Computer Vision Group Edge Detection Giacomo Boracchi 5/12/2007 [email protected]

Computer Vision Group

Edge DetectionGiacomo Boracchi

5/12/2007

[email protected]


Edge Detection in Images

Goal: Automatically find the contour of objects in a scene.

What For: Edges are significant descriptors, useful for classification, compression…


Edge Detection in Images

What is an object?

It is one of the goals of computer vision to identify objects in scenes.


Edges May Have Different Sources

Object Boundaries Occlusions Textures Shading


What is an Edge

Lets define an edge to be a discontinuity in image intensity function.

Several Models• Step Edge• Ramp Edge• Roof Edge• Spike Edge

They can bethus detected asdiscontinuitiesof image Derivatives


Differentiation and convolution

Recall

Now this is linear and shift invariant. Therefore, in discrete domain, it will be represented as a convolution

xfxf

x

f0

lim


Differentiation and convolution

Recall

Now this is linear and shift invariant, so must be the result of a convolution.

We could approximate this as

(which is obviously a convolution with Kernel

; it’s not a very good way to do things, as we shall see)

xfxf

x

f0

lim x

xfxf

x

f n

1

11


Finite Difference in 2D

yxfyxf

x

yxf ,,lim

,0

yxfyxf

y

yxf ,,lim

,0

x

yxfyxf

x

yxf mnmn

,,, 1

x

yxfyxf

y

yxf mnmn

,,, 1

11

1

1

Discrete Approximation

Horizontal

Convolution Kernels

Vertical


Finite differences

11* II x

1

1*II y

I


Noise

Simplest noise model• independent stationary

additive Gaussian noise• the noise value at each pixel

is given by an independent draw from the same normal probability distribution

Issues• this model allows noise values

that could be greater than maximum camera output or less than zero

• for small standard deviations, this isn’t too much of a problem - it’s a fairly good model

• independence may not be justified (e.g. damage to lens)

• may not be stationary (e.g. thermal gradients in the ccd)


Finite differences and noise

Finite difference filters respond strongly to noise• obvious reason: image noise results in pixels

that look very different from their neighbors

The more relevant is the noise, the stronger the response


Finite differences and noise

What is to be done?• intuitively, most pixels in images look quite a lot

like their neighbors.• this is true even at an edge; along the edge

they’re similar, across the edge they’re not.• suggests that smoothing the image should help,

by forcing pixels different to their neighbors (=noise pixels?) to look more like neighbors.


Finite differences responding to noise

Increasing noise ->(this is zero mean additive gaussian noise)








Smoothing reduces noise

Generally expect pixels to “be like” their neighbors• surfaces turn slowly• relatively few reflectance changes

Generally expect noise processes to be independent from pixel to pixel


Smoothing reduces noise

Generally expect pixels to “be like” their neighbors• surfaces turn slowly• relatively few reflectance changes

Generally expect noise processes to be independent from pixel to pixe

Implies that smoothing suppresses noise, (i.i.d. noise!)

Gaussian Filtering• the parameter in the symmetric Gaussian• as this parameter goes up, more pixels are involved in the average• and the image gets more blurred• and noise is somehow suppressed


The effects of smoothing Each row shows smoothingwith gaussians of differentwidth; each column showsdifferent realisations of an image of gaussian noise.

Low - Pass


Classical Operators : Prewitt

11

11

11

1

1

11

Smooth Differentiate

101

101

101

111

000

111

111

111

Horizontal

Vertical


Classical Operators: Sobel

11

22

11

1

1

11

Smooth Differentiate

101

202

101

121

000

121

121

121

Horizontal

Vertical


Gaussian Filter

2

22

2 2exp

2

1,

yx

yxG

2

22

2 2

11exp

2

1,

kjki

jiH

array 1212 is , where kkjiH


Sobel Edge Detector

Detecting Edges in Image

Image I

101

202

101

121

000

121

*

*

Idx

d

Idy

d

22

Idy

dI

dx

d

ThresholdEdges

any alternative ?

Discrete Derivatives Gradient NormsThreshold


Quality of an Edge Detector

Robustness to Noise Localization Too Many/Too less Responses

Poor robustness to noise Poor localization Too many responses

True Edge


Canny Edge Detector Criteria

Good Detection: The optimal detector must minimize the probability of false positives as well as false negatives.

Good Localization: The edges detected must be as close as possible to the true edges.

Single Response Constraint: The detector must return one point only for each edge point.


Canny Edge Detector

Basically• Convolution with derivative of Gaussian• Non-maximum Suppression• Hysteresis Thresholding


Canny Edge Detector

Smooth by Gaussian

Compute x and y derivatives

Compute gradient magnitude and orientation

IGS *2

22

2

2

1

yx

eG

Tyx

T

SSSy

Sx

S

22yx SSS

x

y

S

S1tan


Canny Edge Operator

IGIGS ** T

y

G

x

GG

T

Iy

GI

x

GS

**


Canny Edge Detector

xS

yS

I


The Gradient Orientation

Like for continuous function, the gradient at each pixel points at the steepest intensity growth direction.

The gradient norm indicates the inclination of the intensity growth.

Matlab…..


Canny Edge Detector

I

22yx SSS

25 ThresholdS


We wish to mark points along the curve where the magnitude is biggest.We can do this by looking for a maximum along a slice normal to the curve(non-maximum suppression). These points should form a curve. There arethen two algorithmic issues: at which point is the maximum, and where is thenext one?

Non-Maximum Suppression - Idea


We wish to mark points along the curve where the magnitude is biggest.We can do this by looking for a maximum along a slice normal to the curve(non-maximum suppression). These points should form a curve. There arethen two algorithmic issues: at which point is the maximum, and where is thenext one?

Non-Maximum Suppression – Idea (II)


Non-Maximum Suppression - Threshold

Suppress the pixels in ‘Gradient Magnitude Image’ which are not local maximum

edgean tonormaldirection thealong

in of neighbors theare and Sx,yy,xy,x

otherwise0,,&

,, if,

, yxSyxS

yxSyxSyxS

yxM

yx ,

yx,

yx ,


An Overview on Threshold

According to the way the Threshold T is used they are divided into• Global Threshold• Local (or) Adaptive Threshold

According to the output they can be classified in • Binary Threshold• Hard Threshold• Soft Threshold


Non-Maximum Suppression

22yx SSS M

25ThresholdM


Hysteresis Thresholding

Use of two different threshold High and Low for • For new edge starting point• For continuing edges

In such a way the edges continuity is preserved



If the gradient at a pixel is above ‘High’, declare it an ‘edge pixel’.

If the gradient at a pixel is below ‘Low’, declare it a ‘non-edge-pixel’.

If the gradient at a pixel is between ‘Low’ and ‘High’ then declare it an ‘edge pixel’ if and only if it is connected to an ‘edge pixel’ directly or via pixels between ‘Low’ and ‘ High’.



M 25ThresholdM

15

35

Low

High


Examples

an image its detected edges


In matlab..

Canny edge detector is implemented with theedge.m command


A brief overview on Morphological Operators in Image Processing

Giacomo Boracchi

5/12/2007

[email protected]


An overview on morphological operations

Erosion, Dilation

Open, Closure

We assume the image being processed is binary, as these operators are typically meant for refining “mask” images.


AND operator in Binary images

Images From Digital Image Processing II edition Gonzalez & Woods Chapter 9


OR in Binary Images


Erosion

General definition: Nonlinear Filtering procedure that replace to each pixel valuethe maximum on a given neighbor

As a consequence on binary images

E(x)=1 iff the image in the neighbor is constantly 1

This operation reduces thus the boundaries of binary images

It can be interpreted as an AND operation of the image and the neighbor overlapped at each pixel


Erosion Example


Erosion Example


Dilation

General definition: Nonlinear Filtering procedure that replace to each pixel valuethe minimum on a given neighbor

As a consequence on binary images

E(x)=1 iff at least a pixel in the neighbor is1

This operation grows fat the boundaries of binary images

It can be interpreted as an OR operation of the image and the neighbor overlapped at each pixel


Dilation Example


Dilation Example


In matlab

They are performed using the

bwmorph.m script passing the name of the operation as a parmeter

Examples…


Open and Closure

Open Erosion followed by a Dilation

Closure Dilation followed by an Erosion


Open

Open Erosion followed by a Dilation• Smoothes the contours of an object• Typically eliminate thin protrusions


Open

Figure to Open,

Structuring element a Disk


Open

First Erode,


Open

Then Dilate,

Open results, the bridge has been eliminate from the first erosion and cannot be replaced by the dilation.

Edges are smoothed

Corners are rounded


Closure

Closure Dilation followed by an Erosion• Smoothes the contours of an object, typically creates• Generally fuses narrow breaks


Open

Figure to Open,

Structuring element a Disk


Close

First Dilate,


Close

Then Erode,

Close results, the bridge has been preserved

Edges are smoothed

Corners are rounded

Documents

Computer Vision Group Edge Detection Giacomo Boracchi 5/12/2007 [email protected]