EDGE DETECTION IN COMPUTER VISION SYSTEMS

PRESENTATION BY :

ATUL CHOPRA

JUNE 28 2004

EE-6358 COMPUTER VISION

UNIVERSITY OF TEXAS AT ARLINGTON

CONTENTS:

Introduction One-dimensional Formulation Finding Optimal detectors by numerical

Optimization Detectors for step edges. An efficient approximation Noise estimation and Threshold

Cont.

Two or more dimensions Need for multiple widths Conclusion References

INTRODUCTION

Edge Detection: Simplify the analysis on the image by drastically reducing the amount of data to be processed , while at the same time it

preserves the useful structural information about edge boundaries. Criterions for edge detection :

Low error rate Well localized edge points

Need to add third criterion to circumvent the possibility of multiple response to single edge

Reference: A computational approach to Edge detection , John Canny,

1986 IEEE

One dimensional Formulation

We will assume 2-D edges have a constant cross-section in some direction.

Performance criterion are as follows Good Detection 1) Low probability of failing to mark real edge 2) Low probability of falsely marking non edge point.

Good Localization: The points marked as edge points by the operator should be as close as possible to the centre of true edge.

Only one response to single edge

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

MATHEMATICAL FORMULATION OF PERFORMANCE CRITERIONS Detection and Localization criterions

Let, impulse response of filter : f(x)

Let, edge be : G(x)

Assume edge is centered at x=0

The response of the filter to the edge at its centre is given by a convolution

Reference: A computational approach to Edge detection

John Canny, 1986 IEEE

a.) First criterion SNR

Assume the filter has finite response bounded by [-W,W]

The root mean squared noise will be given by

Now the first criterion , the output signal-to-noise ratio (SNR) is given by

Reference: A computational approach to Edge detection, John Canny, 1986 IEEE

b.)Second criterion Localization For localization we need some measure

which increases as localization increases. We will use the reciprocal of root-mean

squared distance of the marked edge from the centre of true edge.

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Contd.

Root mean squared distance is

Localization is defined as reciprocal of

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

0x

Maximizing the product

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

The designing problem is reduced to maximize

the product of localization and SNR .

c.) Third criterion Multiple responses. When maxima are close together it is difficult

to separate the step from noise We need to obtain an expression for the

distance between adjacent noise peaks .

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Contd.

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

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Contd.

Expected number of noise maxima in region of width 2W is given by

kx

WNn

22

max

Fixing k fixes the number of noise maxima that could lead to false response

Finding optimal detectors by Numerical Optimization

It is impossible to find an optimal filter f which maximizes the SNR localization product in presence of multiple response constraint.

If the function f is discrete the computational problem is reduced to calculation of four inner terms .

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Contd.

Penalty function: non-zero values when one of the constraints is violated

We then find f which maximizes

)()(*)( fPfonLocalizatifSNR ii

Detector for Step Edges.

Formulation of the SNR and Localization criterion

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Contd.

)(1

)(

)(

)/()(

'' fw

f

fwf

wxfxf

w

w

w

A spatially scaled filter is formed as show below

Contd.

The uncertainty principle

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Gaussian Edge Detector[2]

In real images step edges are not perfectly sharp

Images are severely corrupted by noise The type of linear operator that provides the

best compromise between noise immunity and localization while retaining the advantages of Gaussian filter is

FIRST DERIVATIVE OF GAUSSIAN

First derivative of Gaussian[2] This operator corresponds to smoothing an

image with Gaussian function and then computing the gradient.

The operator is symmetric along the edge and antisymmetric perpendicular to edge (i.e. along the direction of gradient.)

Sensitive to edge in direction of steepest change , but insensitive to the edge.

Acts as smoothing operator in the direction of edge

Canny edge detector -an Efficient Approximation The optimal operator is similar to the first

derivative of Gaussian

The reason for doing this is that there are very efficient ways to compute the 2-D extension of filter if it can be represented as some derivative of Gaussian.

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Noise Estimation

Weiner Filtering is used to estimate the noise component in the image

Requires knowledge of autocorrelation function of two components i.e. noise and response due to step edges .

Requires knowledge of autocorrelation function of combined signal.

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Contd.

Global histogram estimation is used to estimate the noise strength after noise component has been optimally separated.

Noise response should be Gaussian While step edge response should be composed of large

values occurring very infrequently . If we take a histogram of the filter values , we should

find that the position of low percentiles (less than 80%) will be determined mainly the noise energy. [1]

Thresholding

Broken edge Caused by the operator output fluctuating above and below the threshold along the length of contour.

Streaking Single threshold scheme and limitation Possible solution, used by Pentland with

Marr-Hildreth zero-crossing

Two or more dimensions[1]

Definition of edge direction. Edge direction i.e. the direction of the tangent to the

contour that the edge defines in 2-D.

In 2-D an edge has one position coordinate and an orientation.

Contd.

Detection of edges in 2-D[1] 2-D mask for orientation is created by convolving a

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

Projection function is Gaussian with same deviation as that of detection function.[1]

The need for multiple widths

Choosing the width so as to give best detection/localization tradeoff in particular direction.

SNR will be different for each edge in the image [1]

Feature Synthesis approach

Contd.

Reference: A computational approach to Edge detection , John Canny, 1986 IEEE

Feature Synthesis Mark edges from smallest operator. Synthesis large operator outputs from these edges

(convolve with Gaussian normal to the edge direction) Compare the actual operator outputs with these

synthesized outputs. Additional edges are marked only if the large operator

has significantly larger response.

Synopsis

Edge detection criterions and mathematical formulation

Numerical optimization technique to find optimal operators.

Detection and localization tradeoff Impulse response of the optimal operator-first

derivative of Gaussian.

Contd.

Adaptive thresholding according to noise estimation

Feature synthesis

Reference

[1] A computational approach to Edge detection , John Canny, 1986 IEEE

[2] Machine vision, Ramesh Jain