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Geometric rosette patterns analysis and generation
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Original article
Geometric rosette patterns analysis and generation Abdelbar Nasri a,∗ , Rachid Benslimane a , Aziza El Ouaazizi b a b
Laboratory of Data Transmission and Processing, Sidi Mohamed Ben Abdellah University, EST, Route d’Imouzzer, BP 2427, Fès, Morocco Laboratory of Engineering Sciences, Sidi Mohamed Ben Abdallah University, FP Taza, BP 1223, Taza Gare, Morocco
a r t i c l e
i n f o
Article history: Received 24 February 2016 Accepted 22 December 2016 Available online xxx Keywords: Star pattern Adjacency graph Rotational symmetry Islamic geometric pattern Genetic algorithm Tiling-based approach
a b s t r a c t The geometric rosettes, which are the most known design elements of the Islamic rosette patterns, are usually tessellated in a concealed composition structure. To understand and reveal this structure, we propose first to detect and characterize its basic geometric rosettes by using techniques of computer vision and image analysis. Then, the analysis of the spatial arrangement of the detected rosettes, characterized by their respective orders, will reveal the underlying tiling and the mesh grid, together with the harmonious proportions of the design elements. These results are used in turn to generate new innovative and authentic rosette patterns, by using the extracted geometric rosettes and new tile motifs constructed in the basis of an adaptation of the well-known polygonal technique. The performances of the proposed method to reveal the spatial composition of a rosette pattern are tested by the ability to extract its geometric rosettes and by the exact extraction of its underlying composition structure. Finally, the innovative character of the proposed generative method is shown through the creation of new periodic and quasi-periodic patterns characterized by their authenticities and sophistication. © 2017 Elsevier Masson SAS. All rights reserved.
1. Introduction The artisans of Islamic world continue to perpetuate the practice and the development of Islamic Geometric Patterns (IGP) by preserving the long established tradition consisting to reveal their practical knowledge to a few apprentices. However, as the time passes, the traditional methods of transmission of this ancestral art may be threatened or disrupted. In this context, its preservation as well as its development will be impacted. Now re-finding or renewing these innovative methods is a concern for today’s researchers in the field; this paper is concerned with the use of computer science and related technologies to approach the related problems. The analysis IGPs shows that they often take the form of a division of the plan into star-like motifs. These commonly used patterns are known as the Islamic star patterns. The geometric rosette is among the many different star-like motifs used in Islamic art, which stands out as distinctively “Islamic”. As shown in Fig. 1i, a geometric rosette is defined by 3 concentric circles that pass respectively by the set of points a, b and c. The symmetry order, which is the main distinguishing characteristic of the geometric rosette, is connected with the number of points a, b or c. Fig. 1ii and iii illustrates two
∗ Corresponding author. E-mail address: [email protected] (A. Nasri).
geometric rosettes characterized respectively by 12 and 16 folds symmetry. Fig. 2 gives examples of three types of spatial arrangement of geometric rosettes and stars (design elements). These geometric patterns show that the design elements are tessellated in complex geometrical arrangements, usually in tiles of a tiling, which is used as an underlying structure for the spatial composition. Often, the craftsmen influence the way in which a composition is perceived by emphasizing pattern elements (star/rosette motifs, tiling grid) by the use of the color. Fig. 2a shows that the rosette motifs, together with the used tiling are highlighted. However, Fig. 2b highlights only the motifs; the used tiling is not explicitly revealed. Otherwise, it is obvious to see that for all the star patterns, rosette motifs and star motifs are the visual anchors that appear in all the compositions. Thus, this paper will focus first on the extraction and characterization of these motifs. The spatial representation of the extracted patterns is then operated by using adjacency graphs. Finally, the analysis of this adjacency graph leads to the determination of some pattern features to be used in the characterization of the pattern and in the process to generate new patterns. The rest of the paper is organized as follows: Section 2 is concerned with related works focusing methods of analysis and generation of IGPs. Section 3 presents our proposed methods for respectively analyzing and generating a rosette pattern. The proposed analysis method is concerned with respectively the
http://dx.doi.org/10.1016/j.culher.2016.12.013 1296-2074/© 2017 Elsevier Masson SAS. All rights reserved.
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Fig. 1. Examples of geometric rosettes: (i) synthetic rosette according to A.J Lee [13] a: outer points, on circum-circle; b: outer midpoints, on outer mid-circle; c: inner midpoints, on inner mid-circle; d: inner points, on in-circle; o: center of rosette; (ii, iii): geometric rosettes of Zellij style in Fez – Morocco, (ii) rosette with 12 petals, (iii) rosette with 16 petals. R. Benslimane.
Fig. 2. Examples of three types of spatial arrangement of geometric rosettes: Zellig patterns in Fez-Morocco. Photos: R. Benslimane.
extraction and recognition of the design elements, the discovering of the underlying spatial composition structures and their representation. Based on the outputs of the analysis phase, the proposed generative method gives a step by step procedure to generate new periodic and a periodic rosettes patterns. Experimental results and discussion are presented in Section 4, and finally, conclusion and perspectives are drawn in Section 5. 2. Related works Analytical and generation approaches of IGPs can be divided into three main categories: the tiling-based approach, the symmetry groups based approach and the strand-based approach: • symmetry groups based approach: methods of this approach represent a periodic pattern with its underlying lattice and with its symmetry group [1,2]; • strand-based approach: methods of this approach consider a periodic pattern as drawing lines (Strands); which are linked segments at V-model corners [3,4]; • tiling-based approach: this tiling approach relies mainly on the analysis and generation of ornaments by considering the character of the motifs inscribed in polygons which form the tiling. In Islamic design traditions, tiling is used as the underlying structures for a composition. A tiling is an arrangement of polygons
that can contain design elements. For the star patterns, which are the most sophisticated ones in Islamic art, the well-known design elements are the geometric rosettes. There are two categories of tiling: tiling using polygons that are periodic and those that are non-periodic. A periodic tiling is a tiling of the Euclidean plane with periodic symmetry. It is proven that all 2D periodic patterns extended by two linearly independent translational vectors can be classified, depending on the type of network or lattice, into five types of unit cells and seventeen symmetry groups [5]. One of the first Western studies of the tiling-based approach was published by E.H. Hankin [6]. The Hankin’s technique, known also as polygons in contact technique, was a reference for several works. Kaplan in [7,8] provided a method for rendering Islamic Star Patterns based on Hankin’s Polygonal technique. The method builds star patterns from a tiling of the plan and a small number of intuitive parameters. J. Bonner more recently proved the historical use of this construction technique from his examination of the Topkapi Scroll. Bonner called this polygonal technique. The use of this technique generates a multitude of patterns from a single underlying polygonal tessellation. He proposed in [9] and [10] to create patterns from systematic and non-systematic geometric constructions. He classified systematic patterns into five distinctive systems: the 3-46-12 system of regular polygons, the 4-8 system A, the 4-8 system B, the 5-10 system, and the 7-14 system. The non-systematic patterns are characterized by more than a single region of higher order star-forms. Castera in [11] summed up the rules for the Moroccan Zellij patterns construction, when he introduced the concept of skeleton. The skeleton is an underlying structure, made up of the alternation of two tiles: the octagonal star and an irregular hexagon called “Saft”. Fig. 2a gives an example of skeleton. The point is that the skeleton follows the outline of a network similar to a tiling. This particular tiling is constituted by polygons and different star shapes. In this tiling-based approach, motifs inscribed in the polygons in contact can be considered as separate mini-compositions. Recognizing and characterizing these design elements and their spatial arrangement opens the way to a better understanding of the design process. This paper proposes, first, to detect these motifs by using an improved version of our recent developed method [12]. Then, we propose to represent the spatial relationship between the detected motifs by an adjacency graph. From the analysis of this spatial representation, we finally extract the hidden tiling, the repetitive unit cell and other pattern settings. An original generative method, based on the analysis outputs, is finally proposed to create new rosette patterns.
3. Proposed method for rosette pattern analysis and generation In this paper, we propose a new method to analyze and generate a rosette pattern based on two stages, the first stage consist to analyze an input image of a rosette pattern by using image analysis techniques. The outcomes of this analysis stage (adjacency graph, hidden tiling, and other pattern settings) are used in turn in the second stage for the generation of new rosette patterns.
3.1. Proposed method for rosette pattern analysis In [12], we proposed an original method to detect and characterize rosettes of a geometric rosette by performing the following stages:
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3.1.2. Rosettes detection by using a Genetic algorithm In an x – y Cartesian coordinate system, a circle with center ‘O’ of coordinates (a, b) and radius r is the set of all points (x, y) such that: 2
f (x, y) = (x − a)2 + (y − b) − r 2 = 0 This equation can be written in parametric form using the trigonometric functions sine and cosine as:
x = a + r cos
∈ [0, 2]
y = b + sin
Fig. 3. Flowchart of the proposed method for rosette pattern analysis.
• image pattern segmentation by using a global threshold operation; • rosette center detection by using a genetic algorithm that detect the optimal circle passing by the maximum points of the pattern binary image; • determination of the detected rosette order by using Frieze expansion technique. As shown in Fig. 1i, and according to A.J. Lee [13], a rosette is characterized by sets of interest points (a, b, c, d) that correspond to corners. The points of each set are radially distributed around the center of the rosette. Thus, to improve the rosette detection phase, it is better to take into consideration the sets of corners (a, b, c, d) instead of all the points generated from the segmentation operation considered in our previous method. For this purpose, we propose in this paper to start the process of the rosette detection by detecting first the corners of the pattern image. To identify circles delimiting the rosettes of the pattern, a genetic algorithm is then applied to the detected corners. It is important to underline the fact that there exists more than one rosette or star in a star pattern. To detect all these stars or rosettes, the proposed method detects the corresponding circles sequentially by applying an iterative procedure. The corners participating to the extracted circle at the current iteration are removed from the image permitting the detection of the rest of circles corresponding to the other rosettes to be detected at the next iterations. To determine the order of each rosette, the area of the detected rosette is cropped and then transformed on a Frieze expansion representation. Finally, the spatial arrangement of the detected rosettes is represented by an adjacency graph. In this purpose, each detected rosette is represented by a graph node and the relationship between two rosettes/stars is represented by an arc. Fig. 3 illustrates the proposed method for rosette pattern analysis; details of each step are presented in the next sub-sections. 3.1.1. Corner detection A large number of corner detectors have been proposed in the literature. They can be grouped broadly into three categories: Gray-level based methods [14–19], contour based methods [20–24] and parametric model based methods [25–27]. In this paper, we use the Förstner method presented in [23] and its implementation described in [24,25,28]. The choice of this descriptor has been motivated by the comparison studies proposed in [27] and also by the results obtained on images of our tested database. Figure A.1 from the supplementary material [29] gives examples of corner detection made by the Förstner operator.
where is a parametric variable in the range 0 to 2, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis. To find the circle passing by the maximum of corners, we propose to use a genetic algorithm, which can achieve to the optimal circle parameters (a, b, r) by maximizing the following criterion:
U (a, b, r) =
B (x, y) ds f
where B (x, y) is the corner point (x, y). The individuals of our genetic algorithm will represent circles by three coded genes a, b and r that we shall code in binary to its chromosome structure. Individual Chromosome
a ␣1 ␣2 . . .
b ˛1 ˛2 . . . ˛i
r ˛1 ˛2 . . . ˛l
where: ␣i is a binary value and i, j, l are the numbers of bits coding respectively a, b and r. The GA starts with a randomly generated population of N pairs of individuals, and in each generation, a new population of the same size is generated from the current population by applying operators, termed selection, crossover and mutation, which mimic the corresponding processes of natural selection. The algorithm is terminated if no improvement in the fitness value of the best individual for a fixed number of iterations and the best chromosome is taken to be the optimal one. The old population of GA is completely replaced by the offspring population after reproduction, crossover, and mutation. In this way, it is possible that the best chromosome in the old population disappears in the current generation. A reasonable improvement is reached by preserving the best string obtained so far in a separate location outside the population so that the algorithm may report the best value found, among all solutions inspected during the whole process. In the present work, we have used the elitist strategy of [30] where the best string in the previous iteration is copied into the current population. The genetic algorithm is stopped if no improvement in the best individual during 200 generations is give. Figure A.2 in supplementary material [29] and Fig. 9c give examples of detected circles by using genetic algorithm with the following setting parameters: • • • •
population size Ni = 40; one randomly selected crossover with probability Pc = 0.75; randomly mutation with probability Pm = 0.35; stochastic remainder roulette rule selection with elitist strategy.
The proposed genetic algorithm is applied n times depending on the number of rosettes and stars forming the analyzed pattern. For a current iteration, one circle (center, radius) can be detected and the interest points that participated to its formation are then removed. The next iteration performs the genetic algorithm on the rest of the interest points. This iterative process is stopped if no circle is detected in the current iteration.
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Fig. 4. Illustration of order detection and different cases of the motif circumscribed by the detected circles.
3.1.3. Rosette characterization: determination of the geometric rosette symmetry order Rosettes are rotationally symmetrical patterns, so a simple test of symmetry with the frieze expansion technique allows to check if the finding pattern circumscribed in the detected circle is a rosette or not. Fig. 4 gives an illustration of this technique. Extensive explanations are given in our work [12]. This test can discriminate between the possible following cases: • circles that correspond to rosettes: circles circumscribing symmetrical motifs. In this case, to determine the rosette order, Frequency analysis using Discrete Fourier Transform is applied to the Frieze expansion pattern; • circles that are no rosettes: circles circumscribing nonsymmetrical motifs. These circles may correspond to overlapping rosettes. 3.1.4. Analysis and representation of the spatial composition of a rosette pattern The occurring structures underlying a rosette pattern composition are the mesh grid and the tiling. A mesh grid represents the spatial arrangement of main rosettes (identical rosettes having the highest symmetry order and the greatest radius). However, the tiling represents the spatial arrangement of satellite rosettes (rosettes having the lower symmetry order and the smallest radius). The green network presented in Fig. 5 represents the grid of the pattern. This grid underlies the spatial arrangement of the main rosettes. The red polygons represent the tiling, in which the satellite rosettes are located at their corners. By taking into consideration both the occurring composition structures, it will be easy to identify the relationships between
adjacent rosettes (neighboring rosettes). Properties of the adjacent rosettes define the type of the underlying structure to be identified. Therefore to identify respectively grid and tiling, we have to identify couples of rosettes Xi and Xj , which are adjacent and respectively characterized by the following properties P1 and P2 : • P1 : Xi and Xj are the main rosettes; • P2 : Xi and Xj are the satellite rosettes. It is important to underline that the main rosette is circumscribed by n satellite rosettes. This local composition, named ‘constellation’, involves the definition of the following third property: • P3 : Xi is a main rosette and Xj is a satellite rosette. The number ‘n’ of the satellite rosettes is a divisible number of the main rosette order. The satellite rosettes are centered on the vertices of the n sides tile. In addition to the known relationship between the number of the satellite rosettes and the main rosette symmetry order, the following relationships can be defined: • the relationship between the radius and the order of the main rosette; • the relationship between main rosette parameters (order and radius) and the number of the satellite rosettes. The inter-rosettes relationships can be represented by an adjacency graph or by an adjacency matrix ‘M’ whose coefficients 0 and 1 are defined as follows [31,32]:
Mi,j =
1 if Xi RXj
0 otherwise
(Xi R Xj ) if Xi and Xj are neighbors and are defined by a property Pi . Thus, to represent a rosette pattern by an adjacency graph, we propose the following three-stage method:
Fig. 5. Example of composition using 10-fold rosette as main rosette and 10-fold stars as satellites motifs.
• stage 1: labeling rosettes according to their parameters X (center, order, radius). It is important to underline that the mismatch between circles and interests points, as illustrated in Fig. 6, is due to the fact that the circle is a continuous function while the points of interest belong to a discrete grid; • stage 2: constructing an adjacency matrix: Connecting neighboring rosettes Xi and Xj characterized by the property P2 (Xi and Xj have the same order and the same radius r1 ) with applying the following test:
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Fig. 6. Flowchart of the proposed method for the spatial composition analysis and representation by an adjacency graph.
◦ if the distance d (Xi , Xj ) = ws ± → Mij = 1, ◦ else → Mij = 0, where ws is the most occurred distance calculated from the map of distance Mapd (i, j) = d (Xi , Xj ); • stage 3: representing the adjacency matrix by its corresponding adjacency graph. The flowchart of this proposed method is illustrated in Fig. 6. Fig. 5 gives an example of a Moroccan Zellij pattern for which the tiling is revealed by connecting the 10-fold stars, considered here as satellites motifs of a 10-fold main rosette. As illustrated in Fig. 5, the arrangement of the main rosette centers reveals a hidden mesh grid defined by a smallest region or a repetitive unit cell (Bravais lattice). When translated by 2 lattice vectors, the unit cell can cover the entire plane without overlapping. The underlying grid may be rectangular or centered rectangular (cf. Fig. 5) depending on whether the color attribute is taken into consideration or not. Generally, there are five distinct Bravais lattices: square, rectangular, centered rectangular, hexagonal (rhombic) and oblique. As shown in Fig. 5, the spatial arrangement of the satellite rosettes reveals a polygonal distribution of their centers. The revealed underlying polygonal tessellation is known as the tiling. The analysis of this spatial composition can be supplemented by metric parameters defining the right proportions between the constituent parts of a geometric pattern. These parts are harmoniously proportioned in relation to each other and in relation to the whole. To discover the inner logic of proportion used by the Islamic designers, we propose to establish the relationships between the satellite rosettes attributes (r1 radius of symmetry and order o1 ), the main rosettes attributes (r2 , o2 ) and the attributes of the principal polygon circumscribing the main rosette (Sn , Rn ). Referring to the illustration of the Fig. 7, it is easy to see that: Rn =
Sn 2 sin
180 = r1 + r2 n
n is the number of sides of the circumscribing polygon. Sn is the length of a side: Sn = 2r1 + d d is the distance between two consecutive satellite rosettes. 3.2. Proposed generative method The proposed generative method adopts an adapted version of the polygonal technique proposed by J. Bonner. The proposed
Fig. 7. Parameters defining the relationship between central rosette and its satellite rosettes.
method covers the Eastern Islamic geometric patterns as well as the Moroccan and Andalusian ones which are designed by the traditional technique named ‘Tastir’ [33]. Otherwise, the main difference with the Bonner’s method concerns the procedure adopted to design motifs of the considered tiles. Therefore, the proposed method takes into consideration two cases: • the first one concerns the generation of new patterns by considering other spatial arrangements of the tiles, extracted from the underlying tiling of the analyzed pattern. In this case, the generation process consists to tessellate the plane with the extracted tiles in different ways. For the tiles extraction, we use the algorithm proposed in [34,35]. At this stage, the tessellation of the extracted tiles is a hand drawing. A clear opportunity exists to automate this pattern generation step algorithmically; • the second case concerns the generation of new patterns by tessellating the extracted tiles and additional ones selected among those of the corresponding Bonner’s polygonal system. As mentioned in related work, J. Bonner has defined five distinctive systems: the 3-4-6-12 system of regular polygons, the 4-8 system A, the 4-8 system B, the 5-10 system, and the 7-14 system. The generation process in this case is based in the following five steps: ◦ extraction and identification of the tiles of the tiling. These polygonal elements of tiling, in the logic of adjacency graph, are the shorter closed path of the planar graph [34,35], ◦ finding the Bonner’s polygonal system corresponding to the identified tiles, ◦ selecting the tiles to be used for the pattern to be created, ◦ constructing the template motif of each selected tile. For this purpose, the proposed method is based on two stages:
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the unit cell is not addressed in this paper. This issue is already addressed in our recent work [36,37]. It is easy to see in Fig. 9j that the design polygonal elements of the original pattern presented in Fig. 9a are the decagon motif and the concave hexagon motif presented in Fig. 10.
Fig. 8. Construction process of interstitial motif.
– construction of the polygonal motif. The proposed method consists to center n satellite rosettes on the corners of the considered tile and then to take into consideration the relationship between two neighboring rosettes. This relationship can be adjacent, disjoint or in overlapping. Table A.1 in supplementary material [29] gives results of these relationships, – construction of the interstitial motif by filling the gap (the space left blank into the polygonal motif). The construction of this motif is commonly used for all generative methods using tiling approach. The most known method to fill the gap is the inference algorithm proposed by Kaplan in [8]. For the Moroccan Zellij style patterns, considered in this work, the situation is easy, and does not require the use of such a sophisticated solution as the Kaplan inference algorithm. For the proposed method, two solutions can be considered depending on the gap’s size. If this size is less than that of one rosette, then the extension rule is applied. Else, the size of the gap is certainly larger than that of one or two overlapped rosettes. In this case, the extension rule can be applied after one or two overlapped rosettes are placed on the gap. The extension rule consists to extend each edge eik of a free rosette’s spike (Pi ) and cut it off at point Cij when it meets another extended edge ejk’ of a free spike Pj . If the edges eik (k = 1,2) and ejk (k’ = 1,2) are collinear; connect Pi and Pj at the middle of the segment Pi , Pj . Thus, the segments Pi Cij and Pj Cij will be added to the template motif. Fig. 8 gives an illustration of this process; ◦ tessellating the polygonal motifs: combination of polygonal motifs extracted from the analyzed pattern and those constructed in the previous step. 4. Experimental results The performances of the proposed method are tested here by using 3 Moroccan Zellij patterns. These performances are evaluated, first, by testing the ability of the analysis phase to detected and characterize the geometric rosettes together with the extraction of the underlying tiling of the patterns. Then, to show the high performance of the proposed method to create new innovative patterns from the analysis phase some new innovative patterns created by applying the proposed that we create by following the proposed method. 4.1. Experiment 1: example of patterns generated by the use of the 5-10 system polygons 4.1.1. Results of the analysis stage Fig. 9 shows the obtained results by applying respectively the operation of corners detection, circles/rosettes detection, rosettes characterization and the spatial composition (grid and tiling detection). Figure A.7 in supplementary material shows the Graphic User Interface (GUI) of the program that allows the implementation of different steps of the proposed method. The symmetry analysis of
4.1.2. Results of the generation process The analysis phase shows as illustrated in Fig. 9 that the underlying tiling of the analyzed pattern (Fig. 9a) is formed by a tessellation of a decagon and a concave hexagon (Fig. 9k). Fig. 10 represents the standard polygonal elements of the 5-10 polygonal system proposed by Bonner, which is the system of the analyzed pattern. This figure shows that the standard polygonal elements of the 5-10 system are adorned by the insertion of rosettes at their corners. Combination of the tiles extracted from the underlying tiling of the pattern and a selection of polygonal elements of its polygonal system allows the creation of a variety of new rosette patterns. To show these combinatory possibilities, we propose to generate new periodic patterns by tessellating the long hexagon motif (cf. Fig. 10) and the two polygonal motifs extracted from the original pattern (decagon and concave hexagon motifs presented in Fig. 9k). To do so, the long hexagon motif must be generated by drawing the polygonal and the interstitial motifs (cf. Fig. 11). The polygonal motif construction follows the process described in the previous section, depending on the relationship between each two neighboring rosettes. This relationship may express two adjacent stars or two disjoint stars D2 (Supplementary material, Table A.1), depending on their distance d: • case 1: d = 0 (adjacent stars, Supplementary material, Table A.1); • case 2: d (r1 ) = 4r1 cos − = 4r1 cos 2 (disjoint stars 5 2 n1 D2, Supplementary material, Table A.1). To manage the overlapping between stars/rosettes of the polygonal motif (case of the thin rhombus, Figure A.3 in supplementary material), the overlapping cases “O1” or “O2” are used. To fill the gap with a size larger than that of one or two overlapped stars (case of the octagon, Figure A.3 in supplementary material), the overlapping cases “O1” or “O2” and the case “D1” of disjoint stars can be used. Table A.1 in supplementary material [29] illustrates all these cases of relationship between 10-fold rosettes connected by one and two spikes (Ra and Rb). Otherwise, if the size of the gap is less than that of one star (cf. Fig. 11), the extension technique presented in the previous section is used. Fig. 11 illustrates step by step the generation process of the long hexagon template motif. Figure A.3 in supplementary material illustrates the template motifs of Moroccan Zellij style corresponding to three tiles belonging to 5-10 Bonner system, obtained by applying our proposed method. To generate the concave hexagonal motif (cf. Fig. 9k), “O1Ra” and “ARa” rules of the Supplementary material, Table A.1 are used. The long hexagon and the wide rhombus motifs are generated by using “D1Rb” and “ARa” (cf. Fig. 11 and Figure A.3 in supplementary material). The Thin rhombus motif (cf. Figure A.3 in supplementary material) is generated by the use of “O2Ra” and “ARa”. Finally, the octagon motif (cf. Figure A.3 in supplementary material) is generated by the use of “D1Rb”, “ARa” and “O2Rb”. By using the constructed template motif of the long hexagon together with the concave hexagon and decagon motifs extracted from the analyzed pattern (cf. Fig. 9a), Fig. 12a and b illustrate two examples of new generated periodic patterns belonging to the p2 mm and c2 mm crystallographic groups [5]. Figure A.4 in supplementary material in [29] illustrates these patterns in a high resolution.
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Fig. 9. Application of our process to analyze rosette pattern.
To show the performance of the proposed method to generate non-periodic patterns, rarely used in Zellij style, Fig. 13a and b presents two examples of new quasi -periodic patterns. The first example is based on the Penrose tiles (Figure A.5 in supplementary material) with angles 72◦ (wide rhombus) and 36◦ (thin rhombus), arranged on a five-fold symmetry [38–40]. The second is based on decagonal element (Figure A.4 in supplementary material) arranged on a decagonal quasi-periodic tiling [40–42]. Based on a construction by Petra Gummelt, an East German mathematician, a new way of constructing Penrose tiling was discovered in 1996 by Hyeong-Chai Jeong [42]. Gummelt showed that it was possible to construct a quasi-periodic tiling using only a decagon, though it is not a conventional tiling in the sense that neighboring tiles overlap (Fig. 13b).
4.2. Experiment 2: examples of pattern generated by the use of 4-5-20 tiling By this experiment, we want to underline the importance for defining the parameters characterizing the proportion harmony of the design elements, together with the distance between two neighboring satellite rosettes (Figs. 14 and 15): • constellation case 2r1 180 ; 1, R20 = 2 sin
1:
d = 0 n1 = 10, n2 = 20, r1 =
20
• constellation case 2: d = 4r1 cos 20, r1 = 1, R20 =
180
2r1+d 2 sin
2
−
n1
, n1 = 10, n2 =
20
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Fig. 10. The standard polygonal elements from the 5-10 system represented by adjacency graph based on connectivity of rosette region.
Fig. 13. Quasi-periodic pattern: a: quasi-periodic rosette pattern based on Penrose elements arranged on a five-fold symmetry; b: quasi-periodic rosette pattern based on decagonal elements arranged on a decagonal quasi-periodic tiling.
Fig. 11. Generation of the long hexagon motif.
Fig. 14. Constellation based on the 4-5-20 tiling case of d = 0.
4.3. Experiment 3: examples of patterns generated from the use of the 4-8-A system polygons and from design elements of the underlying tiling of a Zellij pattern Fig. 12. Periodic geometric rosette pattern: a: p2 mm pattern; b: c2 mm pattern.
The generation of a new central rosette for Moroccan style (Fig. 15) is based on the manual procedure used by [11]. This procedure infers the central rosette from the extension of the spikes of the satellite stars.
Fig. 16 presents the standard polygonal elements from the 4-8-A system proposed by J. Bonner, in which circles circumscribing the satellite rosettes are placed. See Figure A.6 in supplementary material [29]. By using the original pattern presented in Fig. 16a, the three design elements which are the Small octagon, the Trapezoid and the square presented in Fig. 16b lead to generate the design patterns
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The results of this study could contribute to alleviating the preservation needs of this cultural heritage and the development of innovation in the process of generating new ornaments. The perspective of this work is to automate all the steps of the proposed generative method algorithmically. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.culher. 2016.12.013. References
Fig. 15. Constellation case of d = / 0.
Fig. 16. a: processed image from database; b: tiling elements extracted from input image (a); c: tiling elements generated by composition or inference of extracted elements; d and e: new pattern generated by composition of extracted and constructed elements.
illustrated in Fig. 16d and e by respectively repeating the motif obtained by the union of the trapezoid motif and its vertical mirror motif, respectively, the small octagonal motif and the small square. 5. Conclusion This paper proposed a new system allowing an analytical representation of rosette patterns, which in turn is applied to generate new ones. For the analytical representation, we proposed an automatic method using image analysis techniques. The method starts by using genetic algorithm to detect the geometric rosettes of the pattern, which are its design elements. The order of each detected geometric rosette is then calculated by using a frequency analysis of the Frieze expansion function. The last step of this analysis process analyzes the spatial arrangement of the identified geometric rosettes. Results of this analysis concern the spatial composition structures (the repetitive unit cell, the mesh grid and the underlying tiling) and the metric parameters defining the right proportions between the constituent parts of a geometric pattern. The performances of the proposed analysis method were tested by the analysis of complex Moroccan Zellij patterns. Finally, by the use of the extracted tiling and a selection of polygonal elements of the Bonner’s polygonal systems, a generative method is proposed to create a variety of new rosette patterns. These patterns cover periodic patterns as well as periodic ones.
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Original article
Geometric rosette patterns analysis and generation Abdelbar Nasri a,∗ , Rachid Benslimane a , Aziza El Ouaazizi b a b
Laboratory of Data Transmission and Processing, Sidi Mohamed Ben Abdellah University, EST, Route d’Imouzzer, BP 2427, Fès, Morocco Laboratory of Engineering Sciences, Sidi Mohamed Ben Abdallah University, FP Taza, BP 1223, Taza Gare, Morocco
a r t i c l e
i n f o
Article history: Received 24 February 2016 Accepted 22 December 2016 Available online xxx Keywords: Star pattern Adjacency graph Rotational symmetry Islamic geometric pattern Genetic algorithm Tiling-based approach
a b s t r a c t The geometric rosettes, which are the most known design elements of the Islamic rosette patterns, are usually tessellated in a concealed composition structure. To understand and reveal this structure, we propose first to detect and characterize its basic geometric rosettes by using techniques of computer vision and image analysis. Then, the analysis of the spatial arrangement of the detected rosettes, characterized by their respective orders, will reveal the underlying tiling and the mesh grid, together with the harmonious proportions of the design elements. These results are used in turn to generate new innovative and authentic rosette patterns, by using the extracted geometric rosettes and new tile motifs constructed in the basis of an adaptation of the well-known polygonal technique. The performances of the proposed method to reveal the spatial composition of a rosette pattern are tested by the ability to extract its geometric rosettes and by the exact extraction of its underlying composition structure. Finally, the innovative character of the proposed generative method is shown through the creation of new periodic and quasi-periodic patterns characterized by their authenticities and sophistication. © 2017 Elsevier Masson SAS. All rights reserved.
1. Introduction The artisans of Islamic world continue to perpetuate the practice and the development of Islamic Geometric Patterns (IGP) by preserving the long established tradition consisting to reveal their practical knowledge to a few apprentices. However, as the time passes, the traditional methods of transmission of this ancestral art may be threatened or disrupted. In this context, its preservation as well as its development will be impacted. Now re-finding or renewing these innovative methods is a concern for today’s researchers in the field; this paper is concerned with the use of computer science and related technologies to approach the related problems. The analysis IGPs shows that they often take the form of a division of the plan into star-like motifs. These commonly used patterns are known as the Islamic star patterns. The geometric rosette is among the many different star-like motifs used in Islamic art, which stands out as distinctively “Islamic”. As shown in Fig. 1i, a geometric rosette is defined by 3 concentric circles that pass respectively by the set of points a, b and c. The symmetry order, which is the main distinguishing characteristic of the geometric rosette, is connected with the number of points a, b or c. Fig. 1ii and iii illustrates two
∗ Corresponding author. E-mail address: [email protected] (A. Nasri).
geometric rosettes characterized respectively by 12 and 16 folds symmetry. Fig. 2 gives examples of three types of spatial arrangement of geometric rosettes and stars (design elements). These geometric patterns show that the design elements are tessellated in complex geometrical arrangements, usually in tiles of a tiling, which is used as an underlying structure for the spatial composition. Often, the craftsmen influence the way in which a composition is perceived by emphasizing pattern elements (star/rosette motifs, tiling grid) by the use of the color. Fig. 2a shows that the rosette motifs, together with the used tiling are highlighted. However, Fig. 2b highlights only the motifs; the used tiling is not explicitly revealed. Otherwise, it is obvious to see that for all the star patterns, rosette motifs and star motifs are the visual anchors that appear in all the compositions. Thus, this paper will focus first on the extraction and characterization of these motifs. The spatial representation of the extracted patterns is then operated by using adjacency graphs. Finally, the analysis of this adjacency graph leads to the determination of some pattern features to be used in the characterization of the pattern and in the process to generate new patterns. The rest of the paper is organized as follows: Section 2 is concerned with related works focusing methods of analysis and generation of IGPs. Section 3 presents our proposed methods for respectively analyzing and generating a rosette pattern. The proposed analysis method is concerned with respectively the
http://dx.doi.org/10.1016/j.culher.2016.12.013 1296-2074/© 2017 Elsevier Masson SAS. All rights reserved.
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Fig. 1. Examples of geometric rosettes: (i) synthetic rosette according to A.J Lee [13] a: outer points, on circum-circle; b: outer midpoints, on outer mid-circle; c: inner midpoints, on inner mid-circle; d: inner points, on in-circle; o: center of rosette; (ii, iii): geometric rosettes of Zellij style in Fez – Morocco, (ii) rosette with 12 petals, (iii) rosette with 16 petals. R. Benslimane.
Fig. 2. Examples of three types of spatial arrangement of geometric rosettes: Zellig patterns in Fez-Morocco. Photos: R. Benslimane.
extraction and recognition of the design elements, the discovering of the underlying spatial composition structures and their representation. Based on the outputs of the analysis phase, the proposed generative method gives a step by step procedure to generate new periodic and a periodic rosettes patterns. Experimental results and discussion are presented in Section 4, and finally, conclusion and perspectives are drawn in Section 5. 2. Related works Analytical and generation approaches of IGPs can be divided into three main categories: the tiling-based approach, the symmetry groups based approach and the strand-based approach: • symmetry groups based approach: methods of this approach represent a periodic pattern with its underlying lattice and with its symmetry group [1,2]; • strand-based approach: methods of this approach consider a periodic pattern as drawing lines (Strands); which are linked segments at V-model corners [3,4]; • tiling-based approach: this tiling approach relies mainly on the analysis and generation of ornaments by considering the character of the motifs inscribed in polygons which form the tiling. In Islamic design traditions, tiling is used as the underlying structures for a composition. A tiling is an arrangement of polygons
that can contain design elements. For the star patterns, which are the most sophisticated ones in Islamic art, the well-known design elements are the geometric rosettes. There are two categories of tiling: tiling using polygons that are periodic and those that are non-periodic. A periodic tiling is a tiling of the Euclidean plane with periodic symmetry. It is proven that all 2D periodic patterns extended by two linearly independent translational vectors can be classified, depending on the type of network or lattice, into five types of unit cells and seventeen symmetry groups [5]. One of the first Western studies of the tiling-based approach was published by E.H. Hankin [6]. The Hankin’s technique, known also as polygons in contact technique, was a reference for several works. Kaplan in [7,8] provided a method for rendering Islamic Star Patterns based on Hankin’s Polygonal technique. The method builds star patterns from a tiling of the plan and a small number of intuitive parameters. J. Bonner more recently proved the historical use of this construction technique from his examination of the Topkapi Scroll. Bonner called this polygonal technique. The use of this technique generates a multitude of patterns from a single underlying polygonal tessellation. He proposed in [9] and [10] to create patterns from systematic and non-systematic geometric constructions. He classified systematic patterns into five distinctive systems: the 3-46-12 system of regular polygons, the 4-8 system A, the 4-8 system B, the 5-10 system, and the 7-14 system. The non-systematic patterns are characterized by more than a single region of higher order star-forms. Castera in [11] summed up the rules for the Moroccan Zellij patterns construction, when he introduced the concept of skeleton. The skeleton is an underlying structure, made up of the alternation of two tiles: the octagonal star and an irregular hexagon called “Saft”. Fig. 2a gives an example of skeleton. The point is that the skeleton follows the outline of a network similar to a tiling. This particular tiling is constituted by polygons and different star shapes. In this tiling-based approach, motifs inscribed in the polygons in contact can be considered as separate mini-compositions. Recognizing and characterizing these design elements and their spatial arrangement opens the way to a better understanding of the design process. This paper proposes, first, to detect these motifs by using an improved version of our recent developed method [12]. Then, we propose to represent the spatial relationship between the detected motifs by an adjacency graph. From the analysis of this spatial representation, we finally extract the hidden tiling, the repetitive unit cell and other pattern settings. An original generative method, based on the analysis outputs, is finally proposed to create new rosette patterns.
3. Proposed method for rosette pattern analysis and generation In this paper, we propose a new method to analyze and generate a rosette pattern based on two stages, the first stage consist to analyze an input image of a rosette pattern by using image analysis techniques. The outcomes of this analysis stage (adjacency graph, hidden tiling, and other pattern settings) are used in turn in the second stage for the generation of new rosette patterns.
3.1. Proposed method for rosette pattern analysis In [12], we proposed an original method to detect and characterize rosettes of a geometric rosette by performing the following stages:
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3.1.2. Rosettes detection by using a Genetic algorithm In an x – y Cartesian coordinate system, a circle with center ‘O’ of coordinates (a, b) and radius r is the set of all points (x, y) such that: 2
f (x, y) = (x − a)2 + (y − b) − r 2 = 0 This equation can be written in parametric form using the trigonometric functions sine and cosine as:
x = a + r cos
∈ [0, 2]
y = b + sin
Fig. 3. Flowchart of the proposed method for rosette pattern analysis.
• image pattern segmentation by using a global threshold operation; • rosette center detection by using a genetic algorithm that detect the optimal circle passing by the maximum points of the pattern binary image; • determination of the detected rosette order by using Frieze expansion technique. As shown in Fig. 1i, and according to A.J. Lee [13], a rosette is characterized by sets of interest points (a, b, c, d) that correspond to corners. The points of each set are radially distributed around the center of the rosette. Thus, to improve the rosette detection phase, it is better to take into consideration the sets of corners (a, b, c, d) instead of all the points generated from the segmentation operation considered in our previous method. For this purpose, we propose in this paper to start the process of the rosette detection by detecting first the corners of the pattern image. To identify circles delimiting the rosettes of the pattern, a genetic algorithm is then applied to the detected corners. It is important to underline the fact that there exists more than one rosette or star in a star pattern. To detect all these stars or rosettes, the proposed method detects the corresponding circles sequentially by applying an iterative procedure. The corners participating to the extracted circle at the current iteration are removed from the image permitting the detection of the rest of circles corresponding to the other rosettes to be detected at the next iterations. To determine the order of each rosette, the area of the detected rosette is cropped and then transformed on a Frieze expansion representation. Finally, the spatial arrangement of the detected rosettes is represented by an adjacency graph. In this purpose, each detected rosette is represented by a graph node and the relationship between two rosettes/stars is represented by an arc. Fig. 3 illustrates the proposed method for rosette pattern analysis; details of each step are presented in the next sub-sections. 3.1.1. Corner detection A large number of corner detectors have been proposed in the literature. They can be grouped broadly into three categories: Gray-level based methods [14–19], contour based methods [20–24] and parametric model based methods [25–27]. In this paper, we use the Förstner method presented in [23] and its implementation described in [24,25,28]. The choice of this descriptor has been motivated by the comparison studies proposed in [27] and also by the results obtained on images of our tested database. Figure A.1 from the supplementary material [29] gives examples of corner detection made by the Förstner operator.
where is a parametric variable in the range 0 to 2, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis. To find the circle passing by the maximum of corners, we propose to use a genetic algorithm, which can achieve to the optimal circle parameters (a, b, r) by maximizing the following criterion:
U (a, b, r) =
B (x, y) ds f
where B (x, y) is the corner point (x, y). The individuals of our genetic algorithm will represent circles by three coded genes a, b and r that we shall code in binary to its chromosome structure. Individual Chromosome
a ␣1 ␣2 . . .
b ˛1 ˛2 . . . ˛i
r ˛1 ˛2 . . . ˛l
where: ␣i is a binary value and i, j, l are the numbers of bits coding respectively a, b and r. The GA starts with a randomly generated population of N pairs of individuals, and in each generation, a new population of the same size is generated from the current population by applying operators, termed selection, crossover and mutation, which mimic the corresponding processes of natural selection. The algorithm is terminated if no improvement in the fitness value of the best individual for a fixed number of iterations and the best chromosome is taken to be the optimal one. The old population of GA is completely replaced by the offspring population after reproduction, crossover, and mutation. In this way, it is possible that the best chromosome in the old population disappears in the current generation. A reasonable improvement is reached by preserving the best string obtained so far in a separate location outside the population so that the algorithm may report the best value found, among all solutions inspected during the whole process. In the present work, we have used the elitist strategy of [30] where the best string in the previous iteration is copied into the current population. The genetic algorithm is stopped if no improvement in the best individual during 200 generations is give. Figure A.2 in supplementary material [29] and Fig. 9c give examples of detected circles by using genetic algorithm with the following setting parameters: • • • •
population size Ni = 40; one randomly selected crossover with probability Pc = 0.75; randomly mutation with probability Pm = 0.35; stochastic remainder roulette rule selection with elitist strategy.
The proposed genetic algorithm is applied n times depending on the number of rosettes and stars forming the analyzed pattern. For a current iteration, one circle (center, radius) can be detected and the interest points that participated to its formation are then removed. The next iteration performs the genetic algorithm on the rest of the interest points. This iterative process is stopped if no circle is detected in the current iteration.
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Fig. 4. Illustration of order detection and different cases of the motif circumscribed by the detected circles.
3.1.3. Rosette characterization: determination of the geometric rosette symmetry order Rosettes are rotationally symmetrical patterns, so a simple test of symmetry with the frieze expansion technique allows to check if the finding pattern circumscribed in the detected circle is a rosette or not. Fig. 4 gives an illustration of this technique. Extensive explanations are given in our work [12]. This test can discriminate between the possible following cases: • circles that correspond to rosettes: circles circumscribing symmetrical motifs. In this case, to determine the rosette order, Frequency analysis using Discrete Fourier Transform is applied to the Frieze expansion pattern; • circles that are no rosettes: circles circumscribing nonsymmetrical motifs. These circles may correspond to overlapping rosettes. 3.1.4. Analysis and representation of the spatial composition of a rosette pattern The occurring structures underlying a rosette pattern composition are the mesh grid and the tiling. A mesh grid represents the spatial arrangement of main rosettes (identical rosettes having the highest symmetry order and the greatest radius). However, the tiling represents the spatial arrangement of satellite rosettes (rosettes having the lower symmetry order and the smallest radius). The green network presented in Fig. 5 represents the grid of the pattern. This grid underlies the spatial arrangement of the main rosettes. The red polygons represent the tiling, in which the satellite rosettes are located at their corners. By taking into consideration both the occurring composition structures, it will be easy to identify the relationships between
adjacent rosettes (neighboring rosettes). Properties of the adjacent rosettes define the type of the underlying structure to be identified. Therefore to identify respectively grid and tiling, we have to identify couples of rosettes Xi and Xj , which are adjacent and respectively characterized by the following properties P1 and P2 : • P1 : Xi and Xj are the main rosettes; • P2 : Xi and Xj are the satellite rosettes. It is important to underline that the main rosette is circumscribed by n satellite rosettes. This local composition, named ‘constellation’, involves the definition of the following third property: • P3 : Xi is a main rosette and Xj is a satellite rosette. The number ‘n’ of the satellite rosettes is a divisible number of the main rosette order. The satellite rosettes are centered on the vertices of the n sides tile. In addition to the known relationship between the number of the satellite rosettes and the main rosette symmetry order, the following relationships can be defined: • the relationship between the radius and the order of the main rosette; • the relationship between main rosette parameters (order and radius) and the number of the satellite rosettes. The inter-rosettes relationships can be represented by an adjacency graph or by an adjacency matrix ‘M’ whose coefficients 0 and 1 are defined as follows [31,32]:
Mi,j =
1 if Xi RXj
0 otherwise
(Xi R Xj ) if Xi and Xj are neighbors and are defined by a property Pi . Thus, to represent a rosette pattern by an adjacency graph, we propose the following three-stage method:
Fig. 5. Example of composition using 10-fold rosette as main rosette and 10-fold stars as satellites motifs.
• stage 1: labeling rosettes according to their parameters X (center, order, radius). It is important to underline that the mismatch between circles and interests points, as illustrated in Fig. 6, is due to the fact that the circle is a continuous function while the points of interest belong to a discrete grid; • stage 2: constructing an adjacency matrix: Connecting neighboring rosettes Xi and Xj characterized by the property P2 (Xi and Xj have the same order and the same radius r1 ) with applying the following test:
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Fig. 6. Flowchart of the proposed method for the spatial composition analysis and representation by an adjacency graph.
◦ if the distance d (Xi , Xj ) = ws ± → Mij = 1, ◦ else → Mij = 0, where ws is the most occurred distance calculated from the map of distance Mapd (i, j) = d (Xi , Xj ); • stage 3: representing the adjacency matrix by its corresponding adjacency graph. The flowchart of this proposed method is illustrated in Fig. 6. Fig. 5 gives an example of a Moroccan Zellij pattern for which the tiling is revealed by connecting the 10-fold stars, considered here as satellites motifs of a 10-fold main rosette. As illustrated in Fig. 5, the arrangement of the main rosette centers reveals a hidden mesh grid defined by a smallest region or a repetitive unit cell (Bravais lattice). When translated by 2 lattice vectors, the unit cell can cover the entire plane without overlapping. The underlying grid may be rectangular or centered rectangular (cf. Fig. 5) depending on whether the color attribute is taken into consideration or not. Generally, there are five distinct Bravais lattices: square, rectangular, centered rectangular, hexagonal (rhombic) and oblique. As shown in Fig. 5, the spatial arrangement of the satellite rosettes reveals a polygonal distribution of their centers. The revealed underlying polygonal tessellation is known as the tiling. The analysis of this spatial composition can be supplemented by metric parameters defining the right proportions between the constituent parts of a geometric pattern. These parts are harmoniously proportioned in relation to each other and in relation to the whole. To discover the inner logic of proportion used by the Islamic designers, we propose to establish the relationships between the satellite rosettes attributes (r1 radius of symmetry and order o1 ), the main rosettes attributes (r2 , o2 ) and the attributes of the principal polygon circumscribing the main rosette (Sn , Rn ). Referring to the illustration of the Fig. 7, it is easy to see that: Rn =
Sn 2 sin
180 = r1 + r2 n
n is the number of sides of the circumscribing polygon. Sn is the length of a side: Sn = 2r1 + d d is the distance between two consecutive satellite rosettes. 3.2. Proposed generative method The proposed generative method adopts an adapted version of the polygonal technique proposed by J. Bonner. The proposed
Fig. 7. Parameters defining the relationship between central rosette and its satellite rosettes.
method covers the Eastern Islamic geometric patterns as well as the Moroccan and Andalusian ones which are designed by the traditional technique named ‘Tastir’ [33]. Otherwise, the main difference with the Bonner’s method concerns the procedure adopted to design motifs of the considered tiles. Therefore, the proposed method takes into consideration two cases: • the first one concerns the generation of new patterns by considering other spatial arrangements of the tiles, extracted from the underlying tiling of the analyzed pattern. In this case, the generation process consists to tessellate the plane with the extracted tiles in different ways. For the tiles extraction, we use the algorithm proposed in [34,35]. At this stage, the tessellation of the extracted tiles is a hand drawing. A clear opportunity exists to automate this pattern generation step algorithmically; • the second case concerns the generation of new patterns by tessellating the extracted tiles and additional ones selected among those of the corresponding Bonner’s polygonal system. As mentioned in related work, J. Bonner has defined five distinctive systems: the 3-4-6-12 system of regular polygons, the 4-8 system A, the 4-8 system B, the 5-10 system, and the 7-14 system. The generation process in this case is based in the following five steps: ◦ extraction and identification of the tiles of the tiling. These polygonal elements of tiling, in the logic of adjacency graph, are the shorter closed path of the planar graph [34,35], ◦ finding the Bonner’s polygonal system corresponding to the identified tiles, ◦ selecting the tiles to be used for the pattern to be created, ◦ constructing the template motif of each selected tile. For this purpose, the proposed method is based on two stages:
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the unit cell is not addressed in this paper. This issue is already addressed in our recent work [36,37]. It is easy to see in Fig. 9j that the design polygonal elements of the original pattern presented in Fig. 9a are the decagon motif and the concave hexagon motif presented in Fig. 10.
Fig. 8. Construction process of interstitial motif.
– construction of the polygonal motif. The proposed method consists to center n satellite rosettes on the corners of the considered tile and then to take into consideration the relationship between two neighboring rosettes. This relationship can be adjacent, disjoint or in overlapping. Table A.1 in supplementary material [29] gives results of these relationships, – construction of the interstitial motif by filling the gap (the space left blank into the polygonal motif). The construction of this motif is commonly used for all generative methods using tiling approach. The most known method to fill the gap is the inference algorithm proposed by Kaplan in [8]. For the Moroccan Zellij style patterns, considered in this work, the situation is easy, and does not require the use of such a sophisticated solution as the Kaplan inference algorithm. For the proposed method, two solutions can be considered depending on the gap’s size. If this size is less than that of one rosette, then the extension rule is applied. Else, the size of the gap is certainly larger than that of one or two overlapped rosettes. In this case, the extension rule can be applied after one or two overlapped rosettes are placed on the gap. The extension rule consists to extend each edge eik of a free rosette’s spike (Pi ) and cut it off at point Cij when it meets another extended edge ejk’ of a free spike Pj . If the edges eik (k = 1,2) and ejk (k’ = 1,2) are collinear; connect Pi and Pj at the middle of the segment Pi , Pj . Thus, the segments Pi Cij and Pj Cij will be added to the template motif. Fig. 8 gives an illustration of this process; ◦ tessellating the polygonal motifs: combination of polygonal motifs extracted from the analyzed pattern and those constructed in the previous step. 4. Experimental results The performances of the proposed method are tested here by using 3 Moroccan Zellij patterns. These performances are evaluated, first, by testing the ability of the analysis phase to detected and characterize the geometric rosettes together with the extraction of the underlying tiling of the patterns. Then, to show the high performance of the proposed method to create new innovative patterns from the analysis phase some new innovative patterns created by applying the proposed that we create by following the proposed method. 4.1. Experiment 1: example of patterns generated by the use of the 5-10 system polygons 4.1.1. Results of the analysis stage Fig. 9 shows the obtained results by applying respectively the operation of corners detection, circles/rosettes detection, rosettes characterization and the spatial composition (grid and tiling detection). Figure A.7 in supplementary material shows the Graphic User Interface (GUI) of the program that allows the implementation of different steps of the proposed method. The symmetry analysis of
4.1.2. Results of the generation process The analysis phase shows as illustrated in Fig. 9 that the underlying tiling of the analyzed pattern (Fig. 9a) is formed by a tessellation of a decagon and a concave hexagon (Fig. 9k). Fig. 10 represents the standard polygonal elements of the 5-10 polygonal system proposed by Bonner, which is the system of the analyzed pattern. This figure shows that the standard polygonal elements of the 5-10 system are adorned by the insertion of rosettes at their corners. Combination of the tiles extracted from the underlying tiling of the pattern and a selection of polygonal elements of its polygonal system allows the creation of a variety of new rosette patterns. To show these combinatory possibilities, we propose to generate new periodic patterns by tessellating the long hexagon motif (cf. Fig. 10) and the two polygonal motifs extracted from the original pattern (decagon and concave hexagon motifs presented in Fig. 9k). To do so, the long hexagon motif must be generated by drawing the polygonal and the interstitial motifs (cf. Fig. 11). The polygonal motif construction follows the process described in the previous section, depending on the relationship between each two neighboring rosettes. This relationship may express two adjacent stars or two disjoint stars D2 (Supplementary material, Table A.1), depending on their distance d: • case 1: d = 0 (adjacent stars, Supplementary material, Table A.1); • case 2: d (r1 ) = 4r1 cos − = 4r1 cos 2 (disjoint stars 5 2 n1 D2, Supplementary material, Table A.1). To manage the overlapping between stars/rosettes of the polygonal motif (case of the thin rhombus, Figure A.3 in supplementary material), the overlapping cases “O1” or “O2” are used. To fill the gap with a size larger than that of one or two overlapped stars (case of the octagon, Figure A.3 in supplementary material), the overlapping cases “O1” or “O2” and the case “D1” of disjoint stars can be used. Table A.1 in supplementary material [29] illustrates all these cases of relationship between 10-fold rosettes connected by one and two spikes (Ra and Rb). Otherwise, if the size of the gap is less than that of one star (cf. Fig. 11), the extension technique presented in the previous section is used. Fig. 11 illustrates step by step the generation process of the long hexagon template motif. Figure A.3 in supplementary material illustrates the template motifs of Moroccan Zellij style corresponding to three tiles belonging to 5-10 Bonner system, obtained by applying our proposed method. To generate the concave hexagonal motif (cf. Fig. 9k), “O1Ra” and “ARa” rules of the Supplementary material, Table A.1 are used. The long hexagon and the wide rhombus motifs are generated by using “D1Rb” and “ARa” (cf. Fig. 11 and Figure A.3 in supplementary material). The Thin rhombus motif (cf. Figure A.3 in supplementary material) is generated by the use of “O2Ra” and “ARa”. Finally, the octagon motif (cf. Figure A.3 in supplementary material) is generated by the use of “D1Rb”, “ARa” and “O2Rb”. By using the constructed template motif of the long hexagon together with the concave hexagon and decagon motifs extracted from the analyzed pattern (cf. Fig. 9a), Fig. 12a and b illustrate two examples of new generated periodic patterns belonging to the p2 mm and c2 mm crystallographic groups [5]. Figure A.4 in supplementary material in [29] illustrates these patterns in a high resolution.
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Fig. 9. Application of our process to analyze rosette pattern.
To show the performance of the proposed method to generate non-periodic patterns, rarely used in Zellij style, Fig. 13a and b presents two examples of new quasi -periodic patterns. The first example is based on the Penrose tiles (Figure A.5 in supplementary material) with angles 72◦ (wide rhombus) and 36◦ (thin rhombus), arranged on a five-fold symmetry [38–40]. The second is based on decagonal element (Figure A.4 in supplementary material) arranged on a decagonal quasi-periodic tiling [40–42]. Based on a construction by Petra Gummelt, an East German mathematician, a new way of constructing Penrose tiling was discovered in 1996 by Hyeong-Chai Jeong [42]. Gummelt showed that it was possible to construct a quasi-periodic tiling using only a decagon, though it is not a conventional tiling in the sense that neighboring tiles overlap (Fig. 13b).
4.2. Experiment 2: examples of pattern generated by the use of 4-5-20 tiling By this experiment, we want to underline the importance for defining the parameters characterizing the proportion harmony of the design elements, together with the distance between two neighboring satellite rosettes (Figs. 14 and 15): • constellation case 2r1 180 ; 1, R20 = 2 sin
1:
d = 0 n1 = 10, n2 = 20, r1 =
20
• constellation case 2: d = 4r1 cos 20, r1 = 1, R20 =
180
2r1+d 2 sin
2
−
n1
, n1 = 10, n2 =
20
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Fig. 10. The standard polygonal elements from the 5-10 system represented by adjacency graph based on connectivity of rosette region.
Fig. 13. Quasi-periodic pattern: a: quasi-periodic rosette pattern based on Penrose elements arranged on a five-fold symmetry; b: quasi-periodic rosette pattern based on decagonal elements arranged on a decagonal quasi-periodic tiling.
Fig. 11. Generation of the long hexagon motif.
Fig. 14. Constellation based on the 4-5-20 tiling case of d = 0.
4.3. Experiment 3: examples of patterns generated from the use of the 4-8-A system polygons and from design elements of the underlying tiling of a Zellij pattern Fig. 12. Periodic geometric rosette pattern: a: p2 mm pattern; b: c2 mm pattern.
The generation of a new central rosette for Moroccan style (Fig. 15) is based on the manual procedure used by [11]. This procedure infers the central rosette from the extension of the spikes of the satellite stars.
Fig. 16 presents the standard polygonal elements from the 4-8-A system proposed by J. Bonner, in which circles circumscribing the satellite rosettes are placed. See Figure A.6 in supplementary material [29]. By using the original pattern presented in Fig. 16a, the three design elements which are the Small octagon, the Trapezoid and the square presented in Fig. 16b lead to generate the design patterns
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The results of this study could contribute to alleviating the preservation needs of this cultural heritage and the development of innovation in the process of generating new ornaments. The perspective of this work is to automate all the steps of the proposed generative method algorithmically. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.culher. 2016.12.013. References
Fig. 15. Constellation case of d = / 0.
Fig. 16. a: processed image from database; b: tiling elements extracted from input image (a); c: tiling elements generated by composition or inference of extracted elements; d and e: new pattern generated by composition of extracted and constructed elements.
illustrated in Fig. 16d and e by respectively repeating the motif obtained by the union of the trapezoid motif and its vertical mirror motif, respectively, the small octagonal motif and the small square. 5. Conclusion This paper proposed a new system allowing an analytical representation of rosette patterns, which in turn is applied to generate new ones. For the analytical representation, we proposed an automatic method using image analysis techniques. The method starts by using genetic algorithm to detect the geometric rosettes of the pattern, which are its design elements. The order of each detected geometric rosette is then calculated by using a frequency analysis of the Frieze expansion function. The last step of this analysis process analyzes the spatial arrangement of the identified geometric rosettes. Results of this analysis concern the spatial composition structures (the repetitive unit cell, the mesh grid and the underlying tiling) and the metric parameters defining the right proportions between the constituent parts of a geometric pattern. The performances of the proposed analysis method were tested by the analysis of complex Moroccan Zellij patterns. Finally, by the use of the extracted tiling and a selection of polygonal elements of the Bonner’s polygonal systems, a generative method is proposed to create a variety of new rosette patterns. These patterns cover periodic patterns as well as periodic ones.
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