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Discovering Canvas Orientation of Van Gogh Paintings ∗

Nirali Desai This work is produced by OpenStax-CNX and licensed under the † Creative Commons Attribution License 3.0

1 Art Matters In Vincent Van Gogh's lifetime, he sold a single painting. Today, he is one of the most famous artists of all time. He created over 900 paintings in a short career that lasted only a decade. Art historians studying Van Gogh need the help of technology to analyze paintings and determine the probability of them being counterfeits.

The letters left behind by Van Gogh indicate the rolls of canvas he received for painting

were delivered by his brother. Professor Don Johnson has worked closely with the Van Gogh Museum in Amsterdam to determine which paintings came from the same bolts of canvas.

Using signal processing

allows us to determine the location of a painting on a roll of canvas used by Van Gogh, which would help art historians and their study of Van Gogh's works.

2 Van Gogh's Canvas Preparation Van Gogh used canvases brought to him by his brother from a canvas priming company. Van Gogh would get a 2.10 m by 10 m roll and cut smaller portions of the canvas to use for his paintings. The canvas was stretched by rst nailing the top side to a board. Then, the bottom side was secured to another board using a hook and lace method. The canvas was primed, removed from the boards, rolled up, and sold. Since the nails were more closely spaced together than the hooks, we can analyze whether a painting was cut from the top or bottom of the original sheet of canvas. Notable concerns arise when a middle group is formed that does not match with either of the narrow spacings of nails or the wider spacings of the hook-and-lace mechanism.

A representation of the setup of a canvas while being primed. The black dots at the top edge are nails. The bottom edge has hooks that are laced to the bottom board. ∗ Version

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3 Basis for Clique Formation Cliques, by denition, represent a group of paintings that come from the same bolt of canvas.

A bolt of

canvas is approximately 100 m of canvas. The canvas is made in bolts and then cut into 10 m rolls to be primed. Don Johnson sorted and separated Van Gogh's works into cliques based on thread density, angle measurements dependent on the departure of horizontal and vertical threads from the coordinate axes, and thread count. Our project involves the analysis of these cliques by examining the location of specic paintings in each clique. In order to determine where a painting was cut from a given roll of canvas, we sort paintings by analyzing the distance between nails that were present during the priming process for the roll of canvas. It is important to understand bolts and rolls are dierent. Bolts are 100 m of canvas while rolls are 10 m of canvas cut from a bolt. Canvas is created in bolts and later cut and primed in rolls. Since Don Johnson looked at the weaves of canvas, he was able to sort the paintings into which bolt they came from. Later, we group some pdf plots into the "left" and "right" sides of a clique. Left and right are words Don Johnson uses to describe which side of the bolt the painting came from. We are sorting the painting into "top" and "bottom" clusters. Top and bottom refer to canvas rolls. Top means the painting was near the nail side while bottom means the painting was closer to the hook side. All paintings that are on one side of a bolt are not necessarily on the same side for priming. Once the canvas was cut in 10 m rolls, it could have been ipped around in the process of priming.

4 The K-Means Algorithm K-means is an algorithm that can be used to sort data into a given number of clusters. It rst estimates an initial centroid for each cluster. Then, it assigns each data point to a cluster based on the centroid. The centroids are then recalculated for each cluster based of the data points that fall within each one. The data is reassigned, and the centroids are recalculated until they no longer change. When implementing the k-means algorithm for this project, we did not code the function by hand and instead relied on MATLAB's built in k-means function for our calculations. When clustering, the value of k is decided by experimentation. This means identifying a value of k for which clusters appear to be most spatially dense. We rst tried to calculate the k-means of cliques by clustering with the probability density function of variable ysmooth. Ysmooth represents a discrete pdf lled with the nail spacings of a whole clique, which is interpolated by convolving with the Hanning window, and then normalized. However, we then calculated the k-means of dierent cliques by clustering according to the mean and standard deviation of nail spacings in the selected clique. With this, we obtained more meaningful results because clusters created using this method are more spatially dense. As such, we are choosing to cluster according to mean and standard deviation.

5 Coded Implementation We worked with 4 dierent MATLAB les while trying to implement the k-means algorithm: Files from Professor Don Johnson: nail_data: Contains data of 5 dierent cliques with varying numbers of paintings. Data structures are organized by cliques, painting name, and nail spacings.

The clique and nail spacing of each painting is

gathered by nail_data. nailprint: Prints out formatted data from nail_data. Paintings are organized by cliques and then paintings present within each clique. Subsequent data is representative of spacings. nailplot: Calling nailplot will always print out Figures 15 and 16. nailplot: Calling nailplot will always print out Figures 15 and 16. Figure 15 is a histogram of paintings colored by their clique, and then sorted into left and right. Left is indicative of hook-and-lace implementation, while right indicates a use of nails at the edge of a painting.

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Figure 16 has separate histograms for each clique, and plots nail spacing (cm) vs. percentage of nails at that spacing for the given clique. The histograms for each clique also delineate right from left, but only for the paintings in the specied clique. Figure 17 and 18 will be printed out if a clique is specied in a command line. Figure 17 prints out the nail spacing for every painting in the specied clique on separate histograms. Nail spacing (cm) vs. frequency of nail spacing for all paintings sorted into the left group. Figure 18 is identical to Figure 17, except specic to the right group. Our le: nailcluster: 1. Convolved vector of nail spacings for each painting with normal pdf, and then used Hanning window to interpolate. While this code was not used in the actual clustering, it was important in obtaining the mean and standard deviation parameters of each clique. 2. Used the normalized pdf vector to obtain mean and standard deviation parameters for clustering. 3. Performed a k-means clustering of each clique by implementing the kmeans function built into MATLAB. 4. Calling nailcluster will print out 3 gures:

a) The rst gure will be a plot of Frequency vs. Spacings. Frequency represents the number of occurrences of each width of nail or hook-and-lace spacing. b) The second gure will be a plot of the pdfs for the left side of each clique. This separates the data by each painting within a given clique. c) The third gure will be a plot of the pdfs for the right side of each clique and separates data in the same way as the second gure.

6 Clustered Paintings by Spacings When running the code for Clique 1, we obtain the following results:

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The pdf plots for the left side of Clique 1:

The pdf plots for the right side of Clique 1:

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Clique 1 is an interesting clique. After interpolating the values for nail spacing, convolving them to form a normalized pdf, and performing k-means clustering, we plot the value for spacings as shown above. The plot indicates that there exists a middle, unknown, group of nail spacings which occur on any given side for a painting. The green cluster could potentially be classied as noise. More data may give rise to a distinct new clique but with our current data, we can't be sure. If a third cluster did exist, Professor Don Johnson's theory is that Van Gogh might have received canvas rolls primed at a dierent facility. This facility, instead of using the recognizable nail and hook-and-lace methods, would have primed the canvas in similar ways for both the top and the bottom of the frame. When running the code for Clique 3, we obtain the following results:

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The pdf plots for the left side of Clique 3:

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The pdf plots for the "right" side of Clique 3:

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Clique 3 could be classied as a boring clique because the analysis of nail spacings, and hook-and-lace spacings for the paintings in this clique was consistent with predicted spacings. Clustering via the k-means algorithm results in relatively clean groupings. The unknown cluster is an outlier because only one painting appears in this cluster. The pdf plots analyzing specic paintings matches up with the overall data and our expectations for this clique. We obtained the following results for Clique 6:

The pdfs for the left side of Clique 6:

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The pdfs for the right side of Clique 6:

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Clique 6 could also be classied as a boring clique because Clustering via the kmeans algorithm results in relatively clean groupings. The unknown cluster is an outlier because only one painting appears in this cluster. The pdf plots analyzing specic paintings matches up with the overall data and our expectations for this clique.

7 Conclusions We contributed to Don Johnson's research project by applying signal processing in the form of the k-means algorithm to the canvases of origin for specically grouped paintings by Vincent Van Gogh. The methods we used are not limited to Van Gogh's paintings and could be applied to other artists' work done on canvas prepared in similar ways.

8 Poster This is our poster from the presentation.

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9 Future Work and References Professor Don H. Johnson, Department of Electrical and Computer Engineering, Rice University, [email protected] Eva Dyer, Department of Electrical and Computer Engineering, Rice University, [email protected] Johnson, Don H., Ella Hendriks, and C. Richard Johnson. Art Matters:

"Interpreting Canvas Weave Matches."

International Journal for Technical Art History 5 (2013):

53-61.

Web.

8 Dec.

2013.

http://artmattersjournal.org/images/past_issues/volume_5/05-AM-Johnson- Hendriks-v2.pdf Krichevsky, Raphail E., Victor K. Tromov. The Performance of Universal Encoding. IEEE Transactions on Information Theory 27, No. 2 (March, 1981): 199-206. Matteucci, Matteo. "K-Means Clustering." A Tutorial on Clustering Algorithms. N.p., n.d. Web. 25 Nov. 2013. http://home.deib.polimi.it/matteucc/Clustering/tutorial_html/kmeans.html. "Van Gogh Fun Facts." The Van Gogh Gallery. Templeton Reid, LLC, 15 Jan. 2013. Web. 17 Dec. 2013. http://www.vangoghgallery.com/misc/fun_facts.html.

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