The Enigma of Complexity
TThe Enigma of Complexity
Jon McCormack − − − , Camilo CruzGambardella − − − , and Andy Lomas − − − SensiLab, Monash University, Melbourne, Australia
[email protected] [email protected] https://sensilab.monash.edu Goldsmiths, University of London, London, UK [email protected] https://andylomas.com
Abstract.
In this paper we examine the concept of complexity as itapplies to generative art and design. Complexity has many different, dis-cipline specific definitions, such as complexity in physical systems (en-tropy), algorithmic measures of information complexity and the field of“complex systems”. We apply a series of different complexity measuresto three different generative art datasets and look at the correlationsbetween complexity and individual aesthetic judgement by the artist (inthe case of two datasets) or the physically measured complexity of 3Dforms. Our results show that the degree of correlation is different for eachset and measure, indicating that there is no overall “better” measure.However, specific measures do perform well on individual datasets, indi-cating that careful choice can increase the value of using such measures.We conclude by discussing the value of direct measures in generative andevolutionary art, reinforcing recent findings from neuroimaging and psy-chology which suggest human aesthetic judgement is informed by manyextrinsic factors beyond the measurable properties of the object beingjudged.
Keywords:
Complexity · aesthetic measure · generative art · generativedesign · evolutionary art · fitness measure. “The number of all the atoms that compose the world is immense butfinite, and as such only capable of a finite (though also immense) numberof permutations. In an infinite stretch of time, the number of possiblepermutations must be run through, and the universe has to repeat itself.Once again you will be born from a belly, once again your skeleton willgrow, once again this same page will reach your identical hands, onceagain you will follow the course of all the hours of your life until that ofyour incredible death.”—Jorge Luis Borges, The doctrine of cycles , 1936 a r X i v : . [ c s . N E ] F e b J. McCormack et al.
Complexity is a topic of endless fascination in both art and science. Forhundreds of years scholars, philosophers and artists have sought to understandwhat it means for something to be “complex” and why we are drawn to com-plex phenomena and things. Today, we have many different understandings ofcomplexity, from information theory, physics, and aesthetics [6, 8, 32, 37].In this paper we again revisit the concept of complexity, with a view tounderstanding if it can be useful for the generative or evolutionary artist. Theapplication of complexity measures and their relation to aesthetics in generativeand evolutionary art are numerous (see e.g. [14] for an overview). A number ofresearchers have tested complexity measures as candidates for fitness measuresin evolutionary art systems for example. Here we are interested in the value ofcomplexity to the individual artist or designer, not the system (or though thatmay benefit too). Put another way, we are asking what complexity can tell usabout an individual artist’s personal aesthetic taste or judgement, rather thanthe value of such measures in general.A long held intuition is that visual aesthetics are related to an artefact’sorder and complexity [2, 15, 22]. From a human perspective, complexity is oftenregarded as the amount of “processing effort” required to make sense of anartefact. Too complex and the form becomes unreadable, too ordered and onequickly looses interest. Birkhoff [4] famously formalised an aesthetic measure M = O/C , the ratio of order to complexity [4], and similar approaches have builton this idea. To mention some examples, Berlyne and colleagues, defined visualcomplexity as “irregularities in the spatial elements” that compose a form [30],which lead to the formalisation of the relationship between pleasantness andcomplexity as an “inverted-U” [2]. That is, by increasing the complexity of anartefact beyond the “optimum” value for aesthetic preference, its appeal starts todecline [35]. Another example is Biderman’s theory of “geons”, which proposesthat human understanding of spatial objects depends on how discernible its basicgeometric components are [3, 30] Thus, the harder an object is to decomposeinto primary elements, the more complex we perceive it is. This is the basis forsome image compression techniques, which are also used as a measure of visualcomplexity [16].More recent surveys and analysis of computational aesthetics trace the his-tory [9, 12] and current state of research in this area [14]. Other approachesintroduce features such as symmetry as a counterbalance to complexity, situat-ing aesthetic appeal somewhere within the range spanning between these twoproperties [30]. The most recent approaches combine measures of algorithmiccomplexity with different forms of filtering or processing to eliminate noise butretain overall detail [16, 38].Multiple attempts to craft automated methods for the aesthetic judgementof images have made use of complexity measures. Moreover, some of these showencouraging results. In this paper we test a selection of these methods on threedifferent image datasets produced using generative art systems. All of the imagesin these datasets have their own “aesthetic” score as a basis for understandingthe aesthetic judgements of the system’s creator. he Enigma of Complexity 3
Computational methods used to calculate image complexity are based on thedefinitions of complexity described the previous section (Section 1): the amountof “effort” required to reproduce the contents of the image, as well as the wayin which the patterns found in an image can be decomposed. Some methodshave been proposed as useful measures of aesthetic appeal, or for predicting aviewer’s preference for specific kinds of images. In this section we outline theones relevant for our research.In the late 1990s Machado and Cardoso proposed a method to determineaesthetic value of images derived from their interpretation of the process thathumans follow when experiencing an aesthetic artefact [20]. In their method theauthors use a ratio of
Image Complexity – a proxy for the complexity of the artitself – to
Processing Complexity – a proxy of the process humans use to makesense of an image – as a representation of how humans perceive images.In 2010, den Heijer and Eiben compared four different aesthetic measures ona simple evolutionary art system [11], including Machado and Cardoso’s
ImageComplexity / Processing Complexity ratio, Ross & Ralph’s colour gradient bellcurve, and the fractal dimension of the image. Their experiments demonstratedthat, when used as fitness functions, different metrics yielded stylistically differ-ent results, indicating that each assessment method biases the particular imagefeatures or properties being evaluated. Interestingly, when interchanged – whenthe results evolved with one metric are evaluated with another – metrics showeddifferent affinities, suggesting that regardless of the specificity of each individualmeasure, there are some commonalities between them.
To try and answer our question about the role and value of complexity measuresin developing generative or evolutionary art systems, we compared a variety ofcomplexity measures on three different generative art datasets, evaluating themfor correlation with human or physical measures of aesthetics and complexity.
We tested a number of different complexity measures described in the literatureto see how they correlated with individual evaluations of aesthetics. We firstbriefly introduce each measure here and will go into more detail on specificmeasures later in the paper.
Entropy ( S ): the image data entropy measured using the luminance histogram(base e ). Energy ( E ): the data energy of the image. Contours ( T ): the number of lines required to describe component boundariesdetected in the image. The image first undergoes a morphological binarisa-tion (reduction to a binary image that differentiates component boundaries)before detecting the boundaries. J. McCormack et al.
Euler ( γ ): the morphological Euler number of the image (effectively a count ofthe number of connected regions minus the number of holes). As with the T measure, the image is first transformed using a morphological binarisation. Algorithmic Complexity ( C a ): measure of the algorithmic complexity of theimage using the method described in [16]. Effectively the compression ratioof the image using Lempil-Ziv-Welch lossless compression. Structural Complexity ( C s ): measure of the structural complexity, or “noise-less entropy” of an image using the method described in [16]. Machardo-Cardoso Complexity ( C mc ): a complexity measure used in [21],without edge detection pre-processing. Machardo-Cardoso Complexity with edge detection ( C Emc ): the C mc mea-sure with pre-processing of the image using a Sobel edge detection filter. Fractal Dimension ( D ): fractal dimension of the image calculated using thebox-counting method [7]. Fractal Aesthetic ( D a ) aesthetic measure similar to that used in [10], basedon the fractal dimension of the image fitted to a Gaussian curve with peak at1.35. This value is chosen based on an empirical study of aesthetic preferencefor fractal dimension.While each of these measures is in some sense concerned with measuring im-age complexity, the basis of the measure for each is different. Entropy ( S ) and Energy ( E ) measures are based on information theoretic understandings of com-plexity but concern only the distribution of intensity, while Contours ( T ) and Euler ( γ ) try to directly count the number of lines or features in the image, some-what in line with perceptual notions of complexity. Lakhal et. al’s AlgorithmicComplexity ( C a ) and Machardo & Cardoso’s Complexity ( C mc ) measures usealgorithmic or Kolmogrov-like understandings of complexity, relying on imagecompression algorithms to proxy for visual complexity. Lakhal et. al also definea Structural Complexity measure ( C s ) designed to address the limitations of al-gorithmic complexity measures in relation to high frequency noise or many finedetails. This is achieved by a series of “course-graining” operations, effectivelylow-pass filtering the image to remove high frequency detail in both the spatialand intensity domains. Finally, the fractal methods recognise self-similar featuresas proxies for complexity. They are based on past analysis of art images thatdemonstrated relationships between fractal dimension and aesthetics [7, 31, 36]. For the experiments described in this paper, we worked with three differentgenerative art datasets (Figure 1). As the goal of this work was to understandthe effectiveness of complexity measures in actual generative art applications,we wanted to work with artistic systems of demonstrated success, rather thaninvented or “toy” systems often used in this research. This allows us to under-stand the ecological validity [5] of any system or technique developed. Ecologicalvalidity requires the assessment of creative systems in the typical environmentsand contexts under which they are actually developed and used, as opposed he Enigma of Complexity 5a b c
Fig. 1.
Example images from the Lomas (a), Line Drawing (b) and 3D DLA Forms (c)datasets. to laboratory or artificially constructed settings. It is considered an importantmethodology for validating research in the creative and performing arts [13].Additionally, all the datasets are open access, allowing others to validate newmethods on the same data.
Dataset 1: Andy Lomas’ Morphogenetic Forms
This dataset [28] consistsof 1,774 images generated using a 3D morphogenetic form generation system, de-veloped by computer artist Andy Lomas [18,19]. Each image is a two-dimensionalrendering (512 ×
512 pixels) of a three-dimensional form that has been algorith-mically “grown” from 12 numeric parameters. The images were evolved usingan
Interactive Genetic Algorithm (IGA)-like approach with the software
SpeciesExplorer [18,19]. As the 2D images, not the raw 3D models are evaluated by theartist, we perform our analysis similarly.The dataset contains an integer numeric aesthetic rating score for each form(ranging from 0 to 10, with 1 the lowest and 10 the highest, 0 meaning a failurecase where the generative system terminated without generating a form or theresult was not rated). These ratings were all performed by Lomas, so represent hispersonal aesthetic preferences. Additionally, each form is categorised by Lomasinto one of eight distinct categories (these were not used in the experimentsdescribed in this paper).
Dataset 2: DLA 3D Prints
This dataset [27] consists of 2,500 3D formscreated using a Differential Line Algorithm (DLA) based method [1]. Multipleclosed 2D line segments develop over time. At each time-step the geometry iscaptured and forms a sequential z-layer in a 3D form. After several hundred time-steps, the final 3D form is generated, suitable for 3D printing (Figure 2). Eachimage is 600 ×
600 pixels resolution. Images in this set are 3D line renderingsof the final form, from a perspective projection and orthographic projection inthe xy plane. In the experiments described here we tested both the top-down J. McCormack et al.
Fig. 2.
Example 3D printed from from the DLA 3D Prints dataset. orthographic images and perspective images, finding the perspective images gavebetter results and so are the ones reported here.Rather than human-designated aesthetic measures, this dataset has a phys-ically computed complexity measure. This measure is based on two geometricaspects of the 3D form: convexity (how much each layer deviates from its con-vex hull) and the quartile coefficient of dispersion of angles between consecutiveedges that make up each layer in the 3D form. These measures are calculated foreach layer (weighted equally) and the final measure is the mean of all the layersin the form. This physical complexity measure appears to be a reasonable proxyto the visual complexity of the forms generated by the system.
Dataset 3: Line Drawings
A set of 53 line drawings generated using an agent-based method based on the biological principles of niche construction [24, 26].Each image is 1024 × , Our preliminary investigations showed that some measures are sensitive to pa-rameter settings. The structural complexity measure ( C s ) has two parame-ters: r cg , a course-grain filter radius (in pixels), δ ∈ [0 , .
5] a threshold fordetermining the black to white pixel ratio, η ∈ [0 ,
1] (white if η ≤ δ , grey if δ < η ≤ − δ , black for η > − δ ). In the original study, the authors [16] used he Enigma of Complexity 7 values ( r cg , δ ) = (7 , .
23) for one set of test images (abstract textures generatedby Fourier synthesis) and (13 , .
12) for the second set (abstract random boxesplaced using an inverse of the fractal box counting method) for 256 ×
256 res-olution images. For the experiments described her we used ( r cg , δ ) = (5 , . Original Image r = 2 r = 200Fractal Dimension 1.864 1.845 Fig. 3.
The effect of different adaptive binarisation radii on an image from the Lomasdataset
For the fractal dimension measurements (
D, D a ), images are pre-processedusing a local adaptive binarisation process to convert the input image to a binaryimage (typically used to segment the foreground and background). A radius, r , is used to compute the local mean and standard deviation over (2 r + 1) × (2 r + 1) blocks centered on each pixel. Values above the mean of the r -rangeneighbourhood are replaced by 1, others by 0. Figure 3 shows a sample imagefrom the Lomas dataset (left) with binary versions for r = 2 (middle) and r = 200 (right). Higher values of r tend to reduce high frequency detail andresult in a lower fractal dimension measurement. For the DLA 3D prints andLine Drawing datasets, which are already largely comprised of lines, the valueof r has negligible effect on the measurement.Our Fractal Aesthetic Measure ( D a ) is defined as: D a ( i ) = exp ( − ( D ( i ) − p ) σ ) , (1)where p is the peek preference value for fractal dimension and σ the width ofthe preference curve. D a returns a normalised aesthetic measure ∈ [0 , p, σ ) = (1 . , . C mc )) is defined as: C mc ( i ) = RM S ( i, f ( i )) × s ( f ( i )) s ( i ) , (2) J. McCormack et al. where i is the input image, RM S a function that returns the root mean squarederror between it’s two arguments, f a lossy encoding scheme for i and s a functionthat returns the size in bytes of its argument. For the lossy encoding schemewe used the standard JPEG image compression scheme with a compression levelof 0.75 (0 is maximum compression).
For each dataset we computed the full set of complexity measures (Section 3.1)on every image in the dataset, then computed the Pearson correlation coefficientbetween each measure and the human assigned aesthetic score (Lomas and LineDrawings datasets) or physically calculated complexity measure (DLA 3D Printsdataset).
Table 1.
Lomas Datatset: Pearson’s correlation coefficient values between image mea-surements and aesthetic score ( Sc ). The C mc complexity measure (bold) has the highestcorrelation with aesthetic score for this dataset. In all cases p -values are < × − . S E T γ C a C s C mc C Emc
D D a Sc S E − .
989 1 T . − .
375 1 γ − .
423 0 . − .
999 1 C a . − .
945 0 . − .
495 1 C s . − .
874 0 . − .
659 0 .
940 1 C mc . − .
732 0 . − .
589 0 .
907 0 .
860 1 C Emc . − .
699 0 . − .
602 0 .
869 0 .
907 0 .
930 1 D − .
352 0 .
452 0 . − . − . − .
052 0 .
223 0 .
257 1 D a . − . − .
318 0 . − . − . − . − . − .
931 1Sc 0 . − .
590 0 . − .
536 0 .
757 0 . . .
774 0 . − .
389 1
The results are shown for each dataset in Tables 1 (Lomas), 2 (DLA 3DPrints) and 3 (Line Drawings) with the highest correlation measure shown inbold.As the tables show, a different complexity measure performed best for eachdataset. For the
Lomas dataset there is a strong correlation (0.873) betweenthe artist assigned aesthetic score and the C mc complexity measure, and that all We adopted this measure as it specifically deals with complexity as defined in [22].Machardo & Cardoso also define an aesthetic measure as the ratio of image complex-ity to processing complexity [20], as used by den Heijer & Eiben in their comparisonof aesthetic measures [10].he Enigma of Complexity 9
Table 2.
DLA 3D Prints Datatset: Pearson’s correlation coefficient values betweenimage measurements and physically computed complexity score ( Sc ). The C s structuralcomplexity measure (bold) has the highest correlation with aesthetic score for thisdataset. S E T γ C a C s C mc C Emc
D D a Sc S E − .
995 1 T . − .
880 1 γ . − . − .
363 1 C a . − .
956 0 . − .
106 1 C s . − .
892 0 . − .
204 0 .
942 1 C mc . − .
935 0 . − .
197 0 .
969 0 .
950 1 C Emc . − .
935 0 . − .
188 0 .
965 0 .
954 0 .
999 1 D . − .
949 0 .
869 0 .
012 0 .
896 0 .
801 0 .
898 0 .
895 1 D a − . − . − . − . − . − . − . − . − .
972 1Sc 0 . − .
726 0 . − .
066 0 . . .
704 0 .
706 0 . − .
434 1
Table 3.
Line Drawing Datatset: Pearson’s correlation coefficient values between imagemeasurements and aesthetic score ( Sc ). The Contours T measure (bold) has the highestcorrelation with aesthetic score for this dataset. S E T γ C a C s C mc C Emc
D D a Sc S E − .
910 1 T . − .
677 1 γ − .
559 0 . − .
000 1 C a . − .
934 0 . − .
541 1 C s . − .
717 0 . − .
474 0 .
618 1 C mc . − .
690 0 . − .
233 0 .
592 0 .
761 1 C Emc . − .
811 0 . − .
312 0 .
712 0 .
822 0 .
927 1 D . − .
807 0 . − .
431 0 .
640 0 .
835 0 .
867 0 .
914 1 D a − .
434 0 . − .
323 0 . − . − . − . − . − .
942 1Sc 0 . − . . − .
564 0 .
218 0 .
364 0 .
267 0 .
199 0 . − .
457 10 J. McCormack et al. the algorithmic and structural complexity measures are highly correlated. Thisis to be expected since they all involve image compression ratios. It is furtherhighlighted in Figure 4, which shows a plot of aesthetic score vs C mc (a) and C s vs C mc (b). The banding in 4a is due to the aesthetic scores being integers. Aclear non-linear relationship between the complexity measures C s and C mc canbe seen in 4b. Aesthetic Score vs C mc C a vs C mc a b Fig. 4.
Plots for the Lomas dataset showing the relationship between aesthetic scoreand C mc (a) and C a vs C mc (b). Also of note is that fractal measures performed the worst of the measurestested. This seems to be confirmed visually: while certainly the images are com-plex (many are composed of 1 million or more cells) and have patterns at differentscales, the patterns are not self-similar.
Sc vs C s Sc vs D a a b Fig. 5.
Plots for the DLA 3D Prints dataset showing the relationship between physicalcomplexity score ( Sc ) and C s (a) and D a (b).he Enigma of Complexity 11 For the
DLA 3D Prints the most highly correlated measure was structuralcomplexity ( C s ) with a correlation of 0.774. The structural complexity aimsto give a “noiseless entropy” measure by filtering high frequency spatial andintensity details. Given that the images are composed of many hundreds of thinlines stacked on top of each other, there is a significant amount of high frequencyinformation, hence filtering is likely to give a better measure of real geometricdetails in each form. As can be seen in Figure 5 a clear correlation can be seenbetween the physical complexity ( Sc ) and Structural Complexity measure ( C s ).Again we note that the fractal measures ( D, D a ) had the lowest correlation andthat all the algorithmic complexity measures are highly correlated. As shownin Figure 5b however, there appears to be a kind of bifurcation and clusteringin the relationship between Sc and D a , indicating a more complex relationshipbetween fractal dimension and complexity. a b Fig. 6.
Thumbnail grid of the entire Line Drawing dataset, ordered with increasingaesthetic score (lowest top left, highest bottom right) (a) and ordering by structuralcomplexity ( C s ). As the size of the dataset is relatively small in comparison with theothers, the images can be shown in the figure. The
Line Drawing dataset exhibited quite different results over the pre-vious two. Here the Contours ( T ) measure had the highest, but only moderate,correlation with artist-assigned aesthetic scores (0.565). Given the nature of thedrawings, measures designed to capture morphological structure seem most ap-propriate for this dataset. It is also interesting to note that the algorithmic com-plexity measures perform relatively poorly in this case. The original basis for thedrawings came from the use of niche construction as a way to generate density Readers should not draw any direct relation between the terms “structural” and“physical” in relation to complexity used here. Structural refers to image structures,whereas physical refers to characteristics of the 3D form’s line segments.2 J. McCormack et al. variation in the images. The dataset contains images both with and without theuse of niche construction, and generally those with niche construction are morehighly ranked than those without. Figure 6 shows the entire dataset ordered interms of artist-assigned aesthetic score (a) and structural complexity (b). Thedrawings with niche construction are easy to see as they are more highly rankedthan those without. The structural complexity measure has greater difficulty indifferentiating them (b).With this in mind, we ran an additional image measure on this dataset thatlooks at asymmetry in intensity distribution (
Skew ). Since the niche constructionprocess results in contrasting areas of high and low density it was hypothesisedthat this measure might be able to better capture the differences. This measurehad a correlation of 0.583 ( p = 4 . × − ), so better than any of the othermeasures, but still only mildly correlated. Our results show that there appears to be no single measure that is best toquantify image complexity in the the context of generative art. Hence it seemswise to select a measure most appropriate to the style or class of imagery orform being generated.It is also important to point out that, in general, computer synthesised im-agery and in particular images generated by algorithmic methods, have impor-tant characteristics that differ from other images, such as photographs or paint-ings. Apart from any semantic differences or differentiation between figurativeand abstract, intensity and spatial distributions in computer synthesised imagesdiffer from real world images. This is one reason why we selected datasets thatare specific to the application of these measures (generative art and design),rather than human art datasets in general, for example.The rational for this research was to further the question: how can com-plexity measures be usefully employed in generative and evolutionary art anddesign?
Based on the results presented in this paper, our answer is that – ifchosen appropriately – they can be valuable aids in course-level discrimination.Additionally, they are quite quick to compute and work without prior trainingor exposure to large numbers of examples or training sets. So, for example, theycould be helpful in filtering or ranking individuals in an IGA or used to helpclassify or select individuals for further enhancement. However they are insuffi-cient as fully autonomous fitness measures – the human designer remains a vitaland fundamental part of any aesthetic evaluation.
In Section 1 we discussed possible relationships between complexity measuresand aesthetics. It is worth reflecting further here on this relationship and thelong-held “open problem” for evolutionary and generative art of quantifyingaesthetic fitness [23]. he Enigma of Complexity 13
In the last decade or so, the biggest advances in the understanding of compu-tational and human aesthetic judgements have come from (i) large, open accessdatasets of imagery with associated human aesthetic rankings and (ii) psycho-logical and neuroscience discoveries on the mechanisms of forming an aestheticjudgement and what constitutes aesthetic experience.In a recent paper, Skov summarised aesthetic appreciation from the per-spective of neuroimaging [33]. Some of the key findings included neuroscientificevidence suggesting that “aesthetic appreciation is not a distinct neurobiologicalprocess assessing certain objects, but a general system, centered on the mesolim-bic reward circuit, for assessing the hedonic value of any sensory object” [33].Another important finding was that hedonic values are not solely determinedby object properties. They are subject to numerous extrinsic factors outside theobject itself. Similar claims have come from psychological models [17]. Thesefindings suggest that any algorithmic measure of aesthetics which only considersan object’s visual appearance ignores many other extrinsic factors that humansuse to form an aesthetic judgement. Hence they are unlikely to correlate stronglywith human judgements.Our results appear to tally with these findings. Complexity measures, care-fully chosen for specific styles or types of generative art can capture some broadaspects of personal aesthetic judgement, but they are insufficient alone to fullyreplace human judgement and discretion. Using other techniques, such as deeplearning, may result in slightly better correlation to individual human judge-ment [29], however such systems require training on large datsets which can betedious and time-consuming for the artist and still do not do as well as thetrained artist’s eye in resolving aesthetic decisions.
Making and appreciating art is a shared human experience. Computers canexpand and grow the creative possibilities available to artists and audiences.The fact that humans artists are successfully able to create and communicateartefacts of shared aesthetic value indicates some shared concept of this valuebetween people and cultures. Could machines ever share such concepts? Thisremains an open question, but evidence suggests that achieving such a unitywould require consideration of factors beyond the quantifiable properties of ob-jects themselves.In this paper we have examined the relationship between complexity mea-sures and personal or specific understandings of aesthetics. Our results suggestthat some measures can serve as crude proxies for personal visual aesthetic judge-ment but the measure itself needs to be carefully selected. Complexity remainsan enigmatic and contested player in the long-term game of computational aes-thetics.
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