Art, Geography/GIScience, and Mathematics: A Surprising Interface

Abstract

Do any of the known synergies existing between either geography and art or mathematics and art bridge all three of these disciplines? The geo-humanities and the math-humanities literatures describe only these two individual synergies. A new quantitative geography methodology exploits a sophisticated mathematical concept to analyze remotely sensed satellite images, which, when extended to artistic paintings, indeed spans all three disciplines. The organizing concept is spatial autocorrelation, or the tendency for dis/similar colors and their intensities to cluster in paintings. This article summarizes demonstrations of this contention, with specific applications to da Vinci, Monet, and Rembrandt paintings. Its principal contribution is that, for high geographic resolution digital versions of paintings, a replication constructed with judiciously selected and combined spatial autocorrelation components remarkably closely corresponds with a digital copy of its original source, further generalizing certain recent findings reported in the literature.

地理与艺术、数学与艺术的协同作用, 是否将这三个学科链接起来？地理人文学和数学人文学的研究, 仅仅描述了各自的协同作用。定量地理学采用新的方法, 利用复杂的数学概念去分析遥感卫星影像。将这种新方法扩展到艺术绘画, 就能够涵盖这三个学科。这种方法基于空间自相关性的概念, 即, 绘画中（不）相似颜色及其强度的聚集趋势。本文总结了这一论点的证据, 将其应用于达芬奇、莫奈和伦勃朗的绘画作品。本文的主要贡献是, 针对高空间分辨率的数字版绘画, 通过认真选择和组合空间自相关分量去复制绘画, 能达到与原始数字版本非常接近的效果。本文还进一步概括了文献中的最新研究成果。

¿Pueden cualquiera de las conocidas sinergias que existan entre la geografía y el arte, o entre las matemáticas y el arte, entrelazar a todas las tres disciplinas? Las literaturas relacionadas con las geo-humanidades y las de las matemáticas-humanidades describen meramente estas dos sinergias individuales. Una nueva metodología de la geografía cuantitativa toma ventaja de un concepto matemático sofisticado para analizar imágenes satelitales obtenidas mediante teledetección, que, cuando se aplica a las pinturas artísticas, en vedad abarca las tres disciplinas. El concepto organizador es la autocorrelación espacial, o la tendencia al clúster de los colores di/similares y sus intensidades en las pinturas. El artículo resume las demostraciones de este planteamiento, con aplicaciones específicas a las pinturas de da Vinci, Monet y Rembrandt. Su principal contribución es que, para las versiones digitales de alta resolución geográfica de las pinturas, una réplica construida con componentes de autocorrelación espacial juiciosamente seleccionados y combinados, de modo notable se corresponde muy cercanamente con una copia digital de su fuente original, generalizando todavía más ciertos hallazgos recientes de los cuales se informa en la literatura.

Key Words:

• eigenvector
• geographic visualization
• painting
• spatial autocorrelation
• spatial weights matrix

关键词：:

• 特征向量
• 地理可视化
• 绘画
• 空间自相关性
• 空间权重矩阵。

Palabras clave::

• autocorrelación espacial
• matriz de ponderaciones espaciales
• pintura
• vector propio
• visualización geográfica

Notes

Notes

1 Matrix decompositions are useful mathematical tools for reducing any matrix to its constituent parts. Perhaps the most frequently employed of these decompositions is the calculation of eigenfunctions, or eigenvalue–eigenvector pairs. A dirigible/airship/blimp/zeppelin/three-dimensional ellipse—the largest are 194 feet long, 67 feet high, and 63 feet wide—furnishes an illuminating simple example of this scenario. Randomly sampling a very large number of (z, y, z) coordinates from its hull, where their three-axes centroid is at the center of the aircraft, followed by calculating the six pairwise coordinates’ correlation coefficients, allows the construction of a 3-by-3 matrix. The three eigenfunctions of this matrix are as follows: The largest eigenvalue quantifies the relative length of the craft, measured from its centroid, with the accompanying eigenvector establishing the foundational three-dimensional axis for this length; the second-largest eigenvalue quantifies the relative height of the craft, measured from its centroid, with the accompanying eigenvector establishing an axis orthogonal to length for it; and, the third-largest eigenvalue quantifies the relative width of the craft, measured from its centroid, with the accompanying eigenvector establishing an axis orthogonal to both length and height for it. Abudayah, Alomari, and Sander (2018) furnished another illustration, albeit more technical, concerning Sudoku puzzle solutions. Furthermore, both Gould (1967) and Tinkler (1972) furnished eigenfunction interpretations from a geographer’s perspective.

2 Footnote 1 describes a geometric interpretation of eigenfunctions. Shifting the matrix for decomposition to a spatial weights matrix changes row–column cell entries to indicators (i.e., 0–1) of whether or not row and column labeling areal units are adjacent or neighbors; it indicates which nearby locations experience direct correlation in a given spatial autocorrelation setting. Accordingly, eigenvalues furnish a global index of spatial autocorrelation degree, from maximum positive to maximum negative, for a particular spatial weights matrix—this is analogous to length in Footnote 1. The corresponding eigenvectors—each has n entries, one for and affiliated with its respective row–column label location—represent distinct map patterns composed of varying numbers and sizes of geographic clusters of values. After Legendre and his colleagues, Griffith (2021b) classified them as global, regional, and local clusters of similar attribute values for positive spatial autocorrelation. They convert to geographic clusters of contrasting values for negative spatial autocorrelation.

3 Clearly, digital images are not the original paintings they represent. The capturing and encoding of paintings as digital images (e.g., methods, resolution, format) obviously influence a rendered digital representation, and in turn certain analyses of it. This article focuses on color, particularly as seen by the naked eye. As such, the conversion of a painting to a digital image by means of spectroscopy should have minimal distortion on a color composition analysis (Delaney et al. 2016).

4 The contribution here does not necessarily concern image compression or reduction, but rather properties of spatial autocorrelation embodied in paintings. Griffith (2021a) presented some of the more informative statistical ones that help successfully differentiate between paintings; these same quantities could allow satellite remotely sensed image differentiation, too. This is a topic for future research, as is expanding the set of spatial autocorrelation properties examined. Meanwhile, today’s software industry prevailing image compression methods tend to be from computer science, and lack serious spatial conceptualizing. For example, lossy image compression, a powerful image-management tool, is a type of digital image data reduction technique that reduces computer file size for storage or transmission by reorganization coupled with the discarding of some information, which, in turn, can affect decompressed image quality. JPEG compression is perhaps the most common algorithm used to achieve this size reduction. Its initial step can involve subsampling of color information (less resolution for chroma than for luma information) without a significant loss of visible image information. Its most frequent implementation is 2-by-2 subsampling, which partitions an image into 2-by-2-pixel blocks and only stores the average color information for each block. This subsampling is more commonly performed with video than still images. Next, converted image data with subsampled color components are divided into 8-by-8-pixel blocks, and then each of these blocks is transformed with the discrete cosines transformation (DCT)—sixty-four before-transformation pixel values become sixty-four after-transformation coefficients representing the spatial frequencies for vertical and horizontal orientation. One outcome of this DCT is an averaging of nearby pixel colors. In the ensuing step, these sixty-four coefficients are quantized (i.e., most likely, higher rather than lower frequency coefficients are converted to zero) to achieve data reduction. At this point, the data are reordered from low to high frequencies in a zig-zag fashion across each 8-by-8 block. Accordingly, for storage purposes, kx0 can be substituted for k individual 0s at the end of a sequence of coefficients (i.e., this reordering and coding is labeled Huffman-encoding, an entropy-based algorithm that relies on an analysis of the frequency of symbols in an array), accomplishing image size reduction. The zig-zag sequence tends to optimize k. Alternatively, WebP replaces the averaging of color information with a prediction for each pixel conditional on the fragments surrounding it. The data inserted into the resulting compressed image are the differences between predicted and observed color. A preponderance of empirical applications suggests that many of these predictions are accurate, yielding a difference of zero. Consequently, achieving compression involves replacing large numbers of stored zeroes with a single symbol, which by happenstance might exploit spatial autocorrelation. In contrast, geo-oriented spatially informed methods exist (e.g., Bian and Xie 2004), at least in the academic literature, but are not the topic of this article.

5 One discovery with this analytical eigenvector solution is that most computer software packages, which calculate eigenfunctions numerically, do not necessarily yield a linearly independent set of theoretical eigenvectors (although numerically they are correct). Rather, they yield some that are linear combinations of their theoretical eigenvector counterparts (these linear combinations vary from one software package algorithm to another) to ensure their pairwise orthogonality property. This technical point is noteworthy only for analysis replication purposes.

Additional information

Notes on contributors

Daniel A. Griffith

DANIEL A. GRIFFITH is an Ashbel Smith Professor in the Geography and Geospatial Information Sciences Program, University of Texas at Dallas, Richardson, TX 75080. E-mail: [email protected]. His research interests include spatial statistics and econometrics, quantitative urban economic geography and regional science, and urban public health.