This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Signal Processing 93 (2013) 573–578 Contents lists available at SciVerse ScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro The Abstract Expressionists and Les Automatistes: A shared multi-fractal depth? J.R. Mureika a,n, R.P. Taylor b a Department of Physics, Loyola Marymount University, Los Angeles, CA 90045, USA b Department of Physics, University of Oregon, Eugene, OR 97403, USA article info abstract Article history: Statistical analysis of abstract paintings is becoming an increasingly important tool for Received 15 August 2011 understanding the creative process of visual artists. We present a multifractal analysis Received in revised form of ‘poured’ paintings from the Abstract Expressionism and Les Automatistes move- 16 March 2012 ments. The box-counting dimension (D0) is measured for the analyzed paintings, as is Accepted 2 May 2012 the associated multifractal depth DD¼D ÀD , where D is the asymptotic dimension. Available online 2 June 2012 0 N N We investigate the role of depth by plotting a ‘phase space’ diagram that examines the Keywords: relationship between D0 and DN. We show that, although the D0 and DN values vary Multifractal between individual paintings, the collection of paintings exhibit a similar depth, Art suggesting a shared visual characteristic for this genre. We discuss the visual implica- tions of this result. & 2012 Elsevier B.V. All rights reserved. 1. Introduction esthetics. However, despite the millions of words written about this body of work through the years, the artistic In 1945, Jackson Pollock started to perfect a radically significance behind these complex swirls of paint new approach to painting. Abandoning physical contact remained the source of fierce debate in the art world. with the canvas, he dipped his brush in and out of a can One of the central questions within this debate concerned and poured the fluid paint onto horizontal canvases. The the degree of variability between paintings by different uniquely continuous paint trajectories served as artistic artists: for example, are there shared visual characteris- ‘fingerprints’ of his motions through the air. Over the next tics between Pollock’s paintings and those of Les Auto- decade, he generated vast abstract works featuring com- matistes? Examples of their respective works are shown plex patterns formed across many scales—from the width in Figs. 1 and 2. of the canvas down to finest speck of paint. Fractal analysis techniques hold great promise for both In doing so, he became one of the leaders of the the academic and artistic communities, since these tech- Abstract Expressionist movement, which shifted the focus niques serve to identify underlying visual signatures in an of the art world from its traditional base of Paris to New artwork. The fractal dimension DF of an artwork can be York. Pollock’s form of Abstract Expressionism inspired regarded as a preliminary indicator of complexity in a the Quebec-based Les Automatistes who also adopted the pattern: lower fractal dimensions are a measure of shal- pouring style of painting. Art theorists now recognize the low complexity, while higher fractal dimensions (i.e. ‘‘drip and pour’’ style as a revolutionary approach to those which approach the dimension of the embedding space) demonstrate high complexity. That is, a line has fractal dimension DF¼1, while a wrapping-curve that n densely fills the plane has dimension D -2. Within this Corresponding author: Tel.: þ1 310 338 7809; F fax: þ1 310 338 5816. scheme, fractal paintings are quantified by DF values in E-mail address: [email protected] (J.R. Mureika). the range 1oDF o2, where paintings with DF values 0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2012.05.002 Author's personal copy 574 J.R. Mureika, R.P. Taylor / Signal Processing 93 (2013) 573–578 Fig. 2. The blob structure of Tumulte (1973) by Les Automatistes. problem to the multifractal arena and extracted the full spectrum of D values for q¼0toq-N [4,5]. Based on this initial research, we and a number of other groups have since shown diverse fractal analysis techniques to be useful approaches to quantifying the visual complexity of Pollock’s poured patterns [6–24]. It is critical to note a fundamental limitation common to all of these fractal analysis techniques [9,10]. Along with nature’s fractals, art works are examples of physical fractals and consequently have inherent upper and lower cut-offs beyond which their fractal geometry either can- not be resolved or does not exist. The magnification range of nature’s physical fractals can be surprisingly small – the typical range is only 1.25 orders – and this is an inevitable quality of fractal art also [3]. For this reason, we adopt the term ‘‘effective dimension’’ to highlight this limited-range quality. Nevertheless, the Dq values extracted from the multifractal analysis still quantify the associated visual complexities of the pattern [25,26]. The importance of comparing and contrasting para- meters extracted from multiple analysis techniques was emphasized early in this field’s development [2]. More recently, the employment of ‘phase space’ diagrams, generated by plotting different parameters along several axes, was highlighted as an efficient approach to identify- Fig. 1. Reflections of the Big Dipper (1947), by Jackson Pollock. Image progression shows section of (a) original painting, and (b) blob structure ing the trademark ‘parameter space’ for given paintings for the black pigment layer. [22]. Here, we build on earlier multi-fractal comparisons of Pollock’s patterns with those of Les Automatistes by constructing a phase space diagram based on the box- closer to 1 appearing simple and sparse, and paintings counting dimension D0 and the asymptotic dimension with DF values closer to 2 appearing more rich and DN. This plot is used to investigate the multifractal depth, intricate. The multifractal spectrum takes a closer look which is defined as DD¼D0 ÀDN for the art works. We at this visual relationship by considering an infinite family show that, although the D0 and DN values vary between of fractal dimensions {Dq, q¼0, 1, 2 y } that yield key individual paintings, the collection of paintings exhibit a information about the degree to which complexity is similar depth, suggesting a shared visual characteristic for manifest in a pattern. the ‘poured’ genre of art. The focus of previous fractal Motivated by these concepts, one of us (R.P.T.) studies has been the search for variations between art employed an established fractal analysis called the ‘box- works by different artists, with the aim of developing counting’ technique to extract D0 values for Pollock’s novel authenticity techniques. In contrast, the work pre- poured paintings [1–3]. The fractal character of Pollock’s sented here serves as a reminder that identifying shared work was confirmed by J.R.M., who also extended the qualities is equally important for art history. Author's personal copy J.R. Mureika, R.P. Taylor / Signal Processing 93 (2013) 573–578 575 2. Multifractal analysis of art construction of the painting. The largest patterns, pre- sumably created by wide-swing arm motions and move- 2.1. Fractals and multifractals ment of the artist, are characterized by Dq (small q), while the finer brush motions of the wrist and fingers yield the The ‘‘box-counting’’ method is a well-established tech- large q dimensionality. This is a similar interpretation to nique for extracting the fractal dimension D0 for a fractal the bi-fractal behavior discussed in [1–3]. pattern. In this approach, digitized images of paintings are A convenient measure of the complexity depth of a covered with a computer-generated mesh of identical multifractal is defined as squares (or ‘‘boxes’’). The statistical scaling qualities of DD ¼ D0ÀD1 ð1Þ the pattern are then determined by calculating the pro- portion of squares occupied by the painted pattern and Patterns with a rich multifractal structure will have the proportion that are empty. This process is then large DD, while those with little or no variation will show repeated for meshes with increasingly small square sizes. the opposite. A monofractal is characterized by DD¼0. Reducing the square size is equivalent to looking at the The multifractal spectrum of dimensions is readily pattern at finer magnification. In this way, it is possible to calculated by a modified box counting algorithm. As compare the pattern’s statistical qualities at different described above, the pattern is covered by N(e) boxes of magnifications. Specifically, the number of squares, n(e), scale size e, of which only n(e) actually contain the that contained part of the painted pattern can be counted pattern. These contribute to the multifractal moments and this is repeated as the size, e, of the squares in the [27,28], mesh was reduced. The largest size of square is chosen to XNðeÞ n ðeÞ Xðq,eÞ¼ ½p ðeÞq, p ðeÞ¼ i ð2Þ match the canvas size (for a Pollock painting this is i i NðeÞ typically e2.5 m) and the smallest is chosen to match i ¼ 1 the finest paint work (e1 mm).