Wrong Colored Vermeer: Color-Symmetric Image Distortion

Hendrik Richter HTWK Leipzig University of Applied Sciences D–04251 Leipzig, Germany. Email: [email protected].

June 30, 2021

Abstract Color symmetry implies that the colors of geometrical objects are assigned according to their symmetry properties. It is defined by associating the elements of the symmetry group with a color permutation. I use this concept for generative art and apply symmetry-consistent color distortions to images of paintings by Johannes Vermeer. The color permutations are realized as mappings of the HSV color space onto itself.

Color Symmetry

Symmetry is an ubiquitous concept in many branches of mathematics, most clearly visible, for instance, in geometry or group theory or mathematical physics. Symmetry is even more frequently present in visual arts, for example in architecture or painting or drawing. Thus, juxtaposing mathematics and art immediately suggest dealing with symmetry. In the following, I focus on a special kind of symmetry: color symmetry. Suppose we have a geometrical object with some symmetry properties. This implies that the object in invariant under certain transformations. A complete set of these transformations yields the symmetry group G of the geometrical object. Further assume that the geometrical object is (or can be) colored. Thus, color symmetry means that the coloring of the object is consistent with the symmetry group. More specifically, an element g ∈ G is called a color symmetry if g is associated with a color permutation over a color set [7]. In other words, g permutes the colors consistently by a color permutation assigning which elements of G change and which elements of G preserve colors, see for instance [1, 2, 4, 5, 6] for explicitly addressing the arXiv:2106.15179v1 [cs.CV] 29 Jun 2021 topic of color symmetry in generative art. In principle, a color symmetry may change all colors or even preserve all colors. From an artistic point of view, however, color symmetry is more interesting if we have a permutation which mixes color-changing and color-preserving elements. Take as a very simple example a 2D rectangular object as in Figure 1. The symmetry group of the ◦ rectangle is the dihedral group D2 of order 4. A rotation α turns the rectangle counter-clockwise 180 , the reflections βh and βv mirror the rectangle in the horizontal and vertical axis. The composed symmetries are from left to right, as for instance βhβv = α is a horizontal reflection followed by a vertical reflection, which 2 2 2 equals a rotation. With the identity e = α = βh = βv and βhβv = βvβh = α, we obtain the symmetry group G = (e, α, βh, βv). The reflection axes divide the image into 4 sections, north-east (ne), north-west (nw), south-east (se) and south-west (sw). Each section can be mapped into any other section by the elements of the symmetry group. A rotation maps (nw,ne) ↔ (se,sw), while for a horizontal and a vertical reflection the mapping is (nw,ne) ↔ (sw,se) and (nw,sw) ↔ (ne,se), respectively. The sections in Figure 1 are further subdivided into subsections, which can be individually colored. Thus, the rectangles show simple color symmetries. For instance, in the left image in Figure 1 we have triangular subsections and a rotation α is associated with preserving the colors yellow (y) and purple (p) and changing blue (b) to orange (o)

1 and orange to blue. Using Cauchy’s two-line notation, we may describe the color permutation pα associated  o b y p  with the rotation α by pα = b o y p . Both reflections change all colors: blue ↔ yellow, orange ↔ purple, purple ↔ blue and yellow ↔ orange. The color permutations pβh and pβv associated with the reflections (horizontal and vertical) can be expressed similarly by a two-line notation. Color symmetry may likewise apply to other schemes of subdividing the sections of a rectangle, for instance bubble-like or chessboard-like subsections. The middle and the right image of Figure 1 shows further examples of color symmetry where the horizontal (middle image) and vertical (right image) reflection preserve some colors, while the rotation changes all colors. The schemes of subdividing the sections may be different in the geometrical form (triangle, circle, rectangle, etc.) or in the granularity. The granularity is expressed by the number of subsection Λ. For instance, we have a granularity of Λ = {8, 12, 16} with the 8, 12 and 16 subsections in the images in Figure 1 (from left to right). In the following I propose to take the granularity to the level of the pixels composing the image. In other words, each section can have as many subsections as there are pixels in it. Thus, for any image composed of pixels color symmetries can be designed, for instance to pre-existing source images such as paintings.

Figure 1: Simple color symmetry.

Let us consider images composed of pixels where each pixel pk has a color ck in a HSV color space: ck = (hk, sk, vk). We start with focusing on the hue hk, assuming that the saturation sk and the value (brightness) vk remain constant for now. The method is equally applicable to the other elements of the HSV color space and in the visual results also such mappings are used. We define a color permutation associated with the symmetry group by a map f(hk), which is a mapping of the hue component of the HSV color space onto itself. A similar mapping of the color space has recently been used to visualize number-theoretic functions with portraits of mathematicians [8]. Here, I use general algebraic functions solely to achieve certain color distortion effects. Thus, by using such a map f(hk) and for rectangular source images, a color symmetry can be imposed by a color permutation as follows. For each subsection `, ` = 1, 2,..., Λ, we set a map f(hk, `), which maps the hue component of the k-th pixel in subsection `, and thus changes the color of the pixel. If the color symmetry stipulates that the color is preserved, the map is the identity f(hk, `) = hk. For instance, a color permutation associated with the rotation α can be expressed by a two-line notation as

 fi(hk,`p)  pα = ∀{`p, `q} ∈ Λα, `p 6= `q, (1) fj (hk,`q) where Λα is the set of rotation-symmetric subsections. A rotation maps (nw,ne) ↔ (se,sw). Thus, the color permutation (1) implies that the hue hk of all pixels pk in the `p-th subsection in either nw or ne take the colors specified by fi(hk) while all pixels pk in the rotation-symmetric `q-th subsection in either se or sw take the colors specified by fj(hk), and vice versa. As this means that for any pair of colors fi(hk) and hj(hk), a rotation is taking the color fi(hk) to the color hj(hk), such a color symmetry is even a transitive coloring [7]. If, as a simplification, the upper map in (1) is the identity fi(hk, `p) = hk, then the color of the `p-th subsection are identical to the source image, while the color of the symmetric subsection changes to fj(hk, `q). For the reflections, the color permutations can be defined likewise. Such a definition of a color permutation provides an algorithmic template for symmetry-consistent color distortions of source images.

2 Figure 2: Vermeer’s Girl with a Pearl Earring, 1665 (upper left) and The Glass of Wine, 1661 (middle left) and Girl with a Red Hat, 1666 (lower left) with three color distortions each.

Visual Results

In this section, the color permutations associated with the rectangular symmetry group are applied and visual results are presented. As source images three paintings by Johannes Vermeer (1632-1675) are taken. Apart from reasons of artistic taste these paintings are particularly suitable for the color symmetric image distortions I intended to study. Many Vermeer paintings feature fine-scaled and strong contrasts in color, as for instance analytically shown for the Girl with a Pearl Earring [3, 9]. The maps varying the hue con- sequently create a substantial variety of color shifts in the sections affected by color symmetry, particularly for the color maps involving trigonometric functions. Thus, fine-scaled and strong contrasts in color lead to substantial color distortions. Moreover, if at least the pixels’ saturation or brightness are preserved, the “color symmetric” sections of the image resembles the original image, but we also have a kind of alienation or distancing effect in terms of color. Results√ of the color changing distortions are shown in Figure 2. As color maps the functions f1(hk) = 0.45| sin ( 2 · 20πhk)| + 0.55| sin (20πhk)|, f2(hk) = 0.5(1 + sin (40πhk)) and h3(hk) = 4hk(1−hk), f4(hk) = 4hk(hk −1)+1, f5(hk) = 0.15(1+cos (40πhk))+0.5hk and fn(hk) = nhk mod 1 are used. In addition, there are some instances where similar functions are employed to map value or saturation, for instance to change the dark background in the upper images of Figure 2. The images obtained still resemble the Vermeer paintings, but also leave behind the impression of somehow wrong colors. Figure 3 presents color symmetric image distortions using the Vermeer paintings. In each image, either a rotation, or a horizontal or a vertical reflection is realized. Subsections as in Figure 1 are used to create symmetric color effects, namely triangular (Girl with a Pearl Earring), chessboard-line (The Glass of Wine) and bubble-like (Girl with a Red Hat) subsection. Thus, works of generative art emerge in which can be seen as quotations, homages and reinterpretations of Vermeer’s famous works.

3 Figure 3: Wrong colored Vermeer: Color-symmetric image distortions using the images in Fig. 2.

References

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[2] V. Adanova, S. Tari. “Beyond Symmetry Groups: A Grouping Study on Escher’s Euclidean Ornaments.” Graphical Models 83, 2016, 15–27.

[3] J. K. Delaney, K. A. Dooley, A. van Loon, et al. “Mapping the Pigment Distribution of Vermeer’s Girl with a Pearl Earring.” Heritage Science 8, 2020, 4.

[4] F. A. Farris. “Natural Color Symmetry.” Bridges Conference Proceedings, Waterloo, Ontario, Canada, July 27–31, 2017, pp. 131–138. http://archive.bridgesmathart.org/2017/bridges2017-131.html

[5] H. Richter. “Generating Symmetry and Symmetry Breaking in Sand-Bubbler Patterns.” In: Bongard, J., et al. (eds) Proc. Artificial Life Conference 2020 (ALIFE 2020), 2020, pages 509–517, MIT Press, Boston, MA. https://www.mitpressjournals.org/doi/abs/10.1162/isal_a_00246

[6] H. Richter. “Designing Color Symmetry in Stigmergic Art.” arXiv, 2106.14773. 2021. https://arxiv. org/abs/2106.14773

[7] R. L. Roth. “Color Symmetry and Group Theory.” Discrete Mathematics 38, 1982, 273–296.

4 [8] D. Spector. “Visualizing (Number Theoretic) Functions with Portraits.” Bridges Conference Proceedings, Online, Aug. 1–5, 2020, pp. 395–398. http://archive.bridgesmathart.org/2020/bridges2020-395. html

[9] A. van Loon, A. Vandivere, J. K. Delaney, et al. “Beauty Is Skin Deep: The Skin Tones of Vermeer’s Girl with a Pearl Earring.” Heritage Science 7, 2019, 102.

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