Jackson Pollock, Mural

Murual is an example of Pollock's drip oeuvre, executed by pouring and dripping paint onto the canvas on the floor. There are a large number of such works of doubtful authorship, and indeed outright fakes, and computer image analysis has been explored as a method to aid in the authentication of such works. The following is based on:

• Mohammad Irfan and David G. Stork, "Multiple visual features for the computer authentication of Jackson Pollock's drip paintings: Beyond box counting and fractals," SPIE Electronic Imaging: Machine vision applications II, 2009 (pdf)
• David G. Stork, "Learning based authentication of Jackson Pollock's drip paintings," SPIE Newsroom, May 2009 (pdf)
• Mahmoud AlAyyoub, Mohammad T. Irfan and David G. Stork, "Machine learning of multi-feature visual texture classifiers for the authentication of Jackson Pollock's drip paintings," in Computer vision and image analysis of art II, David G. Stork, Jim Coddington and Anna Bentkowska-Kafel (eds.), 2011 (pdf)

The debate over Fractals

Richard Taylor, a professor of physics and painter with a masters degree in art, pioneered the use of fractals for the authentication of Pollock's drip paintings in the mid-1990s. A fractal is a mathematical structure that exhibits self-similarity: the shape of the figure at one scale is nearly the same as that at a different scale. Taylor and his colleagues used a "box counting" algorithm to estimate the fractal properties of Pollock's works. The details need not concern us here, but the basic method is to divide up the painting into boxes of different size and count the number of boxes that contain the image of any paint. For large boxes each box contains some paint, and thus the proportion of boxes containing paint is 100%. But as the boxes are made smaller and smaller, an ever increasing percentage will not have paint. The plot of this proportion (on a log-log scale) yields the fractal dimension. Taylor and his colleagues gave evidence that genuine Pollock's yielded plots that had two segments with slightly different slopes, due (they believed) to Pollock's large arm gestures and to the physics of splattering paint.

This work was criticised, primarily by Jones-Smith and her colleagues, on three grounds:

1. Non-uniqueness: The most serious criticism was the empirical evidence that images quite unlike Pollocks works could nevertheless exhibit the same "fractal" characteristics. In fact, an angular computer image could have such properties.
2. Non-fractality: Jones-Smith et al claimed that the range of box sizes used in Taylor's analyses was too small to infer true fractal properties.
3. Disruption due to occlusion: A related claim was that higher layers of paint in Pollock's works partially obscured lower layers, and thus any fractal properties of the lower layers would be disrupted, incorrectly estimated from the (visible) image.

Our responses to these criticisms are as follows:

1. Non-uniqueness: Theoretically, even if a feature, taken alone, is uninformative, it can be informative when used in conjunction with other features... even if those features are themselves also uninformative. The famous exclusive-OR problem is one such example where feature f1 is uninformative and f2 is uninformative, but f1 XOR f2 classifies the patterns perfectly. Moreover, we showed empiracally that multiple features lower the error rate, even when one of the features is nearly uninformative.
2. Non-fractality: While this criticism may be valid, it is also irrelevant. In pattern recognition, the precise name of a feature is not important, just that it be specifiable. (Of course, too, one must check that it provides use in a full classifier, including a multi-feature classifier.) So we can ignore the term "fractal" and call the computed feature the "box-counting statistic," or whatever we like.
3. Disruption due to oclusion: Just as in the case of non-fractality, the visual feature computed from an occluded paint layer may be useful anyway; it need not be a "fractal." Moreover, our empirical results show that such a feature may indeed improve accuracy of a classifier for authenticating Pollocks.

In sum, it is clear that a single visual feature, such as a "fractal feature," is insufficient to distinguish genuine from fake Pollocks. After all, the large body of research in the relevant field of texture classification shows that all successful methods rely on multiple visual features. We should not reject the use of a "box-counting statistic" for Pollock authentication studies, but rather augment such a feature with other visual features, and use machine learning methods to integrate them for the most accurate classifier.

Image-based authentication of Pollock's drip paintings is an example of texture classification, and several decades of research shows that multiple features are necessary for accurate texture classification. Irfan, Stork and Coddington were the first to use multiple features for the authentication of Pollock's drip paintings, and showed modest accuracy, but statistically significantly better than chance.