Observation-Driven Adaptive Differential Evolution for ...

Observation-Driven Adaptive Differential Evolution for Robust Bronchoscope 3-D Motion Tracking Xiongbiao Luo and Kensaku Mori Information and Communications Headquarters, Nagoya University

Abstract. This paper proposes an observation-driven adaptive differential evolution (OADE) algorithm for accurate and robust bronchoscope 3dimensional (3-D) motion tracking during electromagnetically navigated bronchoscopy. Two advantages of our framework are distinguished from any other adaptive differential evolution methods: (1) current observation information including sensor measurement and video image is used in the mutation equation and the selection function, respectively, and (2) the mutation factors and crossover rate are adaptively determined in terms of current image information. From experimental results, our OADE method was demonstrated to be an effective and promising tracking scheme. Our approach can reduce the tracking position error from 3.9 to 2.8 mm, as well as the position smoothness from 4.2 to 1.4 mm.

1

Introduction

During bronchoscopy navigation, bronchoscope 3-D motion tracking still remains a challenge to find the camera motion parameters in the reference coordinate system and can be considered as a 6-degree-of-freedom (DOF) optimization problem. To estimate the bronchoscope movement, two major methods (or a combination of them) were published: (1) intensity-based image registration: constructing an optimization function to minimize the pixel difference between real video images and virtual renderings generated from pre-operative images (e.g., CT) [1,2,3,4]; and (2) electromagnetic tracker (EMT): fixing a sensor at the bronchoscope tip and directly measuring its motion [5,6,7,8]. Image-based methods work well but suffer from image artifacts (e.g., motion blurring) and easily get trapped in local minima during optimization. Although the EMT-based approaches have be commercialized to clinical applications [9], their tracking accuracies are deteriorated by respiratory motion and magnetic field distortion. This paper seeks for a more robust and accurate tracking framework to boost the EMT approaches to meet clinical requirements. As one of powerful evolutionary algorithms, differential evolution (DE), which was developed by Storn and Price [10], has been applied as a successful optimization technique to address any complex problems [11]. However, its performance reckons on evolutionary parameters of the mutation factor and the crossover rate. We modified such an algorithm and proposed an observation-driven adaptive differential evolution K.M. Lee et al. (Eds.): ACCV 2012, Part III, LNCS 7726, pp. 259–271, 2013. c Springer-Verlag Berlin Heidelberg 2013

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(OADE) method, which can not only adaptively determine the evolutionary parameters based on intensity information but also add observation information of the EMT sensor and video image to mutate each individual in a population. The main contribution of this work is summarized as follows. First, we modified the mutation equation in the DE algorithm by integrating current observation information, which can control the perturbation velocity and direction of each individual during evolution, augmenting the DE performance. Next, to the best of our knowledge, our OADE framework is a novel application of DE in bronchoscope 3-D motion tracking. We successfully formulated bronchoscope 3-D motion tracking as an OADE-based stochastic optimization process. EMT sensor measurements and bronchoscopic video images can be effectively leaded into OADE to achieve a more robust and accurate tracking method. Additionally, our OADE framework can be suitable to track other endoscope, e.g., colonscope.

2 2.1

Differential Evolution Operations of DE

Basically, the DE method propagates a population of individuals or vectors {Xi,G |Xi,G ∈ D }N i=1 (N is the population size, G is the generation index, and D is the dimension of vector) toward to the global optimum during any stochastic optimization procedures. After initializing the parameters of the population {Xi,G |Xi,G ∈ D }N i=1 , each target vector or individual Xi,G , which is considered as a potential solution to a multi-dimensional optimization problem, will be evolved by performing three operations, as discussed as follows. Mutation. For target vector Xi,G at generation G, its mutant vector Vi,G can be obtained by several frequently used mutation schemes in DEs [11]: ⎧ ⎪ Xri1 ,G + Fi Xri2 ,G − Xri3 ,G ① ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Xbest,G + Fi Xri1 ,G − Xri2 ,G ② ⎪ ⎪ ⎨ Vi,G = Xi,G + Fi (Xbest,G − Xi,G ) + Fi Xri1 ,G − Xri2 ,G ③ , (1) ⎪ ⎪ ⎪ ⎪ Xbest,G + Fi Xri1 ,G − Xri2 ,G + Fi Xri3 ,G − Xri4 ,G ④ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Xr1 ,G + Fi Xr2 ,G − Xr3 ,G + Fi Xr4 ,G − Xr5 ,G ⑤ i i i i i which corresponds to ①∼⑤ mutation strategies of “DE/rand/1”, “DE/best/1”, “DE/target−to−best/1”, “DE/rand/2”, and “DE/best/2”, respectively; these strategies are consistent with a general name or convention: DE/a/b, where DE denotes the standard DE algorithm, a indicates the base vector to be perturbed, and b is the number of difference vectors. The indexes ri1 , ri2 , ri3 , ri4 , and ri5 are mutually exclusive integers chosen randomly from set {1, · · · , i−1, i+1, · · · , N }. The part (Xri1 ,G − Xri2 ,G ) in Eq. 1 represents the difference vector, Xbest,G is the best individual at generation G, and Fi is the mutation factor. Crossover. After mutation, a binomial crossover operation is performed to generate trial vector Ui,G = {u1i,G , · · · , uD i,G } in accordance with target vector 1 D } and mutant vector Vi,G = {vi,G , · · · , vi,G }: Xi,G = {x1i,G , · · · , xD i,G

Observation-Driven Adaptive Differential Evolution

uji,G

=

j vi,G xji,G

if (randj [0, 1] ≤ Cr ) or (j = jrand ) , otherwise

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(2)

where randj [0, 1] is a random number yielded an uniform distribution, Cr is the crossover rate or probability that determines whether uji,G ∈ Ui,G is copied from uji,G ∈ Vi,G , and jrand is randomly selected from set {1, 2, · · · , D}. Selection. The selection operation chooses the better vector or individual for the next generation from Vi,G ∪ Ui,G in terms of their fitness value W (·). For a maximization optimization problem, such an operation is formulated by: Xi,G+1 =

Ui,G Xi,G

if W (Ui,G ) ≥ W (Xi,G ) . otherwise

(3)

Algorithm 1. The DE Algorithm with the Binomial Crossover 1. Control parameters: Fi = constant, Cr = constant, population size N ; 2. At generation G = 0, randomly initializing population P = {Xi,G }N i=1 ; 3. Go to iteration and perform the main body of the DE algorithm: while (termination is unsatisfied) for i = 1 to N do ➊ Mutation Operation: 1 D , · · · , vi,G } for target vector Generate mutant vector Vi,G = {vi,G 1 D Xi,G = {xi,G , · · · , xi,G } using one of mutation strategies in Eq. 1; ➋ Crossover Operation: Generate trial vector Ui,G = {u1i,G , · · · , uD i,G } on the basis of 1 D 1 D vectors Xi,G = {xi,G , · · · , xi,G } and Vi,G = {vi,G , · · · , vi,G } if (randj [0, 1] ≤ Cr ) or (j = jrand ) then j uji,G = vi,G ; else uji,G = xji,G ; end ➌ Selection Operation: Evaluate Xi,G and Ui,G and choose the better vector Xi,G+1 for the next generation according to their fitness value W (·) if W (Ui,G ) ≥ W (Xi,G ) then Xi,G+1 = Ui,G ; else Xi,G+1 = Xi,G ; end end G = G + 1; end while

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The DE algorithm is implemented by three operations above until termination is satisfied. The pseudo-code of DE is generalized in Algorithm 1. 2.2

Remarks On Current DEs

Although DE was proved to be a powerful and easily implemented stochastic optimization algorithm, its performance is controlled by the mutation factor Fi and crossover rate Cr . Several adaptive differential evolution (ADE) methods were proposed and proved to obtain the better performance [12,13,14,15,16]. In our framework, we also compute these parameters adaptively, especially we introduce two mutation factors that base on the individual fitness value or image information. On the other hand, to effectively and successfully solve any dynamic or stochastic optimization problems, two general questions must be considered: (1) how to use current observation information or the temporally/spatially continuous information between two consecutive frames/outputs and (2) how to retain or even enhance the diversity of the population during optimization. In DE, the mutation operation seeks to to answer the first question while the crossover operation deals with the second one. From the point of view of Doucet et al. [17], the optimal solution space of any stochastic optimization problems should be integrated the current observation information into. Unfortunately, none of current DE or ADE algorithms takes the current observation information into account. The mutation operation of DE or ADE perturbs each target vector without any observation information, possibly resulting in the local minima problem. To enhance the performance of DEs, our idea is to incorporate the current observation into the mutation operation, which is further discussed in the next section.

3 3.1

Proposed OADE Framework Problem Statement

Bronchoscope 3-D motion tracking is to determine the bronchoscope location (position and orientation) in the reference frame (in our case, it is the CT coordinate system) using bronchoscopic video images, EMT measurement, and CT slices. It is generally a multi-modal information fusion procedure where several coordinate systems involve, as shown in Fig 1. To fuse these multi-modal information, the relationship of several transformations can be expressed by: CT

TC = CT TEMT EMT TS S TC ,

(4)

where CT TEMT , EMT TS , and S TC represent different transformation relationships among four coordinate systems of bronchoscopic camera, EMT sensor, EMT system, and CT images during bronchoscope 3-D motion estimation. Now our problem is how to determine the transformation CT TC from the bronchoscopic camera coordinate system to the CT coordinate system.


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Fig. 1. Coordinate systems C, S, EM T , and CT respectively denote bronchoscopic camera, EMT sensor, EMT system, and CT space during tracking

3.2

Motion Representation

Bronchoscopic camera motion is characterized by the transformation CT TC that includes translation or position vector CT tC and rotation matrix CT RC . Position vector has three components: CT tC = (CT txC ,CT tyC ,CT tzC ), where CT txC , CT tyC , and CT tzC are the coordinate values of the bronchoscopic camera position in the x-, y-, and z-axes of the CT coordinate system. Otherwise, rotation matrix CT RC is described by a quaternion CT QC with four elements (Q0 , Qx , Qy , Qz ): CT

RC ←→CT QC (Q0 , Qx , Qy , Qz ),

Q20 + Q2x + Q2y + Q2z = 1.

(5)

Therefore, during DE optimization, target individual Xi,G in the population related to 3-D motion CT TC can be represented by a seven-dimensional vector:

CT TC −→ Xi,G = CT tC CT QC = CT txC ,CT tyC ,CT tzC , Q0 , Qx , Qy , Qz . (6) 3.3

OADE-Based Tracking

Our observation-driven adaptive differential evolution (OADE) algorithm consists of the following steps: (1) initialization, (2) mutation, (3) crossover, and (4) determination of bronchoscope camera 3-D pose parameters. The second step is our main point of our proposed method. We modified the mutation strategy by incorporating the current observation information of EMT sensor measurement. We skip discussing the crossover step since it is similar to Algorithm 1 expect the crossover rate computation. In the forth step, we define a fitness function on the basis of the current observation information of video image and evaluate each individual in the population, and eventually obtain the best individual from the updated population as the current estimation of camera 3-D motion pose. Random Initialization. Our OADE algorithm approximates the global optimal solution in 7-D space from a population that is randomly initialized. To initialize the population, we manually register the first real video image to the CT-based virtual rendering image and obtain CT TC and in turn get the initial individual X0,0 . Note that during any stochastic optimization problems, the

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Fig. 2. Illustration of our proposed mutation strategy with three perturbations Ωi (Ek − Ek−1 ), Fib (Xbest,G − Xi,G ), and Fir (Xr1 ,G − Xr2 ,G ) for diversification i

i

diversity of the population in the DE methods plays a positive role in the optimization performance. To enhance the population diversity, we implement a randomization processing in terms of the normal distribution and obtain Xi,0 : Xi,0 = G(X0,0 , πi Ξ),

(7)

where πi is a normally distributed random number and Ξ is a predefined vector. New Mutation Operation. The mutation operation is the key stage of any DE methods. Currently, various research work focused on how to modify the mutation equation to enhance the standard DE performance [18,12,19,13,14,15,16]. Hence, many mutation strategies were proposed, as shown in Eq. 1. The advantages and limitations of different mutation schemes were presented in [11]. In our framework, we modify the strategy of DE/target − to − best/1 (see Eq. 1) that had a good convergence performance due to its usage of the best solution or individual information [19]. However, the usage of the best individual information possibly results in the loss of the population diversity and in turn unreliable or precocious convergence. To address such a limitation, we propose a new mutation scheme called DE/target − to − best/1 with current observation: Vi,G = Xi,G + Ωi (Ek − Ek−1 ) +Fib (Xbest,G − Xi,G ) + Fir (Xri1 ,G − Xri2 ,G ), (8)

observation

where Ek and Ek−1 are EMT sensor measurements at times or frames k and (k − 1). The first mutation factor Ωi determines how much current observation to be reserved, and we set it to be an uniformly distributed random number: Ωi ∈ [0 1]. On the other hand, we adaptively calculate the other two mutation factors Fib and Fir on the basis of the fitness value of Xbest,G and Xi,G : Fib =

2W (Xbest,G ) , (W (Xbest,G ) + W (Xi,G ))

Fir =

2W (Xi,G ) . (W (Xbest,G ) + W (Xi,G ))

(9)


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Algorithm 2. OADE-Based Bronchoscope 3-D Motion Tracking Input: Video sequence, EMT sensor measurements, and CT images Output: A series of camera 3-D pose CT T∗C in the CT coordinate system ➊ Randomly initialize population P = {Xi,G }N i=1 by Eq. 7; for k = 1 to M (Frame or measurement number) do for G = 1 to Gmax do for i = 1 to N do ➋ Mutation operation: 1 D Generate mutant vector Vi,G = {vi,G , · · · , vi,G } for individual 1 D Xi,G = {xi,G , · · · , xi,G } using the mutation strategy in Eq. 8; ➌ Crossover and selection operations: Crossover rate by 10 The same as they are performed in Algorithm 1; end G = G + 1; end ➍ Pose determination: Obtain the population {Xi,Gmax }N i=1 ; Find the best solution X∗i,Gmax from {Xi,Gmax }N i=1 ; Determine and store camera pose CT T∗C ←− X∗i,Gmax ; k = k + 1; end In the new mutation operation, for each individual Xi,G , the observation term (Ek − Ek−1 ) serves as deterministic perturbation that is utilized in terms of the random number Ωi . Due to a combination of current observation information, the perturbation Ωi (Ek − Ek−1 ) is very beneficial to conduct the population to the best solution space, enhancing the exploitation and exploration abilities of the ADE method, as proved in our latter experimental results. On the other side, we introduce two mutation factors Fib and Fir for controlling the best vector Xbest,G and the stochastic difference vector (Xri1 ,G − Xri2 ,G ), especially these factors involve the current observation information of video image that is used to compute their fitness values. Fig 2 illustrates the proposed mutation strategy. During the binomial crossover operation, we automatically update the crossover rate for each individual on the basis of the fitness values Vi,G and Xi,G . Since Cr was suggested within the interval [0 1] for balancing the global and the local searching abilities [15], we can adaptively calculate it by: Cr = (W (Xi,G ) + W (Vi,G )) /2,

(10)

which shows a strategy to control Cr by W (·) relative to image information. Pose Determination. After the mutation and crossover operations, we must evaluate each individual Xi,G and update Xi,G to Xi,G+1 for the next generation. For each vector Xi,G , we define the fitness value function W on the basis of an image quality similarity [20] between the current observation video image IkR and the virtual image IV that was generated from camera 3-D pose parameters

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from Xi,G in the population {Xi,G }N i=1 using volume rendering techniques [21]. Finally, the fitness value W (IkR , IV (Xi,G )) can be computed by: 1 4ϑr,v κr κv W (IkR , IV (Xi,G )) = 1+ 2 , (11) 2 (ϑr + ϑ2v ) (κ2r + κ2v ) where ϑr,v is the correlation between IkR and IV (Xi,G ); ϑr and ϑv are the covariance IkR and IV (Xi,G ); κr and κv are the average value of IkR and IV (Xi,G ). Note that the value of W (IkR , IV (Xi,G )) is within the range of [0, 1]. Based on Eqs. 3 and 11, we update the population {Xi,G }N i=1 at each generation or iteration. However, for any optimization problems, we need to set termination conditions. In our OADE, we define the termination criteria by a constant number of generations or iterations Gmax . After obtaining the final ∗ population {Xi,Gmax }N i=1 , we find the best individual or vector Xi,Gmax with the largest fitness value as the current solution of camera 3-D motion estimation: ∗ ∗ X∗i,Gmax −→ CT T∗C = CT tC CT QC . (12) Finally, our OADE-based motion tracking is summarized in Algorithm 2.

4

Experimental Results

We evaluate our proposed method on a dynamic phantom that can simulate respiratory motion. We investigate several EMT-based bronchoscope 3-D motion tracking methods in the literature: (1) Schwarz et al. [9], directly using absolute EMT sensor outputs; (2) Mori et al. [5], integrating absolute EMT sensor outputs into image registration; (3) Luo et al. [22], combining either absolute or relative (inter-frame) EMT sensor outputs with image registration; (4) Luo et al. [8], utilizing sequential Monte Carlo methods to fuse inter-frame EMT sensor outputs with video image information; (5) our method, as proposed in Section 3. As suggested in [16], we set the population size N=30 due to D = 7 < 10 in our experiments. We manually generated ground truth data with two groups of total 3456 video frames that were registered to know their corresponding camera 3-D motion pose information (position and orientation) in the CT coordinate system. For determining the generation number, we perform ten experiments. Fig. 3 shows the results of tracking error and processing time under different generation number. We choose Gmax = 3 since it balances the accuracy and run-time. Fig. 4 displays the tracking errors of different approaches evaluated on two datasets. The average tracking accuracy of position and orientation of our OADE method was about 2.8 mm and 7.9◦ . However, other methods provide the tracking error at least with 3.9 mm and 8.7◦ . To further quantify the tracking results, we define the tracking smoothness as the Euclidean distance between two consecutive frames. We also compute the similarity index between video image and virtual renderings in terms of Eq. 11. Fig. 5 shows the similarity index, run-time, position and orientation smoothness of tracking results. Table 1 summarizes all tracking results estimated by different methods evaluated on two datasets.


(a) Tracking error

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(b) Run-time

Fig. 3. Tracking accuracy and run-time under different generation numbers

(a) Position error of Group 1

(b) Orientation error of Group 1

(c) Position error of Group 2

(d) Orientation error of Group 2

Fig. 4. Tracking errors from different methods on two groups of datasets

5

Discussion

Generally, our OADE framework for bronchoscope 3-D motion tracking is a more accurate and robust method than that of other available methods. Compared to sequential Monte Carlo methods [8], DEs have more powerful exploitation and exploration abilities since they can maintain the multiplicity of population and augment the capacity of local search, while SMC methods are somewhat

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Table 1. Quantitative comparison of tracking results from different approaches. Each (· · · ) corresponds to position and orientation errors, run-time, similarity index, position and orientation smoothness, respectively (“×” means real-time). Datasets Schwarz et al. [9] Mori et al. [5] Luo et al. [22] Luo et al. [8] Our method

Group 1 (5.2, 11.7, ×, 0.65, 4.8, 5.4) (4.9, 10.7, 0.74, 0.71, 4.8, 4.7) (4.3, 10.5, 1.37, 0.75, 4.3, 2.7) (4.3, 9.60, 1.79, 0.75, 4.6, 2.6) (2.8, 8.12, 1.62, 0.79, 1.4, 1.9)

Group 2 (4.7, 10.4, ×, 0.68, 4.8, 5.4) (4.1, 10.4, 0.69, 0.75, 4.5, 4.7) (3.6, 9.05, 1.13, 0.76, 3.9, 2.7) (3.6, 7.87, 1.83, 0.78, 3.8, 2.6) (2.8, 6.68, 1.64, 0.81, 1.3, 1.8)

constrained on the diversity loss. Due to the premature convergence of DEs, we introduce current observation information to the mutation operation. For the DE and ADE algorithms, only stochastic perturbation is performed in the mutation step, which may prohibits the population to reach the current bronchoscope position and orientation. However, our modified mutation that combines stochastic perturbation with deterministic perturbation of current EMT sensor output can positively guide individuals in the population to approximate the best solution space for the current bronchoscope location. On the other hand, we introduce two evolutionary factors into our mutation step and control the mutant factors and crossover rate adaptively on the basis of their fitness values from current video image, which is also beneficial to avoid unstable optimization results. We must note that the current tracking error is a relative error since we definitely introduce error in ground truth data. As shown in Fig. 5 (a,b), the similarity index of ground truth (black curves) is about 0.87; theoretically, it should be 1.0. It is a very challenge to obtain real ground truth data. Currently, the processing time per frame of our methods was about 1.64 seconds, which is far from the camera video rate 30 frames per second. The main time was spent on computing the fitness value between video and virtual images. We still need to put much effort on improving the computational efficiency in our future work, e.g., using graphics processing unit (GPU) techniques or manifold-based learning methods. We can also change the fitness function that uses without the whole intensity information of one video image to reduce time. In summary, we proposed a new bronchoscope 3-D motion tracking scheme that bases on an observation-driven adaptive differential evolution algorithm that incorporates current sensor measurement and video image and can adaptively refresh its control parameters based on image intensity information during iterations. All experimental results demonstrates that our methods provides a more advantageous tracking performance than state-of-the-art methods. Acknowledgment. This work was partly supported by the Center of Excellence project “Development of Bedside Medical Devices for High Precision Diagnosis of Cancer in Its Preliminary Stage” (01-D-D0806) funded by the Aichi Prefecture, and “Computational anatomy for computer-aided diagnosis and therapy: frontiers of medical image sciences” (21103006) funded by the Grant-in-Aid for Scientific Research on Innovative Areas, MEXT, Japan.


(a) Similarity index of Group 1

(b) Similarity index of Group 2

(c) Run-time of Group 1

(d) Run-time of Group 2

(e) Position of Group 1

(f ) Orientation of Group 1

(g) Position of Group 2

(h) Orientation of Group 2

Fig. 5. Comparison of similarity, run-time, and smoothness of all methods

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References 1. Deligianni, F., et al.: Nonrigid 2-D/3-D registration for patient specific bronchoscopy simulation with statistical shape modeling: Phantom validation. IEEE TMI 25, 1462–1471 (2006) 2. Helferty, J., et al.: Computer-based system for the virtual-endoscopic guidance of bronchoscopy. CVIU 108, 171–187 (2007) 3. Deguchi, D., et al.: Selective image similarity measure for bronchoscope tracking based on image registration. Media 13, 621–633 (2009) 4. Luo, X., et al.: Development and comparison of new hybrid motion tracking for bronchoscopic navigation. Media 16, 577–596 (2012) 5. Mori, K., Deguchi, D., Akiyama, K., Kitasaka, T., Maurer Jr., C.R., Suenaga, Y., Takabatake, H., Mori, M., Natori, H.: Hybrid Bronchoscope Tracking Using a Magnetic Tracking Sensor and Image Registration. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3750, pp. 543–550. Springer, Heidelberg (2005) 6. Gergel, I., et al.: Particle filtering for respiratory motion compensation during navigated bronchoscopy. In: Proc. SPIE MI 2010, vol. 7625, p. 76250W (2010) 7. Soper, T.D., et al.: In vivo validation of a hybrid tracking system for navigation of an ultrathin bronchoscope within peripheral airways. IEEE TBME 57, 736–745 (2010) 8. Lu´ o, X., Reichl, T., Feuerstein, M., Kitasaka, T., Mori, K.: Modified Hybrid Bronchoscope Tracking Based on Sequential Monte Carlo Sampler: Dynamic Phantom Validation. In: Kimmel, R., Klette, R., Sugimoto, A. (eds.) ACCV 2010, Part III. LNCS, vol. 6494, pp. 409–421. Springer, Heidelberg (2011) 9. Schwarz, Y., et al.: Real-time electromagnetic navigation bronchoscopy to peripheral lung lesions using overlaid CT images: the first human study. Chest 129, 988– 994 (2006) 10. Storn, R., Price, K.V.: Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11, 341–359 (1997) 11. Das, S., et al.: Differential evolution: A survey of the state-of-the-art. IEEE TEC 15, 4–31 (2011) 12. Liu, J., Lampinen, J.: A fuzzy adaptive differential evolution algorithm. Soft Comput.: Fusion Found., Methodologies Applicat. 9, 448–462 (2005) 13. Brest, J., et al.: Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems. IEEE TEC 10, 646–657 (2006) 14. Teo, J.: Exploring dynamic self-adaptive populations in differential evolution. Soft Comput.: Fusion Found., Methodologies Applicat. 10, 673–686 (2006) 15. Qin, A.K., et al.: Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE TEC 13, 398–417 (2009) 16. Zhang, J., Sanderson, A.C.: JADE: Adaptive differential evolution with optional external archive. IEEE TEC 13, 945–958 (2009) 17. Doucet, A., et al.: On sequential Monte Carlo sampling methods for bayesian filtering. Statistics and Computing 10, 197–208 (2000) 18. Price, K.V., Storn, R.M., Lampinen, J.A.: Differential evolution: A practical approach to global optimization. Springer, New York (2005) 19. Mezura-Montes, E., et al.: Modified differential evolution for constrained optimization. In: Proc. IEEE Congr. Evol. Comput., pp. 25–32 (2006)


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20. Wang, Z., Bovik, A.C.: A universal image quality index. IEEE Signal Processing Letters 9, 81–84 (2002) 21. Lacroute, P.: Real-time volume rendering on shared memory multiprocessors using the shear-warp factorization. In: Proc. IEEE Sym. on Para. Ren., pp. 15–22 (1995) 22. Luo, X., et al.: Towards hybrid bronchoscope tracking under respiratory motion: evaluation on a dynamic motion phantom. In: Proc. SPIE MI 2010, vol. 7625, p. 76251B (2010)