CHAPTER 3 Algorithm
3.7 Rendering Approaches to Op Art Results
After observing a great quality of square-based Op Art works drawn by artists, we select two rendering approaches to render our experiment results. The first type refers to the Vasarely illusion. This illusion is the perception of an oblique light cross which is like a ‘X” along the diagonals of concentric squares [2]. But this cross does not really exist. It has been incorporated into many Op Art paintings, as shown in Figure 3.12(a). Several research studies discussed the Vasarely illusion such as Tsofe et al. [20] and Troncoso et al [19]. The artworks of this type render the object squares by a luminance gradient, as shown in Table 3.1. The squares 𝐺𝑖,𝑗 which is a cluster center is tinted by the color of index 1. Other squares 𝐺𝑚,𝑛 are tinted by the color of index ‖𝑚 − 𝑖‖ + ‖𝑛 − 𝑗‖. If the index is greater than the greatest index 8, we use index 8 as the index. We can create more colors for obvious perception of layers by interpolating the colors in Table 3.1. The second type of our rendering approach is
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inspired by the Vasarely’s Harlequin. We stored the colors in a table. Then apply these colors to our results. For each row in Harlequin, the squares are tinted by four colors, two of them are for the object squares and the others are for the background squares, as shown in Figure 3.12(b).
(a) (b)
Figure 3.12: (a) The Op Art work Arcturus, by Vasarely. (b) A part of Harlequin, by Vasarely.
Table 3.1: The colors of a luminance gradient are rendered in our experiment results.
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Chapter 4
Experimental Results
We present the implementations and results in this chapter. The input sources are 3D triangle meshes, and the outputs are 2D square-based Op Art images. All the 3D models we used can be found from Google 3D Warehouse. These models are post-processed to show their depth maps, as shown in Figure 4.1. Even a skilled Op artist may spend several days drawing a painting. By using our rendering system, each result would be generated only in a few seconds.
Figure 4.1: Depth maps of our 3D models.
In our system, users need to define some parameters for generating the desired Op Art images. Figure 4.3 compares four experimental results with different parameters 𝑆𝑀𝑎𝑥 and 𝑑𝑀𝑎𝑥. The corresponding values of each result were defined in Table 4.1. Figure 4.3(a) was produced by the greatest value of 𝑆𝑀𝑎𝑥. So the inflated
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effect of squares around each cluster center is more obvious than those in Figure 4.3(b)-(d). Figure 4.3(b) and Figure 4.3(c) used the same 𝑆𝑀𝑎𝑥. Figure 4.3(c) looks smoother, especially the squares near to the cluster centers because the parameter 𝑑𝑀𝑎𝑥 is smaller.
(a) (b)
(c) (d)
Figure 4.3: Op Art results with different values of parameters.
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Table 4.1: The parameters of each experiment results. (𝒅𝑴𝒊𝒏 = 𝟏 , 𝑺𝑴𝒊𝒏= 𝟎.)
Figure 4.4 shows the result of the pig model. We first show the depth map of the model and its corresponding clustering results, the different clusters are represented by different colors, as shown in Figure 4.4(b). Then we render the object squares of the pig model by a luminance gradient which creates the Vasarely illusion. The parameter 𝑁𝐻𝑜𝑟𝑖 and 𝑁𝑉𝑒𝑟𝑡𝑖 of Figure 4.4(d) are greater than those in Figure 4.4(c).
When the number of square increases, there is a seeming lightening at the edges, forming the “X” [2]. We perceive that the Vasarely illusion is clearer with larger 𝑁𝐻𝑜𝑟𝑖 and 𝑁𝑉𝑒𝑟𝑡𝑖.
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(a) (b)
(c) (d)
(e)
Figure 4.4: The rendered result of pig. (a) The depth map. (b) The clustering result.
(c)-(e) The rendered Op Art images.
Figure 4.5 shows the clustering and rendering result of the zebra model. This model has eight clusters, as shown in Figure 4.5(a). We used black and white to tint the zebra because Op Art often uses two highly contrasting colors to render the artworks. Besides, we rotated the squares to make it diagonal, as shown in Figure 4.5(d).
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(a) (b)
(c) (d)
Figure 4.5: The rendered results of zebra. (a)(c) The clustering results of the rendered Op Art images (b) and (d), respectively.
Figure 4.6 compares the result with and without using the fake object squares.
The background squares close to Venus’s loin are regarded as the fake object squares.
So they are deformed along with the object squares, as shown in Figure 4.6(b).
Besides, we compare the results with perturbing the points or not. In Figure 4.6(b), we perturb the points so that the thighs are more perceptible. On the contrary, the result which only used the cluster centers to shift the points is shown in Figure 4.6(c).
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(a) (b) (c)
Figure 4.6: The rendered results of Venus. (a) The clustering results. (b)(c) The rendered Op Art images.
Finally, we compare our result with the Op Art painting The Juggler by Vasarely, as shown in Figure 4.7. The inflated effects on the raised regions look similar. Besides, the background squares left to the loin were deformed analogously. More Op Art experiment results created by our system are shown in Figure 4.8 to Figure 4.10.
Figure 4.7: Compare our result with The Juggler by Vasarely. The left is the rendered result of Venus. The right is Vasarely’s The Juggler.
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Figure 4.8: The rendered bear with perturbing the points or not.
Figure 4.9: The Op Art results rendered by different approaches of the bunny.
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(a) (b)
Figure 4.10: The rendered Op Art result of Hulk. (a) The clustering result. (b) The experiment result.
The corresponding execution times of each Op Art output image were shown in Table 4.2. We observe that the number of object squares and fake object squares dominates the execution time. The greater the number is, the more the execution time it spends. Besides, our system spends most of the time finding the new intersections of each horizontal line and vertical line. Because the points of a square may alter after square deformation and square alignment, as shown in Figure 4.11. The red points in Figure 4.11(a) transfer to the red points in Figure 4.11(b) after square deformation and square alignment. We find the new intersections by calculating the distance between the points of each row and column in the user specified 𝑆𝑀𝑎𝑥. The two points with minimum distance will be the new intersection. When 𝑆𝑀𝑎𝑥 is larger, the displacement of the point is greater. So the range of each row and column that we
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should check is wider, as shown in Figure 4.12. After we find all the new intersections, we can obtain the points in a square.
Table 4.2: The execution times of each experiment results.
(a) (b)
Figure 4.11: The points of a square alter after square deformation and square alignment. (a) Before square deformation. (b)
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Figure 4.12: A new intersection is obtained by finding the smallest distance between the red points. (a) 𝑆𝑀𝑎𝑥 =𝑁𝑊
𝐻𝑜𝑟𝑖× 2.0. (b) 𝑆𝑀𝑎𝑥 =𝑁𝑊
𝐻𝑜𝑟𝑖× 1.0.
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Chapter 5
Conclusion and Future Works
This thesis explores the square-based Op Art and the methods for achieving the desired artistic effects. The inflated and deformed squares create the swelling illusion which is similar to the artworks by Op artists. The input 3D models provide the depth information, so the users do not need to decide the inflated squares manually. We can cluster the squares by making use of the peak-climbing algorithm. Our system provides some parameters for the user preference. To imitate the Op artworks which have some deformed background squares, we create a type of square called fake object square. Besides, user can choose the rendered approach to tint the Op Art image. To avoid too many static squares existing in our result, we perturb the points of each square by the depth value of the adjacency squares. One limitation of our approach is that if the squares which have the smaller depth value are not the raised region of the input 3D model, it would fail to produce a reasonable result.
There are a few directions for further research. Our system differentiates the foreground and background by the rendered approach which tints the two regions by different colors. If we tint the two regions by the same color, the contour of the model may become unclear or disappear. Ideally, the system should identify areas that require greater deformation such as the finer details in an image. We expect it can help to create the illusory contour on the boundary of the model. Besides, we expect the technique introduced in this paper can be extended to create the animation of Op Art.
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