A 3D computer model of the head of Michelangelo's David
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A 3D computer model of the head of Michelangelo's David |
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The dataset we have acquired of Michelangelo's David is large - 2 billion polygons. (For more statistics about the dataset, click here.) We know of no computer that can hold the uncompressed dataset in memory. As a result, we have been compelled to process the statue in pieces, first without color, and first at low resolution. The images above show two computer renderings of a 2mm model of David's head.
The first step in constructing this model was to align the approximately 100 scans of his head taken from different gantry positions. Since gantry movements are not tracked in hardware, alignment was bootstrapped by aligning each scan to one of its neighbors interactively. This was followed by automatic pairwise alignment of scans using a modified iterated-closest-points (ICP) algorithm, and finally by a global relaxation procedure designed to minimize alignment errors across the entire statue. After aligning the scans, we combined them using Brian Curless's volumetric range image processing (vrip) algorithm. The volume contained 2000 voxels on a side, making each voxel 2mm. The resulting model contains 1 million polygons. Since the data was originally acquired at a resolution of 0.29mm, this model represents approximately 1/64th of the available data.
Even at this low resolution, the beauty of Michelangelo's masterpiece is evident. The image at left depicts the classic three-quarter frontal view. The image at right shows the head in strict profile, a view never seen in art books. Doesn't he look like a Roman coin? Although it is theoretically possible to obtain this view of the actual statue, it is impossible in practice, since this is a telephoto (narrow angle-of-view) shot, and the required station point lies outside the museum walls. Note also the direction of gaze in his right eye (in the view at left) and in his left eye (in the view at right). They seem entirely appropriate. Now look at a head-on view. The gaze directions of his two eyes actually diverge pathologically. However, the average viewer can't see this head-on view; it is blocked by David's upraised left arm. I think Michelangelo knew this, so he optimized each eye for its appearance as seen from the side. I haven't seen the placement of David's eyes discussed in the literature, but it is generally accepted that Michelangelo "designed for the profile." This seems like fresh proof. What do you think?
Note added October, 2008: Part of the apparent divergence of David's eyes in the head-on photograph could be attributed to the wide-angle lens employed when taking that shot. However, we have also measured the direction of gaze of his two eyes using our 3D model - by examining the location of the pupil hole within the sphere of the eyeball. The gaze directions indeed diverge, by 21.5 degrees (plus or minus 2 degrees). Thanks to David Koller for performing this analysis.
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Here is a 1mm model of David's right eye, aligned and vripped using the algorithms described above. The model contains 500,000 polygons. Note the chisel marks in front of the sideburn and the drill holes along the bottom of the hairline. The Renaissance drill, a sharp metal bit spun back and forth with the aid of a strung bow (hence its traditional name - "violino"), was often used by sculptors to create deep recesses. Sometimes the webbing between drill holes was knocked out with a chisel, sometimes not.
Despite our best efforts to scan every point on the David, there were recesses that Michelangelo could reach with his chisel or drill that we could not reach with our laser triangulation system. To scan a point, we must simultaneously illuminate it with our laser and see it with our range camera. However, the lines of sight of the laser and camera differ in a triangulation system, by 20 degrees in our case. To fill the holes in this particular model, we used space carving, as described in Brian Curless's Siggraph '96 paper. This algorithm employs the many lines of sight between the observed surface and the laser and camera to determine which parts of space must be empty. The boundary between empty and unseen space forms a new, watertight model of the scene. Given enough lines of sight, the resulting model can be quite good, as evidenced by the image at left. Nevertheless, our model is not perfect. The narrowest recesses are bridged here and there by tiny filament surfaces. In other words, the topological genus of the model is not zero, as it should be. (Don't confuse these filaments with the webbing between drill holes; the filaments are too small to be seen in this rendering.) Of course, for scientific uses it is straightforward to label which surfaces were actually seen by the scanner and which were added during space carving.
Although this 1mm model represents only 1/16th of the (0.29mm) data we have
acquired, it is good enough to begin comparing it to a photograph (shown at
right), taken from approximately (but not exactly) the same viewpoint. Bear in
mind that the photograph also includes color information, which we have also
acquired but not yet mapped onto our model. Features to note include the chip
in the lower eyelid and the shallow pits visible almost everywhere on the face.
These pits are thought by some experts to be naturally occuring air holes in
the marble, possibly enlarged by centuries of weathering. Although our model
is slightly softer in the fine details than the statue, we expect that
higher-resolution models, on which we are currently working, will eliminate
most of this softness.
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Here is an aligned and vripped model of David's left eye at the full resolution of our dataset, 0.29mm. See below for an explanation of the yellow rectangle. At this scale, we believe we are capturing everything Michelangelo did with his chisel. At right is a color photograph for comparison. The viewpoints are similar but not identical.
Unfortunately, at this scale we can also see some noise in the data. Believing that our data should not be this noisy, we asked the National Research Council of Canada (NRC) to scan a small patch of the statue with their laboratory-quality 10-micron scanner. The results of their scan are shown here. Our hypothesis is that this noise arises from scattering of the laser within the crystal structure of the marble, not from noise in our camera or scanning system. If so, then we may have reached the limit of laser scanning technology, at least for marble.
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Here are some closeups of the polygon mesh that constitutes our 3D model of the David. The area shown is a portion of the cornea and pupil of David's left eyeball, corresponding to the yellow rectangle on the shaded rendering above.
The left image is a wire-frame rendering of the data acquired during a single top-to-bottom sweep of the laser. Each polygon in this mesh is a triangle, having three vertices. Each vertex represents a single 3D point measured by the laser. Looking at the left side of the image, it appears that the triangles are arranged in nearly horizontal strips. The lines that separate these strips represent the sets of 3D points measured at consecutive positions of the laser during its downward sweep. The spacing between these lines is 0.29mm. Note that there are gaps in the mesh, corresponding to surfaces that the laser missed due to occlusions. Note also that the mesh is bumpy. This is probably due to laser scattering, as described earlier.
The right image is a wire-frame rendering of the mesh obtained by scanning the eye several times from different directions and combining the scans using Brian Curless's volumetric range image processing (vrip) software. The triangles in this mesh are computed from a volumetric representation of the statue, and their size depends on the resolution of this volume. To avoid aliasing artifacts, this resolution is usually made slightly finer than the resolution of the input data. Note that the triangles are no longer arranged in strips; this is an "irregular" mesh. Note also that the gaps are gone - the surface is now watertight. Finally, note that the noise is somewhat smoothed out - we are averaging together scans taken from different directions, and the scattering on each one is independent.
To see our latest color renderings of the David, click here. If you want to fly around our model of the David, but you don't qualify for a license to download the data, try ScanView: our secure client / server rendering system.
Notice: The images of Michelangelo's statues that appear on this web page are the property of the Digital Michelangelo Project and the Soprintendenza ai beni artistici e storici per le province di Firenze, Pistoia, e Prato. They may not be copied, downloaded and stored, forwarded, or reproduced in any form, including electronic forms such as email or the web, by any persons, regardless of purpose, without express written permission from the project director Marc Levoy. Any commerical use also requires written permission from the Soprintendenza.
Notizia: Questi modelli elaborati al computer, immagini computerizzate, e fotografiche sono proprietà del Progetto Digitale Michelangelo e la Soprintendenza Per I Beni Artistici e Storici per le Province di Firenze, Pistoia e Prato. Non possono essere copiati, scaricati da internet su un file, inviati, o riprodotti in nessuna forma, incluso la posta elettronica o il web, da nessuna persona per nessun motivo, senza un permesso scritto da Marc Levoy, il direttore del progetto. Eventuali usi commerciali esigono anche il permesso scritto dalla Soprintendenza.
