THRESHOLD DECOMPOSITION DRIVEN ADAPTIVE

MORPHOLOGICAL FILTER FOR IMAGE SHARPENING

Tarek A. Mahmoud and Stephen Marshall

Department of Electronic and Electrical Engineering,

University of Strathclyde, 204 George Street, Glasgow, UK, G1 1XW

Keywords: Edge detection, image sharpening, morphological filter, threshold decomposition.

Abstract: A new method is proposed to sharpen digital images. This sharpening method is based on edge detection

and a class of morphological filtering. Motivated by the success of threshold decomposition, gradient-based

operators, such as Prewitt operators, are used to detect the locations of the edges. A morphological filter is

used to sharpen these detected edges. Experimental results demonstrate that the performance of these

detected edge deblurring filters is superior to that of the traditional sharpening filter family.

1 INTRODUCTION

The Chinese proverb “One picture is worth a

thousand words” underestimates the amount of

information contained in a single picture. Most

media (e.g. newspapers, TV, cinema) use pictures

(still or moving) as information carriers. The

tremendous volume of optical information and the

need for its processing and transmission paved the

way to image processing by digital computers (Pitas,

1993).

The past twenty years in particular were

characterized by a massive increase in the speed,

power and availability of digital computers.

Accordingly, one area of information technology

that has grown rapidly is imaging science. This

subject has become increasingly important because

of the growing demand to obtain information about

the structure, composition and behaviour of objects

without the need to inspect them visually

(Blackledge, 1989).

Image sharpening has played an important role in

image processing since the beginning of the digital

image revolution (Pratt, 1978). Thus, many well-

known techniques for image sharpening exist today

and are readily available in most commercial

software packages.

Edge detection is a fundamental tool used in

most image processing applications to obtain

information from the images and frames. This

process detects the boundaries of the objects and

separates them from the background of the image.

Digital image enhancement techniques are

concerned with the improvement of the quality of

the digital image. The principal objective of

enhancement techniques is to process an image so

that the result is more suitable than the original

image for a specific application. Image enhancement

is usually done simultaneously with detection of

features such as edges and peaks. Tools of linear

systems have been used to solve many of the image

enhancement applications. Nowadays, a new

understanding has emerged that linear approaches

are not well suited or even fail to solve problems

involving geometrical aspects of the image. Thus,

there is a need for nonlinear geometric approaches.

A powerful nonlinear methodology that can

successfully address the image sharpening problem

is mathematical morphology (Maragos, 2005).

In this paper, we propose a novel approach for

effectively sharpening blurred images. Our strategy

is that if we can detect the edges and locate their

position in the image, then we are able to increase

the contrast of these edges by applying a

morphological filter at these locations only.

Section 2 introduces the threshold decomposition

and the method used for edge detection.

Morphological filtering for image sharpening is

explained in Section 3. Section 4 will present in

detail the proposed sharpening filter. Then, this

proposed filter is tested on several examples and its

performance is compared with that of traditional

A. Mahmoud T. and Marshall S. (2007).

THRESHOLD DECOMPOSITION DRIVEN ADAPTIVE MORPHOLOGICAL FILTER FOR IMAGE SHARPENING.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 40-45

 SciTePress

sharpener-type filters. Finally, Section 5 contains

some concluding remarks.

2 BACKGROUND

2.1 Threshold Decomposition

Threshold decomposition is a powerful theoretical

tool used in image analysis. Introduced by Fitch in

(Fitch et al., 1984) and later modified by Arce in

(Arce, 1998) and Paredes in (Paredes and Arce,

1999). Consider an integer-valued set of samples

, X

, … , X

forming the vector X = [ X

, X

, … ,

]

where X

∈ {-M, … , -1, 0, 1, … , M}. The

threshold decomposition of X amounts to

decomposing this vector into 2M binary vectors

-M+1

, … , x

, …, x

, where the ith element of x

defined by

⎩

⎨

⎧

≥

<−

mXif1

)

(1)

where T

(.) is referred to as the threshold operator.

The above threshold decomposition is reversible,

such that if a set of threshold signals is given, each

of the samples in X can be exactly reconstructed as

∑

+−=

1Mm

(2)

Thus, an integer-valued discrete-time signal has a

unique threshold signal representation, and vice

versa.

2.2 Edge Detection

Edge detection is a fundamental tool used in most

image processing applications. This process detects

outlines of an object and boundaries between objects

and the background in the image.

An edge-detection filter can be used to improve

the appearance of blurred images or video streams.

The Prewitt operator (Prewitt, 1970), similar to the

Sobel, Robinson and some other operators

approximates the first derivatives of the image. They

are sometimes called compass operators because of

the ability to determine gradient direction. The

gradient is estimated in 4 (for a 3x3 mask) possible

directions with a difference of 90

between each

direction and the other. These 4 operators will be

represented by 4 (3x3 mask) respectively.

⎥

⎦

⎤

⎢

⎣

⎡

−−−

111

DDD

111

2101

⎥

⎦

⎤

⎢

⎣

⎡

−

1D1

⎥

⎦

⎤

⎢

⎣

⎡

−

111

DDD

111

2103

⎥

⎦

⎤

⎢

⎣

⎡

−

1D1

where g = { D

, D

} is the Structuring element

used in the mathematical morphology and will be

explained later.

By the aid of the threshold decomposition

described above, and for each level, the edges are

detected by searching for the 4 masks of the Prewitt

operators. Thus the sharpening filter is applied only

on these detected edges rather than all the pixels of

the image.

3 IMAGE SHARPENING BY

MORPHOLOGICAL

FILTERING

3.1 Introduction to Mathematical

Morphology

Kramer in (Kramer and Bruckner, 1975) defines a

non-linear transformation for sharpening digitized

gray-scale images. The transformation replaces the

gray value at a pixel by either the minimum or the

maximum of the gray values in its neighborhood, the

choice depending on which one is closer in value to

the original gray value.

In mathematical morphology (Serra, 1982), the

transformation that replaces the gray value at a pixel

by the maximum of the gray values in its

neighborhood is known as the gray-scale dilation

image operator.

⊕

[

]

)x(g)(f)x(

−+

∈

∨

(3)

in which function f (x), f : x

∈

→ f (x)

∈

R is the

original image, and g (x), g : x

∈ R

→ g (x)

∈

R is

THRESHOLD DECOMPOSITION DRIVEN ADAPTIVE MORPHOLOGICAL FILTER FOR IMAGE SHARPENING

the structuring element implicitly defining the

weighted neighborhood.

Similarly, the transformation that replaces the

gray value at a pixel by the minimum of the gray

values in its neighborhood is known as the gray-

scale erosion image operator

(f Θ g)

[]

)x(g)(f)x(

−−=

∈

∧

(4)

Note that the dilation operator is extensive:

(f ⊕ g) )x(f)x( ≥ and the erosion operator is

anti-extensive:

(f Θ g) )x(f)x( ≤ .

3.2 Contrast Enhancement

Consider a gray-scale image f (x) and a structuring

element g containing the origin. Kramer in (Kramer

and Bruckner, 1975), and then redefined by

Schavemaker in (Schavemaker et al., 2000), used the

following discrete nonlinear filter to enhance the

local contrast of f by sharpening its edges:

(f) [x] =

⊕

g)[x]

if f [x] >

((f

⊕

g)[x] + (f Θ g)[x])/2

(f Θ g)[x] (5)

if f [x] < ((f

⊕

g)[x] + (f Θ g)[x])/2

⎪

⎩

⎪

⎨

⎧

f [x] otherwise

At each pixel x, the output value of this filter toggles

between the value of the dilation of f by g (i.e., the

maximum of f inside the moving window g centred at

x) and the value of its erosion by g (i.e., the minimum

of f within the same window) according to which

closer to the input value f (x). If the value of the

dilation of f by g equals the value of its erosion by g,

the output value will be the same as the input.

3.3 Structuring Element Selection

In this paper, we will focus on two types of

structuring element in order to sharpen a gray-scale

image.

Firstly, Kramer in (Kramer and Bruckner, 1975)

uses flat structuring elements for sharpening

digitized gray-scale images.

Secondly, Van den Boomgaard in (Van den

Boomgaard et al., 1996) and Schavemaker in

(Schavemaker et al., 2000) have shown that all image

operators using concave structuring elements have

sharpening properties.

In this paper, it will be shown that these two

structuring elements have near sharpening

behaviour, and that the flat structuring elements are

slightly preferable to that of the parabolic concave

structuring ones.

4 THE PROPOSED SHARPENING

FILTER AND EXPERIMENTAL

RESULTS

The sequence of applying the proposed filter will be

explained before introducing the experimental

results. First, the blurred image is digitized by the

threshold decomposition method introduced in

section 2.1. On each level, we search for the 4

possible edge directions explained in section 2.2. We

have two types of structuring elements: the flat

structuring element and the parabolic concave

structuring element. After deciding the type of the

structuring element, the nonlinear discrete filter

introduced in section 3.2 is used to sharpen these

detected edges only, rather than the whole images.

This section presents application results for the

sharpening operator using flat and parabolic concave

structuring elements in the discrete domain. The

performance of the proposed filter is compared with

a number of sharpener-type filters including high-

pass sharpener, modified high-pass sharpener

(Fischer et al., 2002), the lower-upper-middle

(LUM) filter (Hardie and Boncelet, 1993), the

comparison and selection (CS) filter (Lee and Fam,

1987) and the unsharp masking technique.

The normalized mean square error (NMSE) is

used to give a quantitative evaluation on the filtering

results.

Our proposed filter is tested on some examples.

Figures (a) are the Gaussian blurred test images.

Figures (b), (c), (d), (e) and (f) show the sharpened

images after applying the high-pass sharpener,

modified high-pass sharpener, LUM sharpener, CS

sharpener and the unsharp masking respectively. As

mentioned above, there are two types of structuring

elements. Figures (g) show the sharpened images

after applying the proposed edge detected, parabolic

concave structuring element morphological filter,

while Figures (h) show the sharpened images after

applying the proposed, flat structuring element

morphological filter on the detected edges.

Table 1 and Table 2 show the NMSE as a

quantitative comparison between the above

mentioned sharpening techniques. The output of

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

each filter is evaluated by comparing its estimate to

the original image.

Table 1: NMSE for different sharpening-type filters.

Sharpener Filter

Lenna

Peppers

High-pass

sharpener

0.0132 0.0149

Modified high-pass

sharpener

0.0120 0.0138

LUM

sharpener

0.0106 0.0121

sharpener

0.0267 0.0317

Unsharp

masking

0.0127 0.0144

Proposed edge detected

concave structuring

element morphological

filter

0.0082 0.0070

Proposed edge detected

flat structuring element

morphological filter

0.0081 0.0069

Table 2: NMSE for different sharpening-type filters.

Sharpener Filter

Walk Bridge

Girl

High-pass

sharpener

0.0206 0.0137

Modified high-pass

sharpener

0.0187 0.0117

LUM

sharpener

0.0168 0.0113

sharpener

0.0388 0.0271

Unsharp

masking

0.0195 0.0129

Proposed edge detected

concave structuring

element morphological

filter

0.0143 0.0087

Proposed edge detected

flat structuring element

morphological filter

0.0141 0.0087

5 CONCLUSIONS

In this paper, a new image sharpening filter based on

morphological filters was presented. Edges are first

detected through threshold decomposition. Then, we

choose the type of the structuring element from the

flat or the parabolic concave structuring elements.

Both give good results, but the flat structuring

element was found to perform slightly better. Thus,

the threshold decomposition guided adaptive filters

have the ability to sharpen a blurred image.

Experimental results and associated statistics have

indicated that the proposed algorithm provides a

significant improvement over many other well-

known sharpener-type filters in the aspects of edge

and fine detail preservation, as well as minimal

signal distortion.

REFERENCES

Arce, G. R., 1998. A general weighted median filter

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Signal Processing 46 (12), 3195-3205.

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Fischer, M., Paredes, J. L., Arce, G. R., 2002. Weighted

median image sharpeners for the world wide web.

IEEE Trans. Image Processing 11 (7), 717-727.

Fitch, J. P., Coyle, E. J., Gallagher, N. C., 1984. Median

filtering by threshold decomposition. IEEE Trans.

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1188.

Hardie, R. C., Boncelet, C. G., 1993. LUM filters: A class

of rank-order-based filters for smoothing and

sharpening. IEEE Trans. Signal Processing 41 (3),

1061-1076.

Kramer, H. P., Bruckner, J. B., 1975. Iterations of a non-

linear transformation for enhancement of digital

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Lee, Y. H., Fam, A. T., 1987. An edge gradient enhancing

adaptive order statistic filter. IEEE Trans. Acoustic,

Speech, Signal Processing 35, 1061-1076.

Maragos, P., 2005. Morphological filtering for image

enhancement and feature detection, in: Bovik, A. C.

(Eds.), The Image and Video Processing Handbook,

Elsevier Academic Press. pp 135-156.

Paredes, J. L., Arce, G. R., 1999. Stack filter, stack

smoothers, and mirrored threshold decomposition.

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Prentice Hall.

Pratt, W. K., 1978. Digital Image Processing, Wiley. New

York.

Prewitt, J. M., 1970. Object enhancement and extraction.

Picture Processing and Psychopictorics, 75-149.

Schavemaker, J. G., Reinders, M. J., Gerbrands, J. J.,

Backer, E., 2000. Image sharpening by morphological

filtering. Pattern Recognition 33, 997–1012.

Serra, J., 1982. Image Analysis and Mathematical

Morphology, Academic Press. New York.

Van den Boomgaard, R., Dorst, L., Makram-Ebeid, S.,

Schavemaker, J. G., 1996. Quadratic structuring

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P., Schafer, R. W., Butt, M. A. (Eds.), Mathematical

Morphology and its Applications to Image and Signal

Processing, Kluwer Academic Publishers. pp. 147-

154.

THRESHOLD DECOMPOSITION DRIVEN ADAPTIVE MORPHOLOGICAL FILTER FOR IMAGE SHARPENING

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: (a) Blurred Lenna (b) High-pass sharpened

(e) CS sharpened (f) Unsharp mask sharpened.

(g)

(h)

Figure 1: (g) Proposed edge detected concave structuring

element morphological filter (h) Proposed edge detected

flat structuring element morphological filter.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: (a) Blurred Peppers (b) High-pass sharpened

(e) CS sharpened (f) Unsharp mask sharpened.

(g)

(h)

Figure 2: (g) Proposed edge detected concave structuring

element morphological filter (h) Proposed edge detected

flat structuring element morphological filter

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: (a) Blurred Rectangle (b) High-pass sharpened

(e) CS sharpened (f) Unsharp mask sharpened.

(g)

(h)

Figure 3: (g) Proposed edge detected concave structuring

element morphological filter (h) Proposed edge detected flat

structuring element morphological filter.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: (a) Blurred Girl (b) High-pass sharpened

(e) CS sharpened (f) Unsharp mask sharpened.

(g)

(h)

Figure 4: (g) Proposed edge detected concave structuring

element morphological filter (h) Proposed edge detected

flat structuring element morphological filter

THRESHOLD DECOMPOSITION DRIVEN ADAPTIVE MORPHOLOGICAL FILTER FOR IMAGE SHARPENING