2013 Annual IEEE India Conference (INDICON)
Quantum Inspired Meta-heuristic Algorithms for Multi-level Thresholding for True Colour Images Sandip Dey
Siddhartha Bhattacharyya
Ujjwal Maulik
Department of Information Technology Department of Information Technology Department of Computer Science Camellia Institute of Technology RCC Institute of Information Technology & Engineering, Jadavpur University Madhyamgram,Kolkata-700129, India Beliaghata, Kolkata-700015, India Kolkata-700032, India Email: dsandip
[email protected] Email:
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Abstract—In this article, the particle swarm optimization and differential evolution algorithms inspired by the intrinsic principles of quantum mechanics are presented. These quantum versions of meta-heuristic algorithms, namely quantum inspired particle swarm optimization and quantum inspired differential evolution for multi-level thresholding have been designed to find optimal thresholds of colour images at different levels by exploiting Kapur’s entropy as an objective function. The average fitness and the standard deviation of the fitness values are reported. The test results over two test images at different levels certify the efficacy of the proposed methods with reference to precision, computational time, and durability over their classical counterparts. At last, a statistical measure, t-test has been performed among the four methods (two quantum methods and two classical methods) taking two methods in a single grasp to ascertain the supremacy of the results. Keywords—image segmentation, multilevel thresholding, Kapur’s entropy, colour image thresholding, particle swarm optimization, differential evolution, statistical significance test
I.
I NTRODUCTION
As far as the different image segmentation techniques are concerned, thresholding has proven itself as the most admired and well accepted effective tool in this regard. Thresholding is simple to implement and provides most accurate results. It is used efficiently to discriminate objects from the background image. It can also be applied to separate objects from objects having dissimilar gray-levels. In these days, it has found an extensive use in diverse application arena which may include pattern recognition, image processing, image analysis and variant computer vision activities. In the computer world, image segmentation means segregation of image into number of homogeneous regions called classes. If image thresholding is performed for exactly two classes, it is called bi-level image thresholding while for multi-level image thresholding, number of classes exceed two [1]. An extensive investigation on various segmentation approaches have been presented in the literature [2] [3]. Presently, a number of classical methods [4] are available to meet at an apposite thresholding measure for gray-level images. Of late, the idea to renovate the classical method by uniting the concepts of quantum computing with it, was well appreciated. The aforesaid quantum inspired algorithms employ the inherent attributes of quantum computing that leads to minimize its execution time and makes the execution faster [5] [6]. Factorization of large numbers [7] and search
databases [8] may be good examples in this context, which can establish the capability of quantum computers over their classical counterparts. Meta-heuristic algorithms typically intend to uncover the apposite solutions of given combinatorial optimization problems. These are basically generation based process which explore and exploit its search space to find optimal solution of any given problem within the diverge computational time. The application areas of meta-heuristic algorithms may vary from simple local searching process to complicated searching procedures that may require some additional intelligence to solve. Some examples of meta-heuristic methods are presented in [9] [10]. Gao et al. have presented a quantum-based particle swarm algorithm for multilevel image thresholding [11]. A survey over image thresholding, meta-heuristic methods and quantum-behaved algorithms have been reported in their research paper [11]. In this paper, the features of quantum mechanics have been coupled with two basic meta-heuristic techniques to design two quantum version of meta-heuristic algorithms. The proposed Quantum Inspired Particle Swarm Optimization (QIPSO) and Quantum Inspired Differential Evolution (QIDE) algorithms have been developed to optimize colour images for multi-level thresholding using Kapur’s entropy [12] as an objective function. A comparison study has been conducted between each of the proposed method and their conventional counterparts. Lastly, t-test, an eminent statistical significance test, is performed to find the preeminence of the above four methods [13]. II.
BASIS OF Q UANTUM C OMPUTING
Quantum computing (QC) has gained significant attention to the researchers from the last few years. QC can be pertinent to be incorporated in such an arrangement where information can be processed, stored and even sanitized in a Hilbert space. The idea of QC has been originated from the theory of quantum physics [6]. Theoretically, let us assume that a wave function, |ψ > is at hand in Hilbert space and also there exists a group of states ϕj for a quantum system. |ψ > can be expressed by the equation given by ∑ |ψ >= ck |ϕk > (1) k
where, |ψ > is the linear superposition (linear combination) of the basis states, ϕj . Each ck is complex number satisfying
978-1-4799-2275-8/13/$31.00 ©2013 IEEE
the unitary condition given by ∑ ck = 1
(2)
k
Coherence, decoherence and entanglement are some basic features of QC. When the basic states are in superposed form upholding a constant phase relationship between them, it is quantum coherence. The reverse process is called decoherence which subsides the so called phase relationship to destroy the superposed form. It transforms the combined states into state |ϕj > with a probability ck [14]. III.
K APUR ’ S M ETHOD A S O BJECTIVE F UNCTION F OR O PTIMIZATION
Kapur’s function [12] can be used as an objective function in image thresholding optimization problem. It aims to determine appropriate number of optimal threshold {θ1 , θ2 , · · · , θK−1 } from gray-scale/color images. Kapur’s function is a entropy based optimization technique which maximizes the sum of entropies for each individual class. Formally, it is described as follows [12]. V = ς {θ1 , θ2 , · · · , θK−1 } = 2
K−1 ∑
Yk
(3)
k=0
where C ={C1 , C2 , . . . , CK } represents the set of K number of classes and ∑ ∑ pi pi pi = ni /N , ωk = pi , Yk = − ln (4) ωk ωk i∈Ck
i∈Ck
where, ni and N represent ith level pixel frequency and total number of pixels of an image respectively. pi denotes the probability of ith pixel. Note that a colour image is a combination of three different gray-scale images defined as red, green and blue. Each of these colour has the pixel intensity values in the range [0, 255]. The number of optimum thresholds of an image can be obtained by maximizing V.
image. Subsequently, a real random number between (0,1) is assigned to each pixel encoded in P OP U LAT ION using the concept of qubits to produce P OP U LAT ION ′ . After that, P OP U LAT ION ′ passes through an quantum interference to create P OP U LAT ION ′′ . QIPSO finds an user defined number of thresholds as pixel intensities using some probability constraints. A D dimensional space is taken into account to represent each particle in the P OP U LAT ION as a point. At the first generation, a group of particles is randomly invoked and afterwards their velocities and locations are updated in the successive generations using the formulae given by vjt+1 (D) = ϖ ∗ vjt (D) + ι1 ∗ rand(0, 1) ∗ (ptj (D) − wjt (D)) +ι2 ∗ rand(0, 1) ∗ (ptg (D) − wjt (D)) wjt+1 (D) = wjt (D) + vjt+1 (D) The j th particle in the swarm is symbolized as W = (wi1 , wi2 , · · · , wiD ). In QIPSO, wjt (D) and vjt (D) are the current location and its respective velocity of the j th particle at generation t. The best location of each particle is monitored regularly for each generation. ptj (D) is the best location of j th particle visited so far upto generation t whereas, ptg (D) is recognized as globally measured best location of that particle upto the same generation. In swarm, the index of the best particle for red, green and blue component are recorded in bR , bG and bB respectively. In this algorithm, rand signifies a real random number in [0, 1]. ι1 and ι2 are two positive real acceleration constants in [0, 1]. ϖ is the inertia weight. The details of the proposed QIPSO is presented in Algorithm 1. 1) Complexity analysis: The worst case time complexity analysis of the proposed QIPSO for multilevel thresholding of colour image is given below. 1)
A. Quantum Inspired Particle Swarm Optimization In 1995, Kennedy and Eberhart first introduced a population-based optimization technique, popularly known as particle swarm optimization (PSO) [15]. In the recent years, PSO has been successfully implemented by many researchers to resolve various function optimization problems. Birds flock together and work accordingly for sustaining their life. Their behaviour was meticulously observed and have been inflicted to build this optimization strategy. They always strive to explore some fanatical search space to fly. It has been observed that they always hunt the revisited paths efficiently. In this article, the notion of quantum mechanics have been properly coupled with this meta-heuristic algorithm to create a new quantum inspired particle swarm optimization (QIPSO) algorithm for multilevel thresholding for true colour images. In QIPSO, initially a population, P OP U LAT ION is formed by selecting the pixel intensities randomly from the colour image. P OP U LAT ION consists of three different set of particles of size Z each for three different colour component of the colour image. The length of√each particle in P OP U LAT ION was taken as M = M1 × M2 , where M1 and M2 stand for the width and height of the
2) 3) 4) 5) 6) 7)
The population size of the particles of red, green and blue component in the colour image is O(Z). Since a colour image is the combination of three different colour mixtures, the total size of population in P OP U LAT ION becomes 3 × Z = U (say). The time complexity to produce initial particles of population, P OP U LAT ION in QIPSO is O(U √ ×M). Note that, length of each of particle is M = M1 × M2 where, M1 and M2 signify the width and height of an image, respectively. The time complexity for allocating real value to each pixel encoded as population of particles turns into O(U × M). The complexity becomes O(U × M) for conducting quantum interference in QIPSO. Again, the time complexity to create P OP U LAT ION ∗ turns out to be O(U × M). For evaluating the fitness in QIPSO, time complexity becomes O(U × K). Finally, at each generation, swarm is manipulated. So, the time complexity for this manipulation turns into O(U × M). Hence, the time complexity to execute QIPSO for a predefined number of generations becomes O(U × M × Gen) where, Gen signifies the number of generations to be executed.
Algorithm 1: Quantum Inspired Particle Swarm Optimization for multilevel thresholding for colour image Input:Number of generation: Gen Size of each particle for red, green and blue components: Z Acceleration coefficients: ι1 and ι2 Inertia weight: ϖ Number of thresholds: K Result:Optimal threshold values for red: θR Optimal threshold values for green: θG Optimal threshold values for blue: θB 1: Pick pixel values at random to generate a pool of initial particles for red, green and blue, P OP U LAT ION , where, P OP U LAT ION contains three different set of Z particles for√each colour. Note that the length of each particle is denoted by M = M1 × M2 , where, M1 and M2 represent the width and height of an image respectively. 2: The concept of qubits is applied to assign real random number between (0,1) to each pixel encoded in P OP U LAT ION . Let it produces P OP U LAT ION ′ . 3: P OP U LAT ION ′ undergoes an quantum interference to produce P OP U LAT ION ′′ . 4: Find K number of thresholds as pixel values from each different set of particles for red,green and blue in P OP U LAT ION . It should satisfy corresponding value in P OP U LAT ION ′′ > rand(0, 1). Let it creates P OP U LAT ION ∗ . 5: Use equation (3) to evaluate fitness of each particle in P OP U LAT ION ∗ . 6: Save the best particle bR ∈ P OP U LAT ION ′′ for red, bG ∈ P OP U LAT ION ′′ for green and bB ∈ P OP U LAT ION ′′ for blue, their respective thresholds in TR , TG and TB ∈ P OP U LAT ION ∗ . The corresponding fitness values for red, green and blue are recorded in FR , FG and FB respectively. 7: Store the sum of FR , FG and FB in F . 8: repeat {Manipulation of swarm in POPULATION”} 9: for all j ∈ P OP U LAT ION ′′ do 10: For red, green and blue, the best positions from the three individual set of particles in P OP U LAT ION ′′ and the index of the best particles in bR , bG , bB ∈ P OP U LAT ION ′′ are documented. 11: vjt+1 (D) = ϖ ∗ vjt (D) + ι1 ∗ rand(0, 1) ∗ (ptj (D) − wjt (D)) + ι2 ∗ rand(0, 1) ∗ (ptg (D) − wjt (D)). 12: wjt+1 (D) = wjt (D) + vjt+1 (D). 13: The steps 3 and 4 are repeated for updating P OP U LAT ION ′ and P OP U LAT ION ∗ . 14: Compute the fitness of each particle in P OP U LAT ION ∗ using equation (3). 15: The best particles for red, green and blue in bR1 , bG1 , bB1 ∈ P OP U LAT ION ′′ , their respective thresholds in TR1 , TG1 and TB1 ∈ P OP U LAT ION ∗ and finally, the corresponding fitness values in FR1 , FG1 and FB1 are recorded. 16: Save the sum of FR1 , FG1 and FB1 in FN . 17: Keep the utmost fitness value between F and FN in F . The particles ensuing the maximum fitness and their respective threshold values for red, green and blue are also updated in bR , bG , bB ∈ P OP U LAT ION ′′ and TR , TG and TB ∈ P OP U LAT ION ∗ respectively. 18: end for 19: until it reaches a predefined number of generations, Gen 20: Report the optimum threshold values θ = TR for red, θ = TG for green and θ = TB for blue.
Therefore, the overall worst case time complexity (summing up the above steps taken as a whole) for the proposed QIPSO happens to be O(U × M × Gen). B. Quantum Inspired Differential Evolution Storn and Prince first introduced Differential Evolution (DE), a population-based optimization technique [16]. The optimization of real valued functions was the first inspiration to develop DE. In recent years, it became the first choice for the developers to use in various application areas specially in engineering or statistics. Due to its simplicity and powerfulness, it
can be easily implemented to optimize different objective functions, which are very difficult or even analytically impossible to solve. In this article, a new Quantum Inspired Differential Evolution (QIDE) for multilevel thresholding for true colour images is presented. The theory of quantum mechanics have been applied in this meta-heuristic optimization technique to develop QIDE. In QIDE, firstly, the pixel intensities are randomly selected from the colour image to generate a population, P OP U LAT ION . Like QIPSO, P OP U LAT ION contains three different group of vectors, each of size Z for the three colour components of the image. Each√ vector in P OP U LAT ION possesses a length of M = M1 × M2 , where M1 and M2 are the width and height of the image. Afterwards, the theory of qubits are used to assign a real random number between (0,1) to each pixel encoded in P OP U LAT ION to create P OP U LAT ION ′ , which later endures an quantum interference to produce P OP U LAT ION ′′ . Some probability constraint are used in QIDE to get the user defined number of thresholds as the pixel intensities. The fitness of each vector in P OP U LAT ION ′′ is recorded in R. The fitness of the best vectors each for red, green and blue component are recorded in FR , FG and FB and their sum is stored in F for later use. The successive use of mutation and crossover are the cause of population diversity at each generation. Three different vectors κ1 , κ2 and κ3 ∈ [1, Z] are picked up at random from P OP U LAT ION ′′ to perform quantum mutation. Note that, κ1 ̸= κ2 ̸= κ3 ̸= i where, Z signifies the number of participating vectors and i is the particular vector individual in P OP U LAT ION ′′ . The first vector is summed up with the net weighted difference of second and third vectors multiplied by H resulting new mutant vector, P OP U LAT ION ′′ . Here, H is called scaling factor, a real number selected from [0, 1]. All the positions in each participating vector in P OP U LAT ION ′′ may undergo crossover subject to some basic conditions. A particular location, i in a mutant vector goes for crossover if i ̸= r and rand(0, 1) > Pc . Note that, Pc and r represent a predefined crossover probability and a randomly generated number from [1, M], respectively. After quantum mutation and quantum crossover, new vectors are selected from the pool of vectors to form the population for the next generation. For this purpose, the fitness of each vector in P OP U LAT ION ′′ is recorded in S. The fitness of the best vectors each for red, green and blue component are recorded in FR1 , FG1 and FB1 and sum of their best fitness value is kept in FN . The new population, P OP U LAT ION ′′ is filled by each vector, i that possesses better fitness among Ri and Si . QIDE stop executing when it reaches a predefined generation number, Gen. After every generations, the values of F and FN are compared and the best threshold values are stored. The outline of QIDE is described elaborately in Algorithm 2. 1) Complexity analysis: The details of worst case time complexity analysis of the proposed QIDE for multilevel thresholding of colour image is given below. The first five steps of time complexity analysis have been already described in the above subsection. The time complexities for the remaining parts of the proposed method are discussed below. 1)
The time complexities for quantum mutation, quantum crossover and the selection of particles for the next generation happen to be O(U × M).
Algorithm 2: Quantum Inspired Differential Evolution for multilevel thresholding for colour image Input:Number of generation: Gen Size of each particle for red, green and blue components: Z Scaling factor: H Crossover probability: Pc Number of thresholds: K Result:Optimal threshold values for red: θR Optimal threshold values for green: θG Optimal threshold values for blue: θB 1: Select pixel values randomly to produce a pool of initial vectors for red,green and blue, P OP U LAT ION , where, P OP U LAT ION have three individual set of Z vectors each for different colour. Here, the √ length of each vector is denoted by M = M1 × M2 , where, M1 and M2 are the width and height of an image respectively. 2: Allocate real random value between (0,1) to each pixel encoded in P OP U LAT ION using the notion of qubits. Let it creates P OP U LAT ION ′ . 3: P OP U LAT ION ′ goes for an quantum interference to create P OP U LAT ION ′′ . 4: repeat 5: Locate K number of thresholds as pixel values from each individual set of vectors for red,green and blue in P OP U LAT ION satisfying corresponding value in P OP U LAT ION ′′ > rand(0, 1). Let it produces P OP U LAT ION ∗ . 6: Compute the fitness of each vector in P OP U LAT ION ∗ using the equation (3). 7: Record the individual fitness value of each vector of P OP U LAT ION ∗ in R. Save the respective thresholds of the best vectors each for red, green and blue in TR , TG and TB ∈ P OP U LAT ION ∗ and their corresponding fitness values in FR , FG and FB respectively. 8: Keep the sum of FR , FG and FB in F . 9: BKU P = P OP U LAT ION ′′ . 10: for all i ∈ P OP U LAT ION ′′ do 11: for all kth location in i do 12: Pick up three different natural numbers κ1 , κ2 and κ3 randomly from [1, Z] where, κ1 ̸= κ2 ̸= κ3 ̸= k. 13: P OP U LAT ION ′′ (j) = BKU P (κ1 ) + H ∗ (BKU P (κ2 ) − BKU P (κ3 )) 14: end for 15: end for {Apply quantum crossover in POPULATION} 16: for all i ∈ P OP U LAT ION ′′ do 17: for all kth location in i do 18: Pick up a natural number η randomly from [1, Z]. 19: if (k ̸= η and rand(0, 1) > Pc ) then ′′ = BKU P . 20: P OP U LAT IONik ik 21: end if 22: end for 23: end for {Apply selection mechanism in POPULATION”} 24: Repeat steps 5 and 6 to compute the fitness of each vector in P OP U LAT ION ∗ using the equation (3). 25: Store the individual fitness value of each vector of P OP U LAT ION ∗ in S. The respective thresholds of the best vectors each for red,green and blue and their corresponding fitness values are documented in TR1 , TG1 and TB1 ∈ P OP U LAT ION ∗ and in FR1 , FG1 and FB1 respectively. 26: Keep the sum of FR1 , FG1 and FB1 in FN . 27: Compare the values of F and FN and keep the utmost fitness value in F . For each red, green and blue, update the respective thresholds of the best vectors in TR , TG and TB ∈ P OP U LAT ION ∗ . 28: for all i ∈ P OP U LAT ION ′′ do 29: if (Ri > Si ) then 30: P OP U LAT ION ′′ (i) = BKU Pi 31: end if 32: end for 33: P OP U LAT ION ′′ again experience an quantum interference. 34: until it reaches a predefined number of generations, Gen 35: The optimum threshold values θ = TR for red, θ = TG for green and θ = TB for blue are reported.
2)
Therefore, the time complexity to run QIDE for a predefined number of generations is O(U × M × Gen). Here, Gen signifies the number of generations to be executed.
Hence, aggregating the above discussions, the overall worst case time complexity in QIDE turns out to be O(U × M × Gen).
IV.
E XPERIMENTAL R ESULTS
The proposed QIPSO and QIDE algorithms for multi-level thresholding for true colour images have been examined to optimize two colour images. For this purpose, the entropy based method proposed by Kapur has been employed as an objective function. The selected test images are Lena (intensity range varies in [53,255] for red component, in [0,243] for green component and in [30,220] for blue component) and Peppers (intensity range varies in [6,255] for red component, in [0,239] for green component and in [0,255] for blue component) of dimension 512 × 512, each. QIPSO and QIDE maximize the objective function V, as shown in equation (3) for optimizing the selected test images. The original test images are shown in Figs. 1(a) and (b), respectively. Each method has been executed for 10 different runs. The best result of each method is listed in Table I for K = 2, 3, 4 and 5. The table includes the number of classes (K), optimal threshold values for red (θR ), for green (θG ) and for blue (θB ), the fitness value (Vbest ), and the execution time (t) (in seconds). The conventional particle swarm optimization (PSO) and differential evolution (DE) have also been executed for 10 different runs on the same set of images using the same fitness function. The best result for each of the conventional method have also been listed in Table I for same levels. In Table II, the authors have reported the mean fitness (Vavg ) and the corresponding standard deviation (σ) over 10 runs for two proposed methods and two conventional methods. Moreover, a two-tailed t−test has been carried out at 5% confidence level and the corresponding results among two alternate methods are documented in Table III. For K = 2, each method reports the optimal threshold values for lena as 167, 140 and 131 and for peppers as 99, 123 and 98, respectively for red, green and blue components of colour images. Since all methods find similar threshold values for both the images in all individual runs, the best fitness value, (Vbest ) and the average fitness value, (Vavg ) sound equal for each method. Hence, the value of σ becomes 0 for each method. Like K = 2, these four methods also outperform almost equally for K = 3. Although, there are few changes found in results in some occasions. Hence, this outcomes a petite change in Vavg and σ values for both images. It was observed that the values of optimal threshold varies for all the methods when K = 4 and 5. Therefore, the values of mean fitness and standard deviation differ in all methods. Table II confirms that QIPSO is the most time efficient method among others. This fact is also established by statistical test shown in Table III. The results of t-test proves the superiority of the proposed quantum methods over the corresponding conventional methods. The authors have used Toshiba Intel(R) Core (TM) i3, 2.53GHz PC with 2GB RAM for the experimental purpose. The images of QIPSO and QIDE after thresholding are depicted in Figs. 2 and 3, respectively for each level of thresholding. As the thresholds of PSO and DE are same or close to their corresponding quantum versions, only the thresholded images for quantum inspired algorithms have been reported here.
TABLE I.
B EST R ESULTS OBTAINED
FOR
QIPSO, PSO, QIDE
AND
DE
FOR MULTI - LEVEL THRESHOLDING OF COLOUR IMAGES
Lena K 2 3 4 5
QIPSO θR 167 124,172 116,163,214 114,159,177,210
θG 140 76,154 43,112,168 57,75,115,151
θR 99 93,160 63,108,157 54,88,123,143
θG 123 67,147 67,127,172 38,79,129,188
θR 167 123,192 117,159,203 109,137,168,213
θG 140 73,145 39,91,160 65,90,120,166
θR 99 97,157 58,97,162 66,108,144,238
θG 123 70,141 76,131,187 51,95,136,181
K 2 3 4 5
QIPSO
K 2 3 4 5
θB 98 57,107 43,93,155 46,87,120,159 QIDE
K 2 3 4 5
θB 131 95,139 90,126,174 74,109,130,177
θB 131 92,142 78,124,160 79,100,178,196 QIDE θB 98 56,105 57,105,142 24,67,121,154
PSO
Vbest 26.6464 36.3568 45.0167 52.4867
t θR 25.32 167 26.21 132,188 27.13 131,169,212 27.55 84,154,190,221 Peppers
θG 140 76,156 60,132,188 37,87,174,208
Vbest 26.9378 37.1412 45.8218 53.7731
t θR 24.09 99 26.19 86,162 27.33 55,106,143 27.49 76,100,146,194 Lena
θG 123 77,152 49,111,164 79,128,175,209
Vbest 26.6464 36.3630 45.0008 52.1019
t θR 86.28 167 101.12 139,194 116.17 107,136,189 128.17 79,99,180,227 Peppers
θG 140 71,155 57,89,141 34,69,99,156
Vbest 26.9378 37.1117 45.8652 53.9110
t 90.315 109.46 117.20 130.16
θG 123 81,149 38,136,182 38,59,127,175
θB 131 94,150 79,125,174 118,137,170,195 PSO θB 98 58,107 24,80,116 42,85,110,150 DE θB 131 96,135 89,148,190 59,83,135,181 DE
θR 99 98,161 55,98,160 64,113,157,200
θB 98 56,114 34,103,153 54,101,135,172
Vbest 26.6464 36.3511 44.8164 51.4777
t 39.15 41.10 42.23 45.18
Vbest 26.9378 37.1039 45.5546 53.4157
t 38.16 39.10 42.22 44.07
Vbest 26.6464 36.3533 44.6888 51.5233
t 105.15 155.10 187.01 193.09
Vbest 26.9378 37.0668 45.6195 53.8134
t 110.13 142.04 190.25 197.15
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 2. Thresholded test images for K = 2, 3, 4, 5 in (a)-(d), for Lena and, (e)-(h), for Peppers, respectively, after exercising QIPSO for multi-level thresholding
V.
TABLE II. M EAN FITNESS (Vavg ) AND STANDARD DEVIATION (σ) OF QIPSO, PSO, QIDE AND DE FOR MULTI - LEVEL THRESHOLDING OF
C ONCLUSION
In this paper, Quantum Inspired Particle Swarm Optimization (QIPSO) and Quantum Inspired Differential Evolution
COLOUR IMAGES
Lena K 2 3 4 5 K 2 3 4 5
(a) Fig. 1.
QIPSO
Vavg 26.6461 36.3394 44.8200 51.9988
σ 0 0.018 0.090 0.186
QIPSO Vavg σ 26.9377 0 37.0793 0.026 45.7009 0.070 53.5727 0.158
PSO
Vavg 26.6461 36.3362 44.7814 51.5488
σ 0 0.020 0.125 0.214 Peppers
PSO Vavg σ 26.9376 0 37.0686 0.033 45.5870 0.205 53.3696 0.241
QIDE Vavg σ 26.9375 0 36.3190 0.054 44.7809 0.090 51.5745 0.207
DE Vavg σ 26.9375 0 36.2517 0.079 44.7466 0.122 51.5926 0.255
QIDE Vavg σ 26.9376 0 37.0725 0.031 45.6875 0.059 53.5775 0.171
DE Vavg σ 26.9375 0 37.0406 0.095 45.5651 0.215 53.4368 0.281
(b)
Original test images (a) Lena and (b) Peppers
(QIDE) for multi-level thresholding for True Color Images are presented. In these algorithms, Kapur’s entropy is used
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 3. Thresholded test images for K = 2, 3, 4, 5 in (a)-(d), for Lena and, (e)-(h), for Peppers, respectively, after exercising QIDE for multi-level thresholding TABLE III.
R ESULTS OF TWO - TAILED t- TEST FOR MULTI - LEVEL THRESHOLDING OF TEST IMAGES
Lena A&D B&D C&D C&B NaN NaN NaN NaN 0.0449 0.0466 0.0447 0.0436 0.0370 0.0349 0.0341 0.0365 0.0281 0.0272 0.0353 0.0344 Peppers K A&B A&C A&D B&D C&D C&B 2 NaN NaN NaN NaN NaN NaN 3 0.0456 0.0448 0.0437 0.0462 0.0437 0.0455 4 0.0294 0.0264 0.0275 0.0323 0.0268 0.0283 5 0.0241 0.0345 0.0255 0.0270 0.0265 0.0251 A:→ QIPSO, B:→ PSO, C:→ QIDE, D:→ DE, NaN:→ No result K 2 3 4 5
A&B NaN 0.0443 0.0365 0.0286
A&C NaN 0.0451 0.0368 0.0279
as an objective function to find optimal threshold values for two colour images. Optimal threshold values of three different colour components of the selected colour images, their corresponding fitness values and finally, the execution time for each method at varied level have been reported. Mean fitness and standard deviation of the results establish the behavioural outlook of each method. Moreover, t-test, has been performed to find the superiority among the participating methods. The results show that each method is equally efficient at lower level of thresholding. The proposed quantum inspired algorithms are more effectual than their classical counterparts for higher level of thresholding. Furthermore, QIPSO takes least time for its execution among others. However, the multiobjective version of quantum inspired algorithms on colour image are future aspect of research. This is the route of motivation for future work.
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