Informatica 39 (2015) 105-113 105 Differential Evolution Control Parameters Study for Self-Adaptive Triangular Brushstrokes Aleš Zamuda and Uroš Mlakar Faculty of Electrical Engineering and Computer Science, University of Maribor Smetanova ulica 17, SI-2000 Maribor, Slovenia E-mail: ales.zamuda@um.si, uros.mlakar@um.si Keywords: differential evolution, evolutionary computer vision, evolutionary art, image-based modeling, self-adaptation, triangular brushstrokes Received: December 1, 2014 This paper proposes a lossy image representation where a reference image is approximated by an evolved image, constituted of variable number of triangular brushstrokes. The parameters of each triangle brush are evolved using differential evolution, which self-adapts the triangles to the reference image, and also self-adapts some of the control parameters of the optimization algorithm, including the number of triangles. Experimental results show the viability of the proposed encoding and optimization results on a few sample reference images. The results of the self-adapting control parameters for crossover and mutation in differential evolution are also compared to results with keeping these parameters constant, like in a basic differential evolution algorithm. Statistical tests are furthermore included to confirm the improved performance with the self-adaptation of the control parameters over the fixed control parameters. Povzetek: V članku je predlagana izgubna predstavitev slike, kjer je referenčna slika aproksimirana z evoluirano sliko, ki je sestavljena iz spremenljivega števila potez trikotniškega čopiča. Parametre vsake poteze čopiča optimiramo s pomočjo diferencialne evolucije, ki samoprilagaja trikotniške poteze na referenčno sliko in prav tako samoprilagaja nekatere krmilne parametre samega optimizacijskega algoritma, vkljucno s številom trikotnikov. Rezultati poizkusov kažejo primernost predlagane metode in rezultati optimizacije so prikazani za vec izbranih referencnih slik. Rezultati samoprilagodljivih krmilnih parametrov za diferecialno evolucijo so primerjani tudi z rezultati, kjer so ti parametri nespremenljivi, kot je to primer pri osnovnem algoritmu diferencialne evolucije. Dodatno so podani še statisticni testi, ki nadalje potrjujejo izboljšanje kakovosti pristopa ob samoprilagajanju krmilnih parametrov v primerjavi s pristopom z nespremenljivimi krmilnimi parametri. 1 Introduction In this paper, evolvable lossy image representation utilizing an image compared to its evolved generated counterpart image, is proposed. The image is represented using a variable number of triangular brushstrokes [7], each consisting of triangle vertices coordinates and color parameters. These parameters for each triangle brush are evolved using differential evolution [13, 4], which self-adapts the control parameters, including the proposed self-adaptation for the number of triangles to be used. Experimental results show the viability of the proposed encoding and evolution convergence for lossy compression of sample images. Since this paper is an extended version of [8], new additional results are included, where the experiments results with fixed control parameters for differential evolution are included to check and demonstrate the self-adaptation mechanism influence on results. The results show clear superiority of the proposed approach with the self-adaptive control parameters over the approach where its control parameters are fixed. The approach presented is built upon and compared with [7], by addressing and also extending the original challenge. Namely, the challenge introduced in [7] uses triangles in trying to build an approximate model of an image [7]. The triangle is an efficient brush shape for this challenge, since it covers more pixels than a single point, and also allows overlaying and blending of colors over several regional surface pixels, which lines can not. Also, an arbitrary triangle shape is less constrained than any further point-approximated shape, and also other shapes can be built by combining several triangles. Instead of genetic programming in [7], in this paper differential evolution is used with a fixed size tree-like chromosome vector, which is cutoff self-adaptively to form codon and anti-codon parts of the chromosome. Also, our approach uses a modified challenge, where we can reconstruct the model for the reference image solely using the evolved model without using the reference image, whereas the [7] needs the reference image when drawing pixels to the canvas in deciding which pixels match the reference image for accepting them into the evolved canvas. Also, in this paper the triangle brushstroke encoding differs and is proposed especially designed for an efficient DE encoding. 106 Informatica 39 (2015) 105-113 A. Zamuda et al. In the following section, related work is presented, then the proposed approach is defined. In Section 4, the experimental results are reported. Section 5 concludes the paper with propositions for future work. 2 Related Work In this section, related work on evolutionary computer vision, evolutionary art, image representation, and evolutionary optimization using differential evolution, are presented. These topics are used in the proposed method, defined in the next section. 2.1 Image-Based Modeling, Evolutionary Computer Vision, and Evolutionary Art Image-based approaches to modeling include processing of images, e.g., two-dimensional, from which after segmentation certain features are extracted and used to represent a geometrical model [10]. For art drawings modeling, automatic evolutionary rendering has been applied [2, 12]. Heijer and Eiben evolved pop art two-dimensional scalable vector graphics (SVG) images [6] and defined genetic operators on SVG to evolve representational images using SVG, and also to evolve new images, different from source images, leading to new and surprising images for pop-art. Bergen and Ross [3] interactively evolved vector graphics images using genetic algorithm, where solid-coloured opaque or translucent geometric objects or mosaic tile effects with bitmap textures were utilized; they considered the art aspect of the evolved image and multiple possible outcomes due to evolution stochastics and concluded to investigate vector animation of the vectorized image. In [14] animated artwork is evolved using an evolutionary algorithm. Then, Izadi et al. [7] evolved triangular brushstrokes challenge using genetic programming for two-dimensional images, using unguided and guided searches on a three or four branch genetic program, where roughly 5% similarity with reference images was obtained on average per pixel. In this paper, we build upon and compare our new approach with [7], by addressing and also extending this challenge. After extending the challenge, we optimize it using DE, which is described in the next section. 2.2 Evolutionary Optimization Using Differential Evolution Differential evolution (DE) [13] is a floating-point encoding evolutionary algorithm for continuous global optimization. It has been modified and extended several times with various versions being proposed [5]. DE has also been applied to remote sensing image subpixel mapping [18], image thresholding [11], and for image-based modeling using evolutionary computer vision to reconstruct a spatial procedural tree model from a limited set of two dimensional images [16, 15]. DE mechanisms were also compared to other algorithms in several studies [17]. Neri and Tirronen in their survey on DE [9] concluded that, compared to the other algorithms, a DE extension called jDE [4], is superior to the compared algorithms in terms of robustness and versatility over a diverse benchmark set used in the survey. Therefore, we choose to apply jDE in this approach. The original DE has a main evolutionary loop where a population of vectors is computed within each generation. For one generation, counted as g, each vector xj, Vi € {1,..., NP} in the current population of size NP, undergoes DE evolutionary operators, namely the mutation, crossover, and selection. Using these operators, a trial vector (offspring) is produced and the vector with the best fitness value is selected for the next generation. For each corresponding population vector, mutation creates a mutant vector vijS+i ('rand/1' [13]): vi,g+1 + F (xr2]9 xr3,g ), (1) where the indexes r1, r2, and r3 are random and mutually different integers generated in from set {1,..., NP}, which are also different from i. F is an amplification factor of the difference vector, mostly within the interval [0,1]. The term xr2,g -xr3,g denotes a difference vector, which is named the amplified difference vector after multiplication with F. The mutant vector vi g+1 is then used for recombination, where with the target vector xi g a trial vector ui,j,g+1 is created, e.g., using binary crossover: vi,j,g+1, if rand(0, 1) < CR ui,j,g+1 = ^ or j = jrand, ; j „ otherwise, (2) where CR denotes the crossover rate, Vj € {1,..., D} is a j-th search parameter of D-dimensional search space, rand(0,1) € [0,1] is a uniformly distributed random number, and jrand is a uniform randomly chosen index of the search parameter, which is always exchanged to prevent cloning of target vectors. The original DE [13] keeps the control parameters fixed, such as F = 0.5 and CR = 0.9 throughout optimization. However, the jDE algorithm, which is a modification of the original DE, self-adapts the F and CR control parameters to generate the vectors vijg+1 and ui,g+1, corresponding values Fi and CRi, Vi € {1,..., NP} are updated prior to their use in the mutation and crossover mechanisms: p I Fi + randi x Fu if rand2 < ti , (3) i,n+1 1 Fig otherwise, CR i,n+i {rand% CRi,g if rand^ < T2, otherwise, (4) where {rand1,..., rand4} € [0,1] are uniform random floating-point numbers and t1 = t2 =0.1. Finally, the selection operator evaluates and compares the trial to current x Differential Evolution Control Parameters Study for. Informatica 39 (2015) 105-113 107 vector and propagates the fittest: x i,g+i Ui ,g+1 if / (Ui,g+l) 1 zx,y zx,y y=0 x=0 255 x RxRy Ay-1 Rx-1 \ E E 1 - zBB,y 1 y=0 x=0 255 x RxRy (17) 1000 Generation where Tk over (x, y) denotes each triangle being rendered over the pixel (x, y) such that bk,x,y contains the rendered pixels of a brushstroke. Triangles defined possibly over the edges of image canvas are drawn by clipping away pixels outside of the canvas area. The initialization of a genotype is such that the cx,cy, a1, a2, bY, 6Cb, bCr, TjL, and TU are initialized uniform randomly to integer values within their respective definition intervals, while r is kept as a floating-point. All parameters are however evolved as floating-point scalar values in DE. 3.3 Evaluation Evaluation of the phenotype image Zi to be compared against a reference image Z* is as follows. A reference image Z* is represented as RGB-encoded colored pixels integer values in layers Z* = {(zR,y, , zB,y)}. To obtain a difference assessment value, the following comparison metric is used for comparing an evolved image Z = Z, to Z*: Figure 2: Fitness convergence, for best runs of each test image. 4 Experiments The following experiments assess the viability of the approach on different control parameters, each with several independent runs. The parameter sets are as follows: the DE population size NP = {25,50,100} and Tmax = {10, 20,..., 150}, thereby for each run RNi ={0, 1, ..., 51} this counts for total of 45 parameter sets, i.e., 2340 independent runs. The NP and Tmax are fixed during one run. The maximum number of function evaluations (MAXFES) used is same as with [7], MAXFES is 105. For image rendering, basic GDI+ is used. 4.1 Obtained Results The obtained fitness values at the MAXFES termination of 105, over different parameters of Tmax and NP, are seen in Tables 1 and 2. The best values obtained overall for an image are marked in bold underlined text font. The fitness convergence graphs for these best runs are seen in Figure 2, where after the initialization, the fitness is roughly below 40 (i.e., 40% similarity with reference), then drops below 15 for all test images and even further to slightly above 6 for two of them. The convergent obtained results depend on the MAXFES used being same as with [7], but also NP and Tmax, as reported below. From Tables 1 and 2, we choose to report further evolved images up to MAXFES of 106 with all images. The best approximated images after MAXFES of 106 are shown in the Figure 3 which shows the evolution of the four images. In each line of Figure 3, the best fitting vectors upto MAXFES of 106 in generations g = {0,100,200,400, 700, 1200, 2000}, and the final generation, are shown, then the rightmost the corresponding reference image. Figure 4 shows for each test image, dynamics of the number of triangle brushes in current best vector during generations, displaying varying convergent best Ti values across images. Our approach searches for a representative image model and the values obtained such as 6.77, can roughly be compared to the 4.83 of [7]. Such representation of the problem k 0 500 1500 2000 Differential Evolution Control Parameters Study for. Informatica 39 (2015) 105-113 109 Figure 3: The evolved and the reference images (self-adaptive F and CR). also makes our NP parameter have higher value, since we have no guided search and the problem is therefore more general. Also, our approach does not use a dynamically re-allocatable morphable variable-size tree structure as in genetic programming encoding, inspite it rather uses a fixed size vector and limits its brushstrokes set by two simple bounds, making the approach faster for execution. For comparison purposes and since this paper is an extended version of [8], following additional comparison is included. The algorithm is run again with fixed control parameters F = 0.5 and CR = 0.9 in DE, all other settings are kept same as with the proposed above approach. Further, the results in Tables 1 and 2 are statistically tested using t-test with alpha = 0.001, against the null hypothesis, that the results obtained with fixed control parameters F = 0.5 and CR = 0.9 in DE, do not statistically differ. The symbol * with the values in bold text font signifies that the self-adaptive F and CR parameters approach results are significantly better and the symbol * with values in italicized text font signifies that the fixed parameters approach results are significantly better. Comparing the statistics on the varied NP and Tmax settings, DE with changing F and CR is 164 times better, 13 times worse, and 3 times with no significant performance difference, compared to the DE with F = 0.5, CR = 0.9. The Figure 5, the best DE run with F = 0.5, CR = 0.9, nonetheless still shows self-adaptation of the Ti parameter - this is an additional indicator that the performance difference lines in the changing of the F and CR control parameters, which, compared to fixed values, improve the approach performance if they are self-adaptive. Visually, the performance difference is observed from the rendered images in Figure 6, showing superiority of the proposed approach with self-adaptive control parameters over the approach using fixed control parameters. The Figure 7 shows fitness convergence of the best evaluated vector of the best DE run with F = 0.5, CR = 0.9, this Liberty Palace Vegetables Baboon 1000 Generation Figure 4: Number of brushstrokes in best vector, for best runs of each test image, self-adaptive F and CR parameters. Liberty.............. Palace ——— Vegetables .......... Baboon ................ 5- 400 600 Generation Figure 5: Number of brushstrokes in best vector, for best runs of each test image, F = 0.5, CR = 0.9. 80 70 60 - 50 40 30 20 0 500 1500 2000 30 25 20 0 0 200 800 1000 110 Informatica 39 (2015) 105-113 A. Zamuda et al. Table 1: Obtained fitness over Tmax and NP: test instances Liberty and Palace Liberty Palace Tmax Best Worst Average STD Best Worst Average STD 25 10 8.29 11.99 9.93096+ 0.8233 8.69 13.69 10.1362+ 0.9655 25 20 8.03 13.14 10.0935+ 1.0845 7.83 11.5 9.12173+ 0.8092 25 30 8.41 13.74 10.0525+ 1.1712 7.52 11.1 8.97942+ 0.7992 25 40 8.13 12.81 10.4408+ 1.1416 7.34 11.36 8.91788+ 0.8922 25 50 8.49 13.37 10.6767+ 1.1768 7.65 12.53 8.87442+ 0.9788 25 60 7.95 14.65 10.9858+ 1.4284 7.9 11.88 8.99673+ 0.8761 25 70 8.28 14.21 11.4075+ 1.3630 7.79 13.17 9.50327+ 1.0482 25 80 8.72 15.89 11.7554+ 1.6330 7.97 12.34 9.43558+ 0.9765 25 90 8.84 16.24 12.1342+ 1.6608 8.41 13.54 9.82+ 1.2756 25 100 9.01 16.74 12.4798+ 1.7521 8.62 12.96 9.83635+ 0.8869 25 110 8.07 16.78 12.7412+ 1.7849 9.01 14.42 10.4119+ 1.2468 25 120 9.67 16.14 12.8467+ 1.7359 8.93 15.13 10.3858+ 1.3149 25 130 10.16 17.96 13.2692+ 1.7193 9.02 14.2 10.2858+ 1.0292 25 140 9.29 17.99 13.7029+ 1.7886 8.29 13.51 10.7779+ 1.0299 25 150 10.82 18.56 14.0373+ 1.6573 9.89 14.91 11.1206+ 1.0586 50 10 7.51 9.69 8.45077+ 0.4198 7.43 11.84 8.68058+ 0.8825 50 20 6.78 8.99 7.80173+ 0.4987 7.1 11.39 8.79173+ 0.9592 50 30 6.89 9.17 7.81788+ 0.5119 7.53 12.58 9.75654+ 1.1186 50 40 6.77 9.87 8.0375+ 0.6578 8.27 12.24 10.0575+ 0.9537 50 50 7.08 10.61 8.39923+ 0.7056 7.97 13.14 10.3338+ 1.1009 50 60 7.15 10.4 8.67115+ 0.7472 8.59 12.49 10.7817+ 1.0754 50 70 7.46 10.9 9.1025+ 0.8666 7.58 12.8 10.7744+ 1.1086 50 80 7.6 11.4 9.47981+ 0.8689 9.15 13.11 11.3802+ 1.0178 50 90 8.05 12.65 9.67346+ 0.9115 9.97 13.41 11.5227+ 0.9315 50 100 8.75 11.75 10.0152+ 0.7824 8.55 13.62 11.4356+ 0.9923 50 110 8.93 13.63 10.6356+ 0.9682 9.32 13.77 12.0712+ 0.9579 50 120 9.22 13.01 10.7502+ 0.9840 9.77 14.21 12.429+ 0.8972 50 130 9.42 12.59 11.0527+ 0.7707 11.37 14.07 12.7387+ 0.6134 50 140 9.99 13.39 11.5719+ 0.7815 9.69 15.5 12.9317+ 0.9708 50 150 10.2 14.56 12.2633+ 1.0702 9.58 15.36 12.8092+ 1.1717 100 10 7.1 9.12 7.98596+ 0.4241 7.91 13.88 10.9573+ 1.8019 100 20 6.85 9.77 7.83962+ 0.5360 8.86 14.59 12.1117+ 1.2862 100 30 7.15 11.8 8.49077+ 1.1563 9.59 16.15 12.9098+ 1.0589 100 40 7.22 13 8.86327+ 1.1092 9.65 14.97 13.2477+ 1.1543 100 50 7.41 12.75 9.34846+ 1.3939 11.01 15.52 13.8606+ 0.9750 100 60 8.06 12.97 9.77731+ 1.1539 11.5 16.14 14.1856+ 1.1234 100 70 8.67 13.28 10.1954+ 1.3722 10.77 16.32 14.3629+ 1.1713 100 80 8.73 14.48 11.0929+ 1.4093 10.98 17.06 14.9348+ 1.1679 100 90 9.04 14.92 11.3594+ 1.3483 11.1 16.8 15.104+ 1.2586 100 100 9.4 16.13 11.6604+ 1.4952 10.8 17.62 15.36 1.2330 100 110 10.17 15.68 12.3365+ 1.5685 13.01 17.86 16.0202+ 0.9744 100 120 10.26 15.45 12.3358+ 1.5076 11.07 17.99 15.6113^ 1.6455 100 130 10.22 16.19 13.2212+ 1.6108 12.33 18.37 16.4085+ 1.3168 100 140 11.42 16.65 13.7808+ 1.5502 11.64 18.35 16.1229+ 1.4990 100 150 11.35 18.68 14.6113+ 1.9726 10.11 18.34 16.2929 2.0056 Differential Evolution Control Parameters Study for. Informatica 39 (2015) 105-113 111 Table 2: Obtained fitness over Tmax and NP: test instances Vegetables and Baboon Vegetables Baboon Tmax Best Worst Average STD Best Worst Average STD 25 10 14.13 17.21 15.7269+ 0.7148 15.02 18.59 16.381 0.7128 25 20 12.56 18.03 14.5658+ 0.9850 13.44 17.12 15.3815+ 0.8129 25 30 12.33 15.98 13.9215+ 0.8475 12.99 19.03 15.0204+ 1.1150 25 40 11.62 16.21 13.674+ 1.0436 11.99 16.85 14.4342+ 1.0135 25 50 12.16 17.08 13.88+ 1.0726 11.39 17.62 14.4573+ 1.2299 25 60 11.64 17.88 13.6438+ 1.2155 11.74 17.51 14.8038+ 1.2229 25 70 11.29 17.15 13.9056+ 1.3790 11.88 17.9 14.6267+ 1.3495 25 80 11.61 16.6 14.0871+ 1.3881 12.11 17.13 14.3606+ 1.2815 25 90 11.63 17.96 14.1062+ 1.4428 11.93 19.41 14.6644+ 1.5269 25 100 11.34 17 14.4533+ 1.4694 11.7 18.77 14.7642+ 1.7438 25 110 11.74 19.66 14.6085+ 1.7664 12.02 19.11 15.0046+ 1.7605 25 120 12.26 17.91 14.7737+ 1.5726 12.2 18.5 15.6467+ 1.6086 25 130 12.1 19.75 14.6338+ 1.9283 13.01 19.5 15.4254+ 1.5505 25 140 11.94 19.01 14.7635+ 1.6282 12.64 19.37 15.8235+ 1.8458 25 150 12.82 18.7 14.6487+ 1.3015 13.13 20.17 15.7952+ 1.6923 50 10 13.03 15 14.0723+ 0.4674 13.86 16.52 14.9192+ 0.5494 50 20 11.66 13.26 12.4644+ 0.3184 11.8 14.54 13.271+ 0.5569 50 30 11.12 13.59 12.2425+ 0.6528 11.59 13.62 12.5506+ 0.5732 50 40 10.94 14.1 12.1848+ 0.6656 11.1 13.84 12.3137+ 0.6090 50 50 11.04 13.92 12.2946+ 0.7609 11.34 14.36 12.4075+ 0.6304 50 60 11.29 15.86 12.5506+ 0.9222 11.25 14.1 12.3662+ 0.6161 50 70 11.18 15.21 12.6104+ 0.8682 11.54 14.57 12.5437+ 0.6510 50 80 11.32 15.26 12.8619+ 0.7658 11.07 15.56 12.9473+ 0.8087 50 90 11.84 15.28 13.0077+ 0.8038 11.32 16.2 12.857+ 1.0291 50 100 11.72 15.8 13.5058+ 0.9565 11.85 15.72 13.2658+ 0.7972 50 110 12.02 15.92 13.5204+ 0.8750 11.98 15.56 13.4275+ 0.7805 50 120 11.9 16.87 13.829+ 1.1151 12.43 15.66 13.5106+ 0.7265 50 130 12.51 15.97 14.094+ 0.8855 12.64 16.32 14.085+ 0.8259 50 140 12.16 17.07 14.8198+ 1.2154 12.54 16.31 14.15+ 0.8865 50 150 13.11 17.98 14.9838+ 1.2072 13.08 18 14.8765+ 1.0178 100 10 12.56 16.19 13.9815+ 0.8083 13.49 16.19 14.5367+ 0.5672 100 20 11.84 16.45 13.4704+ 1.0483 12.02 15.87 13.8244x 0.8747 100 30 11.83 17.64 13.9133+ 1.3335 12 15.76 13.7206+ 0.9727 100 40 12.01 17.95 14.6354+ 1.3660 11.63 17.01 13.6467+ 1.3582 100 50 11.87 17.35 14.9156+ 1.4272 11.99 17.48 14.1658+ 1.5554 100 60 12.32 18 15.21+ 1.5119 12.12 17.46 14.5021 + 1.4517 100 70 12.13 18.05 15.6513+ 1.2457 12.12 17.16 14.3881+ 1.3782 100 80 12.9 18.86 16.2008+ 1.4121 12.13 17.56 14.8656+ 1.4214 100 90 12.32 20.04 16.3233+ 1.7789 12.25 18.66 15.2558+ 1.5144 100 100 12.98 20.55 16.7275+ 1.7119 13.09 18.42 15.5398+ 1.5064 100 110 13.76 20.18 17.2896+ 1.5242 13 19.62 15.84+ 1.6164 100 120 13.12 20.62 17.626+ 1.5807 13.34 19.58 16.4725+ 1.5223 100 130 13.52 20.12 17.9052 1.3516 13.84 19.6 16.9367+ 1.7362 100 140 14.08 20.52 18.216+ 1.6975 14.3 21 17.4387+ 1.7372 100 150 14.97 21.19 19.1221 1.2128 14.75 21.13 17.9488+ 1.6872 112 Informatica 39 (2015) 105-113 A. Zamuda et al. Figure 6: The evolved and the reference images, F = 0.5, CR = 0.9. time with NP =100 and therefore maximum generation number of 1000. The attained values tend to converge towards Tmax, but results are worse since the different Tmax, seen from Figures 4 and 5. 5 Conclusion This paper presents an evolvable lossy image representation, approximating an image by comparing it to its evolved generated counterpart image. The image is represented using a variable number of triangular brushstrokes, each consisting of a triangle position and color parameters. These parameters for each triangle brush are evolved using differential evolution, which self-adapts the control parameters for mutation and crossover. Also, the proposed DE extension splits the DE vector in the codon and anticodon parts, where the triangles material is used only from the codon part, adjusting the genetic tree center and its borders, together with the number of triangle brushstrokes to be rendered. Experimental results show the viability of the proposed encoding and evolution convergence for the lossy representation of reference images, where fitness is displayed dependent on the population size, maximal number of function evaluations allowed, maximal number of triangles used in image representation, and different input reference images. While analyzing the NP and Tmax, moreover in this paper, we have shown that the self-adaptive jDE control parameters handling mechanism is preferable to the fixed control parameters mechanism from the original DE. Future work can include increasing MAXFES, addressing different encoding aspects, evolutionary operators, control-parameters update, Euclidean distance for colors comparison, and more case studies on input images with different properties. 400 600 Generation Figure 7: Fitness convergence, for best runs of each test image, F = 0.5, CR = 0.9. Acknowledgement This work is supported in part by Slovenian Research Agency, project P2-0041. References [1] B. D. Ackland, N. H. Weste (1981) The edge flag algorithm - a fill method for raster scan displays, IEEE Transactions on Computers, vol. 100, no. 1, pp. 4148. [2] P. Barile, V. Ciesielski, M. Berry, K. Trist, (2009) Animated drawings rendered by genetic programming, Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), pp. 939-946. [3] S. Bergen, B. J. Ross (2012) Automatic and interactive evolution of vector graphics images with genetic algorithms, The Visual Computer, vol. 28, no. 1, pp. 35-45. 0 200 800 1000 Differential Evolution Control Parameters Study for. Informatica 39 (2015) 105-113 113 [4] J. Brest, S. Greiner, B. Boskovic, M. Mernik, V. Zumer (2006) Self-Adapting Control Parameters in Differential Evolution: A Comparative Study on Numerical Benchmark Problems, IEEE Transactions on Evolutionary Computation, vol. 10, no. 6, pp. 646657. [5] S. Das, P. N. Suganthan (2011) Differential Evolution: A Survey of the State-of-the-art, IEEE Transactions on Evolutionary Computation, vol. 15, no. 1, pp. 4-31. [6] E. den Heijer, A. E. Eiben (2012) Evolving pop art using scalable vector graphics, Evolutionary and Biologically Inspired Music, Sound, Art and Design, Springer, pp. 48-59. [7] A. Izadi, V. Ciesielski, M. Berry (2011) Evolutionary non photo-realistic animations with triangular brushstrokes, AI2010: Advances in Artificial Intelligence, Springer, pp. 283-292. [8] U. Mlakar, J. Brest, A. Zamuda (2014) Differential Evolution for Self-adaptive Triangular Brushstrokes, Proceedings of the Student Workshop on Bioinspired Optimization Methods and their Applications (BIOMA), pp. 105-116. [9] F. Neri, V. Tirronen (2010) Recent Advances in Differential Evolution: A Survey and Experimental Analysis, Artificial Intelligence Review, vol. 33, no. 1-2, pp. 61-106. [10] L. Quan (2010) Image-Based Modeling, 1st edition, Springer. [11] S. Rahnamayan, H. R. Tizhoosh (2008) Image thresholding using micro opposition-based Differential Evolution (Micro-ODE), Proceedings of the World Congress on Computational Intelligence (WCCI), pp. 1409-1416. [12] J. Riley, V. Ciesielski (2010) Fitness landscape analysis for evolutionary non-photorealistic rendering, Proceedings of the Congress on Evolutionary Computation (CEC), pp. 1-9. [13] R. Storn, K. Price (1997) Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization, vol. 11, pp. 341-359. [14] K. Trist, V. Ciesielski, P. Barile (2010) Can't see the forest: Using an evolutionary algorithm to produce an animated artwork. Arts and Technology, Springer, pp. 255-262. [15] A. Zamuda, J. Brest (2014) Vectorized procedural models for animated trees reconstruction using differential evolution, Information Sciences, vol. 278, pp. 1-21. [16] A. Zamuda, J. Brest, B. Boskovic, V. Zumer (2011) Differential Evolution for Parameterized Procedural Woody Plant Models Reconstruction, Applied Soft Computing, vol. 11, no. 8, pp. 4904-4912. [17] K. Zielinski, R. Laur (2007) Stopping criteria for a constrained single-objective particle swarm optimization algorithm, Informatica, vol. 31, no. 1, pp. 51-59. [18] Y. Zhong, L. Zhang (2012) Remote sensing image subpixel mapping based on adaptive differential evolution, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 42, no. 5, pp. 1306-1329.