DIGITAL IMAGE PROCESSING
Image Enhancement
(Spatial Filtering 2)
Contents
In this lecture we will look at more spatial filtering
techniques
Spatialfiltering refresher
Sharpening filters
1stderivative filters
2nd derivative filters
Combining filtering techniques
Spatial Filtering Refresher
Origin x
a b c r s t
d
g
e
h
f
i
* u
x
v
y
w
z
Original Image Filter
Simple 3*3 Pixels
e 3*3 Filter
Neighbourhood
eprocessed = v*e +
r*a + s*b + t*c +
u*d + w*f +
y Image f (x, y) x*g + y*h + z*i
The above is repeated for every pixel in the
original image to generate the smoothed image
Sharpening Spatial Filters
Previously we have looked at smoothing filters which
remove fine detail
Sharpening spatial filters seek to highlight fine detail
Remove blurring from images
Highlight edges
Sharpening filters are based on spatial differentiation
Spatial Differentiation
Differentiation measures the rate of change of a
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
function
Let’s consider a simple 1 dimensional example
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
A
Spatial Differentiation
B
1st Derivative
The formula for the 1st derivative of a function is as
follows:
f
f ( x 1) f ( x)
x
It’s just the difference between subsequent values and
measures the rate of change of the function
1st Derivative (cont…) Image Strip
8
7
6
5
4
3
2
1
0
5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7
1st Derivative
0 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0
8
6
4
2
0
-2
-4
-6
-8
2nd Derivative
The formula for the 2nd derivative of a function is as
follows: 2
f
f ( x 1) f ( x 1) 2 f ( x)
x
2
Simply takes into account the values both before and
after the current value
2nd Derivative (cont…) Image Strip
8
7
6
5
4
3
2
1
0
5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7
-1 0 0 0 0 1 0 6 -12 6
2nd0 0 1
Derivative 1 -4 1 1 0 0 7 -7 0 0
10
-5
-10
-15
Sharpening Spatial filters
f
f ( x 1) f ( x)
x
2 f
f ( x 1) f ( x 1) 2 f ( x)
x 2
First order is non-zero along the
entire ramp
Second order is non-zero only in
the beginning and end
First order – thick edges
Second order – fine edges
Second order – stronger response
to fine detail
Sharpening Spatial filters
Sharpening filters using first- second order derivatives
First derivative:
1. Zero in flat segments (constant gray level)
2. Non-zero at start of a gray-level step/ ramp
3. Non-zero along ramps
Second derivative:
1. Zero in flat areas
2. Non-Zero at start and end of step/ramps
3. Zero along ramps of constant slope
The aim is to develop a technique which can identify changes (of
different nature) in the gray levels
1st & 2nd Derivatives
Comparing the 1st and 2nd derivatives we can
conclude the following:
1st order derivatives generally produce thicker edges
2nd order derivatives have a stronger response to fine
detail e.g. thin lines
1st order derivatives have stronger response to grey
level step
2nd order derivatives produce a double response at
step changes in grey level
Using Second Derivatives For Image
Enhancement
The 2nd derivative is more useful for image
enhancement than the 1st derivative
Stronger response to fine detail
Simpler implementation
We will come back to the 1st order derivative later on
The first sharpening filter we will look at is the
Laplacian
Isotropic
Oneof the simplest sharpening filters
We will look at a digital implementation
Various situations encountered for derivatives
Various situations encountered for derivatives
Various situations encountered for derivatives
The Laplacian
The Laplacian is defined as follows:
f f
2 2
f 2 2
2
x y
where the partial 2nd order derivative in the x
direction is defined as follows:
f
2
f ( x 1, y) f ( x 1, y) 2 f ( x, y)
x
2
and in the y direction as follows:
f
2
f ( x, y 1) f ( x, y 1) 2 f ( x, y)
y
2
The Laplacian (cont…)
So, the Laplacian can be given as follows:
f [ f ( x 1, y) f ( x 1, y)
2
f ( x, y 1) f ( x, y 1)]
4 f ( x, y)
We can easily build a filter based on this
0 1 0
1 -4 1
0 1 0
The Laplacian (cont…)
Applying the Laplacian to an image we get a new
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
image that highlights edges and other discontinuities
Original Laplacian Laplacian
Image Filtered Image Filtered Image
Scaled for Display
But That Is Not Very Enhanced!
The result of a Laplacian filtering is not
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
an enhanced image
We have to do more work in order to
get our final image
Subtract the Laplacian result from the
original image to generate our final Laplacian
Filtered Image
sharpened enhanced image Scaled for Display
g ( x, y) f ( x, y) f 2
Laplacian Image Enhancement
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
- =
Original Laplacian Sharpened
Image Filtered Image Image
In the final sharpened image edges and fine detail
are much more obvious
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Laplacian Image Enhancement
Variant of Laplacian
Simplified Image Enhancement
The entire enhancement can be combined into a single
filtering operation
g ( x, y) f ( x, y) f
2
f ( x, y) [ f ( x 1, y) f ( x 1, y)
f ( x, y 1) f ( x, y 1)
4 f ( x, y)]
5 f ( x, y) f ( x 1, y) f ( x 1, y)
f ( x, y 1) f ( x, y 1)
Simplified Image Enhancement (cont…)
This gives us a new filter which does the whole job for
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
us in one step
0 -1 0
-1 5 -1
0 -1 0
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Simplified Image Enhancement (cont…)
Variants On The Simple Laplacian
There are lots of slightly different versions of the
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Laplacian that can be used:
0 1 0 1 1 1
Simple Variant of
1 -4 1 1 -8 1
Laplacian Laplacian
0 1 0 1 1 1
-1 -1 -1
-1 9 -1
-1 -1 -1
Unsharp Masking and Highboost filtering
The process of Unsharp making consists of the following steps:
Blur the original image
Subtract the blurred image from the original (the result is called the
mask)
Add the mask to the original image
_
Let f ( x, y ) denote the blurred image _
The unsharp masking is: g mask ( x, y) f ( x, y) f ( x, y)
To obtain the output: g ( x, y) f ( x, y) k * g mask ( x, y)
When ‘k>1’ the process is called Highboost filtering
Unsharp Masking and Highboost filtering
1st Derivative Filtering
Implementing 1st derivative filters is difficult in practice
For a function f(x, y) the gradient of f at coordinates (x,
y) is given as the column vector:
f
Gx x
f f
G y
y
1st Derivative Filtering (cont…)
The magnitude of this vector is given by:
f mag(f )
G G2
x
2
y
1
2
1
f f 2
2 2
x y
For practical reasons this can be simplified as:
f Gx Gy
1st Derivative Filtering (cont…)
There is some debate as to how best to calculate these
gradients but we will use:
f z7 2 z8 z9 z1 2 z2 z3
z3 2 z6 z9 z1 2 z4 z7
which is based on these coordinates
z1 z2 z3
z4 z5 z6
z7 z8 z9
Sobel Operators
Based on the previous equations we can derive the Sobel
Operators
-1 -2 -1 -1 0 1
0 0 0 -2 0 2
1 2 1 -1 0 1
To filter an image it is filtered using both operators the
results of which are added together
Sobel Example
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
An image of a
contact lens which
is enhanced in
order to make
defects (at four
and five o’clock in
the image) more
obvious
Sobel filters are typically used for edge detection
Directional Derivative
Let f(x, y) be a function mapping two real numbers to
a real number (intensity values).
For the directional derivative of f along the x axis,
we use notation df/dx.
Vertical edges correspond to points in g with high df/dx.
For the directional derivative of f along the y axis,
we use notation df/dy.
Horizontal edges correspond to points in g with high dg/dy.
Approximating df/dx via Filtering
In the discrete domain df/dx is approximated by
filtering with the right kernel:
dx = [-1 0 1;
-2 0 2;
-1 0 1];
dxgray = abs(filter2(gray, dx));
Interpreting filter2(gray, dx):
Resultsfar from zero (positive and negative) correspond to
strong vertical edges.
These are mapped to high positive values by abs.
Results
close to zero correspond to weak vertical edges, or no
edges whatsoever.
Result: Vertical/Horizontal Edges
gray = read_gray('data/myhand.bmp');
dx = [-1 0 1;
-2 0 2;
-1 0 1];
dy = dx’; % dy is the transpose of dx
dxgray = abs(imfilter(gray, dx, 'symmetric', 'same'));
dygray = abs(imfilter(gray, dy, 'symmetric', 'same'));
gray dxgray dygray
(vertical edges) (horizontal edges)
Summary
In this lecture we looked at:
Sharpening filters
1stderivative filters
2nd derivative filters
Combining filtering techniques
Combining Spatial Enhancement Methods
Successful image enhancement is
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
typically not achieved using a
single operation
Rather we combine a range of
techniques in order to achieve a
final result
This example will focus on
enhancing the bone scan to the
right
Combining Spatial Enhancement Methods
(cont…)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
(a)
Laplacian filter of
bone scan (a)
(b)
Sharpened version of
bone scan achieved (c)
by subtracting (a)
and (b) Sobel filter of bone
scan (a) (d)
Combining Spatial Enhancement Methods
(cont…)
Result of applying a (h)
power-law trans. to
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Sharpened image (g)
which is sum of (a)
and (f) (g)
The product of (c)
and (e) which will be (f)
used as a mask
(e)
Image (d) smoothed with
a 5*5 averaging filter
Combining Spatial Enhancement Methods
(cont…)
Compare the original and final images
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
