03 edgedet slides

8/2/2019 03 Edgedet Slides

1/24

Basic image processing: edge detection
http://find/http://goback/


2/24

Edge detection problem

Edge detection is sometimes an abused notion in image processing. Partof the complication lies in the difficulty in defining precisely what ismeant by an edge. A set of different examples of edges are shown below:
http://find/


3/24

Gradient-based methods

Gradient-based methods are often used to find edges in images. Consider

for example a 1-D function f(x) the derivative

f(x) =df

dx

takes on values with a large magnitude at the points where the function

has a high rate of change. It is reasonable to expect that these pointswill correspond to edges in the function.

The generalisation of f(x) to a 2-D function f(x, y) is the gradient

f(x, y) =f(x, y)

x

ix +f(x, y)

y

iy,

where ix and iy are unit vectors in the x and y directions respectively.The magnitude off(x, y) is first computed, and is then compared to athreshold to find candidate edge points.
http://find/


4/24

Nondirectional edge detector

An edge detection system based on a function such as |f(x, y)| is calleda nondirectional edge detector, since such functions favour no particulardirection. However, if for example we use |f(x, y)/x| instead of|f(x, y)| then one obtains a directional edge detector, which in thiscase only detects edges in the vertical direction and does not respond tohorizontal edges.

For a 2-D sequence f(n1, n2), the partial derivatives may be replaced by adiscrete approximation such as

f(x, y)

x [f(n1, n2) f(n1 1, n2)]/T

orf(x, y)

x [f(n1 + 1, n2) f(n1 1, n2)]/(2T).
http://find/


5/24


6/24

Horizontal gradient example

An example of an image and the estimated horizontal derivative is shownbelow:

50 100 150 200 250

50

100

150

200

250

50 100 150 200 250

50

100

150

200

250


7/24

Noise reduction

Derivatives calculated using the method described are typically quitenoisy estimates off(x, y)/x. This can be improved by averaging the

derivative over several samples, for example

f(x, y)

x [f(n1 + 1, n2 + 1) f(n1 1, n2 + 1)]

+ [f(n1 + 1, n2) f(n1 1, n2)]

+ [f(n1 + 1, n2 1) f(n1 1, n2 1)].

This is equivalent to calculating the horizontal derivative by convolvingwith the kernel

0

+1 1

+1

+1

1

1

0

0
http://find/


8/24

Horizontal gradient reduced noise

An example of applying this process to the previous image is shown

below:

50 100 150 200 250

50

100

150

200

250
http://find/


9/24


10/24

Standard edge detectors

Examples of common filters for detecting edges in the vertical andhorizontal direction are

Roberts:

h1 =

0 11 0

, h2 =

1 00 1

.

Prewitt:

h1 =1 0 11 0 11 0 1

, h2 = 1 1 10 0 01 1 1

. Sobel:

h1 =1 0 12 0 21 0 1

, h2 = 1 2 10 0 01 2 1

.In these relationships h1 corresponds to a filter for detecting horizontaledges, and h2 vertical edges. The formulations differ in terms of their

properties, and in some cases their computational complexity.
http://find/


11/24

Nondirectional edge detectors

Nondirectional edge detectors can be developed using a discreteapproximation to |f(x, y)|, such as

|f(x, y)| =

f(x, y)x

2+

f(x, y)y

2.

Clearly this just involves forming some combination of the edges in thehorizontal and vertical directions.
http://find/


12/24

Laplacian-based methods

The objective of an edge detection algorithm is to locate regions wherethe intensity is changing rapidly. For 1-D functions and gradient-basedmethods, f(x) is considered large when its magnitude |f(x)| is largerthan a threshold. Another way is to assume |f(x)| large whenever it

reaches a local extremum, or when f

(x) has a zero crossing.Declaring zero-crossing points as edges results in a large number ofpoints being declared edge points. Since the magnitude of f(x) is notconsidered, any small ripple in f(x, y) generates a detected edge. Due tothis sensitivity to noise, the application of noise reduction is a necessary

prerequisite.


13/24

Estimating the Laplacian

A generalisation of2f(x)/x2 to a 2-D function f(x, y) for purposes ofedge detection is the Laplacian

2f(x, y) = (f(x, y)) = 2f(x, y)x2

+ 2f(x, y)y2

.

For a 2-D sequence f(n1, n2) the partial derivatives can be replaced bysome form of second-order differences. As in the case of gradient-based

methods, these differences can be represented by a convolution off(n1, n2) with the impulse response of a filter h(n1, n2). For example,approximating the derivative by

f(x, y)

x fx(n1, n2) = f(n1 + 1, n2) f(n1, n2),

the term 2f(x, y)/x2 can be approximated by

2f(x, y)

x2 fxx(n1, n2) = fx(n1, n2) fx(n1 1, n2)

= f(n1 + 1, n2) 2f(n1, n2) + f(n1 1, n2).


14/24

Estimating the Laplacian (2)

With the derivatives in the y-direction approximated similarly, theestimate of the Laplacian is

2f(x, y) fxx(n1, n2) + fyy(n1, n2)

= f(n1 + 1, n2) + f(n1 1, n2) + f(n1, n2 + 1)

+ f(n1, n2 1) 4f(n1, n2).
http://goforward/http://find/http://goback/


15/24

Laplacian edge detector example

An estimate of the Laplacian for the image shown previously is

50 100 150 200 250

50

100

150

200

250

Locating edges now involves finding zero crossings of this function.

L l


16/24

Laplacian operators

Some examples of discrete Laplace operators are

h = 0 1 01 4 1

0 1 0

, h = 1 1 11 8 11 1 1

.

P i f d
http://find/


17/24

Perception of edges

Do you think that either of the two methods presented coincide with theway we perceive edges?

Ed li ki


18/24

Edge linking

Obtaining an estimate of the image gradient is only part of the problemin edge detection, where the final goal is a binary image indicating thepresence of edges. There are many different ways of linking regions ofhigh gradient into distinct edges.


19/24


20/24


21/24

E ample Bottles (2)


22/24

Example: Bottles (2)

Estimates of the vertical and horizontal edge gradients are shown below:



23/24


Standard methods for locating the circular rim of the bottle were applied,and an edge direction mask of the following form was applied to theresulting edge estimates:


24/24

03 edgedet slides

Documents