COM2304: Intensity Transformation and Spatial Filtering – III Spatial Filters for Sharpening

COMPUTER GRAPHICS & IMAGE PROCESSING
COM2403
Intensity Transformation and Spatial Filtering – III
Spatial Filters for Sharpening
K.A.S.H.Kulathilake
B.Sc. (Hons) IT (SLIIT), MCS (UCSC), M.Phil (UOM), SEDA(UK)
Rajarata University of Sri Lanka
Faculty of Applied Sciences
Department of Physical Sciences

Learning Outcomes
COM2304 - Computer Graphics & Image
Processing
• At the end of this lecture, you should be
able to;
– describe sharpening through spatial filters.
– Identify usage of derivatives in Image
Processing.
– discuss edge detection techniques.
– compare 1st & 2nd order derivatives used for
sharpening.
– Apply sharpening techniques for problem
solving.
2

Sharpening Spatial Filters
• The principle objective of sharpening is to highlight
transitions in intensity.
• In other terms, sharpening spatial filters seek to highlight
fine detail
– Remove blurring from images
– Highlight edges
• Sharpening filters are based on spatial differentiation.
• The strength of the response of a derivative operator is
proportional to the degree of intensity discontinuity of
the image at the point at which the operator is applied.
• Hence, image differentiation enhances edges and other
discontinuities ( such as noise) and deemphasizes areas
with slowly varying intensities.
Processing
3

Sharpening Spatial Filters (Cont…)
• As the initial approach, it is better to get an idea about
certain properties of first order and second order
derivatives in the digital context.
• To simplify the explanation, it has been focused
attention on one dimensional derivatives.
• We are interested in the behavior of these derivatives
in areas of
– Constant intensity,
– At the onset and end of discontinuities (step and ramp
discontinuities), and
– Along the intensity ramps.
Processing
4

• Differentiation measures the rate of change of
a function.
• In digital context, values are finite and the
maximum possible intensity change is also
finite.
• The shortest distance over which that change
can occur is between the adjacent pixels.
• To understand this let’s discuss an example.
Processing
5

A B
Intensity transition
6
Processing

• The formula for the 1st derivative of a function is as
follows:
• It is just the difference between subsequent values
and measures the rate of change of the function.
• It should satisfy following conditions:
– Must be zero in the areas of constant intensity.
– Must be nonzero at the onset of an intensity step or
ramp.
– Must be nonzero along the ramp.
Processing
7
)()1( xfxf
x
f




Sharpening Spatial Filters (Cont…)Image Strip
0
1
2
3
4
5
6
7
8
1st Derivative
-8
-6
-4
-2
0
2
4
6
8
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 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0
8
Processing

• The formula for the 2nd derivative of a function is as follows:
• Simply takes into account the values both before and after the
current value.
• It should satisfy:
– Must be zero in the areas of constant intensity.
– Must be nonzero at the onset and end of an intensity
step or ramp.
– Must be zero along the ramp of constant slope.
Processing
9
)(2)1()1(2
2
xfxfxf
x
f




Sharpening Spatial Filters (Cont…)Image Strip
0
1
2
3
4
5
6
7
8
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
2nd Derivative
-15
-10
-5
0
5
10
-1 0 0 0 0 1 0 6 -12 6 0 0 1 1 -4 1 1 0 0 7 -7 0 0
Zero crossing
Zero crossing
property is
quite useful
for locating
edges
10
Processing

• The 2nd derivative is more useful for image
enhancement than the 1st derivative because;
– Stronger response to fine detail
– Simpler implementation
Processing
11

The Laplacian Operation
• In this section, we discuss 2D, second order
derivatives used for image sharpening.
• It is an isotropic filter
– It means that the response is independent of the
direction of the discontinuities in the image to
which the filter is applied. / rotation invariant.
Processing
12

The Laplacian Operation (Cont…)
The Laplacian is defined as follows:
where the partial 2nd order derivative in the x direction
is defined as follows:
and in the y direction as follows:
y
f
x
f
f 2
2
2
2
2






),(2),1(),1(2
2
yxfyxfyxf
x
f



),(2)1,()1,(2
2
yxfyxfyxf
y
f



13
Processing

So, the Laplacian can be given as follows:
We can easily build a filter based on this
),1(),1([2
yxfyxff 
)]1,()1,(  yxfyxf
),(4 yxf
0 1 0
1 -4 1
0 1 0
This provides an isotropic
results for rotations in
increments of 90 degrees
0 -1 0
-1 4 -1
0 -1 0
14
Processing

• We can incorporate diagonal directions to
form a laplacian filter;
Processing
15
),1(),1([2
yxfyxff  )]1,()1,(  yxfyxf
),(8 yxf
)]1,1()1,1(  yxfyxf)1,1()1,1([  yxfyxf
1 1 1
1 -8 1
1 1 1
-1 -1 -1
-1 8 -1
-1 -1 -1
This provides an isotropic
results for rotations in
increments of 45 degrees

Applying the Laplacian to an image we get a new image
that highlights edges and other discontinuities
Original
Image
Laplacian
Filtered Image
Laplacian
Filtered Image
Scaled for Display
16
Processing

The result of a Laplacian filtering is
not 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 sharpened enhanced image Laplacian
Filtered Image
Scaled for Display
fyxfyxg 2
),(),( 
17
Processing

Laplacian Image Enhancement: In the final
sharpened image edges and fine detail are much
more obvious
- =
Original
Image
Laplacian
Filtered Image
Sharpened
Image
18
Processing

19
Processing

Image Gradient & Properties
• First derivatives in image processing are implemented using
the magnitude of the gradient.
• Gradient is a tool of choice for finding edge strength and
direction at location (x, y) of an image f and it is defined as a
vector.
• For a function f(x, y) the gradient of f at coordinates (x, y) is
given as the column vector:
Processing
20























y
f
x
f
G
G
y
x
f
This vector has important
geometrical property that it
points in the direction of the
greatest rate of change of f at
location (x, y).

Image Gradient & Properties (Cont…)
• Gradient Operators:
– Obtaining the gradient of an image requires
computing the partial derivatives x and y at every
pixel location in the image.
– Therefore, digital approximation of the partial
derivatives over a neighborhood about a point is
required.
Processing
21
),()1,(
),(
),(),1(
),(
yxfyxf
y
yxf
G
yxfyxf
x
yxf
G
y
x









• There are various gradient operators available
namely; ID mask, Robert Cross, Prewitt, Sobel
• I-D Mask:
– Above equations can be implemented for all
pertinent values of x and y by filtering f(x,y) with
the 1-D masks.
Processing
22
-1
1
-1 1

• Roberts Cross-Gradient Operators:
• When diagonal edge direction is of interest, we need a 2
Dimensional mask.
• This operator cannot compute the edge direction
Processing
23
)(
),(
)(
),(
68
59
zz
y
yxf
G
zz
x
yxf
G
y
x








z1 z2 z3
z4 z5 z6
z7 z8 z9

• Prewitt Gradient Operators:
Processing
24
)()(
),(
)()(
),(
741963
321987
zzzzzz
y
yxf
G
zzzzzz
x
yxf
G
y
x







 z1 z2 z3
z4 z5 z6
z7 z8 z9

• Sobel Gradient Operators:
– Using 2 in the center location provides image
smoothing; hence, it has better noise suppression.
Processing
25
)2()2(
),(
)2()2(
),(
741963
321987
zzzzzz
y
yxf
G
zzzzzz
x
yxf
G
y
x








z1 z2 z3
z4 z5 z6
z7 z8 z9

Processing
26
Original Image Horizontal Gradient Component
Vertical Gradient Component Combined Edge Image

• Note that the coefficients of all the masks
sum to zero, thus giving a response of zero in
areas of constant intensity, as expected of a
derivative operator.
• To filter an image it is filtered using both
operators the results of which are added
together
• All the masks discussed above are used to
determine Gx, and Gy.
Processing
27

• Prewitt and Sobel masks give isotropic results
only for vertical and horizontal edges.
• It is possible to modify the 3*3 masks, so that
they have their strongest responses along the
diagonal directions as shown below;
Processing
28
0 1 1
-1 0 1
-1 -1 0
-1 -1 0
-1 0 1
0 1 1
Modified Prewitt
masks

How Image Gradient Works?
• How Sobel works?
– The Sobel edge detector is a gradient based
method.
– It works with first order derivatives.
– It calculates the first derivatives of the image
separately for the X and Y axes.
– The derivatives are only approximations (because
the images are not continuous).
Processing
29

How Image Gradient Works? (Cont…)
– To approximate them, the following kernels are used for
convolution:
– The kernel on the left approximates the derivative along the X
axis. The one on the right is for the Y axis.
– Using this information, we can calculate the following:
• Magnitude or "strength" of the edge:
• Approximate strength:
• The orientation of the edge:
Processing
30

The magnitude of this vector is given by:
For practical reasons this can be simplified as:
)f( magf
  2
1
22
yx GG 
2
1
22

























y
f
x
f
yx GGf 
31
Processing
It is the value of the rate
of change in the
direction of the gradient
vector.

• The direction of the gradient vector is given by
the angle:
• It is measured with respect to the x- axis.
• The direction of an edge at an arbitrary point
(x, y) is orthogonal to the direction, ᾳ(x, y) of
the gradient vector at the point.
Processing
32







x
y
g
g
yx nta),(

Processing
33
Figure contains zoomed
straight edge segment.
Each square shown here
corresponds to a pixel.
We want to find strength
and direction of the edge
at the point highlighted
with a box?
Example

• Method:
– We need to compute the derivatives of x and y
directions using a 3* 3 neighborhood.
– To get the partial derivatives in the x direction
subtract the pixels in the top row of the
neighborhood from the pixels in the bottom row.
– To get the partial derivatives in the y direction
subtract the pixels in the left column of the
neighborhood from the pixels in the right column.
– Suppose
Processing
34
2/
2/


yf
xf

• M(x,y) at that point
equals to 2+2
• Similarly, the direction
of gradient vector at
the same point equals
to:
• It is same as 1350
measured in positive
direction with respect
to the x-axis.
Processing
35
0
45tan),( 





 
x
y
G
G
yx
https://www.trigonometrytable.com/tan-inverse.php

• Edge at a point is
orthogonal to the gradient
vector at that point.
• So the direction angle of
the edge in this example is
• All edge point in the figure
have the same gradient,
so that entire edge
segment is in the same
direction.
Processing
36
00
4590 

Processing
37

Sobel filters are typically used for edge detection
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
38
Processing

Canny Edge Detection
• The Canny edge detector is a good approximation of the optimal
operator, i.e., the one that maximizes the product of signal-to-noise
ratio and localization.
• Basically, the Canny edge detector is the first derivative of a
Gaussian function.
• The algorithm runs in 5 separate steps:
– Smoothing: Blurring of the image to remove noise.
– Finding gradients: The edges should be marked where the gradients of
the image has large magnitudes.
– Non-maximum suppression: Only local maxima should be marked as
edges.
– Double thresholding: Potential edges are determined by thresholding.
– Edge tracking by hysteresis: Final edges are determined by suppressing
all edges that are not connected to a very certain (strong) edge.
Processing
39
Refer Note

Canny Edge Detection (Cont…)
Processing
40
Canny Edges

Canny Edge Detection (Cont…)
Processing
41
Original Image
Laplacian Edges
Sobel Edges
Canny Edges

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
42
Processing

Combining Spatial Enhancement
Methods
Successful image enhancement
is 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
43
Processing

Methods (Cont…)
Laplacian filter of
bone scan (a)
Sharpened version of
bone scan achieved by
subtracting (a) and (b)
Sobel filter of bone
scan (a)
(a)
(b)
(c)
(d) 44
Processing

Methods (Cont…)
The product of (c) and
(e) which will be used
as a mask
Sharpened image
which is sum of (a)
and (f)
Result of applying a
power-law trans. to
(g)
(e)
(f)
(g)
(h)
Image (d) smoothed with a
5*5 averaging filter
45
Processing

Methods (Cont…)
Compare the original and final images
46
Processing

Reference
• Chapter 03 of Gonzalez, R.C., Woods, R.E.,
Digital Image Processing, 3rd ed. Addison-
Wesley Pub.
Processing
47

Learning Outcomes Revisit
• Now, you should be able to;
– describe sharpening through spatial filters.
– Identify usage of derivatives in Image
Processing.
– discuss edge detection techniques.
– compare 1st & 2nd order derivatives used for
sharpening.
– Apply sharpening techniques for problem
solving.
Processing
48

QUESTIONS ?
Next Lecture – Morphological Image Processing
Processing
49

COM2304: Intensity Transformation and Spatial Filtering – III Spatial Filters for Sharpening

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