A radiometric analysis of projected sinusoidal illumination for opaque surfaces

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Abraham Green
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1 University of Virginia tehnial report CS-21-7 aompanying A Coaxial Optial Sanner for Synhronous Aquisition of 3D Geometry and Surfae Refletane A radiometri analysis of projeted sinusoidal illumination for opaque surfaes Mihael Holroyd Unversity of Virginia Jason Lawrene University of Virginia Todd Zikler Harvard University Abstrat Shifted sinusoidal illumination patterns are useful for appearane apture beause they simultaneously separate loal and non-loal refletions and allow the reovery of surfae geometry. Here we show that the same illumination patterns an be used to estimate the loal surfae refletane (BRDF) as well, provided that an appropriate orretion fator is applied. We derive a losed-form expression for this orretion fator, validate it experimentally, and disuss its impliations. 1 Preliminaries Consider the system in Fig. 1. A thin-lens amera observes a planar surfae path that is illuminated by a ustom light assembly, and this light assembly onsists of a planar Lambertian area soure plaed at the foal plane of another thin lens. The area soure in this light assembly produes radiane patterns that are shifted horizontal sinusoids with fixed frequeny f, amplitude A, and DC offset G. The shifts are represented by a disrete set of phase values: {φ k } k=1...m, so we an write the radiane at a point p = (p, q) on the foal plane of the soure as L k (p, q) = A os(2πf(p + φ k )) + G, k = 1... M. (1) Illumination from the soure is foused at a point x on a planar surfae path, and this path is observed by a thin-lens amera, whih is also foused at x. A pixel (or any square region) on the image plane that is entered at the projetion of x and has dimensions w w measures flux due to the radiane from a neighborhood of the point x on the surfae, and assuming that the amera is a linear devie, the intensity reorded at the pixel is proportional to this flux. Under the sinusoidal illumination of Eq. 1, the pixel response an be written in the form, I k (u) = α(u) os(γt k + φ(u))) + β(u), k = 1... M, (2) where u = Π (x) is the projetion of x (the enter of the pixel), and γ = 2πf(t k+1 t k ), the produt of the spatial frequeny and the displaement of the sinusoid between onseutive shifts. This relation plays a entral role in phase mapping tehniques (e.g. [4]), sine the apparent phase φ(u) provides information about the depth of the surfae along the ray that is bak-projeted from pixel u. Presently, we are interested in the apparent amplitude α(u) sine, as we will show, it provides information about the loal surfae refletane (the BRDF at x) and an be used for refletometry. We show that, in addition to the BRDF, this expression depends on the intrinsi parameters of the lightsoure and amera, as well as their positions and orientations relative to the surfae. 1

2 a b ξ p u A pixel w w A pixel foal plane of lightsoure f l r l ψ l ^ ω l n^ ^ ω v ψ r v f image plane of amera objet surfae x A surf Figure 1: A square pixel in the amera projets, via a planar surfae path, to an oriented parallelogram in the foal plane of a foused light soure. We are interested in the relationship between the amplitude of the translating sinusoidal radiane pattern on the foal plane of the light soure and the apparent amplitude at the amera s pixel. Note that the regions and A pixel are not drawn to sale. 2 Relating projeted and observed sinusoids We obtain flux inident on the image plane from radiane emitted from the foal plane of the soure using three basi relations. Image irradiane from surfae radiane. The image irradiane E (u) that is due to radiane L s (x, ˆω v ) at surfae point x in diretion ˆω v is given by the familiar thin-lens equation (e.g. [1]), E (u) = σ(u)l s (x, ˆω v ). (3) Here, σ(u) models optial fall-off in the amera, whih for a thin lens depends on the area of the amera s aperture, its foal length, and the angle ψ between its optial axis and the ray through u: σ(u) = σ(ψ ) = A os 4 (ψ )/f 2. (See Fig. 1 and Table 1 for summaries of notation.) More generally, this sensitivity funtion an be measured during a radiometri alibration proedure, and we assume it to be known. Emitted radiane from inident irradiane at the surfae. This relation simply follows from the definition of the bi-diretional refletane distribution funtion, or BRDF [3]: L s (x, ˆω v ) = ρ( ˆω l, ˆω v )E s (x). (4) We assume that the sattering properties are statistially uniform over the small area A surf observed by a single pixel, and that following the isolation of loal refletions [2], light transport within the material ours over distanes that are small relative to this area. Surfae irradiane from radiane on the soure foal plane. This relation is less familiar, so we provide a derivation. The basi idea is to ompute the power emitted from the foal plane of the soure toward its lens, and then divide this by the differential surfae area δs to obtain surfae irradiane. The underlying assumption is that δs is in fous, so that all of the power that reahes the lens arrives at δs. Let L be the emitted radiane at the enter of a differential path δp on the foal plane of the soure assembly. The power reeived by the lens is this radiane multiplied by the differential area foreshortened in the diretion of travel and the solid angle subtended by the lens as seen by this small area: A l os ψ l Φ = Lδp os ψ l (fl 2/ os2 ψ l ) = Lδp Al os 4 ψ l. f 2 l

3 Table 1: Summary of notation x sene point u = (u, v) point on image plane of amera p = (p, q) point on foal plane of lightsoure du, dp, ds differential areas on the image plane, soure plane, and surfae ˆn surfae normal ˆω l, ˆω v loal light and view diretions ρ( ˆω l, ˆω v ) BRDF σ(u) amera sensitivity funtion l(x) lightsoure emission funtion f l, f foal lengths of lightsoure and amera ψ l, ψ angles from optial axes of lightsoure and amera r l, r distanes from sene point to lightsoure and amera enters A pixel = w 2 area of square amera pixel A surf area of orresponding region on surfae = ab area of orresponding parallelogram on soure foal plane signed distane that haraterizes the degree of skew in parallelogram A l, A areas of lightsoure and amera apertures Ω l, Ω solid angles subtended by lightsoure and amera as seen by surfae E s ( ), E ( ) irradiane at the surfae, and image plane of amera L( ), L s ( ) radiane emitted at the lightsoure, and surfae Φ( ) power (flux) Sine it is in fous, all of this power arrives at a differential area on the surfae δs, and the surfae irradiane is E s = Φ/δS. The ratio of areas an be obtained by equating the solid angles subtended by δp and δs as seen by the enter of the lens, δp δs = f 2 l r 2 l and ombining these expressions yields the desired relationship: E s = A l os ψ l L (ˆn ˆω l) rl 2. (ˆn ˆω l ) os 3 ψ l, (5) Analogous to the amera sensitivity funtion desribed above, in pratie we generalize this expression to E s = l(x)l(ˆn ˆω l ), (6) where l(x) is an emission funtion that an be measured during a radiometri alibration proess and is assumed to be known. 1 Equipped with Eqs. 3, 4, and 6, we are ready to proeed. The power reeived by the finite pixel area A pixel entered at image point u o (and thus the response of that pixel) is the integral of the image irradiane, whih by Eq. 3 is Φ pixel = σ(u o ) L s (Π 1 A pixel (u))du, where Π 1 ( ) is the bak-projetion of image point u onto the surfae. (In this expression, we have assumed the sensitivity funtion to be onstant over the pixel.) Using an expression analogous to Eq. 5, we hange 1 In fat, as desribed in the main paper, we find it benefiial to inorporate the r 2 l term into this funtion and thus allow it to vary over the three-dimensional working volume: l(x).

4 b a ξ foal plane of lightsoure Figure 2: The soure region orresponding to a single amera pixel, depited in a transformed oordinate system (p, q ) that aligns one side of the region with a oordinate axis (ompare to the left of Fig. 1). The shape of this parallelogram is determined by the lengths of its sides, a and b, and its skew, whih is haraterized by the signed distane. These parameters, along with the position and orientation ξ relative to the horizontal axis of the soure foal plane an be omputed from the known surfae geometry and the parameters of the lightsoure and amera. variables to integrate over the observed area (A surf ) on the surfae instead, and this yields f 2 (ˆn ˆω v ) Φ pixel = σ(u o ) os 3 L s (x)ds, ψ A surf when the pixel is small enough for the angles and distanes (ψ, r 2, ˆω v ) to be onstant over its extent. Finally, the surfae radiane is related to the radiane on the soure foal plane through Eqs. 4 and 6, and making these substitutions along with another hange of variables, ds dp, gives r 2 Φ k pixel = σ(u o )l(p o ) f 2 os 3 ψ l (ˆn ˆω v ) fl 2 os 3 ρ( ˆω l, ˆω v ) L k (p)dp. (7) ψ The last term in this expression is the integral of the sinusoidal radiane pattern (Eq. 1) over an area that is obtained by projeting the square pixel A pixel onto the planar surfae and into the lightsoure. The area is a quadrilateral that, due to the small extent of a single pixel, is relatively unaffeted by perspetive distortion and is well-approximated by a parallelogram, as depited in Fig. 1 and Fig. 2. Any suh parallelogram is ompletely desribed by the lengths of its sides, a and b, it s orientation ξ relative to the horizontal axis of the soure foal plane, the position p o = (p o, q o ) of its enter, and it s skew, whih we parameterize by the signed distane shown in Fig. 2. In the present ase, all of these parameters are determined by the amera and soure parameters and the surfae position and orientation, and sine all of these are known, we an ompute the parallelogram parameters orresponding to any pixel in our amera. In order to ompute the integral over this region, we first hange the oordinate system to be aligned with side b and have one orner as its origin as shown in Fig. 2. We use (p, q ) for these new oordinates, whih allow us to write the integral as L k (p)dp = b b p+a b p r 2 ( G + A os 2πf(p os ξ q sin ξ + p o + φ k ) ) dp dq.

5 By twie using the identity os(sx + t)dx = sin(sx + t)/s, performing trigonometri manipulations, and using the expression sin(x) = sin(πx)/(πx), we obtain L k (p)dp = ab(a sin(af os ξ) sin(bf sin ξ+f os ξ) os(πf(b sin ξ+a os ξ+ os ξ+p o +φ k ))+G). Now, substituting this into Eq. 7, we obtain an expression for the flux Φ k pixel (and thus the pixel response) that is in the desired form of Eq. 2. From this it follows that the apparent amplitude of the observed sinusoid in the amera satisfies α(u o ) σ(u o )l(x)ρ( ˆω l, ˆω v )(ˆn ˆω l )A sin(af os ξ) sin(bf sin ξ + f os ξ), (8) where we have used the fat that the total area of the parallelogram = ab is given by ab = w 2 (ˆn ˆω l)r 2 f 2 l os 3 ψ (ˆn ˆω v )r 2 l f 2 os 3 ψ l. 3 Impliations for refletometry: amplitude loss The result in Eq. 8 onfirms that the apparent amplitude measured at eah amera pixel α(u) (Setion 5.2 in the main paper) is proportional to the produt of the surfae irradiane under point lighting l(x)(ˆn ˆω l ) (reall that 1/r 2 l is aptured by l(x)), the amplitude of the projeted sinusoidal illumination A, the BRDF ρ( ˆω l, ˆω v ), and the amera sensitivity σ(u). In addition, it also predits a less obvious effet that we all amplitude loss whereby the measured response is inversely proportional (via the sin funtions) to the produt of the pixel width w and the frequeny f of the soure radiane pattern in addition to the relative orientations and distanes between x and the amera and soure. In words, if either f or w inrease (holding everything else fixed) the measured amplitude will derease at a rate predited by the produt of the sin funtions in Equation 8 and eventually reah zero this orresponds to the point at whih the sine pattern is no longer visible in the image. Similarly, as the amera approahes a grazing view of the surfae in a diretion perpendiular to that of the sine wave with an overhead soure held fixed or the distane from x to the amera r v inreases, then the measured amplitude will similarly derease. This makes Eq. 8 a useful analyti tool for estimating the lower bounds on f for a partiular experimental setup and, more importantly, it allows onverting the amplitude measured at the amera into measurements of the surfae BRDF. The graphs in Fig. 3 onfirm this effet and validate the analyti model derived above. They show the amplitude measured at a amera pixel as it moves toward the horizon (ˆn ˆω v ) with it s up vetor in the epipolar plane and for a stationary overhead light (ˆn ˆω l ) = 1. For this in-plane onfiguration, the parallelogram redues to a retangle ( = ). The three graphs orrespond to different orientations of the sine wave with respet to the plane of motion. These graphs also inlude preditions by a numerial simulation that agree with our analyti model exatly we have verified that this is true beyond the inplane ase illustrated here. Note that this amplitude loss an be signifiant. In the ase where ξ =, a roughly 2% derease in the amplitude is observed at an elevation angle of 6 degrees whih falls off to roughly 9% at 8 degrees. In pratie, when using measured amplitudes α(u) for refletometry, we orret for this effet by dividing by the terms on the right-hand side in Eq. 8 in order to isolate the BRDF ρ( ˆω l, ˆω v ).

6 1 =, ξ= o 1 =, ξ=45 o 1 =, ξ=9 o amplitude at amera measured our model simulation elevation angle of amera (degrees) amplitude at amera measured our model simulation elevation angle of amera (degrees) amplitude at amera measured our model simulation elevation angle of amera (degrees) Figure 3: Measurements and simulation results onfirming the amplitude loss phenomenon and validating our analyti model. Eah graph shows the amplitude measured at a single amera pixel as the view diretion approahes the horizon (ˆn ˆω v ) with the light soure held fixed overhead (ˆn ˆω l ) = 1. The amera s up vetor remains in the epipolar plane, ausing the parallelogram to redue to a retangle ( = ). For the measurements, we imaged a Spetralon board and orreted for deviations from a perfet Lambertian refletor. We used the parameters of our experimental setup (foal lengths, pixel size, stand-off distanes, et.) to perform a numerial simulation of the amplitude measured at the amera and to evaluate our model in Eq. 8. We observed very lose agreement between measured data, our simulation, and our model. Referenes [1] B.K.P. Horn. Robot vision. MIT Press Cambridge, MA, USA, [2] Shree K. Nayar, Gurunandan Krishnan, Mihael D. Grossberg, and Ramesh Raskar. Fast separation of diret and global omponents of a sene using high frequeny illumination. ACM Transations on Graphis (Pro. SIGGRAPH), 25(3): , 26. [3] FE Niodemus, JC Rihmond, JJ Hsia, IW Ginsberg, and T. Limperis. Geometrial onsiderations and nomenlature for refletane. National Bureau of Standards Monograph 16, [4] V. Srinivasan, H. C. Liu, and M. Halioua. Automated phase-measuring profilometry: A phase mapping approah. Applied Optis, 24: , 1985.

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ow.mit.edu 1.510 Introdution to Seismology Spring 008 For information about iting these materials or our Terms of Use, visit: http://ow.mit.edu/terms. 1.510 Leture Notes 3.3.007