Gaussian higher Order Derivative based Structural Enhancement of Digital Bone X-Ray Images

8/7/2019 Gaussian higher Order Derivative based Structural Enhancement of Digital Bone X-Ray Images

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Gaussian Higher Order Derivative Based Structural

Enhancement of Digital Bone X-ray Images

a,1Raka Kundu,

b,2Ratnesh Kumar,

a,3Biswajit Biswas,

a,4Amlan Chakrabarti

aA. K. Choudhury School of Information Technology,

University of Calcutta, Kolkata - 700 009, India.1

[email protected], 3

[email protected], [email protected]

bNational Institute for The Orthopaedically Handicapped,

B. T. Road, Bon-Hoogly, Kolkata- 700 090, [email protected]

Abstract

A novel method for enhancement of digital X-ray

images of bones is presented in this paper. It has come

to observation that the proposed method based on the

Gaussian higher order derivative shows an appreciable

enhancement of edges in digital X-ray images of

bones that can be used for detection of various bone

deformities as well as for the better understanding of

the bone structure. We have achieved a level of

improvement in distinguishing the bone information

from the other parts of the digital X-ray images.

Keywords: Gaussian function, higher order derivative

operator, digital X-ray image, image enhancement.

1. Introduction

Digital X-ray images are the common form of

electromagnetic radiation image. X-ray images [1]

play a significant role in medical images for the

diagnosis of diseases and deformities of bone in

human body. Edges are the basic features of a given

image and they are the regions of rapid intensitychange i.e. high frequency regions in the spatial

domain. Edges characterize the boundaries, thus

preserving the important structural properties of an

image. Therefore enhancing high-frequency

components of the image can sharpen edges

effectively.

Due to lack of sharpness, digital X-ray images

sometimes do not hold enough information for

medical diagnosis. Here, we have undergone a process

of image enhancement of bone structures in digital X-

ray images. The basic common method of sharpening

[1], [2] is addition or subtraction of Laplacian image

[1] to the original input image. Therefore a study was

carried out on the third order Gaussian operator for

increasing the sharpness of the digital bone X-ray

image. The higher order Gaussian operators are easyto realize but, are very sensitive to noise. So, prior to

application of the higher order Gaussian operator there

is need of smoothing the image by noise removal

filter. Here, in this paper we mainly focus our research

on the derivation of Gaussian higher order derivative

operator and its use in highlighting the regions of

bones of digital X-ray image by detection of

meaningful edges of the image.

The organization of this paper is as follows. Section II

contains the proposed method of our paper work. Here

we have discussed the algorithms. Section IIIcompares and illustrates the results. Concluding

remarks are in Section IV.

2. Methodology

2.1. Proposed Gaussian Operator Algorithm

The formulation of the proposed higher order

derivative is from Gaussian function [3]. The one-

ISSN : 2229-60

Raka Kundu,Ratnesh Kumar,biswajit Biswas,Amlan Chakrabarti, Int. J. Comp. Tech. Appl., Vol 2 (1), 142-146

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dimensional Gaussian function is given by:

2

2

2

)(

2exp*

2

1)(

x

x D (1)

μ = mean, with μ = 0 , σ 2= variance.

2

2

2

2exp*

2

1)(

x

x D

(2)

The two dimensional Gaussian function is the product

of two such Gaussians function one in each dimension.

The two dimensional Gaussian function is:

2

22

2

)(

2exp*

2

1),(

y x

y x D

(3)

For simplicity we drop2

2

1

.

2

22

2

)(

exp),(

y x

y x D

(4)

Now, x

y x D y x D x

)),((),('

2

22

2

)(

2

' exp*),(

y x

x

x y x D

(5)

x

y x D y x D x

x

),((),(

'"

2

22

2

)(

4

22" exp*),(

y x

x

x y x D

(6)

x

y x D y x D x

x

)),((),(

"

'"

2

22

2

)(

6

32'" exp*

3),(

y x

x

x x y x D

(7)

Similarly,

y

y x D y x D

y

y

)),((),(

"

'"

2

22

2

)(

6

32'" exp*

3),(

y x

y

y y y x D

(8)

So,

),(),(),('" '"'" y x D y x D y x D y x

2

22

2

)(

6

3232

exp*33

y x y y x x

(9)

),('"

y x D , the Gaussian third order operator is

convolved with input image to obtain the third order

Gaussian image. Figure 1 and Figure 2 show the input

image and Gaussian third order image respectively.

Figure 1: Original Image

Figure 2: Results of Gaussian operator applied on

Figure 1

Figure 3 is the plot of 1D Gaussian derivative function

for order of 3, it shows the variation of the function

w.r.t. x. It is a zero crossing operator whose

polynomial is (3x σ 2

- x3) ∕ σ

6. This polynomial is

also known as Hermit polynomial [4].

Figure 3 : 1D plot of the 3rd order Gaussian operator

The 3x3 convolution mask of equation (9) with the

adjustment of variance (σ 2) from 0.25 to 0.3 gives the




desirable result for the enhancement of bone structure

of digital X-ray image.

2.2. Image enhancement

The 3x3 window obtained from third order Gaussian

derivative is next used for digital X-ray image

sharpening. Considering the centre pixel of the

obtained window as positive, we add the third order

Gaussian image with the original digital X-ray image.

Resultant image obtained after addition is the edge

sharpened image.

Let,

)],([),(),( 3 y x I y x I y xS (10)

Where, I(x,y) and S(x,y) are the input image and

structural enhanced image respectively. ),(3 y x I is

the third order Gaussian image. Figure4 describes the

proposed method.

),( y x I ),(3

y x I ),( y xS

Figure4: Procedure for the generation of the enhanced

image

The resultant image is significantly sharper than the

original input image and is able to confer the structure

of the bone of the digital image.

3. Results and Discussions

Let us demonstrate the structural enhancement of the

digital X-ray image from our experimental results.

Figure 5 and Figure 6 illustrate the mesh plot of the

input digital X-ray image (Figure 9) and the enhanced

digital X-ray image (Figure 10). If we compare thevalues of Figure 6 with Figure 5 we see that the Figure

6 have much higher peaks for the edges whereas the

non-edges remains almost same as compared to the

random distribution in Figure 5. Thus it can be

visualized from the results that the edge regions of the

bone of the processed image is much sharper than the

original image. This gives a clear view of the structure

of the bone.

Figure 5: Mesh plot of the test image (Figure 9)

Figure 6: Mesh plot of the enhanced image (Figure 10)

The contour plot of the original image and enhanced

image gives more clear idea of the edge enhancement.

The result in Figure 8 shows that the contour of the

bones has been well detected in the enhanced image

compared to the initial input as in Figure 7. This

proves that our algorithm is efficient and serves the

purpose of edge enhancement.




Figure 7: Contour plot of test image (Figure 9)

Figure 8: Contour plot of enhanced image (Figure 10)

A visual assessment of the prior and post processed

digital X-ray images are carried out for comparison.

Where, Figure 9, Figure 11 are the input digital images

and Figure 10, Figure 12 are the higher order Gaussian

applied enhanced images. Figure 10 and Figure 12clearly shows the bone region and its structural

formation which cannot be so clearly seen in the

original digital X-ray image of Figure 9 and Figure 11.

The bones from the soft tissues are easily

distinguishable. The new image formed by addition of

the higher order Gaussian image provides us more

information than the normal digital X-ray image. The

results displayed here are encouraging evidence that

the new process will be helpful in yielding better

informative X-ray images.

Figure 9: Original image

Figure 10: Enhanced image

Figure 11: Original image




Figure 12: Enhanced image

4. Conclusion

This work proposes an efficient technique of

generating enhanced images from the given digital

bone X-ray images. The results show that this is also a

useful technique for identifying contours in bone

images which has a numerous applications in

understanding and diagnosing bone deformities. Our

future work in this line will be segmentation of bone

information from the bone X-ray images and

characterizing the amount of deformities through

quantitative means.

References

[1] Gonzalez, R.C., R.E. Woods, Digital Image Processing,

Publisher – Pearson Education.

[2]Gonzalez, R.C, R.E. Woods, Digital Image Processing

Using Matlab, Publisher – Pearson Education, 2008.

[3]Spock, Gaussian derivatives, Shape and algebraic

structure, pp.53-57.

[4]W.T. Freeman, E.H. Adelson, “The Design and Use of Steerable Filters”, TRANSACTIONS ON PATTERN

ANALYSIS AND MACHINE INTELLIGENCE, Vol.13,

No.9, September 1991, pp. 891-906.

[5]M. Juneja, P.S. Sandhu, “Performance Evaluation of

Edge Detection Techniques for Image in Spatial Domain”,

International Journal of Computer Theory and Engineering,

2009, pp. 614-621.

[6]S. Lakshmi, V. Sankaranarayan, “A Novel Approach for

Edge Detection”, International Journal of Computer Science

and Network Security, Vol.10, No.4, April 2010, pp. 278-

282.

[7]R. Maini, H. Aggarwal, “Study and Comparison of

Various Image Edge Detection Techniques”, International

Journal of Image Processing (IJIP), Volume (3): Issue (1),

pp. 1-12.

[8]S. Sarkar, K. Ghosh, K. Bhaumik, “A Bio-inspired

Interpolation Kernel for Medical Image Processing

Implemented on DSP Processor”, 15th International

Conference on Advanced Computing and Communications,

2007, pp. 159-163.

[9]M.S. Khalid, M.U. Ilyas, M.S. Sarfaraz, M.A. Ajaz,“Bhattacharyya Coefficient in Correlation of Gray-Scale

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April 2006, pp. 56-61.


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Gaussian higher Order Derivative based Structural Enhancement of Digital Bone X-Ray Images