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\author{Don H. Johnson\\[4pt]
Electrical \& Computer Engineering Department\\
Rice University\\
\url{dhj@rice.edu}
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\title{Art Signal Processing}

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\section{Introduction}\label{sec:intro}

\begin{wrapfigure}[20]{o}{2.7in}
\includegraphics[width=3in]{F205_crop.eps}
\caption[]{\label{fig:xray}
Radiograph of van Gogh's \emph{Old Man with a Beard}.
}
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To investigate the history and authenticity of paintings by the great masters, signal processing algorithms can provide new insights.
Our focus here is on x-ray images that can reveal much about what's below the visible surface.
The greater the radiographic-absorbing paint thickness along the beam, the greater the opacity, meaning that x-ray image intensity variations correspond to paint thickness variations.
\figurename~\ref{fig:xray} shows an x-ray taken of a painting by Vincent van~Gogh.
At the proper scale, when the canvas weave is made visible by the thicker ground layer of lead-white paint in the grooves between
canvas threads, the thread density can be determined.
Thread count data is commonly used as evidence for dating, linking pictures from the same canvas roll, and attribution.
We model the canvas weave as a superposition of vertical and horizontal sinusoids.
\begin{multline}\label{eq:model}
c(\vec{x}) =p(\vec{x})\\
+\frac{A}{2}\cdot\big(2+a_h \sin(2\pi \vec{f}_h\cdot \vec{x}+\phi_h)\\
+a_v \sin(2\pi \vec{f}_v\cdot \vec{x}+\phi_v)\big)
\end{multline}

\section{Short-Space Spectral Analysis}
\begin{figure}
\centering
\subfloat[Spectrum of swatch 1]%
{\label{fig:spec1}\includegraphics[width=2in]{swatch1spec.eps}}
\hspace{2cm}
\subfloat[Spectrum of swatch 2]%
{\label{fig:spec2}\includegraphics[width=2in]{swatch2spec.eps}}
\caption[]{\label{fig:spec}
Spectra of swatches taken from \figurename~\ref{fig:xray}.
}
\end{figure}

``Short-space'' Fourier analysis was chosen since the artist's renderings creates in the x-ray what amounts to a non-stationary \emph{additive} pattern to the canvas weave.
The additivity arises because it is the \emph{integral} along the x-ray beam of a object's opacity that determines the x-ray image.
Since paints of different colors have nearly the same opacity, the thickness of the paint determines x-ray image intensity.
Thus, a well-imaged x-ray of a painting should show approximately sinusoidal canvas-related variations throughout a painting, whether the artist paints thinly or uses impasto.%
\footnote{
Several x-rays do not satisfy the criterion of being ``well-imaged.''
In the digitization process, clipping sometimes occurs in heavily painted areas, which means the small intensity variations due to the canvas weave are absent.
Automatic thread counting algorithms must allow for the possibility that \emph{no} thread counts can be measured in some areas of the painting.
}
Weave stretching varies across a painting, further reinforcing local rather than global spectral estimates.
The most important deviation from the ideal model is the painting itself, a phenomenon that is difficult to describe mathematically.
When the painting has ``out of band'' components, they are simply ignored.
In-band components pose a more serious challenge.
In our experience, the spectral approach is very robust and reliable thread-count estimates could be produced for nearly all of the painting surface.

The ``short-space'' spectral analysis algorithm mimics the familiar short-time Fourier analysis used in analyzing non-stationary time series (e.g., speech)~\cite{Rabiner-Schafer-78}.
A square section is extracted from the x-ray, the average subtracted and a window applied.
To obtain spectral detail, the 2D Fourier transform size is much larger than the windowed section.
Spectral resolution is determined by the window size and shape, but accuracy in locating spectral peaks is determined by the transform length (about every $0.1$~thread/cm for the data shown here).
The first swatch's spectrum in \figurename~\ref{fig:spec1} evidences four clear peaks evidenced by the sinusoidal canvas weave pattern of \equationname~\eqref{eq:model}.
Note that the peaks do not lie precisely on the axes;
they are rotated slightly, indicating that the sampling grid was not precisely aligned with the weaving.
Since such precise alignment is impossible, the algorithm searches for spectral peaks within a radius range known for canvas thread counts (between 10 and 20 threads/cm) within small wedges spanning $\pm20^\circ$ centered on the axes as shown in \figurename~\ref{fig:spec2}.
In this easy case, only one peak in the expected spectral range occurs, yielding an unambiguous estimates of vector frequency magnitude in both the horizontal and vertical directions.

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