1 Lih-Jen Kau Multimedia Communication Group National Taipei Univ. of Technology 1/90 Digital Image Processing Chapter 03 – Part II Spatial Filtering Instructor:Lih-Jen Kau(高立人) Department of Electronic Engineering National Taipei University of Technology Dec. 02, 2010 2010/12/02 2/90 Lih-Jen Kau Multimedia Communication Group National Taipei Univ. of Technology An Introduction: Several basic concepts underlying the use of spatial filters for image processing are to be introduced. Spatial filtering is one of the principal tools used in this filed for a broad spectrum of applications. The name “filter” is borrowed from frequency domain processing, which is the topic of Chapter4. Here, “filtering” refers to accepting(passing)or rejecting certain frequency components. For example, a filter that passes low frequencies is called a lowpass filter. The net effect produced by a lowpass filter is to blur(smooth) an image. Fundamentals of Spatial Filtering(1/2)
Microsoft PowerPoint - Chapter03 Part II.pptDigital Image
Processing Chapter 03 – Part II
Spatial Filtering
Dec. 02, 2010
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
An Introduction: Several basic concepts underlying the use of
spatial filters for image processing are to be introduced. Spatial
filtering is one of the principal tools used in this filed for a
broad spectrum of applications. The name “filter” is borrowed from
frequency domain processing, which is the topic of Chapter4.
Here, “filtering” refers to acceptingpassingor rejecting certain
frequency components. For example, a filter that passes low
frequencies is called a lowpass filter. The net effect produced by
a lowpass filter is to blursmooth an image.
Fundamentals of Spatial Filtering1/2
2
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
An Introduction:cont. We can accomplish a similar smoothing
directly on the image itself by using spatial filters. Spatial
filters are usually referred as
Spatial masks kernels templates windows
In fact, as we will see in Chapter4, there is a one-to-one
correspondence between linear spatial filters and filters in the
frequency domain. However, spatial filters offer considerably more
versatility because, as you will see later, they can be used also
for nonlinear filtering, something we cannot do in the frequency
domain.
Fundamentals of Spatial Filtering2/2
2010/12/02 4/90
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
Filter Operation: Fig. 3.28 illustrates the mechanics of linear
spatial filtering using a 3x3 neighborhood. At any point (x, y) in
the image, the response , g(x, y), of the filter is the sum of
products of the filter coefficients and the image pixels
encompassed by the filter.Please see next page for details
The Mechanics of Spatial Filtering1/3
3
The Mechanics of Spatial Filtering2/3
Filter Operation:cont.
Observe that the center coefficient of the filter, w(0,0), aligns
with the pixel at location (x, y). For a mask of size mxn, we
assume that m=2a+1 and n=2b+1, where a and b are positive integers.
This imply that our focus in the following discussion is on filters
of odd size, with the smallest being of size 3x3.
)1,1()1,1(),()0,0( ),1()0,1()1,1()1,1(),(
The Mechanics of Spatial Filtering3/3
Filter Operation:cont. In general, linear spatial filtering of an
image of size MxN with a filter of size mxn is given by the
expression
∑∑ −= −=
An Introduction: Correlation and Convolution are two closely
related concepts that must be understood clearly when performing
linear spatial filtering. Correlation:
The process of moving a filter mask over the image and computing
the sum of products at each location, exactly as explained in the
previous section.
Convolution: The mechanics of convolution are the same, except that
the filter is first rotated by 180°.
The best way to explain the difference between the two concepts is
by example. Please see next pages for a 1-D illustration.
2010/12/02 8/90
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
Spatial Correlation and Convolution2/12
Correlation and Convolution: See Fig. 3.29 Fig. 3.29a shows a 1-D
function, f and a filter, w. Fig. 3.29b shows the starting position
to perform correlation. We notice that there are parts of the
functions that do not overlap.
The solution is to pad f with enough 0s on either side to allow
each pixel in w to visit every pixel in f. If the filter is of size
m, we need m-1 0s on either side of f. Fig. 3.29c shows a properly
padded function.
The first value of correlation is the sum of products of f and w
for the initial position shown in Fig. 3.29c, and the result is 0.
This corresponds to a displacement x=0. To obtain the second value
of correlation, we shift w one pixel location to the righta
displacement of x=1and compute the sum of products. The result
again is 0.
5
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
Spatial Correlation and Convolution3/12
Correlation and Convolution: See Fig. 3.29 cont. In fact, the first
nonzero result is when x=3, in which case the 8 in w overlaps the 1
in f and the result of correlation is 8. Proceeding in this manner,
we obtain the full correlation result in Fig. 3.29g. Note that it
took 12 values of xi.e., x=0, 1, 2,…, 11 to fully slide w past f so
that each pixel in w visited every pixel in f. Fig. 3.29h shows the
cropped correlation resultCrop two points on both side of the
result in Fig. 3.29g.
2010/12/02 10/90
6
Correlation and Convolution: See Fig. 3.29 cont.
Two important points to note from the discussion in the preceding
paragraph.
First, correlation is a function of displacement of the filter. In
other words, the first value of correlation corresponds to zero
displacement of the filter, the second corresponds to one unit
displacement, and so on. The second thing to notice is that
correlating a filter w with a function that contains all 0s and a
single 1 yields a result that is a copy of w, but rotated by
180°.
We call a function that contains a single 1 with the rest being 0s
a discrete unit impulse. So we conclude that correlation of a
function with a discrete unit impulse yields a rotated version of
the function at the location of the impulse.
2010/12/02 12/90
Correlation and Convolution: See Fig. 3.29 cont.
A fundamental property of convolution is that convolving a function
with a unit impulse yields a copy of the function at the location
of the impulse. We saw in the previous paragraph that correlation
yields a copy of the function also, but rotated by 180°.
Therefore, if we pre-rotate the filter and perform the same sliding
sum of products operation, we should be able to obtain the desired
result. As the right column in Fig. 3.29 shows, this indeed is the
case.
Thus, we see that to perform convolution all we do is rotate one
function by 180° and perform the same operations as in correlation.
As it turns out, it makes no difference which of the two functions
we rotate.
7
Correlation and Convolution: Two Dimensional Case Two Dimensional
Correlation: See Fig. 3.30
The preceding concepts extend easily to images. For a filter of
size mxn, we pad the image with a minimum of m-1 rows of 0s at the
top and bottom and n-1 columns of 0s on the left and right.
In this case, m and n are equal to 3, so we pad f with two rows of
0s above and below and two columns of 0s to the left and right, as
Fig. 3.30(b) shows. Figure 3.30(c) shows the initial position of
the filter mask for performing correlation, and Fig. 3.30( d) shows
the full correlation result. Figure 3.30( e) shows the
corresponding cropped result. Note again that the result is rotated
by 180°.
2010/12/02 14/90
8
Two Dimensional Convolution: Fig. 3.30 cont.
For convolution, we pre-rotate the mask as before and repeat the
sliding sum of products just explained. Figures 3.30(f) through (h)
show the result. We see again that convolution of a function with
an impulse copies the function at the location of the impulse. It
should be clear that, if the filter mask is symmetric, correlation
and convolution yield the same result. If, instead of containing a
single 1, image f in Fig. 3.30 had contained a region identically
equal to w, the value of the correlation function (after
normalization) would have been maximum when w was centered on that
region of f.
Thus, as you will see in Chapter 12, correlation can be used also
to find matches between images.
2010/12/02 16/90
Two Dimensional Correlation and Convolution: A Short
Conclusion
Summarizing the preceding discussion in equation form, the
correlation of a filter w(x,y) of size m x n with an image f(x,y),
denoted as w(x,y) f (x,y), is given by the equation below:
(3.4-1)
This equation is evaluated for all values of the displacement
variables x and y so that all elements of w visit every pixel in f
where we assume that f has been padded appropriately.
As explained earlier, a = (m - 1)/2, b = (n - 1)/2, and we assume
for notational convenience that m and n are odd integers.
),( t)(s, y) (x, ),( a
Two Dimensional Correlation and Convolution: A Short
Conclusioncont.
In a similar manner, the convolution of w(x,y) and f (x,y), denoted
by w(x,y) f (x,y), is given by the expression
(3.4-2)
where the minus signs on the right flip f (i.e., rotate it by
180°). Flipping and shifting f instead of w is done for notational
simplicity and also to follow convention. The result is the
same.
∑ ∑ −= −=
Two Dimensional Correlation and Convolution: A Short
Conclusioncont.
Using correlation or convolution to perform spatial filtering is a
matter of preference. All the linear spatial filtering results in
this chapter are based on Eq. (3.4-1). Finally, we point out that
you are likely to encounter the terms, convolution filter,
convolution mask or convolution kernel in the image processing
literature.
As a rule, the above terms are used to denote a spatial filter, and
not necessarily that the filter will be used for true convolution.
Similarly, “convolving a mask with an image" often is used to
denote the sliding, sum-of-products process we just explained, and
does not necessarily differentiate between correlation and
convolution. Rather, it is used generically to denote either of the
two operations.
10
Vector Representation of Linear Filtering 1/2
Vector Representation: When interest lies in the characteristic
response, R, of a mask either for correlation or convolution, it is
convenient sometimes to write the sum of products as
(3.4-3)
where the ws are the coefficients of an mxn filter and the zs are
the corresponding image intensities encompassed by the
filter.
It is implied that Eq. (3.4-3) holds for a particular pair of
coordinates (x, y).
mnmnzwzwzwR +++= ...2211
Vector Representation of Linear Filtering 2/2
Vector Representation: An Example Fig. 3.31 shows a general 3x3
mask with coefficients labeled as above. In this case, Eq. (3.4-3)
becomes
(3.4-4)
where w and z are 9-dimensional vectors formed from the
coefficients of the mask and the image intensities encompassed by
the mask, respectively.
992211 ... zwzwzwR +++=
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
Generating Spatial Filter Masks 1/2
Basic Concepts: Generating an mxn linear spatial filter requires
that we specify mn mask coefficients.
In turn, these coefficients are selected based on what the filter
is supposed to do
keeping in mind that all we can do with linear filtering is to
implement a sum of products.
2010/12/02 22/90
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
Generating Spatial Filter Masks 1/3
An Example: Suppose that we want to replace the pixels in an image
by the average intensity of a 3x3 neighborhood centered on those
pixels. The average value at any location (x, y) in the image is
the sum of the nine intensity values in the 3x3 neighborhood
centered on (x, y) divided by 9. Letting zi, i = 1,2,...,9, denote
these intensities, the average is
∑ =
Lih-Jen Kau Multimedia Communication Group
National Taipei Univ. of Technology
Generating Spatial Filter Masks 2/3
Continuous Functions: In some applications, we have a continuous
function of two variables, and the objective is to obtain a spatial
filter mask based on that function. For example, a Gaussian
function of two variables has the basic form
where σ is the standard deviation and, as usual, we assume that
coordinates x and y are integers. To generate, say, a 3x3 filter
mask from this function, we sample it about its center. Thus, w1 =
h(-1,-1), w2= h(-1,0),…, w9 = h(1, 1).
2
22
Continuous Functions: cont.
An mxn filter mask is generated in a similar manner. Recall that a
2-D Gaussian function has a bell shape, and that the standard
deviation controls the “tightness" of the bell.
13
Smoothing Spatial Filters 1/16
Basic Concepts: Smoothing filters are used for blurring and for
noise reduction. Blurring is used in preprocessing tasks, such as
removal of small details from an image prior to (large) object
extraction, and bridging of small gaps in lines or curves. Noise
reduction can be accomplished by blurring with a linear filter and
also by nonlinear filtering.
2010/12/02 26/90
Smoothing Spatial Filters 2/16
Smoothing Linear Filters: The outputresponseof a smoothing, linear
spatial filter is simply the average of the pixels contained in the
neighborhood of the filter mask. These filters sometimes are called
averaging filters. As mentioned in the previous section, they also
are referred to a lowpass filters.
14
Smoothing Spatial Filters 3/16
Smoothing Linear Filters: cont.
How to do: The idea behind smoothing filters is straightforward. By
replacing the value of every pixel in an image by the average of
the intensity levels in the neighborhood defined by the filter
mask, this process results in an image with reduced “sharp"
transitions in intensities. Because random noise typically consists
of sharp transitions in intensity levels, the most obvious
application of smoothing is noise reduction. However, edges (which
almost always are desirable features of an image) also are
characterized by sharp intensity transitions, so averaging filters
have the undesirable side effect that they blur edges.
2010/12/02 28/90
Smoothing Spatial Filters 4/16
Smoothing Linear Filters: cont.
∑ =
19 1
i iZR
which is the average of the intensity levels of the pixels in the 3
x 3 neighborhood defined by the mask, as discussed earlier.
15
Smoothing Spatial Filters 5/16
Smoothing Linear Filters: cont.
Note that, instead of being 1/9, the coefficients of the filter are
all 1s. The idea here is that it is computationally more efficient
to have coefficients valued 1. At the end of the filtering process
the entire image is divided by 9. An mxn mask would have a
normalizing constant equal to 1/mn. A spatial averaging filter in
which all coefficients are equal sometimes is called a box
filter.
2010/12/02 30/90
Smoothing Spatial Filters 6/16
Smoothing Linear Filters: cont.
The second mask in Fig. 3.32 is a little more interesting. This
mask yields a so called weighted average, terminology used to
indicate that pixels are multiplied by different coefficients, thus
giving more importance (weight) to some pixels at the expense of
others. In the mask shown in Fig. 3.32(b) the pixel at the center
of the mask is multiplied by a higher value than any other, thus
giving this pixel more importance in the calculation of the
average.
16
Smoothing Spatial Filters 7/16
Smoothing Linear Filters: cont.
In practice, it is difficult in general to see differences between
images smoothed by using either of the masks in Fig. 3.32, or
similar arrangements, because the area spanned by these masks at
anyone location in an image is so small.
2010/12/02 32/90
Smoothing Spatial Filters 8/16
Smoothing Linear Filters: cont.
With reference to Eq. (3.4-1), the general implementation for
filtering an MxN image with a weighted averaging filter of size
mxnm and n oddis given by the expression
(3.5-1)
∑∑
∑∑
−= −=
−= −=
Smoothing Spatial Filters 9/16
An Example: Consider Fig. 3.33 All with a square averaging filter
mask
2010/12/02 34/90
Smoothing Spatial Filters 10/16
An Example: Consider Fig. 3.34, an image captured by Hubble Space
Telescope Fig. 3.34c is the result of using the thresholding
function with a threshold value equal to 25% of the highest
intensity in Fig. 3.34b.
18
Smoothing Spatial Filters 11/16
Order-StatisticsNonlinearFilters: Order-statistic filters are
nonlinear spatial filters whose response is based on ordering
(ranking) the pixels contained in the image area encompassed by the
filter, and then replacing the value of the center pixel with the
value determined by the ranking result. The best-known filter in
this category is the median filter, which, as its name implies,
replaces the value of a pixel by the median of the intensity values
in the neighborhood of that pixel. Please note that the original
value of the pixel is included in the computation of the
median.
2010/12/02 36/90
Smoothing Spatial Filters 12/16
Order-StatisticsNonlinearFilters: cont.
Median filters are quite popular because, for certain types of
random noise, they provide excellent noise-reduction capabilities,
with considerably less blurring than linear smoothing filters of
similar size. Median filters are particularly effective in the
presence of impulse noise, also called salt-and-pepper noise
because of its appearance as white and black dots superimposed on
an image.
19
Smoothing Spatial Filters 13/16
Order-StatisticsNonlinearFilters: cont.
The median, ξ, of a set of values is such that half the values in
the set are less than or equal to ξ, and half are greater than or
equal to ξ. In order to perform median filtering at a point in an
image, we first sort the values of the pixel in the neighborhood,
determine their median, and assign that value to the corresponding
pixel in the filtered image. For example, in a 3x3 neighborhood the
median is the 5th largest value, in a 5x5 neighborhood it is the
13th largest value, and so on.
2010/12/02 38/90
Smoothing Spatial Filters 14/16
Order-StatisticsNonlinearFilters: cont.
When several values in a neighborhood are the same, all equal
values are grouped.
For example, suppose that a 3x3 neighborhood has values (l0,20,
20,20,15, 20,20,25,100). These values are sorted as (10, 15, 20,
20,20,20, 20,25, 100), which results in a median of 20.
Thus, the principal function of median filters is to force points
with distinct intensity levels to be more like their neighbors. In
fact, isolated clusters of pixels that are light or dark with
respect to their neighbors, and whose area is less than m2/2
(one-half the filter area), are eliminated by an mxm median filter.
In this case “eliminated" means forced to the median intensity of
the neighbors. Larger clusters are affected considerably
less.
20
Smoothing Spatial Filters 15/16
An Example: Consider Fig. 3.35 Figure 3.35(a) shows an X-ray image
of a circuit board heavily corrupted by salt-and-pepper
noise.
2010/12/02 40/90
Smoothing Spatial Filters 16/16
An Example: To illustrate the point about the superiority of median
filtering over average filtering in situations such as this, we
show in Fig. 3.35b the result of processing the noisy image with a
3x3 neighborhood averaging mask, and in Fig. 3.35c the result of
using a 3x3 median filter. The averaging filter blurred the image
and its noise reduction performance was poor. The superiority in
all respects of median over average filtering in this case is quite
evident. In general, median filtering is much better suited than
averaging for the removal of salt-and-pepper noise.
21
Sharpening Spatial Filters 1/25
Sharpening Spatial Filters: Basic Concepts The principal objective
of sharpening is to highlight transitions in intensity. Uses of
image sharpening vary and include applications ranging from
electronic printing and medical imaging to industrial inspection
and autonomous guidance in military systems. In the last section,
we saw that image blurring could be accomplished in the spatial
domain by pixel averaging in a neighborhood. Because averaging is
analogous to integration, it is logical to conclude that sharpening
can be accomplished by spatial differentiation.
2010/12/02 42/90
Sharpening Spatial Filters 2/25
Basic Concepts: cont.
This, in fact, is the case, and the discussion in this section
deals with various ways of defining and implementing operators for
sharpening by digital differentiation. Fundamentally, 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. Thus, image differentiation enhances
edges and other discontinuities (such as noise) and deemphasizes
areas with slowly varying intensities.
22
Sharpening Spatial Filters 3/25
Foundation: The derivatives of a digital function are defined in
terms of differences. There are various ways to define these
differences. However, we require that any definition we use for a
first derivative
Must be zero in areas of constant intensity Must be nonzero at the
onset of an intensity step or ramp; and Must be nonzero along
ramps.
2010/12/02 44/90
Sharpening Spatial Filters 4/25
Foundation: cont.
Similarly, any definition of a second derivative Must be zero in
constant areas; Must be nonzero at the onset and end of an
intensity step or ramp; and Must be zero along ramps of constant
slope.
Because we are dealing with digital quantities whose values are
finite, the maximum possible intensity change also is finite, and
the shortest distance over which that change can occur is between
adjacent pixels. A basic definition of the first-order derivative
of a one- dimensional function f(x) is the difference
23
Sharpening Spatial Filters 5/25
Foundation: First-order Derivativecont.
A basic definition of the first-order derivative of a one-
dimensional function f(x) is the difference
(3.6-1)
We used a partial derivative here in order to keep the notation the
same as when we consider an image function of two variables,
f(x,y), at which time we will be dealing with partial derivatives
along the two spatial axes.
)()1( xfxf x f
Sharpening Spatial Filters 6/25
Foundation: Second-order Derivativecont.
We define the second-order derivative of f(x) as the
difference
(3.6-2)
It is easily verified that these two definitions satisfy the
conditions stated above. To illustrate this, and to examine the
similarities and differences between first- and second-order
derivatives of a digital function, consider the example in Fig.
3.36.See next pages please
)(2)1()1(2
2
Sharpening Spatial Filters 7/25
An Example: Figure 3.36bcenter of the figureshows a section of a
scan lineintensity profile. The values inside the small squares are
the intensity values in the scan line, which are plotted as black
dots above it in Fig. 3.36(a). In Fig. 3.36(a) , the dashed line in
connecting the dots is included to aid visualization. As the figure
shows, the scan line contains an intensity ramp, three sections of
constant intensity, and an intensity step. The circles indicate the
onset or end of intensity transitions. The first- and second-order
derivatives computed using the two preceding definitions are
included below the scan line in Fig. 3.36(b), and are plotted in
Fig. 3.36(c).
2010/12/02 48/90
Sharpening Spatial Filters 8/25
An Example:cont.
When computing the first derivative at a location x, we subtract
the value of the function at that location from the next point. So
this is a “look-ahead" operation. Similarly, to compute the second
derivative at x, we use the previous and the next points in the
computation. To avoid a situation in which the previous or next
points are outside the range of the scan line, we show derivative
computations in Fig. 3.36 from the second through the
penultimate2points in the sequence.i.e., 22
25
Sharpening Spatial Filters 9/25
An Example:cont.
Note that the sign of the second derivative changes at the onset
and end of a step or ramp. In act, we see in Fig. 3.36(c) that in a
step transition a line joining these two values crosses the
horizontal axis midway between the two extremes. This zero crossing
property is quite useful for locating edges, as you will see in
Chapter 10.
2010/12/02 50/90
Sharpening Spatial Filters 10/25
An Example:cont.
Edges in digital images often are ramp-like transitions in
intensity, in which case the first derivative of the image would
result in thick edges because the derivative is nonzero along a
ramp. On the other hand, the second derivative would produce a
double edge one pixel thick, separated by zeros. From this, we
conclude that the second derivative enhances fine detail much
better than the first derivative, a property that is ideally suited
for sharpening images. Also, as you will learn later in this
section, second derivatives are much easier to implement than first
derivates, so we focus our attention initially on second
derivatives.
26
Sharpening Spatial Filters 11/25
Sharpening Spatial Filters 12/25
The Laplacian: Using the Second Derivative for Image
Sharpening
In this part, we consider the implementation of 2-D, second- order
derivatives and their use for image sharpening. The approach
basically consists of defining a discrete formulation of the
second-order derivative and then constructing a filter mask based
on that formulation 1./ 2.. We are interested in isotropicfilters,
whose response is independent of the direction of the
discontinuities in the image to which the filter is applied.
isotropic filters are rotation-invariant, in the sense that
rotating the image and then applying the filter gives the same
result as applying the filter to the image first and then rotating
the result.
27
Sharpening Spatial Filters 13/25
The Laplacian:cont.
It can be shown that the simplest isotropic derivative operator is
the Laplacian, which, for an image f(x,y) of two variables, is
defined as
(3.6-3)
Because derivatives of any order are linear operations, the
Laplacian is a linear operator.
2
2
2
Sharpening Spatial Filters 14/25
The Laplacian:cont.
To express this equation in discrete form, we use the definition in
Eq.3.6-2, keeping in mind that we have to carry a second variable.
In the x-direction, we have
(3.6-4)
(3.6-5)
),(2),1(),1(2
2
Sharpening Spatial Filters 15/25
The Laplacian: Two Dimensional Laplacian Therefore, it follows from
the preceding three equations that the discrete Laplacian of two
variables is
(3.6-6)
),(4)1,()1,(),1(),1( ),(2
yxfyxfyxfyxfyxf yxf
Sharpening Spatial Filters 16/25
Two Dimensional Laplacian:cont.
This equation can be implemented using the filter mask in Fig.
3.37(a), which gives an isotropic result for rotations in
increments of 90°. The diagonal directions can be incorporated in
the definition of the digital Laplacian by adding two more terms to
Eq. (3.6-6), one for each of the two diagonal directions. The form
of each new term is the same as either Eq. (3.6-4) or (3.6-5), but
the coordinates are along the diagonals. Because each diagonal term
also contains a -2f(x, y) term, the total subtracted from the
difference terms now would be -8f(x,y).
29
Sharpening Spatial Filters 17/25
2010/12/02 58/90
Sharpening Spatial Filters 18/25
Two Dimensional Laplacian:cont.
Figure 3.37(b) shows the filter mask used to implement this new
definition. This mask yields isotropic results in increments of
45°. You are likely to see in practice the Laplacian masks in Figs.
3.37(c) and (d). They are obtained from definitions of the second
derivatives that are the negatives of the ones we used in Eqs.
(3.6-4) and (3.6-5). As such, they yield equivalent results, but
the difference in sign must be kept in mind when combining (by
addition or subtraction) a Laplacian-filtered image with another
image.
30
Sharpening Spatial Filters 19/25
Two Dimensional Laplacian:cont.
Because the Laplacian is a derivative operator, its use highlights
intensity discontinuities in an image and deemphasizes regions with
slowly varying intensity levels. This will tend to produce images
that have grayish edge lines and other discontinuities, all
superimposed on a dark, featureless background0 . Background
features can be “recovered” Preservedwhile still preserving the
sharpening effect of the Laplacian simply by adding the Laplacian
image to the original.
2010/12/02 60/90
Sharpening Spatial Filters 20/25
Two Dimensional Laplacian:cont.
As noted in the previous paragraph, it is important to keep in mind
which definition of the Laplacian is used. If the definition used
has a negative center coefficient, then we subtract, rather than
add, the Laplacian image to obtain a sharpened result. Thus, the
basic way in which we use the Laplacian for image sharpening
is
(3.6-7)
where f(x, y) and g(x, y) are the input and sharpened images,
respectively.
The constant c=-1 if the Laplacian filters in Fig. 3.37(a) or (b)
are used, and c=1 if either of the other two filters is used.
)],([),(),( 2 yxfcyxfyxg ∇+=
Sharpening Spatial Filters 21/25
Two Dimensional Laplacian: An Example Figure 3.38(a) shows a
slightly blurred image of the North Pole of the moon. Figure
3.38(b) shows the result of filtering this image with the Laplacian
mask in Fig. 3.37(a). Large sections of this image are black
because the Laplacian contains both positive and negative values,
and all negative values are clipped at 0 by the display.
2010/12/02 62/90
Sharpening Spatial Filters 22/25
Sharpening Spatial Filters 23/25
Two Dimensional Laplacian: An Examplecont.
A typical way to scale a Laplacian image is to add to it its
minimum value to bring the new minimum to zero and then scale the
result to the full [0, L-1] intensity range, as explained in Eqs.
(2.6-10) and (2.6-11).
In addition, you can also shift the value of 0 to 128 and then
perform scaling operation for nonzero pixels.
The image in Fig. 3.38(c) was scaled in this manner. Note that the
dominant features of the image are edges and sharp intensity
discontinuities. The background, previously black, is now gray due
to scaling. This grayish appearance is typical of Laplacian images
that have been scaled properly.
2010/12/02 64/90
Sharpening Spatial Filters 24/25
Two Dimensional Laplacian: An Examplecont.
Figure 3.38(d) shows the result obtained using Eq. (3.6-7) with
c=-1. The detail in this image is unmistakably clearer and sharper
than in the original image. Adding the original image to the
Laplacian restored the overall intensity variations in the image,
with the Laplacian increasing the contrast at the locations of
intensity discontinuities. The net result is an image in which
small details were enhanced and the background tonality was
reasonably preserved.
33
Sharpening Spatial Filters 25/25
Two Dimensional Laplacian: An Examplecont.
Finally, Fig. 38(e) shows the result of repeating the preceding
procedure with the filter in Fig. 3.37(b). Here, we note a
significant improvement in sharpness over Fig. 3.38(d). This is not
unexpected because using the filter in Fig. 3.37(b) provides
additional differentiation (sharpening) in the diagonal directions.
Results such as those in Figs. 3.38(d) and (e) have made the
Laplacian a tool of choice for sharpening digital images.
2010/12/02 66/90
Unsharp Masking and Highboost Filtering1/8
Unsharp Masking: Basic Concept A process that has been used for
many years by the printing and publishing industry to sharpen
images. The method consists of subtracting an unsharp (smoothed)
version of an image from the original image. This process, called
unsharp masking, consists of the following steps:
Blur the original image. Subtract the blurred image from the
original (the resulting difference is called the mask.) Add the
mask to the original.
34
Unsharp Masking and Highboost Filtering2/8
Unsharp Masking: Process Letting denote the blurred image, unsharp
masking is expressed in equation form as follows.
First we obtain the mask:
(3.6-8)
Then we add a weighted portion of the mask back to the original
image:
(3.6-9)
where we included a weight, kk0, for generality.
),( yxf
),(),(),( yxfyxfyxgmask −=
Unsharp Masking and Highboost Filtering3/8
Unsharp Masking:cont.
When k = 1, we have unsharp masking, as defined above. When k >
1, the process is referred to as highboost filtering. Choosing k
< 1 deemphasizes the contribution of the unsharp mask.
35
Unsharp Masking and Highboost Filtering4/8
Unsharp Masking:cont.
Figure 3.39 explains how unsharp masking works. The intensity
profile in Fig. 3.39(a) can be interpreted as a horizontal scan
line through a vertical edge that transitions from a dark to a
light region in an image. Figure 3.39(b) shows the result of
smoothing, superimposed on the original signal (shown dashed) for
reference. Figure 3.39(c) is the unsharp mask, obtained by
subtracting the blurred signal from the original. By comparing this
result with the section of Fig. 3.36(c) corresponding to the ramp
in Fig. 3.36(a), we note that the unsharp mask in Fig. 3.39(c) is
very similar to what we would obtain using a second-order
derivative.
2010/12/02 70/90
Unsharp Masking and Highboost Filtering5/8
Unsharp Masking:cont.
Figure 3.39(d) is the final sharpened result, obtained by adding
the mask to the original signal. The points at which a change of
slope in the intensity occurs in the signal are now emphasized
(sharpened). Observe that negative values were added to the
original. Thus, it is possible for the final result to have
negative intensities if the original image has any zero values or
if the value of k is chosen large enough to emphasize the peaks of
the mask to a level larger than the minimum value in the
original.
Negative values would cause a dark halo around edges, which, if k
is large enough, can produce objectionableresults.
36
Unsharp Masking and Highboost Filtering6/8
Unsharp Masking: An Illustration of unsharp masking and highboost
filtering.
2010/12/02 72/90
Unsharp Masking and Highboost Filtering7/8
Unsharp Masking: An Example Figure 3.40(a) shows a slightly blurred
image of white text on a dark gray background. Figure 3.40(b) was
obtained using a Gaussian smoothing filter see Section 3.4.4of size
5x5 with σ=3. Figure 3.40(c) is the unsharp mask, obtained using
Eq. (3.6-8). Figure 3.40(d) was obtained using unsharp masking with
k=1. This image, i.e., Fig. 3.40(d), is a slight improvement over
the original, but we can do better. Figure 3.40(e) shows the result
of using Eq. (3.6-9) with k=4.5, the largest possible value we
could use and still keep positive all the values in the final
result. The improvement in this image over the original is
significant.
37
Unsharp Masking and Highboost Filtering8/8
Unsharp Masking: An Examplecont.
See Figure 3.40
The Gradient1/17
The Gradient: Using First-Order Derivatives for (Nonlinear) Image
Sharpening cont.
First derivatives in image processing are implemented using the
magnitude of the gradient. For a function f(x, y), the gradient of
f at coordinates (x,y) is defined as the two-dimensional column
vector
(3.6-10)
The Gradient2/17
The Gradient:cont.
This vector has the important geometrical property that it points
in the direction of the greatest rate of change of f at location
(x, y). The magnitude (length) of vector f, denoted as M(x,y),
where
(3.6-11)
is the value at (x,y) of the rate of change in the direction of the
gradient vector.
Note that M(x,y) is an image of the same size as the original,
created when x and y are allowed to vary over all pixel locations
in f.
It is common practice to refer to this image as the gradient image
(or simply as the gradient when the meaning is clear).
22)(),( yx ggfmagyxM +=∇=
The Gradient3/17
The Gradient:cont.
Because the components of the gradient vector are derivatives, they
are linear operators. However, the magnitude of this vector is not
because of the squaring and square root operations. On the other
hand, the partial derivatives in Eq. (3.6-10) are not rotation
invariant (isotropic), but the magnitude of the gradient vector is.
In some implementations, it is more suitable computationally to
approximate the squares and square root operations by absolute
values:
(3.6-12) yx ggyxM +≈),(
The Gradient4/17
The Gradient:cont.
This expression still preserves the relative changes in intensity,
but the isotropic property is lost in general. However, as in the
case of the Laplacian, the isotropic properties of the discrete
gradient defined in the following paragraph are preserved only for
a limited number of rotational increments that depend on the filter
masks used to approximate the derivatives. As it turns out, the
most popular masks used to approximate the gradient are isotropic
at multiples of 90°. These results are independent of whether we
use Eq. (3.6-11) or (3.6-12), so nothing of significance is lost in
using the latter equation if we choose to do so.
2010/12/02 78/90
The Gradient5/17
The Gradient:cont.
As in the case of the Laplacian, we now define discrete
approximations to the preceding equations and from there formulate
the appropriate filter masks. In order to simplify the discussion
that follows, we will use the notation in Fig. 3.41(a) to denote
the intensities of image points in a 3x3 region. For example, the
center point, z5, denotes f(x,y) at an arbitrary location, (x,y);
z1 denotes f(x-1, y-1); and so on, using the notation introduced in
Fig. 3.28.
40
The Gradient6/17
The Gradient:cont.
The Gradient7/17
The Gradient:cont.
As indicated in Section 3.6.1, the simplest approximations to a
first-order derivative that satisfy the conditions stated in that
section are gx = (z8 – z5) and gy = (z6 – z5). Two other
definitions proposed by Roberts [1965] in the early development of
digital image processing use cross differences:
and (3.6-13) )( 59 zzg x −= )( 68 zzg y −=
41
The Gradient8/17
The Gradient:cont.
If we use Eqs. (3.6-11) and (3.6-13), we compute the gradient image
as
(3.6-14)
(3.6-15)
where it is understood that x and y vary over the dimensions of the
image in the manner described earlier.
The partial derivative terms needed in equation (3.6-13) can be
implemented using the two linear filter masks in Figs. 3.41(b) and
(c).
These masks are referred to as the Roberts cross-gradient
operators.
6859),( zzzzyxM −+−≈
2/12 68
The Gradient9/17
The Gradient:cont.
Masks of even sizes are awkward to implement because they do not
have a center of symmetry. The smallest filter masks in which we
are interested are of size 3x3. Approximations to gx and gy using a
3x3 neighborhood centered on z5 are as follows:
(3.6-16)
and
(3.6-17)
=
=
The Gradient10/17
The Gradient:cont.
These equations can be implemented using the masks in Figs. 3.41(d)
and (e). The difference between the third and first rows of the 3x3
image region implemented by the mask in Fig. 3.41(d) approximates
the partial derivative in the x-direction, and the difference
between the third and first columns in the other mask approximates
the derivative in the y-direction.
2010/12/02 84/90
The Gradient11/17
The Gradient:cont.
After computing the partial derivatives with these masks, we obtain
the magnitude of the gradient as before. For example, substituting
gx and gy into Eq. (3.6-12) yields
(3.6-18) ( ) ( ) ( ) ( )741963
The Gradient12/17
The Gradient:cont.
The masks in Figs. 3.41 (d) and (e) are called the Sobel operators.
The idea behind using a weight value of 2 in the center coefficient
is to achieve some smoothing by giving more importance to the
center point (we discuss this in more detail in Chapter 10). Note
that the coefficients in all the masks shown in Fig. 3.41 sum to 0,
indicating that they would give a response of 0 in an area of
constant intensity, as is expected of a derivative operator.
2010/12/02 86/90
The Gradient13/17
The Gradient:cont.
As mentioned earlier, the computations of gx and gy are linear
operations because they involve derivatives and, therefore, can be
implemented as a sum of products using the spatial masks in Fig.
3.41. The nonlinear aspect of sharpening with the gradient is the
computation of M(x, y) involving squaring and square roots, or the
use of absolute values, all of which are nonlinear operations.
These operations are performed after the linear process that yields
gx and gy.
44
The Gradient14/17
The Gradient:cont.
The gradient is used frequently in industrial inspection, either to
aid humans in the detection of defects or, what is more common, as
a preprocessing step in automated inspection. We will have more to
say about this in Chapters 10 and 11. However, it will be
instructive at this point to consider a simple example to show how
the gradient can be used to enhance defects and eliminate slowly
changing background features. In this example, enhancement is used
as a preprocessing step for automated inspection, rather than for
human analysis.
2010/12/02 88/90
The Gradient15/17
The Gradient:cont.
Figure 3.42(a) shows an optical image of a contact lens ,
illuminated by a lighting arrangement designed to highlight
imperfections, such as the two edge defects in the lens boundary
seen at 4 and 5 o'clock. Figure 3.42(b) shows the gradient obtained
using Eq. (3.6-12) with the two Sobel masks in Figs. 3.41(d) and
(e). The edge defects also are quite visible in this image, but
with the added advantage that constant or slowly varying shades of
gray have been eliminated, thus simplifying considerably the
computational task required for automated inspection.
45
The Gradient16/17
The Gradient:cont.
The gradient can be used also to highlight small specs that may not
be readilyvisible in a gray-scale image (specs like these can be
foreign matter , air pocketsin a supporting solution, or
minisculeimperfectionsin the lens). The ability to enhance small
discontinuities in an otherwise flat gray field is another
important feature of the gradient.
2010/12/02 90/90
The Gradient17/17
<< /ASCII85EncodePages false /AllowTransparency false
/AutoPositionEPSFiles true /AutoRotatePages /All /Binding /Left
/CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1)
/CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile
(sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Warning
/CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true
/ConvertImagesToIndexed true /PassThroughJPEGImages true
/CreateJDFFile false /CreateJobTicket false /DefaultRenderingIntent
/Default /DetectBlends true /DetectCurves 0.0000
/ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false
/EmbedAllFonts true /EmbedOpenType false
/ParseICCProfilesInComments true /EmbedJobOptions true
/DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1
/ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100
/Optimize true /OPM 1 /ParseDSCComments true
/ParseDSCCommentsForDocInfo true /PreserveCopyPage true
/PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness
true /PreserveHalftoneInfo false /PreserveOPIComments false
/PreserveOverprintSettings true /StartPage 1 /SubsetFonts true
/TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue
false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [
true ] /AntiAliasColorImages false /CropColorImages true
/ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK
/DownsampleColorImages true /ColorImageDownsampleType /Bicubic
/ColorImageResolution 300 /ColorImageDepth -1
/ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold
1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode
/AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG
/ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1]
/VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1] >>
/JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256
/Quality 30 >> /JPEG2000ColorImageDict << /TileWidth
256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false
/CropGrayImages true /GrayImageMinResolution 300
/GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true
/GrayImageDownsampleType /Bicubic /GrayImageResolution 300
/GrayImageDepth -1 /GrayImageMinDownsampleDepth 2
/GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true
/GrayImageFilter /DCTEncode /AutoFilterGrayImages true
/GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict <<
/QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >>
/GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples
[1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth
256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict
<< /TileWidth 256 /TileHeight 256 /Quality 30 >>
/AntiAliasMonoImages false /CropMonoImages true
/MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK
/DownsampleMonoImages true /MonoImageDownsampleType /Bicubic
/MonoImageResolution 1200 /MonoImageDepth -1
/MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true
/MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1
>> /AllowPSXObjects false /CheckCompliance [ /None ]
/PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000
0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true
/PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ]
/PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier ()
/PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False
/Description << /CHS
<FEFF4f7f75288fd94e9b8bbe5b9a521b5efa7684002000500044004600206587686353ef901a8fc7684c976262535370673a548c002000700072006f006f00660065007200208fdb884c9ad88d2891cf62535370300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c676562535f00521b5efa768400200050004400460020658768633002>
/CHT
<FEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef653ef5728684c9762537088686a5f548c002000700072006f006f00660065007200204e0a73725f979ad854c18cea7684521753706548679c300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002>
/DAN
<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>
/DEU
<FEFF00560065007200770065006e00640065006e0020005300690065002000640069006500730065002000450069006e007300740065006c006c0075006e00670065006e0020007a0075006d002000450072007300740065006c006c0065006e00200076006f006e002000410064006f006200650020005000440046002d0044006f006b0075006d0065006e00740065006e002c00200076006f006e002000640065006e0065006e002000530069006500200068006f00630068007700650072007400690067006500200044007200750063006b006500200061007500660020004400650073006b0074006f0070002d0044007200750063006b00650072006e00200075006e0064002000500072006f006f0066002d00470065007200e400740065006e002000650072007a0065007500670065006e0020006d00f60063006800740065006e002e002000450072007300740065006c006c007400650020005000440046002d0044006f006b0075006d0065006e007400650020006b00f6006e006e0065006e0020006d006900740020004100630072006f00620061007400200075006e0064002000410064006f00620065002000520065006100640065007200200035002e00300020006f0064006500720020006800f600680065007200200067006500f600660066006e00650074002000770065007200640065006e002e>
/ESP
<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>
/FRA
<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>
/ITA
<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>
/JPN
<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>
/KOR
<FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020b370c2a4d06cd0d10020d504b9b0d1300020bc0f0020ad50c815ae30c5d0c11c0020ace0d488c9c8b85c0020c778c1c4d560002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken
voor kwaliteitsafdrukken op desktopprinters en proofers. De
gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe
Reader 5.0 en hoger.) /NOR
<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>
/PTB
<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>
/SUO
<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>
/SVE
<FEFF0041006e007600e4006e00640020006400650020006800e4007200200069006e0073007400e4006c006c006e0069006e006700610072006e00610020006f006d002000640075002000760069006c006c00200073006b006100700061002000410064006f006200650020005000440046002d0064006f006b0075006d0065006e00740020006600f600720020006b00760061006c00690074006500740073007500740073006b0072006900660074006500720020007000e5002000760061006e006c00690067006100200073006b0072006900760061007200650020006f006300680020006600f600720020006b006f007200720065006b007400750072002e002000200053006b006100700061006400650020005000440046002d0064006f006b0075006d0065006e00740020006b0061006e002000f600700070006e00610073002000690020004100630072006f0062006100740020006f00630068002000410064006f00620065002000520065006100640065007200200035002e00300020006f00630068002000730065006e006100720065002e>
/ENU (Use these settings to create Adobe PDF documents for quality
printing on desktop printers and proofers. Created PDF documents
can be opened with Acrobat and Adobe Reader 5.0 and later.)
>> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [
<< /AsReaderSpreads false /CropImagesToFrames true
/ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides
false /IncludeGuidesGrids false /IncludeNonPrinting false
/IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ]
/OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false
/SimulateOverprint /Legacy >> << /AddBleedMarks false
/AddColorBars false /AddCropMarks false /AddPageInfo false
/AddRegMarks false /ConvertColors /NoConversion
/DestinationProfileName () /DestinationProfileSelector /NA
/Downsample16BitImages true /FlattenerPreset <<
/PresetSelector /MediumResolution >> /FormElements false
/GenerateStructure true /IncludeBookmarks false /IncludeHyperlinks
false /IncludeInteractive false /IncludeLayers false
/IncludeProfiles true /MultimediaHandling /UseObjectSettings
/Namespace [ (Adobe) (CreativeSuite) (2.0) ]
/PDFXOutputIntentProfileSelector /NA /PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling
/LeaveUntagged /UseDocumentBleed false >> ] >>
setdistillerparams << /HWResolution [2400 2400] /PageSize
[612.000 792.000] >> setpagedevice