical Optimization It is not possible to find a function f which will

maximize the SNR, Localization product in the presence of the multiple response constraint

If the function f is discrete then the computation is simplified to four inner products

Additional constraints can be included by the use of Penalty Function – it has non-zero value when the constraint is violated

f is then found out for maximizing the following expression

)()(*)( fPfonLocalizatifSNR ii


Detector for Step Edges G(x) = Au-1(x)


Spatial scaling We can scale the

filter width Detection improves

with width, localization degrades with increase in width

The uncertainty principle is

)(1)(

)(

)/()(

'' fw

f

fwf

wxfxf

w

w

w


Efficient approximation Depending on the above principles,

several optimal edge detectors are calculated

Best approximation to the above detectors is the First Derivative of Gaussian

It is chosen because of the ease of computation in 2 dimensions


Noise estimation Important to estimate the amount of noise

in the image to set thresholds Noise component can be efficiently isolated

using Weiner Filtering – requires the knowledge of the autocorrelation of individual components and their cross-correlation

Noise strength is estimated by Global Histogram Estimation


Thresholding Broken edges due to fluctuation of

operator output above and below the threshold – results in Streaking

Use double thresholding to eliminate streaking


Two Dimensional Edge Detection In two dimensions edge has both position

and direction A 2-D mask is created by convolving a

linear edge detection function aligned normal to the edge direction with a projection function parallel the edge direction

Projection function is Gaussian with same deviation as the detection function

The image is convolved with a symmetric 2-D Gaussian and then differentiated normal to the edge direction


Need for multiple widths and Feature Synthesis Width is chosen to give best tradeoff

between detection and localization For different edges, width has to be

different since SNR is different Feature Synthesis

Smaller operator responses are used to predict the larger operator response

New edge points are marked if the actual large operator output varies significantly from the predicted output


Implementation of Canny Edge Detector Step 1

Noise is filtered out – usually a Gaussian filter is used

Width is chosen carefully Step 2

Edge strength is found out by taking the gradient of the image

A Roberts mask or a Sobel mask can be used

GyGxGyGxG 22


Implementation of Canny Edge Detector – contd. Step 3

Find the edge direction

Step 4 Resolve edge direction

GxGy1tan


Canny Edge Detector – contd. Step 5

Non-maxima suppression – trace along the edge direction and suppress any pixel value not considered to be an edge. Gives a thin line for edge

Step 6 Use double / hysterisis thresholding to

eliminate streaking


Canny Edge Detector – contd. Compare the results of Sobel and

Canny


Edge Localization – Subpixel Location Estimation Take Gaussian edge detector output

without non-maxima suppression Error displacement is computed by

taking the weighted mean of distance, along the edge gradient on either side of the edge

n

ii

n

iii

g

dgd

1

1


Edge Detector Performance Criteria

Probability of false edges Probability of missing edges Error in estimation of edge angle Mean square distance of edge estimate

from true edge Tolerance to distorted edges and other

features such as corners and junctions


Figure of Merit Basic errors in a Edge Detector

Missing Edges Error in localizing Classification of noise as Edge

IA : detected edges II : ideal edges d : distance between actual and ideal edges : penalty factor for displaced edges

IA

i iIA dIIFM

121

1),max(

1


Line Detection Lines in an image present another

form of discontinuity It is worthwhile to consider them

separately since they are very frequent

Methods Use of Convolution Mask Hough Transform


Line Detection using Convolution Mask We need to decide the orientation

and width of line to be detected Following masks are for single pixel

wide lines in four directions


Line Detection using Convolution Mask – contd. The mask for

horizontal orientation is applied

Lines are 5 pixels wide, mask is tuned for detecting 1 pixel wide line

Line detector acts as edge detector

Demo

http://homepages.inf.ed.ac.uk/rbf/HIPR2/linedetdemo.htm


Line detection – Hough Transform Method to isolate the shapes from an image Performed after edge detection Not affected by noise or gaps in the edges Technique

Thresholding is used to isolate pixels with strong edge gradient

Parametric equation of straight line is used to map the edge points to the Hough parameter space

Points of intersection in the Hough parameter space gives the equation of line on actual image


Hough Transform - contd Parametric Equation of Straight Line


Hough Transform – contd.

Edge Detected ImageMapping of Edge Points in Hough Parameter Space


Applications Enhancement of noisy images –

satellite images, x-rays, medical images like cat scans

Text detection Mapping of roads Video surveillance, etc.


Applications Canny Edge Detector for Remote

Sensing Images Reasons to go for Canny Edge Detector –

Remote sensed images are inherently noisy

Other edge detectors are very sensitive to noise


Edge map of remote sensed image using Canny


Image enhancement and sharpening using canny


MATLAB Demo Get an image from the internet Apply following operators on the

image and display the results Roberts Sobel Prewitt Laplacian of Gaussian Canny


References Machine Vision – Ramesh Jain, Rangachar Kasturi, Brian G Schunck,

McGraw-Hill, 1995 INTRODUCTION TO COMPUTER VISION

AND IMAGE PROCESSING - by Luong Chi MaiDepartment of Pattern Recognition and Knowledge EngineeringInstitute of Information Technology, Hanoi, Vietnamhttp://www.netnam.vn/unescocourse/computervision/computer.htm

The Hypermedia Image Processing Reference - http://homepages.inf.ed.ac.uk/rbf/HIPR2/hipr_top.htm

A Survey and Evaluation of Edge Detection Operators Application to Medical Images – Hanene Trichili, Mohamed-Salim Bouhlel, Nabil Derbel, Lotfi Kamoun, IEEE, 2002

Using The Canny Edge Detector for Feature Extraction and Enhancement of Remote Sensing Images - Mohamed Ali David Clausi, Systems Design Engineering, University of Waterloo, IEEE 2001

A Computational Approach to Edge Detection – John Canny, IEEE, 1986

mailto:[email protected]

http://www.netnam.vn/unescocourse/computervision/computer.htm

http://homepages.inf.ed.ac.uk/rbf/HIPR2/hipr_top.htm

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