SHINTA P.

Juli 2011

• What are edges in an image?

• Edge Detection

• Edge Detection Methods

• Edge Operators

• Matlab Program

Edges are those places in an image that correspond to object boundaries.

Edges are pixels where image brightness changes abruptly.

Edges are used frequently for segmenting images based on local changes in intencity

 

Spatial Coordinates

Brightness

Convert 2D image into a set of curvesEdge may represents an image and

contain shape information

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

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

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

Change is measured by derivative in 1DBiggest change, derivative has maximum magnitudeOr 2nd derivative is zero.

Calculus describes changes of continuous functions using derivatives; an image function depends on two variables - partial derivatives. A change of the image function can be described by a gradient that points in the direction of the largest growth of the image function.An edge is a property attached to an individual pixel and is calculated from the image function behavior in a neighborhood of the pixel.

Edge detectors •locate sharp changes in the intensity function •edges are pixels where brightness changes abruptly.

•The gradient direction gives the direction of maximal growth of the function, e.g., from black (f(i,j)=0) to white (f(i,j)=255).

•This is illustrated below; closed lines are lines of the same brightness. •Boundary and its parts (edges) are perpendicular to the direction of the gradient.

Image gradient• The gradient of an image:

• The gradient points in the direction of most rapid change in intensity

The gradient direction is given by:

• how does this relate to the direction of the edge?

The edge strength is given by the gradient magnitude

Tentukanlah besarnya gradien dan arah gradien tiap titik pada citra

berikut ini!

1 1 1 1 1 0

2 2 0 0 0 0

2 2 0 0 0 0

2 2 2 1 0 0

1 1 1 1 0 0

1 1 1 0 0 0

1 1 1 1 0 0

1 1 1 1 1 1

How can we differentiate a digital image f[x,y]?Option 1: reconstruct a continuous image, then take

gradientOption 2: take discrete derivative (finite difference)

How would you implement this as a cross-correlation?

Better approximations of the derivatives existThe Sobel operators below are very commonly

used

-1 0 1

-2 0 2

-1 0 1

1 2 1

0 0 0

-1 -2 -1

• The standard defn. of the Sobel operator omits the 1/8 term– doesn’t make a difference for edge detection

– the 1/8 term is needed to get the right gradient value, however

(a): Roberts’ cross operator (b): 3x3 Prewitt operator(c): Sobel operator (d) 4x4 Prewitt operator

Mark edge point onlyNo information about edge orientationWork best with binary imagesPrimary disadvantage:

High sensitivity to noiseFew pixels are used to approximate the

gradient

First form of Roberts Operator

Second form of Roberts Operator

22 ),1()1,()1,1(),( crIcrIcrIcrI

|),1()1,(||)1,1(),(| crIcrIcrIcrI

10

011h

01

102h

Looks for edges in both horizontal and vertical directions, then combine the information into a single metric.

Edge Magnitude = Edge Direction =

111

000

111

y

101

101

101

x

22 yx

x

y1tan

Similar to the Prewitt, with different mask coefficients:

Edge Magnitude = Edge Direction =

121

000

121

y

101

202

101

x

22 yx

x

y1tan

Consider a single row or column of the imagePlotting intensity as a function of position gives a signal

Where is the edge?

Where is the edge? Look for peaks in

This saves us one operation:

Consider

Laplacian of Gaussianoperator

Where is the edge? Zero-crossings of bottom graph

is the Laplacian operator:

Laplacian of Gaussian

Gaussian derivative of Gaussian

Edge magnitude is approximated in digital images by a convolution sum.

The sign of the result (+ or -) from two adjacent pixels provide edge orientation and tells us which side of edge brighter

Masks for 4 and 8 neighborhoods

Mask with stressed significance of the central pixel or its neighborhood

010

141

010

111

181

111

121

242

121

212

141

212

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

I = imread('rice.tif'); BW1 = edge(I,'prewitt'); BW2 = edge(I,'canny'); imshow(BW1) figure, imshow(BW2)Edge function supports six different edge-

finding methods: Sobel, Prewitt, Roberts, Laplacian of Gaussian, Zero-cross, and Canny.