CMPUT 206 - ugweb.cs.ualberta.caugweb.cs.ualberta.ca/~c206/W08/resources/Lecture...Kernel (nxn...

Irene Cheng Jan 2008

CMPUT 206

Introduction to Digital Image Processing


Overview

1. What is a pixel in an image ?

2. How does Photoshop, + human assistance, detect an edge in a picture/photograph ?

3. Behind Photoshop - How does the computer do it ?


“Pixel” as the basic unit

An image is processed in memory as bytes (8bits/bytes)

• Grey scale image [0 – 255] – 1 byte pixel

• Color image [RGB of values 0-255] – 3 bytes pixel


“Pixel” as the basic unit

R 80 116 228 234G 84 81 87 93B 109 101 59 65

(0, 0) X

(256, 256)


Definition of Edges

• What are edges in an image ?– Locations where there is a sudden variation in

the grey or color scale.

• Which image does not contain edges ?

a b c d e

f g h i


Definition of Edges

Use Photoshop “Find Edges” tool:

a b c d e f g h i


Sharpen edges in images – Photoshop Demo

Sharp

Filter – sharper – sharpen edges/sharpen more

Blur

Increase contrast and sharpen edges

High contrast Low contrast


Sharpen edges in images – Photoshop Demo

Bright

Filter – sharper – sharpen edges/sharpen more

Low luminosity

Discover features

Sharpen and image-adjust-curves


Region growing + Edge detection

The Magic Wand Tool in Photoshop


Can Computer do as well as human ?

• Computer applies edge detection technique based on numeric computation, not based on human perception and cognitive skill


A Simple Edge Detector - gradient

• Based on grey scale gradient at a pixel

),1(),1(),( yxfyxfyxgx −−+≈

)1,()1,(),( −−+≈ yxfyxfyxg y

98 98 98 10610 103 110 11651 38 60 11095 68 60 29

110 116 95 65103 110 116 120

100 139 72

-35 39-15 51

-60 -38-35 -5078 3542 56

Assuming threshold T = 50, a pixel is selected if >= T

x

y What is

22yx gg +

117 4036 8785 5244 76

Answer:


Gradient detection (continued)

• Red squares represent selected pixels

98 98 98 10610 103 110 11651 38 60 11095 68 60 29

110 116 95 65103 110 116 120

Assuming threshold = 50

117 4036 8785 5244 76

=+ 22yx gg

Detected edge


Gradient detection (continued)

• The two gradients gx and gy computed at each pixel are regarded as the x and y components of a gradient vector, which has gradient magnitude and direction given by:

22yx ggg +=

= −

x

y

g

g1tanθ

where the orientation θ is measured relative to x axis.Gradient magnitude is sometimes approximated by:

yx ggg +=


Find edges in images – Photoshop Demo

(Try R, G, B in lab)Extract Intensity values

ApplyEdgeDetection

Stylize – find edges


Find edges in images

increasing threshold (Assume Gradient Approach)

T1 T2 T3

Find Edges

By applying different thresholds T, the results are different: -Assign value 255 if < TAssign value 0 if >= T


Apply edge detection kernelsImage (RxC pixels)

Kernel (nxn pixels)

1. Suppose a nxn (n is an odd number) kernel is used and the centre position of the kernel is denoted by kc.

2. Slide the kernel across the image, one pixel at a time.

3. Compute the new value of each pixel, which is under kc, and repeat for all pixels in the image.

4. Note that the process is not in-place (new values are stored in a new image of same dimension.

5. Computation:� For each pixel inside the kernel,

multiply the pixel value with the kernel value.

� Take the sum of the products.6. (a) The border pixel can be duplicated, or

(b) take the mirror values of the border pixels, to fill up the kernel.

R

C

6 (a) 23 23 23 20 18

6 (b) 18 20 23 20 18

Image border


Edge detection kernels

• Noise reduction or elimination

• Edge enhancement

• Edge localization

111

111

111

9

1

0 0 00 99 00 0 0

An example of Low Pass Filtering:

An example of High Pass Filtering:

−−−

−−

−−−

111

181

111

0 0 00 0 00 0 0

Examples ofImage pixels

Concept of filtering


Convolution operation

• Based on convolution operations − compute weighted averages over a 3x3 neighborhood

),(),( yxfhyxg xx ×=

),(),( yxfhyxg yy ×=

where the (Prewitt) kernels are:

−

−

−

=

101

101

101

xh

−−−

=

111

000

111

yh

Note: high order convolution kernels like 5x5, etc. can also be used, but they are more computational expensive


Convolution operation (continued)

−

−

−

=

101

101

101

−−−

=

111

000

111

yh

98 98 98 10610 103 110 11651 38 60 11095 68 60 29

110 116 95 65103 110 116 120

109 9374 46

-41 -18-37 -80

-145 -940 -172

172 68106 189

98 99 9799 98 9799 98 97

Assuming threshold = ±50

(1) Is 98 an edge?

(2) What are the detected pixels?

(3) What will happen if the threshold is ±100? ±150?


Simple Edge Detectors

Result from gradient Result from Prewitt kernels

One edge but thick

Threshold=±50 Threshold=±100 or ±150


(Top Left) Original (Top Right) Apply gradient magnitudes and scale to 0-255 range

(Bottom Left) Apply threshold of 50 to Top Right

(Bottom Right) Apply thresholdOf 150 to Top Right

An Example – Gradient magnitudes

[Reference 1]

Apply threshold T:If < T, value 1If >= T, value 2


Scaling grey values onto another range

• Let Orange, Omin and Omax be the original range, minimum value and maximum value,

i.e. Orange = Omax – Omin

• The Nrange, Nmin and Nmax be the original range, minimum value and maximum value,

i.e. Nrange = Nmax – Nmin

• To map a pixel P from the original range to P’ in the new range:

minmin' NN

O

OPP range

range+×

−=


Advantage of a bigger grey value range before processing

Findedges Threshold

128

Map from 0-100To 0-255

Threshold50


Sobel kernels

• Sobel kernels, which give more weights to on-axis pixels

−

−

−

=

101

202

101

xh

−−−

=

121

000

121

yh

Original image convoluation with hx with hy

Cross from grey to white

From white to grey

+ve value

-ve value

(assign pixel value 0 if <0)


• Problem with edge localization– High threshold may suppress meaningful edges– Low threshold may include unwanted edges– Noise may have high magnitude

Noise in an image

xWith noiseG

radi

ent m

agni

tude

Gradient magnitude is very sensitive to noiseWithout noise


Blurred edges

– The detected edge can be rather broad in the case of diffuse edges, resulting in a thick band of pixels instead of a single point of maximum gradient.

Slightly blurred Heavily blurred

x

Gre

y va

lues

Sharp edge


Sharp & Blurred Edges

Grey values 255

Sharp edge Blurred edge


Rank or “order statistic” filtering

• Non-linear

• In image processing it is usually necessary to perform high degree of noise reduction in an image before performing higher-level processing steps, such as edge detection. A non-linear digital filtering technique is often used to remove noise from images or other signals.


Rank or “order statistic” filtering (continued)

• Compile a list of grey scales in the neighborhood of a given pixel, sort them in ascending order and select a value as the new value– Median filter; any structure that occupies less than

half of the filter’s neighborhood will be eliminated– Minimum filter and Maximum filter– Range filter: output the difference between the

maximum and minimum


Rank or “order statistic” filtering (continued)

– Hybrids of linear and non-linear filters, e.g. α-trimmed mean filter: sorts the neighborhood into ascending order, discards a number of them and outputs the mean of the remaining

(α is the number of values removed, in the range [0, (n2 – 1)/2] from each end of the list)

∑−

+=−

α

αα

2

12 2

1n

i

ifn

• when α = 0: mean filter• when α = (n2 – 1)/2: median filter


Median filter

• Non-linear technique• Consider all pixels inside the filter.• Compile a list of grey values and sort them in ascending

order.• select a value as the new value

– Median filter; any structure that occupies less than half of the filter’s neighborhood will be eliminated

• Have the advantage of non kernel-based; no problem to filter a smaller neighborhood at the corners or sides of the image

• The shape of the filter applied at the corner can give different results, e.g. square vs. cross-shaped


Rank filtering examples

23 20 5 12 8

8

13

18

10

What is the filtered value of the centre pixel after applying a:

1. median filter 122. mean filter 133. maximum filter 234. minimum filter 55. range filter 186. 4-trim filter 127. 0-trim filter 138. 2-trim filter 12


Median filtering example

(a) (b)

What is the filtered value of P in (a) ?What is the filtered value of P in (b) ?

The square and the cross are both median filters

p P


References

1. “Digital Image Processing − a practical introduction using Java” Nick Efford, Pearson Education 2000.

CMPUT 206 - ugweb.cs.ualberta.caugweb.cs.ualberta.ca/~c206/W08/resources/Lecture...Kernel (nxn...

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