Edge enhancement by linear (and nonlinear) filtering Dr. Dileepan Joseph Dept. of Engineering...
-
Upload
keenan-prouse -
Category
Documents
-
view
216 -
download
0
Transcript of Edge enhancement by linear (and nonlinear) filtering Dr. Dileepan Joseph Dept. of Engineering...
Edge enhancement by linear (and nonlinear) filtering
Dr. Dileepan Joseph
Dept. of Engineering Science
University of Oxford, UK
Objectives
Learn what edge enhancement is, why it is useful & how it differs from edge detection
Define linear and nonlinear spatial filtering Design linear filters to either smoothen or
sharpen the edges in an image and show how the two operations are related
Appreciate that human vision enhances edges using local operations
Edge enhancement
The purpose of edge enhancement is to highlight fine detail in an image or to restore, at least partially, detail that has been blurred (either in error or as a consequence of a particular method of image acquisition)
Applications of edge enhancement include electronic printing, medical imaging, industrial inspection, and autonomous target detection in smart weapons
Edge enhancement
Edge enhancement involves sharpening the outlines of objects and features with respect to their background
Edge detection involves isolating the outlines of objects and features
The former is easier to do than the latter
Spatial filtering
Image processing in the spatial domain may be expressed as
g(x,y) = H{f(x,y)} where f is the input
image, g is the output image, and H is an operator on f, defined over some neighbour-hood of pixel (x,y)
Spatial filtering
The neighbourhood of pixel (x,y), for image f, may be expressed as a column vector w(x,y) of pixel values
e.g. Consider a 3 by 3 square neighbourhood centred on the pixel (x,y) of interest
)1,1(
)1,(
)1,1(
),1(
),(
),1(
)1,1(
)1,(
)1,1(
),(
yxf
yxf
yxf
yxf
yxf
yxf
yxf
yxf
yxf
yxw
Spatial filtering
A spatial filter H is linear if (and only if) H{a∙f(x,y)} = a∙H{f(x,y)} H{f1(x,y)+f2(x,y)} = H{f1(x,y)}+H{f2(x,y)}
For any linear spatial filter H, we may write g(x,y) = h•w(x,y) where g is the output image, w is the neighbourhood vector of the input image f, and • is the inner product operator
The column vector h is called a mask and it defines the properties of the linear filter
Spatial filtering
It is easier to visualize linear spatial filtering as an inner product of h and w over the shape of the neighbourhood
e.g. For a 3 by 3 square neighbourhood centred on the pixel (x,y) of interest
g(x,y) =
h1 h2 h3
h4 h5 h6
h7 h8 h9
•
w1 w2 w3
w4 w5 w6
w7 w8 w9
h1w1+h2w2…+h9w9= f(x,y)
Smoothing filter
To understand how to sharpen edges, we first consider how to smoothen them
The simplest way to smoothen an image f is to use the neighbourhood average of pixel values to define the image g
g(x,y) =
1/9 1/9 1/9
1/9 1/9 1/9
1/9 1/9 1/9
•
w1 w2 w3
w4 w5 w6
w7 w8 w9
f(x,y)
Smoothing filter
Middle region of the original image:
8 9 83 58 15
9 11 127 3411 127 34 14
10 13 160 2313 160 23 14
10 19 124 1719 124 17 13
10 39 93 16 14
Middle region of the smoothed image:
9 35 52 57 33
10 48 58 59 23
11 54 5959 58 19
14 53 56 53 16
20 47 49 41 16
Embossing filter
Compared to the original image, edges in the smoothed image are slightly blurred
Thus, the difference between the original and smooth images, which may be derived by spatial filtering, holds edge information
g(x,y) =
−1/9 −1/9 −1/9
−1/9 8/9 −1/9
−1/9 −1/9 −1/9
•
w1 w2 w3
w4 w5 w6
w7 w8 w9
f(x,y)
Embossing filter
Middle region of the original image:
8 9 83 58 15
9 11 127 3411 127 34 14
10 13 160 2313 160 23 14
10 19 124 1719 124 17 13
10 39 93 16 14
Middle region of the embossed image:
-1 -26 31 1 -18
-1 -37 69 -25 -9
-1 -41 101101 -35 -5
-4 -34 68 -36 -3
-10 -8 44 -25 -2
Sharpening filter
The embossed image holds edge inform-ation over a uniform (zero) background
Thus, the sum of the original and embossed images, which may be derived by spatial filtering, will reinforce edges of the former
g(x,y) =
−1/9 −1/9 −1/9
−1/9 17/9 −1/9
−1/9 −1/9 −1/9
•
w1 w2 w3
w4 w5 w6
w7 w8 w9
f(x,y)
Sharpening filter
Middle region of the original image:
8 9 83 58 15
9 11 127 3411 127 34 14
10 13 160 2313 160 23 14
10 19 124 1719 124 17 13
10 39 93 16 14
Middle region of the sharpened image:
7 0 114 59 0
8 0 196 9 5
9 0 255255 0 9
6 0 192 0 10
0 31 137 0 12
Edge enhancement
Without amplification:Emboss = Original − Smooth
Sharp = Original + Emboss
= 2∙Original − Smooth
With amplification A:Sharp = Original + A∙Emboss
= (1+A)∙Original − A∙Smooth
e.g. Consider the mask h of a 3 by 3 square neighbourhood
9
9
9
9
91
9
9
9
9
A
A
A
A
AA
A
A
A
A
h
Edge enhancement
A = 0 A = 1 A = 2
Mach Bands illusion
This image has three sections: on the left, luminance is at a constant high; on the right, luminance is at a constant low; in the middle, it declines at a constant rate
The thin bands seen on either side of the ramp (and named after their discoverer) are illusory
Mach Bands illusion
Sensory tissue is often organized so that ex-citation of any location produces inhibition of surrounding nerves
In human vision, this lateral inhibition enhances edges by producing overshoot and undershoot
Review
Edge enhancement involves sharpening the outlines of objects and features in an image with respect to their background
Image processing in the spatial domain may be expressed as g(x,y) = H{f(x,y)} where f is the input image, g is the output image, and H is a linear or nonlinear operator on f, defined over some neighbourhood of pixel (x,y)
Linear filtering may be expressed by an inner product of a mask and the neighbourhood
Review
Smoothing of edges may be achieved by neighbourhood averaging
Sharpening of edges may be achieved by subtracting a multiple, A, of the neighbour-hood average from a larger multiple, 1+A, of the neighbourhood centre
The Mach Bands illusion may be understood in terms of edge enhancement by lateral inhibition in human vision
Resources
Gonzales and Woods, Digital Image Processing, Second Edition, Prentice Hall, 2002 (get the first two chapters free from http://www.imageprocessingbook.com/)
Matlab Image Processing Toolbox http://www.cquest.utoronto.ca/psych/psy280f/
ch3/mb/mb.html (Mach Bands illusion)
http://www.siggraph.org/education/materials/HyperVis/vision/latinib.htm (lateral inhibition)