A shift-variant BRDF model and analytical solution

Previous authors in the computer graphics and computer vision fields have made a variety of simplifying assumptions to BRDFs [12,13]. For example, the BRDF has been approximated as shift-invariant to compute the forward scattering problem in the frequency domain [16] and the diffuse component of the BRDF has been approximated as a constant (commonly known as the Lambertian term) to obtain solutions to the inverse scattering problem [4]. In addition, the BRDF has been modeled as separable, such as for data compression purposes [17,18]. These assumptions, although convenient for computer graphic related purposes, are not suitable for analysis of the reconstruction.

Fig. 2-6 illustrates two types of BRDFs expected in real scenes. Fig. 2-6 (a) shows a large and slowly varying diffuse component superimposed on a small quasi-shift-invariant specular component. Fig. 2-6 (b) shows a BRDF that contains only a quasi-shift-invariant specular component, but is restricted to a limited angular range.

Since the specular component is considerably wider than the measurement range, the boxed area can be considered

17

as a diffuse component. In the following, we introduce a BRDF model in eq. (2-14) that covers these common shift-variant BRDFs, enabling an analytical solution to eq. (2-9). Such analytical solutions are required to fully understand the errors in reconstruction and the effect of regularization methods. Note that our BRDF model combines Durand’s model [16] for the forward scattering analysis and Shirley's model [19] for matte/specular

materials such as polished woods and tiles. Our model is thus expected to have wide applicability to real-world scattering surfaces.

Fig. 2-6. Analytically modeling two candidate BRDF functions. (a) Large diffuse component and small specular component. (b) Wide specular component with limited measurable range.

Fig. 2-6 (a) is modeled mathematically by including a shift-invariant specular term 𝑓_{𝑠𝑝𝑒𝑐} that is multiplied by a slowly varying window function 𝑓𝑤𝑖𝑛𝑑 to allow for commonly observed changes in specular peak-values as a function of object angle 𝜃_𝑜 [16]. A symmetric diffuse component 𝑓_{𝑑𝑖𝑓𝑓}(𝜃_𝑜) × 𝑓_{𝑑𝑖𝑓𝑓}(𝜃_𝑚) is then added that is separable and reciprocal in terms of the input and output angles [19]:

𝑓𝐵𝑅𝐷𝐹(𝜃𝑜, 𝜃𝑚) = 𝑓𝑤𝑖𝑛𝑑(𝜃𝑜)𝑓𝑠𝑝𝑒𝑐(𝜃𝑚− 𝜃𝑜) + 𝑓𝑑𝑖𝑓𝑓(𝜃𝑜)𝑓𝑑𝑖𝑓𝑓(𝜃𝑚)

In the computer graphics and computer vision fields, the diffuse component is usually expressed as a constant Lambertian term. In our treatment, we allow the diffuse component to take on any functional form, as long as the function is separable with respect to the input and output angles. This generalization allows for a far more complex diffuse term to be expressed. However, the separability constraint implies that no object information is (2-14)

18

conveyed to the measured signal as explained in the subsequent sections. Moreover, if 𝑓_{𝑠𝑝𝑒𝑐} is sufficiently narrow compared to 𝑓_{𝑤𝑖𝑛𝑑} (i.e. 𝑓_{𝑠𝑝𝑒𝑐} is considered to be delta-function like), then 𝑓_{𝑤𝑖𝑛𝑑}(𝜃_𝑜) × 𝑓_{𝑠𝑝𝑒𝑐}(𝜃_𝑚− 𝜃_𝑜) ≅ 𝑓𝑤𝑖𝑛𝑑(𝜃𝑚) × 𝑓𝑠𝑝𝑒𝑐(𝜃𝑜− 𝜃_𝑚) meaning the model has Helmholtz reciprocity considering the symmetricity of the specular and diffuse component.

We first introduce a weighted inner product [29] (denoted by a double bracket in eq. (2-15)) and weighted basis (denoted by a tilde in eq. (2-16)) to simplify the frequency-domain expression of eq. (2-9). This basis is orthonormal with respect to the weighted inner product, so 𝐿_𝑜𝑏𝑗 can be expressed as a linear combination of 𝜑̃_𝑞 as shown in eq. (2-17), where 𝑂̃_𝑞 is a coefficient of 𝐿_𝑜𝑏𝑗 with respect to 𝜑̃_𝑞. (The relation between 𝑂_𝑞

By substituting eq. (2-9) and (2-17) into (2-11) and referring to eq. (2-12), a new frequency domain expression can be obtained in eq. (2-18). Note that the cosine obliquity factor has been cancelled and 𝐵̃_𝑝,𝑞 is simply the 2-D Fourier coefficient of the BRDF. In the following, the system matrix [𝐵̃] contains the elements 𝐵̃_𝑝,𝑞.

{

By substituting the BRDF model of eq. (2-14) into eq. (2-18), the following expression for 𝐵̃_𝑝,𝑞 is obtained:

(2-15)

(2-16)

(2-17)

(2-18)

19

Because we have assumed that the diffuse component is separable, the second term of eq. (2-19) can be written as eq. (2-23), where 𝐷𝑝 is the Fourier coefficient of 𝑓𝑑𝑖𝑓𝑓 defined in eq. (2-24).

Finally, by substituting eq. (2-20) and (2-23) into (2-19), 𝐵̃_𝑝,𝑞 is expressed in the following simple form:

𝐵̃_𝑝,𝑞= 𝑆_𝑝,𝑝𝑊_𝑝−𝑞+ 𝐷_𝑝𝐷_𝑞^∗

Recall that we have assumed that 𝑓𝑤𝑖𝑛𝑑 is a slowly varying function. To make further progress in the analytical solution of eq. (2-18), we approximate it as 𝑓_{𝑤𝑖𝑛𝑑}(𝜃_𝑜) ≅ 1. We then have

20

By substituting eq. (2-27) into (2-18), and dividing both sides by 𝑆_𝑝,𝑝, we have

𝑂̃_𝑝= 𝑀𝑝

𝑆_𝑝,𝑝− ∑ 𝐷𝑝

𝑆_𝑝,𝑝𝐷_𝑞^∗𝑂̃_𝑞

∞

𝑞=−∞

Eq. (2-28) is a discrete form of the Fredholm integral equation of the second kind [30], and it has a closed-form solution because of the separable kernel in the discrete sum. By expressing eq. (2-28) with a constant 𝛼 defined in eq. (2-29) and multiplying both sides by 𝐷_𝑝^∗, eq. (2-30) is obtained.

By calculating the discrete sum of p for both sides of eq. (2-30), the following is obtained:

∑ 𝐷𝑝∗𝑂̃𝑝

The left-hand-side of eq. (2-31) is equal to 𝛼, and therefore this equation can be solved in terms of 𝛼 as expressed in eq. (2-32). The expression for 𝛼 does not include unknown 𝑂̃𝑝, and therefore, an analytical closed-form

21

B. Understanding the structure of the system matrix [𝑩]

From the analytical expressions in the previous section, it is possible to interpret the structure and eigenvalues of the system matrix [𝐵] in eq. (2-13). From the relation between 𝑂𝑞 and 𝑂̃𝑞 in eq. (a3) in appendix A1, 𝐵_𝑝,𝑞 in eq. (2-13) and 𝐵̃_𝑝,𝑞 in eq. (2-18) have the following relation: 𝐵_𝑝,𝑞≅ 𝐵̃_𝑝,𝑞⁄ , where 𝛽𝛽_{0 0} is a constant related to the cosine obliquity factor and is given by eq. (a2) in appendix A1, and therefore, the structure of [𝐵] can be directly seen from [𝐵̃].

Let us first consider the simplest case where 𝑓_{𝑤𝑖𝑛𝑑} is unity. Fig. 2-7 (a) illustrates 𝐵̃_𝑝,𝑞 in eq. (2-27) where it can be seen that the diagonal elements of [𝐵̃] are 𝑆_𝑝,𝑝, or the Fourier coefficients of 𝑓_{𝑠𝑝𝑒𝑐} in eq. (2-21), and the off-diagonal elements are 𝐷𝑝𝐷𝑞∗ or the outer product of the Fourier coefficients of 𝑓𝑑𝑖𝑓𝑓 in eq. (2-24).

Moreover, it can be understood from eq. (2-33) that 𝑆_𝑝,𝑝 can be considered as the approximate eigenvalues of [𝐵̃] because the right-hand-side of eq. (2-33) includes division by 𝑆_𝑝,𝑝. As is commonly known, many actual BRFDs contain specular components whose shape can be approximated by a Gaussian function and whose Fourier coefficients monotonically decrease as a function of frequency [12,13], and therefore, the diagonal elements tend to have near zero values at higher frequencies. The extent of the non-zero portion of these diagonal elements of the matrix indicates the range of angular frequencies that can be recovered without being adversely affected by measurement noise. The length of this recoverable region is Fourier-transform related to the sharpness of the specular component (see fig. 2-7 (b)).

(2-33) (2-32)

22

Second, let us consider a more general BRDF where 𝑓𝑤𝑖𝑛𝑑 is not unity but rather a slowly varying function. From eq. (2-25), the diagonal elements 𝑆_𝑝,𝑝 are seen to be convolved with 𝑊_𝑝−𝑞, leading to a blur in the diagonal elements. Fig. 2-7 (b) shows an experimentally measured system matrix [𝐵] from a coated white paper. The width of the blur in this figure is a measure of the shift-variance of the specular component. In many common scattering materials where 𝑓_{𝑤𝑖𝑛𝑑} is slowly varying and the width of this blur is small, the 𝑆_𝑝,𝑝 terms are still good approximations to the actual eigenvalues.

Fig. 2-7. Understanding the structure of the system matrix [𝑩] . (a) Schematic structure of [𝑩̃]. (b) Experimentally obtained [𝑩].

C. Influence of measurement noise on reconstruction error

In many inverse problem solutions, the relative reconstruction error can be expressed as the product of the system matrix condition number and the inverse of the measurement system signal-to-noise ratio (SNR) [25,32]. In this section, we will derive a similar expression by making use of the analytical solution of the previous section together with a newly defined BRDF parameter which we call the degree of specularity.

We assume the linear noise model shown in eq. (2-34) as a description of the signal 𝐿𝑚𝑒𝑎𝑠 detected by the camera. The measurement noise 𝑋_𝜃𝑚 is assumed to be an additive zero-mean wide-sense stationary (WSS) random process [33].

23

𝐿_{𝑚𝑒𝑎𝑠}(𝜃_𝑚) = ∫ 𝑓_{𝐵𝑅𝐷𝐹}(𝜃_𝑜, 𝜃_𝑚) cos 𝜃_𝑜𝐿_𝑜𝑏𝑗(𝜃_𝑜)𝑑𝜃_𝑜

𝑏 𝑎

+ 𝑋_𝜃𝑚

Since the process is WSS, the noise can be characterized by random variables 𝐴_𝑝 that are the Fourier coefficients (or more generally the Karhunen-Loéve coefficients [33,35]) of 𝑋_𝜃𝑚 with respect to eq. (2-7) and (2-8). When taking the measurement noise into consideration, eq. (2-11) becomes

𝐿𝑚𝑒𝑎𝑠(𝜃𝑚) − 𝑋𝜃_𝑚= ∑ {𝑀𝑝− 𝐴𝑝}𝜑𝑝(𝜃𝑚)

∞

𝑝=−∞

Therefore, an analytical solution considering the measurement noise, denoted 𝑂̃_𝑝^{𝑛𝑜𝑖𝑠𝑒}, can be obtained in eq. (2-35) simply by replacing 𝑀_𝑝 with 𝑀_𝑝− 𝐴_𝑝 from eq. (2-33), where 𝑒̃_𝑝 is an error caused by the noise. shown below, where the p^th coefficient of the power spectral density of the measurement noise is denoted as 𝑃𝑆𝐷𝑝

(note: for definition, see eq. (a15) in appendix A7. In addition, 𝐸[𝐴_𝑝] = 0 and 𝐸 [|𝐴_𝑝|²] = 𝑃𝑆𝐷_𝑝 from

24

By substituting the above equation, eq. (2-32) and (2-33) into eq. (2-36), the relative noise error can be obtained as:

where the constant 𝛾 is defined in eq. (2-38) as the degree of specularity:

𝛾 ≡

The approximation in eq. (2-38) comes from the fact that value of the second term in the numerator can often be approximated as near zero (for further details, see appendix A2). The approximated 𝛾 is totally determined by the BRDF, and has the following properties; it becomes 1 for perfectly specular BRDF i.e. 𝐷_𝑝 = 0 for all 𝑝, and becomes 0 for perfectly diffusive BRDF i.e. 𝑆𝑝,𝑝= 0 for all 𝑝. In section 2.6, it will be experimentally shown that eq. (2-38) can accurately represent the characteristic of BRDFs from a variety of scattering surfaces.

If the noise 𝑋_𝜃𝑚 is white and Poissonian and the mean of the measured signal is sufficiently large, its statistics can be approximated by a white-Gaussian noise with a standard deviation which is a function (generally, the square root) of the signal mean. By defining the standard deviation of 𝑋_𝜃𝑚 as 𝜎(𝑀₀), where 𝑀₀ (average value of 𝐿𝑚𝑒𝑎𝑠 from eq. (2-11)) represents the signal mean, we can make the following approximation:

√𝑃𝑆𝐷𝑝≅ 𝜎(𝑀₀). (Note: For satisfying Parseval’s law, 𝜎(𝑀₀) needs to be divided by √𝑁 when an 𝑁-point DFT is taken for computation.)

D. Intuitive understanding of the reconstruction error

An inspection of eq. (2-37) reveals the following;

(2-37)

(2-38)

25

1) The first term |𝑆_0,0⁄𝑆_𝑝,𝑝| in eq. (2-37) can be interpreted as the condition number [25,32] of a matrix obtained by totally removing the diffuse component from the system matrix [𝐵], assuming 𝑆_𝑝,𝑝 is a

monotonically decreasing function of 𝑝 and truncating the matrix to include all terms less than or equal to 𝑝. This matches our intuition that BRDFs having sharper and/or narrower specular peaks will retain higher frequency components in the object reconstruction. By contrast, BRDFs having more rounded and/or wider specular peaks will lose higher object frequency information. A similar effect was noted in fig. 2-7. In addition, the above-mentioned specular width is measured relative to the domain range [𝑎, 𝑏]. For example, as the range becomes wider, the relative specular width can be seen to be narrower.

2) Instead of the more common definition of the signal-to-noise ratio (or SNR) in eq. (2-39), the second term in eq. (2-37) containing 𝛾 acts as the effective signal-to-noise ratio. We define this in eq. (2-40) as 𝑆𝑁𝑅𝑒𝑓𝑓.

𝑆𝑁𝑅 = 𝑀₀

√𝑃𝑆𝐷𝑝

𝑆𝑁𝑅_𝑒𝑓𝑓≡ 𝑀₀𝛾

√𝑃𝑆𝐷𝑝

The numerator 𝑀0𝛾 represents the part of 𝐿𝑚𝑒𝑎𝑠 originating from the specular component of the BRDF.

This indicates that BRDFs having larger diffuse components result in smaller values of 𝛾 and 𝑆𝑁𝑅_𝑒𝑓𝑓, making the object reconstruction more difficult. In the case of a perfectly diffusive BRDF (𝛾 = 0), the 𝑆𝑁𝑅_𝑒𝑓𝑓 becomes zero, meaning no object information can be retrieved. (Note: We defined the diffuse component as a separable function, and in general, any non-separable component of the BRDFs can convey object information.) In addition, the 𝛾 coefficient depends on the domain range [𝑎, 𝑏]. For example, if the measurable range in fig. 2-6 (b) becomes wider, 𝛾 becomes larger.

(2-39)

(2-40)

26

For categorizing different types of BRDFs from the viewpoint of solving the inverse problem, the two parameters,

|𝑆0,0⁄𝑆𝑝,𝑝| and 𝑆𝑁𝑅𝑒𝑓𝑓, are informative because they directly reflect the reconstruction error which determines the quality of the reconstructed light field 𝐿_𝑜𝑏𝑗.

E. Examples of noise analysis on wavelength and polarization

Observing the scene with longer wavelength light, such as infrared and terahertz wavelengths, can sometimes lead to a BRDF with a narrower specular peak [37]. This reduces |𝑆0,0⁄𝑆𝑝,𝑝| in eq. (2-37) meaning the error is suppressed and higher frequency object information can be preserved.

Moreover, the specular and diffuse reflections each have polarization properties governed by the characteristics of the scattering surface. The 𝑆𝑁𝑅_𝑒𝑓𝑓 can be taken to determine suitable usage of polarizers for better light field reconstruction. If we assume the simplest situation, such as P-polarized specular reflection and un-polarized Lambertian diffuse, it can be shown that the highest 𝑆𝑁𝑅_𝑒𝑓𝑓 can be achieved simply by using a P-polarizer during measurement. Some imaging systems attempt to remove the diffuse component by subtracting a second measurement with S-polarization from that with P-polarization [38]. However, this procedure does not improve the 𝑆𝑁𝑅𝑒𝑓𝑓 because the subtraction adds more noise. See appendix A3 for details.

2.5 Two fundamental regularization methods derived from the analytical solution

A. Fourier truncation regularization

As explained in section 2.4.B, the Fourier coefficients 𝑆𝑝,𝑝 of the specular component serve as approximate eigenvalues of the system matrix [𝐵] in eq. (2-13). Thus, removing small values of 𝑆_𝑝,𝑝 by truncating the higher frequency components of the system matrix [𝐵] can serve as a simple regularizer. Note that

27

this is similar to the commonly known truncation procedure for Fourier series when performing deconvolution (or inverse filtering) and truncation of the SVD when performing general inversions [22,26]. We previously derived an expression for the relative noise error in eq. (2-37). However, this expression is not appropriate for the present purpose since it lacks the error inherent in a truncated Fourier series [29]. Instead, we will derive a general error expression for the mean-squared error (MSE) to include both errors, and then use the first derivative of this MSE with respect to Fourier terms p as a criterion for selecting the best truncation point for regularization.

- Total MSE expressed as a sum of the truncation error and the noise error

𝐿_𝑜𝑏𝑗 is the original object and 𝐿^𝑒𝑟𝑟_𝑜𝑏𝑗 is the reconstructed object defined in eq. (2-42) to include both

By expanding eq. (2-41) (see appendix A4 for details), the total MSE can be obtained in eq. (2-43) as a function of the truncation term 𝑝𝑚𝑎𝑥. The first and second terms represent the truncation error, and the third term represents the measurement noise error.

- The first derivative of MSE as a criterion to determine the best truncation term for regularization (2-41)

(2-42)

(2-43)

28

As can be seen from eq. (2-43) and fig. 2-8 (a), the behavior of the MSE is governed by the two error sources. The truncation error monotonically decreases as 𝑝_𝑚𝑎𝑥 increases and eventually converges to zero, whereas the noise error monotonically increases as 𝑝𝑚𝑎𝑥 increases and eventually diverges. This indicates that the MSE can have a minimum at a particular value 𝑝_𝑚𝑎𝑥 and the first derivative of the MSE can be used to find the best truncation point. By approximating a derivative as a finite difference and an integration as a discrete summation, the discrete Leibniz integral rule is expressed as shown below, where 𝑐 is a constant.

𝑑

By applying the above rule to eq. (2-43), the following equation can be obtained (note: the first line assumes |𝑂̃_𝑝| and |𝑒̃𝑝| are symmetric with respect to 𝑝 = 0 because they are computed from real functions.):

By substituting eq. (2-33) and (2-35) into the above equation, the analytical first derivative of the MSE can be obtained in eq. (2-44), which is completely determined by the known information. Consequently, the best truncation term can be determined by finding the 𝑝̂_𝑚𝑎𝑥 which makes eq. (2-44) near zero. If the measurement noise is white and approximated as Gaussian with a standard deviation 𝜎(𝑀₀), √𝑃𝑆𝐷𝑝_𝑚𝑎𝑥≅ 𝜎(𝑀₀).

Note that eq. (2-43) does not guarantee an unique global minimum and eq. (2-44) may have some oscillations and multiple zeros. In this case, it is reasonable to fit eq. (2-44) with some slow function and take its zero-crossing point as 𝑝̂_𝑚𝑎𝑥.

(2-44)

29

- Relation between the first derivative of MSE and the relative noise error

Let us assume a single delta-function object (i.e. a point source) expressed as 𝐿_𝑜𝑏𝑗(𝜃𝑜) = 𝛿(𝜃𝑜), which leads to 𝑂̃𝑝= 𝑂̃0 for any 𝑝 from eq. 33) and (a3) in appendix A1. Noting that the first term of eq. (2-44) is 𝑂̃_{𝑝𝑚𝑎𝑥}, we can replace this term with 𝑂̃₀. By considering the relation: 𝑥²− 𝑦²= (𝑥 − 𝑦) × (𝑥 + 𝑦), we have

|√𝑃𝑆𝐷𝑝̂_𝑚𝑎𝑥

𝑆_{𝑝̂𝑚𝑎𝑥,𝑝̂𝑚𝑎𝑥}| − |𝑀0− 𝛼𝐷0

𝑆_0,0 | = 0

By referring to eq. (2-37), the following can be obtained from the above equation:

| 𝑆_0,0

𝑆_{𝑝̂𝑚𝑎𝑥,𝑝̂𝑚𝑎𝑥}| |√𝑃𝑆𝐷𝑝̂_𝑚𝑎𝑥

𝑀₀− 𝛼𝐷₀| = 1 = ∆̃_{𝑝̂𝑚𝑎𝑥}

Finally, we see that the Fourier term 𝑝̂_𝑚𝑎𝑥 (the MSE minimizer) corresponds to a Fourier term which makes the relative noise error ∆̃_𝑝 in eq. (2-37) equal to one, as illustrated in fig. 2-8 (b). Therefore, it can be said that the 𝑝̂_𝑚𝑎𝑥 directly reflects the BRDF characteristics, such as 𝑆_𝑝,𝑝, 𝛾 and 𝑆𝑁𝑅_𝑒𝑓𝑓 explained in section 2.4.C and

2.4.D. Although this requires the object to be a delta function, it is reasonable to expect the similar tendency of 𝑝̂_𝑚𝑎𝑥 for more general objects.

Fig. 2-8. (a) Mean-squared error (MSE) as a sum of two error sources. (b) Relation between 𝒑̂_𝒎𝒂𝒙 and ∆̃𝒑 assuming a single delta object.

30

B. Wiener filter regularization

By assuming a perfectly shift-invariant BRDF, a Wiener filter similar to those applied to image restoration can be derived [2,26,27]. In this work, however, we require a more general form of the Wiener filter that can be applied to the shift-variant BRDF of section 2.4. The derivation of the Wiener filter based on the Bayesian theorem [22,33] and the minimization principle [29] is outlined in appendix A5 ~ A7 for completeness (see also [23,24,25]), and the resulting Wiener filter is shown in eq. (2-45), where 𝐵_𝑟,𝑠^∗ is the complex-conjugate of 𝐵𝑟,𝑠 defined in eq. (2-13) (note: 𝑟 and 𝑠 correspond to 𝑝 and 𝑞 in eq. (2-13) respectively), and 𝑂̂𝑞 is the Fourier coefficient of the Wiener filter estimate 𝐿̂ . (Definitions of other coefficients: 𝑀_{𝑜𝑏𝑗 𝑠} from eq. (2-11), 𝐵_𝑝,𝑞 from eq. (2-13), 𝑆_𝑠,𝑠 from eq. (2-21), 𝐷_𝑠 from eq. (2-24), 𝛼 from eq. (2-32), 𝛽₀ from eq. (a2) in

If the measurement noise is white and approximated as Gaussian with a standard deviation 𝜎(𝑀0), eq. (2-45) is simplified to eq. (2-46). The second term is a regularizer defined as the inverse of 𝑆𝑁𝑅_𝑠^{𝑊𝑖𝑒𝑛𝑒𝑟} in eq.

(2-47). We will use these expressions for the experimental verifications.

∑ {𝐵_𝑟,𝑠^∗ ∑ 𝐵_𝑟,𝑞𝑂̂_𝑞

31

2.6 Experimental verification

In this section, the relative noise error and the two regularization methods derived in the previous sections will be experimentally verified with a variety of scattering materials.

A. Experimental setup and BRDF measurements

Fig. 2-9. (a) Experimental set-up (top view). (b) One-dimensional radiometric intensity from LCD used as 𝑳_𝒐𝒃𝒋. (c) and (d) Actual photographs of the set-up in the enclosure.

Fig. 2-9 (a) shows a schematic of the experimental set-up which reproduces the imaging scenario of fig.

1-1 assuming one-dimensional objects, and fig. 2-9 (c) and (d) are actual photographs of the set-up. It consists of shot-noise limited CMOS camera in the visible wavelength (Photometrics Prime) mounted on a computer controlled angular stage, and an LCD monitor. Though the methods and analyses of the previous sections are applicable to the entire light field, we measure only the angular component from a single spatial location in the

32

current experiment since it is sufficient for verifying the above conclusions. For this reason, the camera captures one single point on the scattering surface 𝑦_𝑠 = 𝑐_𝑟= 0 (restricted by eq. (2-4)) from a variety of different camera angles (29 ~ 61 degree range with 0.5 degree resolution) resulting in the measured radiometric intensity 𝐿_{𝑚𝑒𝑎𝑠}(𝜃_𝑚). In addition, rather than use real objects as shown in fig. 1-1, we employ a well-calibrated LCD monitor to generate the object light field distribution 𝐿_𝑜𝑏𝑗(𝜃𝑜) (shown in fig. 2-9 (b)) appropriate for the point on the scattering surface restricted by eq. (2-5). For suppressing ambient and stray light, the enclosure is designed with an interior entirely covered with black felt, and baffles are employed to reduce unwanted reflections.

Since our experiments are designed for one-dimensional objects, the measurement of the BRDF is greatly simplified. The measurement consists of illuminating the scattering surface with a single narrow slit of light from the LCD screen and recording the scattered light with the camera as a function of output angle of the BRDF 𝜃_𝑚. This measurement is repeated many times, each for a different slit location corresponding to various input angles of the BRDF 𝜃𝑜. Fig. 2-10 shows measured BRDFs at three input angles with their corresponding degree of specularity 𝛾 as defined by eq. (2-38) of four scattering materials.

Fig. 2-10 (c) shows how the measurements are fit to the BRDF model of eq. (2-14) to compute our analytical expressions. The window function 𝑓𝑤𝑖𝑛𝑑 is sufficiently slowly varying in all samples that it can be ignored in the analytical expressions and the regularization methods derived in the previous sections. The shift-invariant specular component 𝑓𝑠𝑝𝑒𝑐 is approximated as the area above the dotted line connecting both edges of the solid curve having a peak around the center of the angular domain, and the area below the dotted line represents the diffuse component. As the simplest expression, the separable function of the diffuse component is approximated as 𝑓_{𝑑𝑖𝑓𝑓}(𝜃_𝑜) × 𝑓_{𝑑𝑖𝑓𝑓}(𝜃_𝑚) ≅ (𝑙₁𝜃_𝑜+

𝑙2) × (𝑙1𝜃𝑚+ 𝑙2), and the constants 𝑙1 and 𝑙2 are determined so the expression can fit the dotted line in a least square sense. In addition, the camera's measurement noise was previously measured, and its standard deviation was obtained as a function of signal mean.

33

Fig. 2-10. Experimentally measured BRDF samples: (a) Brushed metal, (b) coated white-paper with highlights #1, (c) coated white-paper with highlights #2, and (d) white paint with a pearl finish applied to glass.

B. Experimental verification of the relative noise error

The relative noise error ∆𝑝 defined in eq. (2-48) (see eq. (2-10) for 𝑂𝑝) was computed from experimentally measured data through the following steps; 1) a sinusoidal intensity pattern of the p^th Fourier component on a bias was reproduced on the LCD screen, 2) the camera data 𝐿𝑚𝑒𝑎𝑠 (including measurement noise) was recorded as a function of camera angle, and 3) eq. (2-13) was then solved for 𝑂𝑝𝑛𝑜𝑖𝑠𝑒 with a truncated system matrix [𝐵] that included only Fourier terms from zero to 𝑝. This procedure was repeated 500 times to generate the standard deviation statistic for 𝑂𝑝𝑛𝑜𝑖𝑠𝑒 and eq. (2-48) was computed for a specific value of p.

∆_𝑝≡

√𝑉𝑎𝑟[𝑂_𝑝^{𝑛𝑜𝑖𝑠𝑒}]

|𝑂₀| , where 𝑂_𝑝^{𝑛𝑜𝑖𝑠𝑒}≡ 𝑂_𝑝+ 𝑛𝑜𝑖𝑠𝑒 𝑒𝑟𝑟𝑜𝑟 _(2-48)

34

Fig 2-11 shows a comparison between the experimentally obtained values of ∆_𝑝 from eq. (2-48) and the theoretically calculated values of ∆̃𝑝 from eq. (2-37) for the four different BRDF samples in fig. 2-10. (Note:

The theoretical ∆̃_𝑝 can be directly compared with the experimental ∆_𝑝, because ∆_𝑝≅ ∆̃_𝑝 from eq. (a3) in appendix A1.) Good agreements can be seen in all of the samples, which shows that the analytical BRDF model

The theoretical ∆̃_𝑝 can be directly compared with the experimental ∆_𝑝, because ∆_𝑝≅ ∆̃_𝑝 from eq. (a3) in appendix A1.) Good agreements can be seen in all of the samples, which shows that the analytical BRDF model

In document Information Extraction from Non-Line-of-Sight Objects. Using Plenoptic Data from Scattered Light A DISSERTATION SUBMITTED TO THE FACULTY (Page 31-119)