Bidirectional scattering distribution functions of maple and cottonwood leaves

Bidirectional scattering distribution functions of mapleand cottonwood leaves

Michael A. Greiner,1,* Bradley D. Duncan,1,3 and Matthew P. Dierking2,4

1Electro-Optics Program, University of Dayton, 300 College Park, Dayton, Ohio 45469-0245, USA2Sensors Directorate, Air Force Research Laboratory (AFRL�SNJM), 3109 P. Street, Building 622,

Wright-Patterson Air Force Base, Ohio 45433-7700, USA3E-mail: [email protected]

4E-mail: [email protected]

*Corresponding author: [email protected]

Received 29 May 2007; accepted 28 June 2007;posted 5 July 2007 (Doc. ID 83472); published 31 August 2007

We present our investigations into the optical scattering properties of both sugar maple (Acer saccarum)and eastern cottonwood (Populus deltoides) leaves in the near-IR wavelength regime. The bidirectionalscattering distribution function (BSDF) describes the fractions of light reflected by and transmittedthrough a leaf for a given set of illumination and observation angles. Experiments were performed tomeasure the BSDF of each species at a discrete set of illumination and observation angles. We thenmodeled the BSDFs in such a way that other researchers may interpolate their values for scattering inany direction under illumination at any angle. © 2007 Optical Society of America

OCIS codes: 290.1350, 290.5820.

1. Introduction

The scattering and transmission properties of treesand forests are topics of great interest in the field ofoptical remote sensing. As leaves dominate the scat-tering of light within a forest canopy, knowledge oftheir optical properties is essential in order to modelthe interaction of photons with the canopy. Opticalproperties of leaves have been the subject of manystudies in recent years. These studies have foundapplications in several fields, including photobiology,agriculture, and remote sensing [1–3].

This paper focuses upon the bidirectional scatter-ing distribution function (BSDF) of sugar maple (Acersaccarum) and eastern cottonwood (Populus del-toides) leaves. The BSDF of a surface is the ratio ofthe scattered radiance to incident irradiance at agiven wavelength. The function is dependent upontwo directional angles: the angle of illumination andthe angle at which light is scattered. The BSDF istypically split into reflected and transmitted compo-nents, which are treated separately as the bidirec-

tional reflectance distribution function (BRDF) andthe bidirectional transmittance distribution function(BTDF).

Few studies have been performed on the scatteringfunctions of individual leaves because it has typicallybeen assumed that these functions are simply Lam-bertian in nature [4,5]. However, measurements havenot always supported this assumption. In fact, somestudies, including our own, have found a strong spec-ular reflection component accompanying the diffuseLambertian scattering for higher illumination angles[6–8].

It has been demonstrated that the wavelength ofthe illuminating laser dramatically changes the scat-tering properties of leaves. Figure 1, for example,depicts the reflectance and transmittance spectra offresh poplar leaves [9]. Although poplar (Populuscanadensis) is not one of the species investigated inthis research, the general trend in its absorptionspectrum is typical of all deciduous leaves and istherefore relevant to our work. Figure 1 is brokeninto three distinct regions: transmittance T, reflec-tance R, and absorptance A, where the absorptance isdetermined from T and R through the relationshipA � 1 � R � T. Notice that the visible region is

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1 September 2007 � Vol. 46, No. 25 � APPLIED OPTICS 6485

characterized by high absorptance. There are alsostrong absorption peaks in the infrared. However,there is a region in the near-IR, between 800 and1300 nm, where absorptance is minimized. Selectinga wavelength in this region therefore provides thelargest amount of light reflected by and transmittedthrough the leaves. It will thus be the goal of thisresearch to characterize the BSDFs of two deciduousleaf species in the local Dayton, Ohio area (i.e., sugarmaple and eastern cottonwood) at a single, near-IRwavelength.

2. Experimental Setup

Measurement of BSDF data from individual leaves isperformed through the use of the goniometric appa-ratus shown in Fig. 2 and depicted schematically inFig. 3. A linearly polarized (at 20° with respect to thevertical), 1064 nanometer pulsed laser (�6 �J pulseenergy) is directed along the axis of a stationary op-tical rail. A beam splitter is used to send half of thebeam to the leaf for scattering and the other half to anenergy level detector. The beam directed towards theenergy detector is first incident upon a 50% reflectiveSpectralon disk. The measured radiant energy re-flected from this disk is simply used to monitor pulse-to-pulse energy fluctuations so that data can later

be normalized with respect to variations in pulse en-ergy.

Two neutral density (ND) filters, mounted on therail after the beam splitter, are used to attenuatethe power of the laser in order to avoid damaging thetransmission and reflection detectors. The ND filtersare also tilted slightly in order to avoid direct reflec-tions into the beam path. Notice, though, that anydeflection of the beam due to the first filter is coun-teracted by the second so that the path of the beamremains along the rail axis.

After attenuation, the laser beam is directed ontothe leaf, which is mounted in a goniometer for mea-surement. The goniometer has two separate, coaxialrotation stages that are motor driven and indepen-dently controlled by a computer that also tracks theleaf’s rotational position relative to the beam path. Asecond optical rail is mounted on one of the rotationstages with reflection and transmission detectors[both were ThorLabs (Newton, New Jersey) PDA255high-speed amplified InGaAs detectors] equidistantfrom the leaf at opposite ends of the rail. The activeregion of the detectors is small enough to give pointmeasurements and avoid angular averaging of thedata. The leaf is mounted above this rail on a secondrotation stage that is driven by a separate motor. Inthis way the illumination angle of the beam on theleaf can be adjusted independently of the observationangles.

Each rotation stage is capable of 360° of rotation,allowing the reflection and transmission to be ob-served at any angle. The rail rotation angle �D isconsidered to be at 0° when the transmission detectoris at its leftmost position and would see the beampassing through the leaf with no deflection. The leafrotation angle �I is considered to be at 0° when theleaf surface is normal to the incident beam.

The motors of both the detector rail and leaf mountare controlled by an automated LabView programthat rotates the detectors and leaf to predeterminedangles. At each set of angles the program takes 256samples from each detector. The samples are thenaveraged, and a mean value is stored for each detec-tor. The program then moves the leaf and�or detec-tors to the next set of angles, and the process isrepeated until all the desired angle permutations areexhausted. Additionally, the entire procedure is re-peated for four different leaves. The four mean values

Fig. 1. Reflectance and transmittance spectra of fresh poplar(Populus canadensis) leaves [9].

Fig. 2. Photograph of the BSDF measurement apparatus.

Fig. 3. Schematic of the BSDF measurement apparatus.

6486 APPLIED OPTICS � Vol. 46, No. 25 � 1 September 2007

for each detector are then averaged into a singlevalue for every angle permutation. The result is theaverage of 1024 samples taken from four differentleaves. The additional data samples are collected inorder to reduce any leaf-to-leaf variance in the finalreadings.

The diameter of the Gaussian beam is optically setto approximately 1 cm to provide a large illuminationfootprint. This is done for two reasons. First, laser

speckle is reduced by increasing the size of the illu-mination beam. Reducing laser speckle in turn re-duces variance in readings made by the detectors.Second, because a leaf’s structure is not homogeneous(i.e., leaves have veins and stems, as well as other finecellular structures running through them), it is im-portant to illuminate a large enough area of the leafto ensure good spatial averaging.

Speckle reduction and spatial averaging are alsoenhanced by periodically translating the leaf withinthe path of the beam. In addition to being mounted ona rotation stage, the leaf is mounted on a verticaltranslation stage, shown in Fig. 2, which is driven ata frequency of approximately 1 Hz. Translating theleaf up and down, coupled with the use of a broadillumination beam, creates a large effective area overwhich the leaf is sampled. As a result, localized leafstructure has minimal effect on the measurements.

3. Data Acquisition

BSDF data was collected from sugar maple and east-ern cottonwood leaves found in the Dayton, Ohio areaduring the weeks of 21 and 28 May 2006. For eachleaf species both the BRDF and BTDF were mea-

Fig. 4. BSDF of (a) a fresh sugar maple leaf, and (b) the sameappreciably dried maple leaf. Light, depicted by the arrow, is in-cident from the left and illuminates the leaf, portrayed by the solidline, at normal incidence.

Fig. 5. Measured BSDF data for sugar maple leaves at �I illumination angles of (a) 0°, (b) 10°, (c) 20°, (d) 30°, (e) 40°, (f) 50°, (g) 60°, (h)70°, and (i) 78°.


sured for incident angles of 0°, 10°, 20°, 30°, 40°, 50°,60°, 70°, and 78°. Illumination angles higher than 78°were not used because the projected beam waist be-comes larger than the physical width of the leaf as itrotates to higher angles. The scan of a single illumi-nation angle then involves taking measurements of

the scattered radiance at many detector angles aboutthe leaf surface. We collected data at detector anglesspanning a range from �85° to �85° about the leafsurface normal, in increments of 5°.

Because scattering properties change as leaves dryout, scan durations must be kept short [9]. Prelimi-nary experiments showed a strong correlation be-tween the freshness of a leaf and its scatteringcharacteristics. For example, Fig. 4(a) shows the

Fig. 6. Measured BSDF data for eastern cottonwood leaves for �I illumination angles of (a) 0°, (b) 10°, (c) 20°, (d) 30°, (e) 40°, (f) 50°, (g)60°, (h) 70°, and (i) 78°.

Fig. 7. Comparison of the sugar maple leaf BSDF and SpectralonBRDF for normal illumination. For clarity, the detector angleshave been adjusted so that the two curves overlap.

Table 1. Sugar Maple Leaf Reflection, Transmission, and AbsorptionCoefficients as a Function of Illumination Angle

�I RL TL AL

0 0.4861 0.4841 0.029810 0.4879 0.4784 0.033720 0.4890 0.4727 0.038330 0.4884 0.4661 0.045640 0.4900 0.4406 0.069450 0.4984 0.4300 0.071660 0.5000 0.4327 0.067370 0.5009 0.4235 0.075678 0.5031 0.4188 0.0781


BSDF of a fresh sugar maple leaf illuminated withnormally incident light. The BSDF of the same leafleft out to dry overnight is shown in Fig. 4(b). In thesefigures, light incident from the left (illustrated by thearrow) illuminates a leaf (bold line) at normal inci-dence. The resulting transmission profile (right-handlobes) and reflection profile (left-hand lobes) are plot-ted in polar coordinates. There are appreciable effectsdue to leaf drying, as can be seen in the difference inthe size of the reflection and transmission lobes be-tween the two plots. In this case, as water in the leafis replaced with air we observed that the reflectanceof leaves increases while the transmittance decreases.Notice that the shape of the two lobes remains rela-tively constant, though the area encompassed by eachchanges appreciably.

Because of the effect of leaf drying on the scatteringprofile, it is necessary to ensure that the leaves re-main fresh during the entire measurement proce-dure. To combat the effects of leaf drying, a single leafwas used to scan only three illumination angles be-fore it was replaced with a new, fresh leaf. As the scanof a single illumination angle takes approximately 25min, limiting the use of a single leaf to less than 90min proved short enough that drying effects were notnoticeable in any leaves, or our final data.

4. BSDF Results

Polar plots of the BSDF data collected from sugarmaple leaves in the manner described above are pro-vided in Fig. 5. The laser beam (depicted by the ar-rows) was incident upon the leaves at �I angles of 0°,

10°, 20°, 30°, 40°, 50°, 60°, 70°, and 78°. A separatesubfigure was made for each illumination angle. Eachsubplot has been normalized such that the sum of theareas of the transmission and reflection lobes isunity. Note that the scaling of the radial axis spans arange from 0 to 0.01 for each subplot, with the excep-tion of 78° illumination that goes from 0 to 0.025.

The shape of the BRDF and the BTDF at normalincidence and low illumination angles appears to belargely Lambertian in nature. The radiated energy islocated mostly along the leaf surface normal, decreas-ing cosinusoidally with the rotation of the detectorangle. In addition, the shape of the BTDF remainsrelatively constant regardless of incident angle. How-ever, as the incident angle increases past 50°, thespecular component of the reflection increases andthe transmission decreases. The glint protrudes onlyslightly from the diffuse component at 50° and be-comes more pronounced as the incidence angle in-creases. At 78°, the specular component dominatesthe diffuse reflection. Similar trends are seen in theeastern cottonwood BSDF data shown in Fig. 6. Be-cause the maple and cottonwood data are so similar,from this point forward only the maple leaf data will

Fig. 8. Absorption coefficient for maple leaves as a function ofillumination angle. The measured data (circles) is fit with thesecond-order polynomial described in Eq. (5).

Fig. 9. Lambertian fit to the BRDF (left) and BTDF (right) for (a)normal illumination and (b) illumination at 70°.

Fig. 10. Separation of the sugar maple leaf diffuse and specularreflection components for 70° illumination. The nonseparatedBRDF data is depicted by the stars and circles, which represent thediffuse and overlapping regions, respectively. A fourth-order poly-nomial is fitted to the diffuse data and represented by the solidcurve.

Table 2. Eastern Cottonwood Leaf Reflection, Transmission, andAbsorption Coefficients as a Function of Illumination Angle

�I RL TL AL

0 0.5414 0.4263 0.032310 0.5451 0.4227 0.032220 0.5500 0.4178 0.032230 0.5519 0.4160 0.032140 0.5548 0.4131 0.032150 0.5596 0.4084 0.032060 0.5457 0.4221 0.032270 0.5836 0.3847 0.031778 0.5801 0.3834 0.0365


generally be discussed in detail. However, final re-sults for both species of leaves will be presentedthroughout this paper.

5. Calculation of the Absorption Coefficient

Before investigating BRDF and BTDF features inmore detail, it is important to determine the absorp-tion coefficient AL��I� as a function of illuminationangle. The method involves measuring the BRDF ofa target with a known reflection coefficient RS andcomparing its area to that of the BRDF and BTDF ofa leaf. A 60% reflective Spectralon disk (i.e., RS

� 0.60 for all �I) was used as the standard of measureto which the scattering distributions of the leaveswere compared. The normal illumination BRDF ofthe Spectralon disc is plotted, for example, along withthe maple leaf BSDF for normal illumination in Fig.7. The area of each function is then calculated byintegrating over the span of detector angles. That is,

the area of the maple leaf BRDF is given by

AR��I� ��90

�270

rL��I, �D�d�D, (1)

where rL��I, �D� is the BRDF data measured at illu-mination angle �I and detector angle �D. Similarly,the area of the BTDF is given by

AT��I� ��90

�90

tL��I, �D�d�D, (2)

where tL��I, �D� is the BTDF.The values of the reflection RL��I� and transmission

TL��I� coefficients are directly related to the areasunder the BRDF and BTDF curves. In particular, theratio of the reflection coefficients for maple leaves andthe Spectralon disk is equal to the ratio of these areasaccording to the relationship:

RL��I�0.60 �

AR��I�AS

, (3)

where AS is the area of the Spectralon BRDF at nor-mal illumination. A similar expression can be writtenfor the transmission coefficient:

TL��I�0.60 �

AT��I�AS

. (4)

Fig. 11. Normalized, reverse Rayleigh fits of the specular reflectiondata for illumination angles of (a) 50°, (b) 60°, (c) 70°, and (d) 78°.

Fig. 12. Fractional specular reflection for maple leaves as a func-tion of incident angle. The measured data, depicted by the circles,is fit with a polynomial.

Fig. 13. Fitted sugar maple leaf BSDF curves for illumination at70°. (a) The diffuse reflection data is fit with a fourth-order poly-nomial while (b) the transmission data is fit with a second-orderpolynomial.

Fig. 14. Sugar maple leaf BRDF models created by adding thediffuse and specular components at (a) 50°, (b) 60°, (c) 70°, and (d)78° illumination.


The absorption coefficient can then be found throughthe expression

AL��I� � 1 � RL��I� � TL��I�. (5)

Tables 1 and 2 present values for the reflection,transmission, and absorption coefficients for mapleand cottonwood leaves as a function of incident angle.Notice that the transmission coefficients decreaseand the reflection coefficients increase with increas-ing illumination angle. This trend is also seen in thepolar plots of the leaf BSDFs in Figs. 5 and 6 wherethe size of the transmission lobe is seen to growsmaller with increasing illumination angle. Anothertrend evident in Tables 1 and 2 is the growth ofabsorption with illumination angle.

The relationship between illumination angle andthe absorption coefficient for maple leaves is illus-trated in Fig. 8. A second-order polynomial was fit tothe data in order to allow the absorption coefficientfor any illumination angle to be calculated. The fol-lowing two equations, then, describe least square fitregression equations for the absorption coefficients ofmaple and cottonwood, respectively,

AL��I� � �4.5156 � 10�6�I2 � 1.0373 � 10�3�I

� 25.415 � 10�3, (6)

AL��I� � 1.1885 � 10�6�I2 � 67.238 � 10�6�I

� 32.7 � 10�3. (7)

If desired, the reflection and transmission coefficientscan be determined for any illumination angle by in-tegrating the BRDF and BTDF models we will de-velop in the following sections.

6. Modeling Specular Reflection

In Fig. 9(a), the maple leaf BRDF and BTDF fornormally incident light are fitted with a cosine curvenormalized to the integrated area of the measureddata. The measured data is depicted by the circles,while the fitted curve is given by the solid curve. The

quality of the fit suggests that both the BRDF andBTDF are accurately modeled by a Lambertian dis-tribution function for normal incidence. Similar re-sults were seen for the other illumination angles lessthan 50°. However, as the illumination angle in-creases beyond 50°, other features become apparent.For example, plotted in Fig. 9(b) are the 70° illumi-nation angle BRDF and BTDF data fitted with nor-malized area cosine curves. While the transmissiondata remains nearly Lambertian, the reflection dataexhibits other features that will be addressed in thissection.

Because of the irregularities in the shape of thereflection curves, it is difficult to model the BRDFdata with a simple distribution. Also, the specularreflection peak at high incident angles makes it diffi-cult to accurately replicate the curves by simply usinghigh-order polynomials. One approach to modeling theBRDF is to separate the specular and diffuse compo-nents. Accurate models individually representing thediffuse and specular profiles can then be found sepa-rately and later added together to produce the fullBRDF model.

Under the assumption that the specular compo-nent is negligible for detector angles less than 40°, theBRDF can be broken into two regions: one corre-sponding only to diffuse reflection ��D � 40�, and theother containing both specular and diffuse elements��D � 40�. Separation of the two components in thelatter region is then performed by fitting a poly-nomial curve to the known diffuse data and inter-polating values within this region. Subtracting theinterpolated diffuse data from the overlapping regionleaves only the specular reflection component.

This process is illustrated in Fig. 10 for maple leafBRDF data taken at the 70° illumination angle. Thenonseparated BRDF data is depicted by the stars andcircles, which in turn represent the diffuse and over-lapping regions, respectively. A fourth-order polyno-mial is fit to the diffuse data and represented by thesolid curve. The specular reflection component is thenfound by subtracting the diffuse reflection fit from themeasured data. This is shown for the 50°, 60°, 70°,and 78° illumination angles in Fig. 11.

As also shown in Fig. 11, we found that the spec-ular data is best fit with a normalized, reversed Ray-leigh distribution of the form,

rs��D� � FS��I��90 � �D��90 � �P�2 exp��

�90 � �D�2

2�90 � �P�2�, (8)

where FS��I� is the fractional specular reflection, �D isthe detector angle, and �P is the angle at which thefunction is a maximum. Interestingly, we found that

Fig. 15. Polynomial fits of the (a) zeroth-, (b) first-, and (c) second-order p coefficients describing the BTDF data for sugar mapleleaves.

Table 3. Sugar Maple Leaf BTDF q Coefficients

q1t q2t q3t q4t q5t

p1t 3.0388 � 10�12 �463.75 � 10�12 23.428 � 10�9 �300.83 � 10�9 �3.4701 � 10�6

p2t 34.837 � 10�12 �7.1221 � 10�9 517.86 � 10�9 �17.93 � 10�6 135.77 � 10�6

p3t �10.371 � 10�9 1.6844 � 10�6 �89.529 � 10�6 1.0167 � 10�3 246.48 � 10�3


a �P value of 75° produced the best fit for all incidenceangles.

The area of the reversed Rayleigh distributionmust be normalized to the fractional area of the spec-ular reflection peak before reconstructing the datawith this model. Dividing the area of the specularreflection peak, ARS��I�, by the total area of the reflec-tion lobe, AR��I�, for a given incident angle �I gives thefractional specular reflection according to

FS��I� �ARS��I�AR��I�

. (9)

As is seen in Fig. 11, the fraction of specular reflectionis dependent on the illumination angle. For illumi-nation angles less than 50°, specular reflection isnegligible and FS��I 50°� � 0. Moreover, at 90°incidence it is assumed that all reflection is specular,yielding FS�90°� � 1. Using these bounds, the frac-tional specular reflection was fit with a third-orderpolynomial as shown in Fig. 12 for maple leaves. Theratio of specular to total reflection for any illumina-tion angle �I (in degrees) can then be written empir-ically as a piecewise continuous function according tothe following equations provided for sugar maple andeastern cottonwood leaves, respectively,

7. Modeling Transmission and Diffuse Reflection

Both the diffuse reflection and the transmission dis-tribution functions are simply fit with polynomialssuch that when added to the Rayleigh model for spec-

ular reflection, the complete BSDF is reconstructed.Figure 13 shows the fitted data at an illuminationangle of 70° for maple leaves. The transmission datais fit with a second-order polynomial, while a fourth-order polynomial is used for the diffuse reflection. Ahigher-order polynomial is needed for diffuse reflec-tion because the structure of the distribution becomessomewhat more complex at higher incident angles.When the specular reflection component is thenadded to the corresponding polynomials modeling dif-fuse reflection, the complete BRDF is reconstructed,as shown for maple leaves in Fig. 14.

8. Leaf Data Interpolation

The ability to accurately estimate the BSDF for anyillumination angle will be a valuable resource. Thusfar, it has been shown that scattering data at themeasured illumination angles can be reconstructedusing simple polynomial fits and, for high illumina-tion angle BRDF’s, a reversed Rayleigh distribution.However, filling in the gaps for intermediate illumi-nation angles requires creating an additional fit tothe modeled data. As discussed in Sections 6 and 7,for a given incidence angle, the polynomial equationsused to describe the BTDF and diffuse BRDF as afunction of both detector �D and illumination �I anglesare given, respectively, by:

t��D, �I� � p1t�D2 � p2t�D � p3t, (12)

rd��D, �I� � p1r�D4 � p2r�D

3 � p3r�D2 � p4r�D � p5r, (13)

FS ��0 �I � 5012.7 � 10�6�I

3 � 1.7 � 10�3�I2 � 77.2 � 10�3�I � 1.19 �I 50, (10)

FS ��0 �I � 50�1.65 � 10�6�I

3 � 922 � 10�6�I2 � 79.2 � 10�3�I � 1.89 �I 50. (11)

Table 4. Sugar Maple Leaf BRDF q Coefficients

q1r q2r q3r q4r q5r

p1r 1.2954 � 10�15 �180.66 � 10�15 6.8481 � 10�12 �132.35 � 10�12 120.16 � 10�12

p2r 25.825 � 10�15 �5.0125 � 10�12 285.57 � 10�12 �3.6431 � 10�9 32.217 � 10�9

p3r �10.817 � 10�12 1.5674 � 10�9 �58.687 � 10�9 1.0776 � 10�6 �34.531 � 10�6

P4r �123.8 � 10�12 23.992 � 10�9 �1.2636 � 10�6 8.0717 � 10�6 �164.4 � 10�6

P5r �6.2703 � 10�9 369.59 � 10�9 �12.501 � 10�6 �257.65 � 10�6 245.19 � 10�3

Table 5. Eastern Cottonwood Leaf BTDF q Coefficients

q1t q2t q3t q4t q5t

p1t 2.5913 � 10�12 �391.07 � 10�12 17.815 � 10�9 �191.88 � 10�9 �34.326 � 10�6

p2t 4.4203 � 10�12 �4.1417 � 10�9 325.17 � 10�9 �4.7745 � 10�6 �168.86 � 10�6

p3t �9.2296 � 10�9 1.499 � 10�6 �75.217 � 10�6 738.91 � 10�6 245.53 � 10�3


where the polynomial coefficients, p, are dependenton the illumination angle. With knowledge of thesecoefficients, the BTDF and diffuse BRDF data can beinterpolated at intermediate detector angles for apreviously examined illumination angle. However, inorder to interpolate BTDF and diffuse BRDF data atintermediate illumination angles it is necessary toexamine the relationship between �I and the value ofeach of the p coefficients. Fitting p as a function of �I

will allow the interpolation of p at intermediate illu-mination angles.

In Fig. 15, the p coefficients of the maple leaf BTDFdata are shown fit with a fourth-order polynomial. Ingeneral, the fourth-order polynomial fits to the p co-efficients for both BTDF and diffuse BRDF data isdefined by a set of five q coefficients according to

pix��I� � q1�I4 � q2�I

3 � q3�I2 � q4�I � q5, (14)

where, for transmission, i � 1, 2, 3 and x � t, and forreflection, i � 1, 2, 3, 4, 5 and x � r. Knowledge of theq coefficient values then allows one to calculate the pcoefficients for any illumination angle through Eqs.(12) and (13). The q coefficients calculated for thetransmission and diffuse reflection sugar maple leafdata are shown in Tables 3 and 4, respectively. Sim-ilarly, values for the eastern cottonwood q coefficientsare provided in Tables 5 and 6.

Using this method of data interpolation we haveexamined the BSDF estimates at each of the anglepermutations used during the data acquisition pro-cedure. The rms errors found between the originallymeasured data and the estimates generated throughthis interpolation method were found to be approxi-mately 2.5% and 1.0% for the BRDF and BTDF, re-spectively, of sugar maple leaves. The correspondingrms errors calculated for eastern cottonwood leaveswere approximately 3.7% and 1.2%.

9. Procedure for Constructing the BSDF

BSDF values at any set of angles ��I, �D� can be ac-curately estimated for both maple and cottonwoodleaves from the information provided herein. Usingthe following procedure, surface fits for both theBRDF and BTDF can be interpolated for illuminationangles spanning a range from 0° to 78°. These areplotted for maple leaves, for example, in Figs. 16 and17, respectively.

(1) Using Eqs. (6) or (7), calculate the absorptioncoefficient, AL��I�.

(2) Using Eqs. (12) and (14), calculate the BTDF,t��D, �I�. The coefficients required for these equations

are found in Tables 3 and 5 for maple and cottonwoodleaves, respectively.

(3) Using Eqs. (13) and (14), calculate the diffuseBRDF, rd��D, �I�. The coefficients required for theseequations are found in Tables 4 and 6 for maple andcottonwood leaves, respectively.

(4) Generate the specular BRDF, rs��D, �I�, usingEq. 8. Normalize the function to the fractional spec-ular reflection calculated from either Eq. (10) or (11)for maple and cottonwood leaves, respectively.

(5) Construct the complete BRDF by adding to-gether the specular and diffuse components. That is,

rL��D, �I� � rd��D, �I� � rs��D, �I�. (15)

Note that the BRDF and BTDF have been normal-ized in such a way that the sum of their areas plus theabsorption coefficient calculated in step (1) is equal tounity. That is,

1 ��90

90

t��D, �I�d�D ��90

270

rL��D, �I�d�D � AL��I�.

(16)

The above steps present a stand-alone method forgenerating the BSDF of maple and cottonwood leavesfor any illumination angle. This method is appropri-ate for use in remote sensing models where probabil-ity density functions (pdfs) describing the scatteringby individual leaves must be considered (e.g., inMonte Carlo simulations). Such models typically re-quire examination of the scattering pdfs in both the

Fig. 16. Two-dimensional BRDF surface fit for sugar mapleleaves.

Table 6. Eastern Cottonwood Leaf BRDF q Coefficients

q1r q2r q3r q4r q5r

p1r 2.1389 � 10�15 �272.67 � 10�15 8.6857 � 10�12 �111.72 � 10�12 1.6618 � 10�9

p2r �59.223 � 10�15 7.1463 � 10�12 �208.54 � 10�12 1.3098 � 10�9 37.969 � 10�9

p3r �18.62 � 10�12 2.5167 � 10�9 �85.345 � 10�9 1.098 � 10�6 �47.705 � 10�6

P4r 479.27 � 10�12 �62.848 � 10�9 2.2671 � 10�6 �23.813 � 10�6 �355.59 � 10�6

P5r 5.4467 � 10�9 �1.4462 � 10�6 62.194 � 10�6 �993.25 � 10�6 261.58 � 10�3


zenith and azimuth angles. While our models havebeen developed based only upon in-plane measure-ments, they may be extended to allow out-of-planeestimates by assuming azimuthal symmetry in boththe diffuse BRDF and the BTDF. We address theasymmetry of the specular component by forcing theazimuth angle to be equal to the angle of incidence inthe case of specular reflection. We furthermore be-lieve our models to be largely independent of polar-ization, though this will require further investigationto verify.

10. Conclusions

The results of our work have shown a strong corre-lation between the scattering properties of sugarmaple and eastern cottonwood leaves and the Lam-bertian distribution function, except when the illu-mination angle increases past 50°. Beyond thisangle, a specular reflection peak emerges and theLambertian model breaks down, resulting in theneed for a new method to model the BRDF of theseleaf species. The method implemented in this worknot only accurately describes the measured BSDFs,but also allows the interpolation of BSDF valuesfor intermediate illumination and detection angles.Our results will prove to be a valuable tool to otherresearchers investigating remote sensing applica-tions where the interaction of laser beams and tree

leaves must be considered. This work will also aidfurther studies that may be performed using out-of-plane measurements where the detectors do notlie in the plane containing the surface normal of theleaf and the incident laser beam.

This effort was supported in part by the U.S. AirForce and Anteon, Inc., of Dayton, Ohio throughcontract F33601-02-F-A581, and by the Ladar andOptical Communications Institute (LOCI) at theUniversity of Dayton. The authors thank Larry Bar-nes, John Schmoll, Tim Meade, and Dave Mohler ofAFRL�SNJM for their invaluable guidance and as-sistance. The views expressed in this article are thoseof the authors and do not reflect the official policy ofthe Air Force, Department of Defense, or the U.S.Government.

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Fig. 17. Two-dimensional BTDF surface fit for sugar mapleleaves.


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Bidirectional scattering distribution functions of maple and cottonwood leaves