On the Optical Accuracy of the Salvator Mundi
TTechnical Article:
On the Optical Accuracy of the
Salvator Mundi
Marco (Zhanhang) Liang (computer scientist, artist, researcher),University of California, Irvine, Dept. of Computer Science,Irvine, CA 92697-3435, USA. Email: < [email protected] > .Michael T. Goodrich (computer scientist, researcher),University of California, Irvine, Dept. of Computer Science,Irvine, CA 92697-3435, USA. Email: < [email protected] > .ORCID:0000-0002-8943-191X.Shuang Zhao (computer scientist, researcher),University of California, Irvine, Dept. of Computer Science,Irvine, CA 92697-3435, USA. Email: < [email protected] > .ORCID:0000-0003-4759-0514. Abstract
A debate in the scientific literature has arisen regarding whether the orb depicted in
Salvator Mundi , which has been attributed by some experts to Leonardo da Vinci, wasrendered in a optically faithful manner or not. Some hypothesize that it was solid crystalwhile others hypothesize that it was hollow, with competing explanations for its apparentlack of background distortion and its three white spots. In this paper, we study the opticalaccuracy of the
Salvator Mundi using physically based rendering, a sophisticated computergraphics tool that produces optically accurate images by simulating light transport in virtualscenes. We created a virtual model of the composition centered on the translucent orb inthe subject’s hand. By synthesizing images under configurations that vary illuminations andorb material properties, we tested whether it is optically possible to produce an image thatrenders the orb similarly to how it appears in the painting. Our experiments show that anoptically accurate rendering qualitatively matching that of the painting is indeed possibleusing materials, light sources, and scientific knowledge available to Leonardo da Vinci circa1500. We additionally tested alternative theories regarding the composition of the orb, suchas that it was a solid calcite ball, which provide empirical evidence that such alternativesare unlikely to produce images similar to the painting, and that the orb is instead hollow. a r X i v : . [ c s . G R ] D ec alvator Mundi is a painting of Christ (see Fig. 1A) dated to ca. 1500 that has been attributedby some experts to Leonardo da Vinci. Although centuries old, it was rediscovered, restored,and authenticated only recently. When it sold in 2017 for $450.3 million, it became the mostexpensive painting ever sold [6].Since its rediscovery, a debate has arisen as to whether the translucent orb in the paintingwas rendered accurately. Even though the painting dates to a period when Leonardo da Vinciwas studying optics, several observers have commented that the orb is not rendered as it shouldwere it a solid glass or crystal orb, which would invert and magnify the subject’s robe behind theorb and obscure the subject’s palm. For example, Isaacson [9] hypothesizes in his biography,
Leonardo da Vinci , that Leonardo deliberately rendered the orb inaccurately, while Kemp [11],who helped authenticate the
Salvator Mundi , writes in
Nature that the orb “glistens with pointsof light” that are not “spherical bubbles found in glass, but are the kind of cavity inclusions(small gaps) that appear in some specimens of rock crystal and calcite.” Kemp further writesthat Leonardo “observed the double refraction produced by calcite.” In a follow-up
Nature correspondence, Noest [14] questions Kemp’s interpretation, noting a lack of optical distortionin the orb, and he hypothesizes that the specks on the orb’s surface were instead painted onthe orb. Kemp [14] replies, “Leonardo did not aspire to represent his subjects as if he werea scientist recording natural phenomena.” Such correspondences have not settled the issue,however, and this debate has continued [2, 12, 20].Fortunately, we can use scientific tools to address such controversies [1, 3, 5, 8, 18, 19, 21].For example, the technique of inverse rendering uses tools from computer graphics and appliedoptics to infer scene information from photographs [4, 13, 16, 22]. Here, although it is not aphotograph, we nevertheless apply a type of inverse rendering to the
Salvator Mundi using aphysically based renderer (PBR), which is a sophisticated Computer Graphics tool designedto simulate the physics of light flow through a three-dimensional virtual scene to producean optically accurate image [15]. By comparing synthetic images produced by a PBR withthe painting, we qualitatively tested various hypotheses regarding the optical accuracy of thepainting with respect to various materials and light sources that the artist might have used. For Although many believe that this painting was created by Leonardo da Vinci, the authorship is still beingdebated. In this paper, we use “
Leonardo? ” to reflect this uncertainty.
Leonardo? . Experimental Setup
We built a virtual scene with an approximated geometry gleaned from the painting andexperimented with a variety of configurations for the orb. Our experiments explored possiblehypotheses regarding material properties of the orb, with particular interest in possibleexplanations for the lack of optical distortion by the orb.
Virtual Scene Setup
Since our main goal was to test hypotheses regarding the material property of the orb, we depictof the scene geometry using a rough approximation for the subject’s body along with moredetailed representations for the orb and the hand holding it. Specifically, we sculpted a three-dimensional relief as a proxy geometry for the subject, and used the painting (with the orbremoved) as a color texture for this relief. Then, we placed a three-dimensional orb as well asthe hand holding this orb in front of the relief model, as shown in Fig. 2. We used the subject’sleft hand as a reference and scaled the orb and relief accordingly to match the relative size andposition to the painting. This results in the orb with 6.8 cm in radius located 25 cm away infront of the subject.Additionally, we fine-tuned our model to further improve its visual quality as follows.We refined the geometry of the orb-holding hand using Maya, a 3D modeling and animationsoftware, to make it touching the orb softly while avoiding any overlap between the two models.We also slightly adjusted the light transmission rate of the orb material (as a dielectric) toresemble the background-darkening effect from the painting. Lastly, to better reproduce theoverall smooth appearance, we applied Gamma correction to both the relief texture and thefinal rendered image.Besides geometry, illumination and viewing configurations are also key to visual ap-pearance. Using visual cues including the brightness gradients on the subject’s face, chest,3nd hands as well as the shadows under the chin, we conclude that the painted scene toinvolve a strong directional light source from the above and introduced a similar light in ourvirtual scene. Specifically, we used a few directions around a main one (instead of using asingle directional light) to slightly soften the shadow boundaries. To reproduce the ambientillumination (which reduces the contrast between well-lit and shadowed areas), we used a dimconstant environmental light. From a photographic point of view, the directional light acts as the“main light” of the scene while the dim environmental light as the “fill light”. Lastly, we useda perspective viewpoint (which is anachronistically called a “camera” in the context of PBR)that is located 90 cm away and points straight at the subject to simulate the typical viewingconfiguration in a studio.
Material Properties of the Orb
Solid or Hollow?
With the virtual scene ready, we tested whether the orb was solid by comparing renderings ofa solid and a hollow orb. As noted by others [9, 14], a solid orb bends light as a convex lenswould, which would invert and magnify the image of the robe behind the orb (e.g., see Fig. 3A).This effect persists regardless of the orb’s material (as long as it is optically denser than air). Ahollow orb, in contrast, does not cause such distortion. As can be seen in Fig. 3B, the folds of therobe are not distorted or inverted by a hollow orb. Based on this comparison, we conclude thatthe orb was most likely hollow (assuming the painting follows the physics of light transport).Knowing that the orb was likely hollow, we experimented on how such an orb alters theshape of lines on the boundary. We found that when a hollow glass ball is placed directly infront of a straight line, relative to the viewing “eye,” the optics imply that we can see that thestraight line appear to pass into the ball without distortion when the center of the ball is on thisline, but this lack of distortion is not true for lines that do not pass through the ball’s center.For example, suppose we were to place a hollow glass ball directly above three parallel straightlines, with a camera viewing from top, As illustrated in Fig. 4A. The line in the middle, whichpasses through the center of the ball from camera’s viewpoint, preserves its continuity on theboundary, while the other two have curved distortions (shown in Fig. 4B). The alignment of4hree things—the straight line, viewing “eye”, and center of the hollow ball—makes it possiblethat a line passes through a hollow ball without interruption. This is true even for solid balls(with arbitrary thicknesses), as illustrated in Fig. 4C. Another key factor in having a connectedline as it is refracted and reflected with respect to a hollow ball is to keep the line straightoutside the ball. If we bend the middle line to the right outside the ball, for example, it becomesdisconnected on the boundary(Fig. 4D).To test whether
Leonardo? may have utilized these optical devices to his composition andplacement of the connected folds, we analyzed the painting, especially the composition andsetup of the folds in the subject’s robe (Fig. 5A). In the painting, the drape of the subject’sclothes introduces five prominent folds that pass through the orb’s upper-right boundary, amongwhich the leftmost one has a cast shadow wider than the others. By drawing straight lines alongthe edge of these folds (as indicated with solid white line in Fig. 5B), we found that, except forthe leftmost fold, the four other folds converge to a point, which turns out to be the center ofthe orb. This convergent point, or center of the orb, is a hollow dot with light grey color andsmaller than three white spots. Moreover, the folds keep straight for a distance outside the orbbefore they curve towards subject’s left shoulder. Based on this realization of this geometry ofthe folds in the painting as essential in having connected folds on orb’s boundary, we refined the3-dimensional relief in our model to assure the folds were straight and radiative. Furthermore,this analysis suggest that
Leonardo? understood these optical properties of hollow balls andhow to avoid distracting optical distortions from the rendering of the folds of the subject’s robe.
Thickness
With the above findings, we further explored the thickness of the orb in the painting bycomparing the smoothness of the folds going across the orb’s boundary. Given the importanceof the position, we aligned the viewing “eye” (camera), the orb’s center, and convergent pointof the folds. As the hollow orb was likely made via glass blowing with the help of a hemispheremold, we modeled it as hollow with a refractive index of 1.51714. With these settings, wedetermined a thickness relating to the size and position of orb. In particular, for a hollow glassorb that is 6.8 cm in radius, and positioned so that the folds of the robe follow lines approachingthe center of the orb (relative to the viewpoint), we determined that the maximum thickness of5he orb could be 1.3 mm without producing noticeable abruptions on the folds that traverse theboundary (Fig. 5C).In the rendering, the leftmost fold, whose extended edge fails to pass through the convergentpoint, is distorted by the boundary, while the four other folds cross the boundary smoothly(Fig. 5D).
Leonardo? chose to blur this distorted edge in the painting, but rendered the othersaccurately, suggesting that the artist was aware of the optics of how straight lines appear when,relative to the viewing “eye,” they cross the boundary of a hollow glass sphere. Interestingly,the cast shadow of the leftmost fold has its outer region, which is further-away and soft-edge,converges to the center, and has a smooth transition across the orb’s boundary (as denoted bydotted line in Fig. 5B).To further illustrate how the thickness and placement of the orb relative to the folds of thesubject’s robe impact how the folds of the robe appear, we altered the orb’s thickness and theorb’s position to show resulting distortions. We first rendered a hollow orb with 2.6 mm inthickness, and found that the folds are intercepted by a dark layer, which is a reflection of theambient environment (i.e., the room in which the subject sits), as shown in Fig. 6A. As for orbposition, we horizontally moved the hollow orb 1 cm to the left relative to the viewing “eye,”and rendered the result, which shows that in this case the folds appear disconnected (Fig. 6B).
Alternative Theories
In addition to exploring practical orb thickness, we used physics-based simulation to test analternative theory regarding the orb. This theory, which was proposed by Martin Kemp, arguesthat the orb is made of solid calcite and that the two distinct contours on the heel of the subject’shand are due to birefringence [11]. We rendered the birefringence of such an orb by averagingtwo solid orb images rendered with an extraordinary index of refraction of 1.486 and an ordinaryindex of refraction of 1.658. Our physics-based simulation (Fig. 7) shows that this is unlikely,since the solidity of the orb would cause visually significant distortion (of the robe and the palmbehind the orb) that is more prominent than the double-contours effect and these effects areclearly absent from the original painting. 6 iscussion
Given our experimental results, a natural question is whether Leonardo da Vinci had accessin 1500 to the materials, light sources, and scientific knowledge of optics represented inconfigurations we used with the Mitsuba PBR to generate images similar to the painting,
Salvator Mundi . Fortunately, besides being an artist, Leonardo was also a scientist who keptcopious notes, much of which have survived, and these notes shed light on this question. Indeed,a recent volume,
Leonardo da Vinci and Optics , is devoted to studying the relationship betweenoptics and the paintings of Leonardo [7].Furthermore, in 1883, Richter [17] published a compilation of Leonardo’s notes, includingdrawings and English translations. These notes show, for example, that Leonardo had anunderstanding of light refraction (e.g., no. 75, Fig. 8A), glass and crystal materials and diffused,direct, and reflected light (e.g., no. 118, Fig. 8B), the relative position of reflections on a roundbody (e.g., no. 134, Fig. 8C), how light can be directed through a “window” (e.g., no. 146,Fig. 8D), reflected colors (e.g., no. 283, Fig. 8E), how to create a semi-diffused light sourceusing paper and a candle (e.g., no. 524, Fig. 8F), and even how light reflects from a concavemirror (e.g., Fig. 8G). Thus, these notes provide evidence that Leonardo had access to thematerials, light sources, and scientific knowledge necessary to create configurations that ourexperiments show can produce images similar to how the orb is rendered in
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Acknowledgments
SuperDasil provided the hand model on BLEND SWAP. This research was supported in part byNSF-BSF grant 1815073. MZL designed the geometric models and performed the experiments.SZ provided supervision, insights regarding conical light paths, and PBR expertise. MTGprovided supervision, originated the research question, and guided research progress. All three9uthors contributed to the writing of the paper. Data for our configurations and simulations canbe found at https://github.com/salvator-mundi/three-white-spots .10igure 1: (A) The
Salvator Mundi (public domain image). (B) A PBR rendering of the
SalvatorMundi using a hollow orb. 11igure 2: Virtual scene setup where an orb-holding hand model is positioned in front of thesubject’s relief, which is textured with a modified version of the painting.Figure 3: (A) Rendering of a solid orb. (B) Rendering of a hollow orb.12igure 4: (A) Scene setup for the experiment, where a glass ball is above three lines and acamera views from the top. (B) A hollow glass ball in front of three straight lines. (C) A solidglass ball in front of three straight lines. (D) A hollow glass ball in front of three lines. Outsidethe ball, the line in the middle bends to the right .13igure 5: (A) The orb from the
Salvator Mundi for reference. (B) Illustration of the folds onorb’s boundary, with white solid lines drawn along folds’ edge, and a cast shadow denoted by adotted line. (C) Rendering of a hollow orb with 1.3 mm in thickness. (D) Upper-right boundaryof the hollow orb from (C). 14igure 6: (A) A hollow orb with 2.7 mm in thickness. (B) A hollow orb with 1 cm shift to theleft, relative to the viewing “eye”.Figure 7: Rendering of a solid calcite orb with birefringence.15