A new quantum edge detection algorithm for medical images

Xiaowei Fu * a,b, Mingyue Dingc, a, Yangguang Suna , Shaobin Chena

a Institute for Pattern Recognition and Artificial Intelligence, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China; b College of Computer Science & Technology, Wuhan University of Science & Technology, Wuhan, Hubei 430065, China; c College of Life Science and Technology, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China.

ABSTRACT In this paper, a quantum edge detection algorithm was proposed for the blurry and complex characteristic of medical images with the elicitation of the basic concept and principle of quantum signal processing. Firstly, based on the pixel qubit and quantum states superposition concept, an image enhancement operator based on quantum probability statistic is presented which combines with gray correlative characteristic of the pixels in 3×3 neighborhood windows. Then, in order to realize edge detection, an edge measurement operator based on fuzzy entropy is adopted to the quantum enhancement image. Experiments showed that this method is more efficient than traditional edge detection methods because it has a better capability of edge detection to medical images, which can extract not only strong edge but also the weak one. Keywords: Quantum signal processing; Edge detection; Fuzzy entropy

1. INTRODUCTION Medical image processing is one of the important branches of computer vision and image processing. With the advances of medical imaging technology, medical images play an important role in disease diagnosis and therapy. Edge detection technology in medical images will greatly help physicians by providing statistic relevant information from the imaging data automatically[1]. There are many applications of edge detection technique in medical images such as outlining of tumors and/or organs, extraction of interested objects like needle or other surgery tools. Because of the complex imaging mechanism, medical images are often blurred with complex characteristics, which are usually of dark background, high noise and low contrast and so on. Therefore, the traditional edge detection methods have difficulty to extract the edges in medical images effectively.

Along with the development of quantum mechanics and information science, quantum signal processing is one of the newest research directions in signal processing domain. Digital image is an important extension of signal processing. The study of image processing based on quantum signal processing is currently the forefront of image processing. Eldar[2,3] brought forward the concepts and theories of quantum signal processing (Quantum Signal Processing, QSP) for the first time in 2001. QSP applies the math foundations of quantum mechanics to signal processing, which is aimed at developing new or modifying existing signal processing algorithms by borrowing from the principles of quantum *fxw_wh0409@wust.edu.cn; phone 13419623308;

MIPPR 2009: Medical Imaging, Parallel Processing of Images, and Optimization Techniques, edited by Jianguo Liu, Kunio Doi, Aaron Fenster, S. C. Chan, Proc. of SPIE Vol. 7497, 749724

© 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.832499

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mechanics and some of its interesting axioms and constraints. Then, Tseng and Hwang[4] proposed a quantum image edge detection algorithm which takes advantage of quantum superposition and quantum state collapse theories by means of random observation. However, the results of edge detection mainly depend on the edge extraction results by the Sobel operator. Li Ying and Jiao Li-cheng[5] proposed an edge detection algorithm based on a quantum evolutionary algorithm. The results depended on the cost function and its computational cost is very high, and easy to fall into the most superior in local areas. Xie Ke-fu[6] proposed a quantum-inspired mathematical morphology operator and applied it to detect images edge. Results showed that it can detect edge of the image corrupted by noise more efficiently than that traditional morphological gradient. However, it computational cost will exponential growth with the increase of the size of structure element in morphology processing. Lou and Ding[7] introduced the motion concept of quantum particles into boundary extraction, but the method was only estimated with a simple postulate and an initial point near the object boundary was needed. These researches provided us some feasible image processing methods based on the quantum theory, which opened up the thought to solve the images processing questions enormously. In this paper, a quantum edge detection algorithm was proposed for the blurry and complex characteristic of medical images with the elicitation of the basic concept and principle of quantum signal processing, which combines with gray correlative characteristic of pixels in 3×3 neighborhood windows. The results indicate that the proposed algorithm is more effective than the traditional methods, which has the better capability and compatibility of edge detection for medical images. The remainder of this paper is organized as follows: in Section 2, we briefly introduce the basic concept and principle of quantum signal processing relating to our algorithm. Section 3 presents our quantum edge detection algorithm. Section 4 shows experimental results. Conclusions and future work are discussed in Section 5.

2. BASICAL THEORY 2.1 Basic theory of QSP In classical information processing, the state 1 or 0 is used to represent information of a binary bit. In QSP[2-4], the bit is called qubit (Quantum Bit). The qubit is the basic memory unit in a quantum computer. The state of a qubit is a superstition of two quantum states |0> and |1>,i.e.,

0 1φ α β= + （1）

where α and β are probability amplitudes of states |0> and |1> , and |α|², |β|² are the measurement probability of state |0> and |1> respectively. They must satisfy the relation of |α|²+|β|²=1; |0> and |1> are the two ground states in one qubit system. In the quantum world, the state of micro particles is indefinite which may be in some state with different probability simultaneously. Once it has carried on the quantum measurement, it will fall into a definite state. In a n- qubit quantum

subsystem, let the state of the ith qubit be 0 1i i ia bφ = + . According to the principle of quantum states superposition,

then, the state of the n qubits quantum subsystem is the tensor product state of n qubits, which is defined as:

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2 1

1 2 1 2 1 2 1 1 20

... ... 0 0 ...0 ... 0 0 ...01 ... ... 1 1 ...1n

n n n n n i bi

a a a a a a b b b b w iψ φ φ φ−

−=

= ⊗ ⊗ ⊗ = + + = ∑ （2）

where the state vector | bi > is the ith ground state in the n-qubit system. bi is the n-bit binary number. iw is the probability amplitudes of the ith ground state; 2| |iw is the probability of the ith ground state. They must satisfy the

normalizing condition：

2 12

0

| | 1n

iiw

−

=

=∑ . （3）

In quantum mechanics, the wave function in Hilbert space is used to describe the kinetic states of micro particles, which manifests the wave-corpuscle duality principle of quantum. The quantum states exist in the whole ground states simultaneously. Similarly, owing to the completeness of a quantum system, an arbitrary quantum state obtained by the combination of ground states is still a vector in a Hilbert space. That is the famous superposition state principle. QSP is aimed at developing new or modifying existing signal processing algorithms by borrowing from the principles, axioms, and constraints of quantum mechanics. 2.2 Definition of a pixel qubit In order to map the pixel gray value in an image space to the quantum system space, the mathematics expression of a pixel qubit is needed with the elicitation of the basic principles of QSP. Let ( , )f m n be a normalized input image, ( , ) [0,1]f m n ∈ is the gray value of the (m, n) pixel. Under the viewpoint of the probability statistics, the (m, n) pixel qubit may be defined as:

( , ) 1 ( , ) 0 ( , ) 1f m n f m n f m n= − + （4）

where |0> and |1> are the two ground states in a single quantum system. They are the corresponding black and white states of the current pixel in a binary image.1 ( , )f m n− and ( , )f m n are the appearance probability of states |0> and |1>.

Apparently, they satisfy the normalizing condition.

3. PROPOSED METHOD 3.1 Image enhancement based on quantum probability statistics Owing to the highly complex imaging mechanics, the medical images are usually of the characteristics such as low contrast, containing noise, more dark, blurry and complex and so on. Theoretically, an ideal edge of medical images has the step change in gray level. However, practical edges are usually blurred, low contrast because of the impact of optics, sampling, and other imaging imperfections. Therefore, the real edges are more like a ramp profile. In order to detect edge effectively, image enhancement for medical images is needed to obtain high contrast and eliminate noise to some degree. In our method, an image enhancement operator based on the quantum probability statistics was proposed firstly, which combines with gray correlative characteristic of pixels in the 3×3 neighborhood window. There are eight ground states in a three qubits system, which are |000>, |001>, |010>, |011>, |100>, |101>, |110> and |111>. According to the three qubits direct product state equation in QSP, it can calculate the probability amplitudes of each ground state in the three qubits system. Our operator sums up the six ground states probabilities of the three qubits subsystems in the horizontal, vertical, 45° and 135° direction respectively, which can not only enhance the information of edges but also retain the information of smoothing areas in medical images. And then, the processed gray value is the maximum of the operator value in the four directions.

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As shown in Table 1, let 1P be the current (m, n) pixel and iP denotes the arbitrary pixel in the 3× 3 neighborhood of 1P .

iF is the normalized gray value of the corresponding pixel, [ ]0,1iF ∈ . In the horizontal direction of the 3 × 3 neighborhood window, the state vector in the three qubits subsystem 8 1 4| PPP > is computed by:

8 1 4 8 1 4 8 1 4 8 1 4 8 1 4| (1 )(1 )(1 ) | 000 (1 )(1 ) 001 (1 ) (1 ) 010 (1 ) 011PPP F F F F F F F F F F FF>= − − − >+ − − + − − + −

32 1

8 1 4 8 1 4 8 1 4 8 1 40

(1 )(1 ) 100 (1 ) 101 (1 ) 110 111 ii

F F F F F F F F F F F F w i−

=

+ =+ − − + − + − ∑ （5）

Tab1e. 1. Location relationship of pixels in the 3×3 neighborhood

where 2| |iw is the probability of the corresponding ground state in three qubits subsystem, composed of 8P 、 1P 、 4P

pixels. Image edges are usually of strong directivity and tight correlation with the neighboring pixels, but the noise is relative isolative in the 3×3 neighborhood. In the horizontal direction, our operator counts the probability of the | 0 *1> ,| |1* 0 > ground states, which have much more relation with the ramp-like edge information; On the other hand, it calculates the probability of the |1*1> ground states, which are related with the basic information in image smoothing

areas, where * denotes 0 or 1. In order to enhance the information in the horizontal direction, according to the probability statistics in the three qubits subsystem, the operator

0OPER ° is defined as:

20

( , ) { | {1,3,4,5,6,7}}iOPER m n SUM w i° = ∈ . （6）

Then the value 0

OPER ° is calculated by:

8 1 4 8 1 4 8 1 4 8 1 4 8 1 4 8 1 40( , ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 )OPER m n F F F F F F F F F F F F F F F F F F= − × − × + − × × + × − × − + × − × + × × − + × ×o

8 4 8 4F F F F= + − × . （7）

Similarly, the operators of the 1P pixel in the 45°, 90° and 135° directions are expressed as follows:

9 5 9 545*OPER F F F F= + −o （8）

6 2 6 290*OPER F F F F= + −o （9）

7 3 7 3135*OPER F F F F= + −o （10）

And then, the maximum of the four directional operators is considered as the output gray value ( , )GF m n of the 1P pixel after the processing, ( , ) [0,1]GF m n ∈ . That ( , )GF m n is enhanced gray value of current pixel, and is defined as:

( , ) { | {0 ,45 ,90 ,135 }}GF m n MAX OPERθ θ= ∈ o o o o . （11）

GF is the enhanced medical image after processing, which not only keeps the basic information of the smooth regions

but also enhances the edge details of medical images effectively. 3.2 Edge measurement operator based on fuzzy entropy The probability of ground states is much natural associated with the information entropy. In order to detect edge, an edge

7P 6P 5P

8P 1P 4P

9P 2P 3P

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measurement operator ( , )H i j based on fuzzy entropy is explored to the quantum enhanced image, where the membership function ( , )m i jμ is depended on the mean of the gray value ( , )m i j in the 3× 3 neighborhoods. The

definition of membership function is as follows: 1( , )

1 ( , ) ( , )m i j GF i j m i jμ =

+ − （12）

And 1 1

1 1

1( , ) ( , )9 k l

m i j F i k j l=− =−

= + +∑ ∑ （13）

where ( , )m i jμ expresses the membership grade of the quantum enhanced gray value ( , )GF i j belonging to the mean of the gray value ( , )m i j in the 3× 3 neighborhoods. The edge measurement operator is:

1 1

21 1

1( , ) ( , ) log ( , )9 m m

k lH i j i k j l i k j lμ μ

=− =−

−= + + + +∑ ∑ . （14）

Owing to the gray change characteristic of image edges, the ( , )H i j is much less if the center pixel ( , )i j is in the smoothing areas since the gray values of neighborhood pixels are more close to the ( , )m i j ; otherwise, it is much greater if the center pixel ( , )i j is at the image edges since the gray values of neighborhood pixels are much more different with

( , )m i j . The edges of medical images can be obtained effectively through the measurement operator ( , )H i j .

4. EXPERIMENTAL RESULTS

In this section, we present some experimental results about the quantum edge detection algorithm, which is compared with some traditional edge detection algorithms under the MATLAB programming environment. Fig.1 and Fig.2 are the test results of edge detection algorithms for a nuclear whole human body skeleton scan and a chest X-ray image

respectively. As shown in Fig.1，(a) is an original image of size 502× 803 with the narrow dynamic range of gray levels

and high noise content, which is difficult to detect edges effectively[1]; (b)-(d) are the correspondingly edge detection results by the Sobel, Prewitt as well as Laplacian operators of the original image respectively, where the mask of the Laplacian is a 3× 3 matrix with the center coefficient equal to -8 and the surrounding ones to 1. (e) is the effect of the proposed quantum edge detection algorithm.

(a) Original image (b) Sobel (c) Prewitt (d) Laplacian (e) Proposed method

Figure. 1. Effect comparison of edge detection for a nuclear whole body skeleton scan

Obviously, the proposed algorithm can obtain the best edge detection effect, which can not only extract the strong

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skeleton edge details such as the ribs, spinal chord, pelvis, and skull but also detect the significant weak edge details including those in the areas around the wrists, palms, ankles and feet. The edge details of the skeletal bone are more legible than that of the traditional edge detection algorithms because an image enhancement based on the quantum probability statistics is firstly taken into considered in the proposed method, which can effect enhance the edge details of the original image.

(a) Original image (b) Sobel (c) Prewitt

(d) Laplacian (e) fuzzy entropy (f) Proposed method

Figure. 2. Effect comparison of edge detection for a chest X-ray image

Fig.2(a) showed an original image of size 573× 340 with the characteristics of more dark and blurry, (b)-(d) showed the correspondingly edge detection results by the Sobel, Prewitt as well as Laplacian operators of the original image; (e) gave the result directly obtained by the edge measurement operator based on fuzzy entropy for the original image; (f) showed the effect of the proposed quantum edge detection algorithm. Fig.2(e) showed some false edges in vertebra areas distinctly and the result is worse than that of our proposed method. The edge details of bone structure in Fig. 2(f) are clearer than the other four images, which is based on our quantum edge detection algorithm.

5. CONCLUSION In this paper, a new quantum edge detection algorithm for medical images is proposed with the elicitation of the basic concept and principle of QSP. Experimental results show that the proposed method is more efficient than traditional edge detection methods because it has a better capability of edge detection to medical images, which can pick up not only strong edge but also weak edge. In addition, this algorithm provides an effective feasible method to edge detection for medical images, which combines the quantum theory with the image processing technology.

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This work is supported by the National Natural Science Foundation of China (60672057，60471012，60804031), and the

Open Foundation of National Key Laboratory of Pattern Recognition of Chinese Academy of Sciences.

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ACKNOWLEDGMENTS

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