A Segmentation Algorithm for Quantum Remote Sensing Image Data
Bi, S. W.^{*} Rao, S. W.
State Key Laboratory of Remote Sensing Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100101, China
Abstract: Quantum remote sensing imaging is a hot topic in the field of the updated remote sensing image data processing. The remote sensing image segmentation is one of the most useful methods among them. Developing a segmentation algorithm in remote sensing image data processing can be based on the concept of quantum theory. The segmentation method proposed in this study is called Bi’s quantum segmentation method. The quantum genetic algorithm used in this method can automatically perform a multithreshold search on the remote sensing image to be segmented, and its search efficiency was verified through simulation experiments. Compared with the traditional method, the segmentation method proposed here can more effectively retain information from the original image, and significantly improve the accuracy and stability of multi threshold image segmentation results. Bi’s quantum segmentation method is a new technique that will facilitate growth in the segmentation processing of remote sensing image data.
Keywords: quantum remote sensing; image segmentation; quantum genetic algorithm; maximum entropy; optimal threshold
1 Introduction
Since Prof. Bi, Siwen proposed the concept of quantum remote sensing (QRS), they go through several study stages, including QRS theory^{[1]}, QRS information^{[2]}, QRS experiments^{[3]}, QRS imaging^{[4]}, QRS calculation^{[5]}, and QRS measurements. For the past ten years, the author and his nine graduate students have successfully studied the QRS image data denoising^{[6]} and enhancement^{[7]} algorithms, which has laid the foundation for the following studies on quantum image processing (such as edge extraction and image segmentation).
Remote sensing image data segmentation is a process in which remote sensing (RS) images are subdivided into uncrossed subregions where the information has the same characteristics, such as its grayscale values, textures, colors, and contrast features. Image segmentation technology can effectively simplify or change the expression of an image and enable the extraction of targets of interest from the image, making the image easier to be understood and analyzed. Image segmentation methods mainly include the threshold, edgebased, and regionbased methods^{[8]}. In this study, we used the threshold method to perform the image segmentation.
The threshold image segmentation method can be either a singlethreshold or multi threshold process. The former is used to determine an optimal threshold for image segmentation, and thus achieve the image binarization. This is simple and effective, particular effective for images with obvious bimodal gray histograms. However, most images, especially RS images, have complex textures and blurred boundaries with large amounts of information. Therefore, the required image targets are not clear. Such image histograms have multiple peaks and undulations. Under this condition, singlethreshold image segmentation can not reflect the image characteristics, and multithreshold segmentation is necessary for subsequent processing. The rapid and efficient determination of the optimal segmentation threshold is the key to the multithreshold image segmentation method.
Common threshold methods include the OTSU algorithm, fuzzy Kmeans algorithm, twopeak method, and Kmeans algorithms^{[8]}. In recent years, the image threshold method based on the maximum entropy and maximum fuzzy entropy has achieved better results than the traditional methods, and therefore has attracted much attention^{[9}^{–}^{10]}. Pun^{[11}^{–}^{12]} first proposed the image threshold segmentation method based on maximum entropy criterion. Kapur et al.^{[13]} further refined this approach. Subsequently, Cheng et al.^{[14] }combined the fuzzy mathematics theory with the maximum entropy and proposed the maximum fuzzy entropy image segmentation method. Based on this, Xu^{[8]} proposed an improved multithreshold image segmentation method and achieved a better effect. These image segmentation methods, based on maximum entropy criterion, often use the fullsearch method to determine the optimal multithreshold combination. Although this method can obtain a global optimal solution, its application is limited due to the complex programming, and timecosting and computing requirements.
The quantum genetic algorithm (QGA) is a quantum intelligent algorithm that combines quantum mechanics theory with the conventional genetic algorithm (CGA)^{[15–19]}. QGA introduces the state vector representation in submechanics into chromosome coding. In the evolution mechanism, a quantum revolving gate is used to realize chromosome evolution. The quantum properties help the QGA to obtain a better population diversity, faster convergence speed and stronger global search ability than traditional genetic algorithms. This new and improved algorithm has a far superior performance to traditional algorithms. Therefore, this study combined the maximum entropy criterion and QGA, to develop a multithreshold segmentation method based on the gray image of the QGA. Results showed that the proposed algorithm improved the reliability and stability of image segmentation and can obtain better segmentation results compared to traditional algorithms.
2 The Quantum Segmentation Algorithm
Both the QRS image denoising and QRS image enhancement use the pixel qubit form mentioned in references [20–21]. The general principle is that supposing g(m, n) is the original image, with a resolution of M´N, and f(m, n) is its normalized image, which can be represented as a twodimensional matrix of M´N:
(1)
The right side of the expression defines a digital image. Each element in the matrix is an image pixel. Image normalization transformation is generally:
(2)
where L_{max }and L_{min} represent the maximum and minimum gray values of the original image, respectively, and f(m, n)∈[0,1], represents the normalized gray value of pixel g(m, n). To realize the mapping of images from gray space to quantum space, the following pixel qubit expression was defined by referring to the probability statistics in reference [21]:
(3)
where and express the state of the black and white points, respectively. The probability amplitudes of their corresponding ground states areand which satisfy the normalized condition. and represent the appearance probability of white and black pixels, respectively.
When g(m, n) = 0.5×(L_{max}+L_{min}), the appearance probabilities of white and black states are the same, while white and black, and light and dark are relative concepts. Even for different images with the same maximum and minimum gray values, the distribution of gray histograms will not be exactly the same. Similarly, image segmentation also uses the principle of quantum superposition states, and its specific definition is as follows.
2.1 Quantum Chromosome Coding
In genetic algorithms, chromosomes are represented by certain values. Common coding methods include binary, decimal, and symbolic, etc. The chromosomes of the QGA are represented by random probability qubit encoding. A quantum chromosome with m qubit genes can be described as:
(4)
where α_{i}^{2}+β_{i}^{2}=1, the i^{th} qubit gene can be defined as [α_{i}, β_{i}], i = 1, 2, .., m by probability amplitude, and such an expression can express any quantum superposition. For example, a threequbit system (the first line means the probability amplitude of “0”, and the second means the probability amplitude of “1”):
(5)
Then its subsystem can be expressed as:
(6)
The above results show that the appearance probability of the four ground states , , , and in the system are 1/8, 1/8, 3/8 and 3/8, respectively, and therefore the quantum subsystem of Equation (6) can represent four kinds of state information simultaneously. A quantum chromosome no longer expresses a certain piece of information, but contains all of the possible information, and any operation on the gene in the chromosome will act on all possible states simultaneously. This ensures that the QGA has a better diversity than the CGA and the QGA population size is much smaller than that of the CGA for the same optimization problem. In addition, a better convergence can be obtained by using quantum chromosome coding. As ortends to 0 or 1, the qubitencoded chromosome will converge to a certain state.
2.2 Quantum Crossing
Crossing is a means that genetic algorithm mimics the genetic recombination of sexual reproduction in nature, enabling the inheritance of superior genes from the previous generation to new individuals and evolved with more complex genetic structures, which can avoid the precocity phenomenon to a certain extent. In CGA, crossing includes singlepoint crossing, multiplepoint crossing, and consistent crossing.
2.3 Quantum Variation
During biological genetics and evolution, some genes in DNA chromosomes may be mutated due to accidental or specific factors during cell division and replication. They will have new biological traits and produce new chromosomes. Mutation refers to the change of one or several genes in a chromosome coding sequence to generate new individuals. Quantum mutation is an effective means to improve the performance of the QGA in searching for an optimal solution. In quantum mechanics theory, the transformation between states is achieved through quantum gates. Common quantum gate transformation matrices include revolving gates, XOR gates, NOT gates, and Hadamard gates. The mutation of a quantum chromosome can also be realized through a quantum gate, using the information of optimal individual to guide chromosome evolution in the population and improve the convergence of the QGA. A common quantum rotation gate is defined as follows:
(7)
where q indicates the rotational variation angle of the quantum rotation gate, whose size and direction are based on a predesigned adjustment strategy. The quantum rotation variation operation is:
(8)
where [α β]′ is a qubit vector of the gene. The operation of quantum rotation mutation is to achieve the transition among states and get the algorithm converge.
2.4 Multithreshold Segmentation Algorithm Flow Using Maximum Entropy and the QGA
The process is as follows:
(1) Initialize evolutionary algebra at t=0 and population Q(t);
(2) Collect statistics regarding image data information entropy;
(3) Calculate the best fitness value of the function and the corresponding entropy to obtain the mean value;
(4) At t=t+1, update the population through quantum mutation and quantum rotation gates;
(5) If the mean entropy reaches maximum, output the best result, otherwise return to step 2).
2.5 Quantum Image Data Segmentation Algorithm Flow Chart
The flow chart of image data segmentation is shown in Figure 1, including initializing population, collecting image data information entropy, individual measurement, population regeneration, and output of the best individual.
Figure 1 Flow chart of image data segmentation using maximum entropy and the quantum genetic algorithm
3 Simulation Experiment and the Results of the Quantum Image Data Segmentation Algorithm
A simulation experiment was conducted on the algorithm using C++ and the openCV library in the Visual Studio 2013 integrated development environment. Segmentation image results (qualitative), data statistics (quantitative), and the average running time (each method was tested for 10 times) were compared for the OTSU (maximum interclass variance) algorithm, Kmeans clustering algorithm, maximum entropy threshold segmentation method and the quantum segmentation method. An image segmentation experiment was conducted on a computerized tomography (CT) scan of a patient’s pathological heart and an aerial image of an airport obtained by RS. The results are shown below.
The results showed that the quantum segmentation algorithm had a better segmentation effect than the traditional methods, such as the OTSU, Kmeans and maximum entropy threshold algorithms. Segmented images had a clearer texture and structure for both the CT of a patient’s pathological heart (Figure 2) and the aerial image of an airport (Figure 3). More importantly, as shown in Tables 1 and 2, the information entropy of the quantum segmentation algorithm was higher (>0.1) than that of the best traditional methods while the running time was almost the same, proving that the quantum method can obtain better segmentation results. Images segmented by this method retained more information from the original image than images segmented using the traditional methods.
(a) Original image (b) OTSU algorithm (c) Kmeans algorithm
(d) Maximum entropy threshold segmentation method

(e) Bi’s quantum segmentation method

Figure 2 Image results for four different segmentation algorithms
Table 1 Comparison of the segmentation results for the four different methods
Segmentation algorithm

Image information entropy after segmentation

Average running time (s)

OTSU algorithm

0.581,895

0.052,065

Kmeans algorithm

0.553,726

0.048,037

Maximum entropy threshold segmentation method

0.727,849

0.062,642

Bi’s quantum segmentation
method

0.921,277

0.832,738

4 Conclusion
Based on the basic principle of maximum entropy image segmentation, a multithreshold image segmentation method based on adaptive maximum entropy has been developed. To improve the multithreshold solving efficiency, this study applied QGA as an RS image segmentation method to determine the optimal threshold combination. This reduced the complexity of programming and improved the accuracy and robustness of multithreshold image segmentation results. To avoid generating invalid multithreshold solutions in the improved QGA, the study used an efficient coding method for binary value solutions^{[22]}, and then defined an adaptive rotation angle when using quantum rotation gates. The results showed that the multithreshold image segmentation method could better segment the target in an image than the traditional segmentation algorithm and more image information was retained for the same number of segmentation thresholds.
(a) Original image (b) OTSU algorithm (c) Kmeans algorithm
(d) Maximum entropy threshold segmentation method

(e) Bi’s quantum segmentation method

Figure 3 Image results for four different segmentation algorithms
Table 2 Comparison of the segmentation results for the four different methods
Segmentation algorithm

Image information entropy after segmentation

Average running time (s)

OTSU algorithm

0.793,13

0.066,112

Kmeans algorithm

0.989,53

0.042,504

Maximum entropy threshold segmentation method

1.234,58

0.056,347

Bi’s quantum segmentation method

1.518,22

0.827,539

5 Prospect
The Quantum remote sensing imaging is a hot topic in the field of remote sensing image data processing. The proposed Bi’s quantum segmentation method may enable the field of remote sensing image data segmentation to develop in new directions. In this study, the adaptive multithreshold search algorithm involved in segmenting remote sensing images improved the automatic search efficiency of the image segmentation multithreshold, and improved the accuracy and stability of multithreshold image segmentation results. In image processing, there are still many problems that need to be globally optimized, such as the determination of the optimal threshold in wavelet denoising and the determination of morphological structural elements. It may be possible to apply a QGA to a related image processing method. However, it should be noted that in the existing QGA, the selection of the number of thresholds and the number of quantum gene positions is artificially set, and other methods can be used to determine their optimal values. In addition, chromosome updates and mutations are usually implemented using quantum revolving gates. Other quantum gates in quantum computing can also be considered to use in evolutionary algorithms, which may further improve the overall optimization efficiency and performance of the algorithm.
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