In addition to the reliance on the human actor, quantitative methods can be brought to bear. X-ray analysis can reveal a painting beneath a painting that can shed light on its origins. Surface analysis of the painting materials is another approach, most famously applied in the investigation of the famous “van Meerghen forgeries.” In this case, the forgery of paintings attributed to Johannes Vermeer was confirmed by dating the paintings according to the proportion of a certain lead isotope in the lead-based paint. An elementary application of differential equations allowed for the actual isotope content to be compared with the expected content had the work been painted in Vermeer's day (2). This technique marked a first use of mathematics in the service of art authentication.

With the advent of powerful digital technology, computational tools may be able to provide new insights into and techniques for the art and science of art authentication. For example, a fractal analysis of Jackson Pollock's drip paintings has revealed interesting relations between the evolution of Pollock's aesthetic and the fractal dimension of his work (3, 4). The analysis also raises the possibility of using fractal dimension to help authenticate work attributed to Pollock. Various techniques from machine learning have been applied to the analysis and classification of “craquelure,” the crack lines that appear over time in a painting (5).

In this paper we present a computational tool for analyzing prints, drawings, and paintings for use in authentication. Specifically, we performed a multiscale, multiorientation image decomposition (e.g., wavelets) of a collection of high-resolution digital scans of a drawing or painting. This decomposition changes the basis from functions maximally localized in space (pixels) to one in which the basis functions are localized not only in space but also in orientation and scale. A familiar analogy comes from sound, where the original sound might be transformed into a vector of local frequency information that reflects how much of each frequency comprises the original sound over a short time window. We constructed a compact model of the statistics from such a multiscale, multiorientation image decomposition, and we looked for consistencies or inconsistencies across different drawings or paintings or within a single work. The latter is the so-called “problem of many hands” in which we try to determine the regions of a collaborative work that have been accomplished by a single artist.

The analysis produced locally oriented spatial frequency data and so suggests that the accompanying model captures the subtle pen and brush strokes characteristic of an artist. Although an imitation§ may be perceptually similar to an original (i.e., very much in the “style of the master”), the subtle differences in stroke can reveal the presence of an imitation. In a sense, this work is a natural successor to the mathematical techniques used for graphology, or handwriting analysis (6), distilling not just the characteristic lines and curves of a painter's literal signature (which is often part of the process of authentication) but, even more, moving toward a characterization of the artist's aesthetic signature, resident within the line and curve of his or her work.

Analogous techniques already have made their way into the literary world, where they fall within the discipline of “stylometry” (7). The problem of classification has been applied to divvying up the attribution of The Federalist Papers between James Madison and Alexander Hamilton (8) and the determination of the authorship of the 15th book in the Oz series (9). Statistical approaches to the question of authentication have surfaced in the analysis of William Shakespeare's sonnets (10). The problem of many hands finds its mirror in a study of the conjectured multiple authorship of the Old Testament (11).

We began by applying our analysis to the problem of authentication of a collection of 13 drawings that have at one time been attributed to the famous draftsman Pieter Bruegel the Elder. We followed with a “many-hands” analysis of a portrait by the great Renaissance painter Perugino. We closed with a synopsis of the tools used in the analysis and described the underlying statistical model. We collected some of the more technical points in an appendix.

Bruegel. The Flemish painter and draftsman Pieter Bruegel the Elder (1525/30–1569) was among the greatest artists of the 16th century. Of particular beauty and fame are Bruegel's landscape drawings. Over time he acquired many imitators; undoubtedly some simply were eager to work in the style of the great master, whereas others surely were hoping to pass off their work as Bruegel's for monetary gain. Some of these followers and imitators were expert enough that after being unmasked (or discovered) they became famous in their own right, e.g., Jacob Savery. Bruegel's work recently has been the subject of renewed study and interest (12–14). As a result, many drawings formerly attributed to Bruegel now are attributed to others (Table 1).

Table 1. Authentic and imitation works of art

The delicate line and shading characterizing these works suggests that their classification according to a wavelet-like analysis may be both appropriate and fruitful. For our analysis we digitally scanned (at 2,400 dots per inch) eight authenticated drawings by Bruegel and five acknowledged Bruegel imitations from 35-mm color slides (Table 1), which were provided courtesy of the Metropolitan Museum of Art, New York (13). These color [red, green, blue (RGB)] images, originally 3,894 × 2,592 pixels, were cropped to a central 2,048 × 2,048-pixel region, converted to grayscale¶ (gray = 0.299 R + 0.587 G + 0.114 B), and autoscaled to fill the full intensity range [0,255]. Fig. 1 shows examples of an authentic drawing and an imitation.

Fig. 1. Authentic works by Bruegel and imitations. An authentic work (Top) and an imitation (Middle) (nos. 6 and 7, respectively, in Table 1). (Bottom) The results of analyzing eight authentic Bruegel drawings (•) and five imitations (). Note (more ...)

Each digital image then was subdivided into 64 nonoverlapping 256 × 256-pixel regions. Each of these subimages then was transformed by using a five-level, three-orientation wavelet-like decomposition (see Methods for details). From this decomposition, a 72 feature vector of coefficient and error statistics was extracted for each subimage (see Methods). Each drawing then corresponded to a set of points in this 72-dimensional space. Authentication was indicated by the distance between these point clouds, with the belief that works by the same artist would be close together, irrespective of content, and that an imitation would be relatively far from the authenticated Bruegels. Thus, we first computed the Hausdorff distance (15) between all pairs of images (see Appendix A). The resulting 13 × 13 distance matrix then was subjected to a multidimensional scaling [(MDS) ref. 16] (see Appendix B). Fig. 1 shows the result of visualizing the original 13 images in a lower-dimensional space as determined by a MDS analysis.

The circles in Fig. 1 correspond to the authentic drawings, and the squares correspond to the imitations. For purposes of visualization, the wire-frame sphere was rendered at the center of mass of the eight authenticated drawings and with a radius set to fully encompass all eight data points (in so doing, we assume knowledge of the authenticated Bruegels). Note that all five imitations fall well outside the sphere. The distances from the authenticated Bruegels to the center of the sphere are 0.34, 0.35, 0.55, 0.90, 0.56, 0.17, 0.54, and 0.85. The distances from the imitations are considerably larger, at 1.58, 2.20, 1.90, 1.48, and 1.33 (the difference in the means of these two distance populations is statistically significant: P < 1^–5, one-way ANOVA). Even in this space of reduced dimension, there is a clear difference between the authentic drawings and the imitations.

Perugino. Pietro di Cristoforo Vannucci (Perugino) (1446–1523) is well known as a portrait and a fresco painter, but perhaps he is best known for his altarpieces. By the 1490s, Perugino maintained workshops in Florence and Perugia, Italy, and he was quite prolific. Shown in Fig. 2 is the painting “Madonna with Child” by Perugino. As with many of the great Renaissance paintings, however, it is likely that Perugino only painted a portion of this work, and apprentices did the rest. To this end, we wondered whether we could uncover statistical differences among the faces of the individual characters.

Fig. 2. How many hands contributed to this painting? (Upper) “Madonna with Child” by Perugino. (Lower) The results of analyzing the Perugino painting. The numbered data points correspond to the six faces (from left to right). Note how the three (more ...)

The painting (at the Hood Museum of Art, Dartmouth College) was photographed by using a large-format camera (8 × 10-inch negative) and drum-scanned to yield a color 16,852 × 18,204-pixel image. As in the previous section, this image was converted to grayscale. The facial region of each of the six characters was manually localized. Each face then was partitioned into nonoverlapping 256 × 256-pixel regions and autoscaled into the full-intensity range (0, 255). This partitioning yielded (from left to right) 189, 171, 189, 54, 81, and 144 regions. The same set of statistics described in the previous section was collected from each of these regions. Also as in the previous section, we computed the Hausdorff distance (see Appendix A) between all pairs of six faces. The resulting 6 × 6 distance matrix then was subjected to MDS (see Appendix B). Fig. 2 shows the result of visualizing the original six faces in a lower-dimensional space as determined by a MDS analysis.

The numbered data points correspond to the six faces (from left to right) in Fig. 2. Note how the three leftmost faces cluster, whereas the remaining faces are distinct. The average distance between faces 1–3 is 0.61, whereas the average distance between the other faces is 1.79. This clustering pattern suggests the presence of at least four distinct hands and is consistent with the views of some art historians (T. Barton Thurber, Hood Museum, Dartmouth College, personal communication).

Our methodology makes use of a decomposition of images by using basis functions that are localized in spatial position, orientation, and scale (e.g., wavelets). These sorts of expansions have proven extremely useful in a range of applications (e.g., image compression, image coding, noise removal, and texture synthesis). One reason for this utility is that such decompositions exhibit statistical regularities that can be exploited (17–19). Described below is one such decomposition and a set of statistics collected from this decomposition.

The decomposition is based on separable quadrature mirror filters (20–22). As illustrated in Fig. 3, this decomposition splits the frequency space into multiple scales and orientations. This decomposition is accomplished by applying separable low-pass and high-pass filters along the image axes, generating a vertical, horizontal, diagonal, and low-pass subband. For example, the horizontal subband is generated by convolving with the high-pass filter in the horizontal direction and the low-pass filter in the vertical direction, the diagonal band is generated by convolving with the high-pass filter in both directions, etc. Subsequent scales are created by subsampling the low-pass subband by a factor of two and recursively filtering. The vertical, horizontal, and diagonal subbands at scale i = 1,..., n are denoted as V_i(x, y), H_i(x, y), and D_i(x, y), respectively. Fig. 4 Right and Left shows a three-level decomposition of the scanned Perugino work, respectively.

Fig. 3. An idealized multiscale and orientation decomposition of frequency space. From top to bottom, levels are 0, 1, and 2. From left to right are low-pass, vertical, horizontal, and diagonal subbands.

Fig. 4. The absolute values of the subband coefficients at three scales and three orientations (the residual low-pass subband is shown in the upper left corner) (Right) for the Perugino painting (Left).

Given this image decomposition, the statistical model is composed of the mean, variance, skewness, and kurtosis of the subband coefficients at each orientation and at scales i = 1,..., n – 2. These statistics characterize the basic coefficient distributions. To capture the higher-order correlations that exist within this image decomposition, these coefficient statistics are augmented with a set of statistics based on the errors in an optimal linear predictor of coefficient magnitude.

As described in ref. 19, the subband coefficients are correlated to their spatial, orientation, and scale neighbors. For purposes of illustration, consider first a vertical band, V_i(x, y), at scale i. A linear predictor for the magnitude of these coefficients in a subset of all possible neighbors may be given by

[1]

Consider again the vertical band, V_i(x, y), at scale i. We constrained the search of neighbors to a 3 × 3 spatial region at each orientation subband and at three scales, namely, the neighbors

[2]

[3]

[4]

[5]

[6]

For an n-level pyramid decomposition, the coefficient statistics consist of 12(n – 2) values, and the error statistics consist of another 12(n – 2) values, for a total of 24(n – 2) statistics. These values represent the measured statistics of an artist's style and are used to classify or cluster drawings or paintings.

As stated above, after the computation of the feature vectors, MDS was used to project the original 72-dimensional feature vectors into a three-dimensional subspace. Features with no discriminating power (e.g., the means) therefore will play no role in the lower-dimensional embedding.

We have presented a computational tool for digitally authenticating or classifying works of art. This technique looks for consistencies or inconsistencies in the first- and higher-order wavelet statistics collected from drawings or paintings (or portions thereof). We showed preliminary results from our analysis of 13 drawings either by or in the style of Pieter Bruegel the Elder as well as a painting by Perugino. We expect these techniques, in collaboration with existing physical authentication, to play an important role in the field of art forensics.

We thank George Goldner and Nadine Orenstein of the Metropolitan Museum of Art and Kathy Hart of the Hood Museum of Art for their assistance and cooperation. D.R. was supported by Air Force Office of Scientific Research Grant F49620-00-1-0280. H.F. was supported by an Alfred P. Sloan Fellowship, National Science Foundation CAREER Grant IIS-99-83806, Department of Justice Grant 2000-DT-CS-K001, and departmental National Science Foundation Infrastructure Grant EIA-98-02068.

The Hausdorff distance is a distance metric defined on two sets of vectors, X and Y. The metric, H(·,·) is defined as

MDS is a popular method to visualize high-dimensional data. Given n vectors , where , the goal of MDS is to find a lower-dimensional embedding for these data that minimally distorts their pairwise distances. Denote the n × n distance matrix as , where d(·,·) is a distance metric in . The most common such metric is Euclidean distance defined as .

Given the pairwise symmetric distance matrix, the classic (metric) MDS algorithm is given by the following steps:

• Let .
• Let B = HAH, where , I_n is a n × n identity matrix, and each component of the n-dimensional vector is 1.
• Compute the eigenvectors, , and corresponding eigenvalues, λ₁,..., λ_n, of matrix B, where λ₁ ≥ λ₂ ≥... ≥ λ_n.
• The new, lower-dimensional representation of the original data, , are then given by where denotes the ith component of the vector, and in our examples m′ = 3.