Simple and effective calculations about spectral power distributions of outdoor light sources for computer vision

Jiandong Tian; Zhigang Duan; Weihong Ren; Zhi Han; Yandong Tang

doi:10.1364/OE.24.007266

1. Introduction

Solar irradiance is changed by atmospheric transmittance effects including absorption, reflecting, and scattering, which causes the spectral power distribution (SPD) of the light that ultimately reaches the Earth’s surface to vary with time and air conditions. As shown in Fig. 1, atmospheric transmittance effects lead to many natural things related to computer vision applications such as the variation of sun and sky appearance, twilight, shadow, haze/fog, and cloud. Therefore, in computer vision, a SPD calculation method of outdoor light sources is useful to deal with the problems caused by different time (or zenith angles) and weather conditions.

Fig. 1 Outdoor appearances vary with time and air conditions.

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Nature light modeling and image illumination processing have received a lot of attention in the computer vision and computer graphics community. Based on the fact that the luminance varies with the different parts of the sky, a physically-based sky luminance model is employed to infer the camera azimuth in [1] and to recover camera focal length in [2]. Lalonde et al. [3] presented how they can geolocate a camera by using the sky appearance and sun position annotated in images. Liu et al. [4] proposed a method to estimate the lighting condition of outdoor scenes through learning a set of images captured at the same sun position. Sunkavalli et al. [5] presented that sunlight and skylight SPDs can be recovered by analyzing a time-lapse video of outdoor scenes. Haber et al. [6] presented an approach to compute the colors of the sky during twilight period before sunrise and after sunset based on the theory of light scattering by small particles. Perez et al. [7] proposed a popular sky model that has been widely used in computer vision and graphics. In general, most of these methods focus on modeling or applying the sky spatial radiance distributions rather than spectral information. Spectral information, i.e., the SPD knowledge of light sources, is useful in some computer vision tasks, such as reflectance recovery [8], camera spectral sensitivities estimation [9], color constancy [10], relighting [11], image rendering [12], and augmented reality [13].

In the computer vision community, two methods are usually applied to estimate SPD illuminations. The first one employs Planck’s blackbody radiation law to approximately calculate the SPDs of outdoor light sources. Finlayson et al. [14] and Tian et al. [15] applied Planck’s blackbody radiation law to approximately estimate the SPDs of daylight and skylight for deriving shadow invariant images and for detecting shadows, respectively. The second one employs daylight characteristic vectors to recover the SPDs of outdoor light sources. Judd et al. [16] proposed the characteristic vector analysis method on 622 samples of daylight measured at 10 nm intervals over the visible range. Their results suggested that most of the daylight samples can be approximated accurately by a linear combination of the three fixed characteristic vectors. Based on Perez sky model [7], Preetham et al. [17] proposed an improved sky model. It suggests an analytical approximation of the five distribution coefficients in Perez sky model to be a linear function of a single parameter, turbidity, by fitting the Perez formulas to the reference images. With another parameter, sun angle, the absolute sky luminance Y as well as the CIE chromaticities x and y of a sky element can be calculated. They also provide a way to convert the outputs to spectral radiance data (See Appendix 5 in [17]). This analytic model of spectral sky-dome radiance is widely-used. Using images captured during a single day, Jung et al. [18] presented an outdoor photometric stereo method via skylight estimation according to the Preetham sky model. Given sky images and viewing geometry as the input, Kawakami et al. [19] proposed a method to estimate the turbidity of the sky by fitting image intensities to the Preetham sky model. Then, the sky spectrum can be calculated. Having the sky RGB values and their corresponding spectrum, the method estimates the camera spectral sensitivities and white balance setting. In the experimental section, we will analyze and compare with the Planck’s blackbody radiation and the Preetham sky model.

In meteorology, there are SPD calculation methods based on analyzing the transmittance functions of absorption and scattering along the path of solar radiation through the atmosphere. The latest versions of two representative SPD calculation models with wavelength resolution are SMARTS2 [20] and MODTRAN [21]. The SPDs calculated by these two methods can approximate real measured ones well. However, they may be too complex to be used in computer vision, because they contain many parameters to be determined and unfortunately most of these meteorological parameters are not readily available for computer vision applications. Bird et al. [22] proposed a simpler method to calculate the SPD of outdoor light sources. Though Bird’s method is much simpler than those in [20, 21], it is still difficult to be used in computer vision. In general, the existing methods are not developed for computer vision. They consider the full spectrum and involve many factors and parameters that have no effects on the visible spectrum. Besides, in all existing methods, the characteristics of human vision are not taken into account. Therefore, they may be more suitable for the research on meteorology than computer vision. Because most computer vision tasks concentrate on analyzing images captured in the visible spectrum, it is possible for us to develop the simpler and effective SPD calculation method that can be easily applied in computer vision tasks, such as illumination processing, shadow removal, and image relighting.

The main contribution of this paper is that we propose a very simple method to calculate the SPDs of sunlight and skylight in the visible spectrum. Different from previous calculations, we first calculate the atmosphere’s absorption, since some light emitted from the sun never reach the Earth’s surface neither as direct sunlight nor as diffuse skylight due to the atmosphere’s absorption. We propose a simple new absorption calculation method and provide the new “total absorption coefficient”. During the development of our method, we pay more attention to the wavelengths near the peaks of the human color matching functions (CMFs), to guarantee the accuracy of color reproduction when using our calculated SPDs to substitute the real ones.

2. A simple method for computing the spectral power distribution of sunlight and skylight

2.1. Light and image

Because our SPDs calculations are for imaging and computer vision, we develop our method based on image formation theories. Figure 2 shows the image formation procedure in outdoor environments. Since both our eyes and conventional cameras are only sensitive to the visible spectrum, all the calculations and analysis in this paper will be done within the spectrum 400–700nm. Given illumination SPD E(λ) and object’s reflectance S(λ), the tristimulus values are,

F_{H} = \int_{400}^{700} E (λ) S (λ) Q_{H} (λ) d λ

where Q_H(λ) denotes the XYZ or sRGB color matching functions in three channels. In this paper, tristimulus values in XYZ will be used in Section 2.4 and those in sRGB will be used in Section 4 or in the experiments in Section 4. The detail about XYZ to sRGB conversion can be found in [23].

Fig. 2 Illustration of the image formation procedure.

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2.2. Absorption of extraterrestrial irradiance passing through the atmosphere

When the extraterrestrial irradiance passes through the atmosphere, it is attenuated by absorption, reflection, and scattering processes, and thereby its spectral composition is changed considerably. Some solar radiation will never reachs the Earth’s surface neither as direct sunlight nor as diffuse skylight due to the atmosphere’s absorption by ozone, nitrogen dioxide, mixed gases, and water vapor. We denote E_oλ as extraterrestrial irradiance at mean earth-sun distance for wavelength λ. After absorption, the irradiance becomes,

E_{i n} = E_{o λ} \cdot T_{o λ} \cdot T_{N λ} \cdot T_{w λ} \cdot T_{u λ}

where T_oλ, T_Nλ, T_wλ, and T_uλ denote the transmittance functions for molecular ozone, nitrogen dioxide, molecular water vapor, and mixed gases, respectively. Extraterrestrial irradiance also varies with the sun-earth distance which is determined by the Day Number of a year. Since this factor is independent of wavelength, it can be taken outside the integration and can be consider as exposure time in Eq. (1). Therefore this factor can be neglected for computer vision application. T_oλ and T_Nλ can be represented by,

T_{o λ} = \exp (- 0.35 a_{o λ} m) (R e f e r t o [24])

T_{N λ} = \exp (- 0.0016 a_{n λ} m) (R e f e r t o [20, 25])

where a_oλ and a_nλ are the absorption coefficients of molecular ozone and nitrogen dioxide, respectively. In [22, 24], T_wλ and T_uλ are given by,

T_{w λ} = \exp [- 0.2385 a_{w λ} W m / {(1 + 20.07 a_{w λ} W m)}^{0.45}]

T_{u λ} = \exp [- 1.41 a_{u λ} m / {(1 + 118.93 a_{u λ} m)}^{0.45}]

where a_uλ and a_wλ are the absorption coefficients of molecular water vapor, and mixed gases, respectively. W is water vapor amount, and its typical value is 1.6. m is air mass that is calculated by,

m = \sec (θ)

θ denotes zenith angle in this paper. Unlike that T_oλ and T_Nλ take noticeable effects in the visible spectrum, T_uλ mainly takes effect on wavelengths longer than 1000nm and in the visible spectrum takes effect in 687nm∼691nm [26]; T_wλ mainly takes effect on wavelengths that are longer than 800nm and in the visible spectrum takes effect in 690nm∼700nm [27]. T_uλ and T_wλ have little effect in the visible spectrum but their computations are more complex than those of T_oλ and T_Nλ. More importantly, T_uλ and T_wλ have different forms with T_oλ and T_Nλ, which brings difficulties for merging these four terms. Therefore, we here want to seek simple and effective expressions for T_uλ and T_wλ. Eq. (5) and Eq. (6) have similar formulas, thus we first merge T_uλ into T_wλ, i.e.,

T_{w λ} (a_{w λ} + ε a_{u λ} | m) = T_{w λ} (a_{w λ} | m) \cdot T_{u λ} (a_{u λ} | m)

Where ε can be determined by,

ε = a r g m i n \sum_{m} {‖ T_{w λ} \cdot T_{u λ} - T_{w λ} (a_{w λ} + ε a_{u λ}) ‖}_{2}

The result is ε = 4.3. We have,

T_{w λ} (a_{w λ} + 4.3 a_{u λ}) \approx T_{w λ} \cdot T_{u λ}

The two attenuations caused by water vapor and mixed gases are merged into a single attenuation coefficient. We define the new term as ${T^{'}}_{_{w λ}}$ ,

{T^{'}}_{w λ} = \exp [- 0.2385 {a^{'}}_{w λ} W m / {(1 + 20.07 {a^{'}}_{_{w λ}} W m)}^{0.45}]

where

{a^{'}}_{w λ} = a_{w λ} + 4.3 a_{u λ}

.

For different m, we find that data counterparts of $[{a^{'}}_{w λ}, l o g ({T^{'}}_{w λ})]$ satisfy a power function. So in the following we write ${T^{'}}_{w λ}$ as the form of T_oλ and T_Nλ. We define a new transmittance function of water vapor and uniform mixture gas as,

T_{w λ}^{n e w} = \exp [p {({a^{'}}_{w λ})}^{q} m]

with,

p, q = a r g m i n \sum_{m} {‖ p {({a^{'}}_{w λ})}^{q} m - \log ({T^{'}}_{w λ}) ‖}_{2}

The results of Eq. (13) are p = −0.055 and q = 0.56. We apply the simple exhaustive search method to solve the optimization problem in Eq. (9) and Eq. (13). We set the sampling step size of λ as 1nm and the step size of ε, p, and q as 0.01. The new attenuation expression of water vapor and mixed gases is approximated as a decreasing exponential function of the combined coefficient.

T_{w λ}^{n e w} = \exp [- 0.055 {({a^{'}}_{w λ})}^{0.56} m]

Then since all the absorption factors can be expressed by a decreasing exponential function, we rewrite Eq. (2) as,

E_{i n} = E_{o λ} \cdot T_{λ}

where,

T_{λ} = \exp (- τ^{'} (λ) \cdot m)

The results in Fig. 3 show that our new simple expression produces similar results compared with the combined effects of the original two terms in the visible spectrum. We name the proposed T_λ as “total absorption transmittance function” and τ′ (λ) as “total absorption coefficient” that is represented by,

τ^{'} (λ) = 0.35 a_{o λ} + 0.0016 a_{n λ} + 0.055 {a^{'}}_{w λ}^{0.56}

Fig. 3 New attenuation and transmittance expressions of water vapor and uniformly mixed gases produce good results as the original expressions.

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The total absorption coefficient can be found in the Appendix (Table 4) and is plotted in Fig. 4, in which a_oλ, a_nλ, a_uλ, and a_wλ are derived from [20] with a_oλ = A_oλ, a_nλ = A_nλ, a_uλ = 100A_gλ, and a_wλ = 100A_wλ. The SPD of extraterrestrial irradiance and an example of its result after absorption at m = 2 is shown in Fig. 5.

Fig. 4 Total absorption coefficient.

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Fig. 5 The SPD of extraterrestrial irradiance reported by World Radiation Center and that after absorption at air mass equals to 2.

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Using our proposed total absorption coefficient and transmittance function, the absorption calculation of ozone, nitrogen dioxide, water vapor, and mixed gases can be considered all together. Eq. (16) is much simpler to use.

2.3. Computing direct sunlight

The direct irradiance at Earth’s surface normal to the direction of the sun can be calculated by,

E_{d λ} = E_{i n} \cdot T_{r λ} \cdot T_{a λ}

where E_in is the SPD of extraterrestrial irradiance after absorption, and T_rλ, T_aλ denote the transmittance functions of Rayleigh scattering and Aerosol scatting, respectively. The transmittance function for the Rayleigh scattering in [28] is adopted in this paper.

T_{r λ} = \exp (- 0.008735 λ^{- 4.08} \cdot m)

The transmittance function for the aerosol scattering in [28] is adopted in this paper.

T_{a λ} = \exp (- β λ^{- α} \cdot m)

where α is wavelength exponent and the value of 1.3 is commonly employed for α. The parameter β is the cleanliness index (turbidity) which varies from 0.0 to 0.4. For clear weather, turbid weather, and very turbid weather, its value is about 0.1, 0.2, and 0.3 respectively [28].

2.4. Computing diffuse skylight

The calculations of diffuse skylight in other works are complicated. Here we propose a very simple method. From Eq. (19) and Eq. (20), we know that T_rλ and T_aλ describe the direct transmittance functions of Rayleigh scattering and Aerosol scattering respectively. Thus (1−T_rλ T_aλ) actually describes the sum of scattered light that can reach the Earth’s surface and that are lost in the atmosphere. So the key point becomes how to determine the proportion of the scattered light that can reach the ground. Therefore, the diffuse skylight on the horizontal surface can be calculated by,

E_{s λ} = E_{i n} \cdot \cos (θ) \cdot (1 - T_{r λ} \cdot T_{a λ}) \cdot κ

where κ accounts for the proportion of the scattered light that can reach the ground.

As shown in Fig. 6, κ should be zenith-dependent. The longer light passes through the atmosphere, the more light will be lost, i.e., less scattered light can reach the ground. We determine κ by fitting CIE ΔE_Lab between the calculated SPDs by Eq. (21) and those from the true measurements by a spectrometer SOC710. In detail, for the ideal white reflectance, S(λ) = 1 is applied in calculation. For the calculated SPD by our method, the XYZ tristimulus values are,

F_{H} = \int_{400}^{700} E_{s λ} S (λ) Q_{H} (λ) d λ .

Fig. 6 Schematic diagram of the solar radiation onto the Earth surface.

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For the measured SPD, the XYZ tristimulus values are,

{F^{'}}_{H} = \int_{400}^{700} {E^{'}}_{s λ} S (λ) Q_{H} (λ) d λ .

The XYZ tristimulus values are converted to CIELAB color space and ΔE_Lab is applied to evaluate the error between E_sλ and ${E^{'}}_{s λ}$ .

Δ E_{L a b} = \sqrt{Δ L^{2} + Δ a^{2} + Δ b^{2}}

κ can be determined by,

κ = a r g m i n Δ E_{L a b}

The reason that we apply XYZ values and ΔE_Lab to determine κ is to keep higher accuracy near the peaks of CMFs. The values of κ with different zenith angles are tabulated in Table 1.

Table 1. Proportion of the scattered light that can reach the ground with different zenith angles

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3. Experiments and comparisons

Figure 7 shows the comparisons of the calculated SPDs by our method with those by SMARTS2 method [20] and Bird’s method [22] in clear weather, i.e. β = 0.1. The results generally denote that there are no significant discrepancies between the calculated SPDs by our simple method and by the other two complex meteorology ones, especially for direct sunlight. To evaluate the influences of the discrepancies on imagery, we simulate sRGB values (see the 2^nd and 4^th columns in Fig. 7) of Xrite colorchecker which contains 24 common colors in our daily life. From the results, it is hard to find differences by our naked eyes on the simulated images under SPDs calculated by the three methods. The last column in Fig. 7 shows the quantitative discrepancies of pixel values of the simulated images. For each color patch, in three channels, max differences versus max values are plotted as percent errors.

Fig. 7 Comparison of the calculated SPDs by our method and by Bird’s method [22] and SMARTS2 method [20] at different zenith angles for β = 0.1. From top to bottom, the zenith angle equals to 20, 30,…,80 degree, respectively. The x-coordinate denotes wavelength (nm) and the y-coordinate denotes irradiance power(W·m⁻²·nm⁻¹). The 2^nd column shows simulated Xrite colorchecker appearances under sunlight and the 4th column is that under skylight. Upper: Our; Middle: Bird’s; Lower: SMARTS2. The last column is pixel value differences (percent). A: Direct sunlight compared with Bird’s method. B: Direct sunlight compared with SMARTS2 method. C: Diffuse skylight compared with Bird’s method. D: Diffuse skylight compared with SMARTS2 method.

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The results in Fig. 7 also show that the SPDs do not vary significantly from 20 degree to 40 degree, which indicates that the outdoor light sources are quite stable during noon. The variation of intensity is larger than that of spectrum. Because variation of intensity can be taken outside the integration and can be put into exposure time in Eq. (1), it has no effect on imagery.

Figure 8 shows the comparison of the calculated SPDs by our method with those by SMARTS2 and Bird’s method under different turbidities. The consistency by the three methods on sunlight is better than that on skylight. We can see that for very clear weather β = 0, the outputs of the three methods are quite similar for skylight. The similarity degrades with increasing β. For very turbid weather, β = 0.3, the three outputs are obvious different. Overall, our results are more close to those of SMARTS2 method in the skylight calculation.

Fig. 8 Comparisons of the calculated SPDs by our method and those by Bird’s method [22] and SMARTS2 method [20] under different turbidities. From top to bottom are β = 0, β = 0.2, and β = 0.3, respectively.

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The comparison on the formula expressions of our method with those of SMARTS2 and Bird’s method are listed in Table 2. From the table, we can see that our expressions are much simpler than those in the other two methods. More importantly, in the other two methods there are many parameters that may be hard to be determined in computer vision.

Table 2. Comparison of the formula expressions of our methods with those of SMARTS2 and Bird’s method

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3.1. Comparison with blackbody radiation

Planck’s blackbody radiation law is usually applied to calculate the SPDs of outdoor light sources. It is well known that extraterrestrial solar irradiance can be approximated by the black-body radiation at 5777 K. However, from Fig. 9, we can see that the extraterrestrial solar irradiance roughly follows the blackbody radiation at 5777 K in the near infrared spectrum while it does not follow blackbody radiation law well enough in the visible region. Figure 10 shows the chromaticity of the blackbody radiation and that of sunlight, skylight, and daylight. We find that the chromaticity of daylight is very near to Planckian locus and so it can be accurately approximated by the blackbody radiation. In contrast, the chromaticity of sunlight and skylight deviate from that calculated by blackbody radiation noticeably, which indicates that noticeable errors occur when blackbody radiation is employed to model the SPDs of sunlight and skylight.

Fig. 9 Using blackbody radiation to approximate the true extraterrestrial data. The right image is the close-up view of the left one.

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Fig. 10 CIE Chromaticity of sunlight, skylight, and daylight are compared with Planckian locus formed by color temperatures from 1000k to 500000k.

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3.2. Comparison with Preetham sky model

The Preetham model [17] also only requires the zenith angle and turbidity as varying parameters. Using Judd’s characteristic vectors, this model can produce radiance spectral information of all points on the skyphere. Details about recovering radiance spectral information from sky chromaticity values based on Judd’s characteristic vectors can be found in the Appendix 5 in [17]. After integration of the incoming light from all the directions in the hemisphere, SPD of skylight irradiance can be calculated. Figure 11 shows the comparison between our method, SMARTS2, Bird’s method, and Preetham method. We can find that, compared with our method, Preetham method is not accurate enough to match the two meteorology methods. This is caused by that some errors may arise when recovering full spectra from sky X, Y, Z values. It only uses three principle basis. Another disadvantage of using the Preetham sky model to recover SPD is that it can only recover skylight SPD while it cannot recover SPD of direct sunlight and daylight.

Fig. 11 Compared with Preetham model. Left: at 30 degree zenith angle. Right: at 60 degree zenith angle.

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4. Applications

4.1. The recovery of camera capture information

We first demonstrate estimating illumination, zenith angle, turbidity, and camera spectral sensitivities from a single 24 color checker image. Rewriting E(λ) in Eq. (1), we have,

F_{H} = \int_{400}^{700} E (θ, T, λ) S (λ) Q_{H} (λ) d λ, H = R, G, B

Jiang’s et al. [29] shows that camera spectral sensitivities (CSS) can be decomposed as Q_H= σ_H·C_H, where σ_H = [σ_H,₁, σ_H,₂] ; C_H=[c_H,₁,c_H,₂]^T is the eigenvector matrix that is precomputed by a dataset including different CSS. Given a 24 color checker image and the reflectance spectrum as shown in Fig. 12, parameters σ_H, θ, T can be recovered by iteratively minimizing the following RMS, and then illumination SPDs and CSS can be recovered.

\sum_{n = 1}^{24} ‖ F_{H}^{n} - \sum_{λ} E (θ, T) \cdot S^{n} \cdot [σ_{H} \cdot C_{H}] ‖, H = R, G, B

Fig. 12 A 24 color checker image captured by a Canon 5D Mark II camera under skylight and its corresponding spectral reflectance.

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Compared with Jiang’s method that can recover CSS and one illuminations-SPD, as shown in Fig. 13, our method can recover CSS and SPDs of three lights, i.e., sunlight, daylight, and skylight simultaneously. More importantly, using our method, sun angle and turbidity can be recovered from the image while Jiang’s method can only recover color temperature.

Fig. 13 The recovered skylight spectrum and the recovered CSS of Canon 5D Mark II. In the second figure, M and E are the abbreviations of ”Measured” and ”Estimated”, respectively.

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4.2. Shadow features for shadow detection

Shadow is an active research topic in computer vision. Its generation and characteristics have close relation to direct sunlight and diffuse skylight. Tian et al. [30] show that the pixel values of a surface illuminated by daylight (in non-shadow region) are proportional to those of the same surface illuminated by skylight (in shadow region). If f_H denote pixel values of a surface in shadow area and F_H denote those of the same surface in non-shadow area, [30] shows,

\frac{F_{H}}{f_{H}} = \frac{\int_{400}^{700} E_{d a y} (θ, T, λ) S (λ) Q_{H} (λ) d λ}{\int_{400}^{700} E_{s k y} (θ, T, λ) S (λ) Q_{H} (λ) d λ} = K_{H}

where

E_{d a y} (θ, T, λ) = E_{s u n} (θ, T, λ) \cdot \cos θ + E_{s k y} (θ, T, λ) .

Since our calculating method can simultaneously calculate SPDs of daylight and skylight. Based on our calculated SPDs, we can easily obtain K_H for any sun angles and turbidity. As shown in Fig. 14, we capture three images at 43 degree sun angle and in clear weather with 0.15 turbidity (estimated by section 4.1). We calculate the normalized K_H in the Canon 5D Mark II RAW images without Gamma correction. We can find these values are quite near to the calculated K_H = [0.69,0.57,0.45] by our calculated SPDs, indicating that our SPD calculation method can predict K_H in shadowed images.

Fig. 14 Our SPD calculation method can predict K_H in shadowed images.

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We also calculated K_H under different sun angles and turbidity to find some rules that will be useful for shadow detection. We list these K_H values in Table 3. From the table we can find that three properties of shadows.

Property 1. K_H satisfy K_R > K_G > K_B
Property 2. K_H decrease as turbidity increases.
Property 3. K_H decrease as sun angle increases.

Table 3. The calculated K_H at different sun angles and under different turbidities

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To test property 1, we collected 563 real-world images downloaded from the web or real-captured. Particularly we carefully captured images at different sun angles and under different weather conditions (including clear sky, little overcast, and heavy clouds that didn’t cover the sun). For each image, shadow regions were manually labeled. We got 3672 shadow & non-shadow counterparts and calculated all the K_H by these counterparts. We find that 90.50% K_H satisfy property 1. Preliminary analysis shows that overexposure, thin shadows, and too large contrast enhancement by post-processing on images usually cause that the calculated K_H do not satisfy property 1.

Since we usually don’t know under what sun angles and turbidity an image is captured, property 2 and 3 are not convenient to be directly used. They may be useful to some inverse problem from images, e.g., using detected shadows to infer sun angle and turbidity. In detail, if we know the CSS of a camera, we can obtain a table like Table 3, and coarsely infer sun angle and turbidity by comparing the K_H in the table and the K_H calculated from the detected shadows. The application in this section shows that our SPD calculation is valuable for finding novel shadow features, which may hard to accomplish by other SPD calculation methods since their methods cannot calculate Eq. (28) and Table 3.

4.3. Deriving intrinsic images

Our SPD calculation method can be used to derive intrinsic images. If we convert the linear RGB to gamma-corrected sRGB vaules (According to [23]), Eq. (28) becomes

\log (F_{H} + 14) = \frac{\log (K_{H})}{2.4} + \log (f_{H} + 14)

From Eq. (30), the following equation holds.

\begin{array}{l} \log (F_{R} + 14) + \log (F_{G} + 14) - β_{1} \cdot \log (F_{B} + 14) \\ = \log (f_{R} + 14) + \log (f_{G} + 14) - β_{1} \cdot \log (f_{B} + 14) \end{array}

where

β_{1} = \frac{\log (K_{R}) + \log (K_{G})}{\log (K_{B})}

For a pixel in an image, Eq. (31) represents a shadow invariant. For an arbitrary pixel and its RGB value vector, (v_R, vG,vB)^T, log(v_R+14)+log(v_G+14) − β₁·log(v_B+14) keeps the same no matter whether the pixel is in shadow region or not. Apparently, a grayscale intrinsic image is obtained. One result is shown in the middle of Fig. 15.

Fig. 15 Intrinsic image result. Left: Original image; middle: Grayscale intrinsic image; right: Color intrinsic image.

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To obtain a color intrinsic image, for a RGB value vector (v_R,v_G,v_B)^T, we first define the following grayscale intrinsic values I₁ according to Eq. (31).

\begin{array}{l} I_{1} = \log (F_{R} + 14) + \log (F_{G} + 14) - β_{1} \cdot \log (F_{B} + 14) \\ = \log (f_{R} + 14) + \log (f_{G} + 14) - β_{1} \cdot \log (f_{B} + 14) \\ = \log (v_{R} + 14) + \log (v_{G} + 14) - β_{1} \cdot \log (v_{B} + 14) \end{array}

Similar to Eq. (33), we can obtain two other grayscale intrinsic values I₂ and I₃,

\begin{array}{l} I_{2} = \log (F_{R} + 14) - β_{2} \cdot \log (F_{G} + 14) + \log (F_{B} + 14) \\ = \log (f_{R} + 14) - β_{2} \cdot \log (f_{G} + 14) + \log (f_{B} + 14) \\ = \log (v_{R} + 14) - β_{2} \cdot \log (v_{G} + 14) + \log (v_{B} + 14) \end{array}

\begin{array}{l} I_{3} = - β_{3} \cdot \log (F_{R} + 14) + \log (F_{G} + 14) + \log (F_{B} + 14) \\ = - β_{3} \cdot \log (f_{R} + 14) + \log (f_{G} + 14) + \log (f_{B} + 14) \\ = - β_{3} \cdot \log (v_{R} + 14) + \log (v_{G} + 14) + \log (v_{B} + 14) \end{array}

where

β_{2} = \frac{\log (K_{R}) + \log (K_{B})}{\log (K_{G})}, β_{3} = \frac{\log (K_{G}) \log (K_{B})}{\log (K_{R})}

A color intrinsic image can be obtained from the unique particular solution of the equation set generated by Eqs. (31), (34), and (35) (More details can be found in [31]). One color intrinsic image result is shown in the right of Fig. 15. When deriving the intrinsic images, K_H are the key parameters. More accurate estimation of K_H can produce better intrinsic image results. As shown in Fig. 16, using K_H computed by the SPDs calculation method proposed in this paper, better intrinsic image results can be obtained. In [31], K_H is just approximately computed from the mean value of some real measured SPDs.

Fig. 16 More accurate estimation of K_H can produce better intrinsic image results. Left: Original images; middle: Intrinsic image results using K_H calculated by Eq. (29) and the SPDs calculation method proposed in this paper. right: Intrinsic image results using K_H introduced in [31] that is calculated from the mean value of some real measured SPDs.

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4.4. Lighting conversion

This application converts an image from one illumination to another. In detail, using our SPD calculation method, we firstly recover reflectance from the original image using the method in [32], and then we can relight the same scene according to Eq. (1) and convert it to sRGB color space. As shown in Fig. 17, we first convert illumination from skylight to daylight at zenith angle equals to 60 degree. Both the color and the intensity of the converted image are quite similar to the real captured one by our naked eyes. The errors of the conversion are shown in Fig. 18. The mean error of pixel value is 9. The max error of pixel value is 36, which occurs at patch No.15 in Red channel. The error mainly arises from that commercial cameras apply some post processing such as tone mapping and gamut mapping on the images. These factors are not considered when we convert RAW images to sRGB JPG images in our calculation.

Fig. 17 Converting illumination from skylight to daylight. Left: Image illuminated by skylight; Middle: The image with illumination converted to daylight. Right: The real image captured in daylight.

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Fig. 18 Errors between the converted image and the real captured one.

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Figure 19 shows two more results of image lighting conversion. The two images in the first row of Fig. 19 show that image lighting conversion from afternoon to noon. The color of the converted image is more similar to the scene at noon than the original one. The second row of Fig. 19 shows image lighting converted from sunset to afternoon. Both the color and the clarity of the converted image are better than that of the original one.

Fig. 19 Two more results of image lighting conversions. Left: Original images; Right: converted images.

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5. Conclusion

Illumination is one of the important components of imaging. Understanding the properties of spectral distributions of outdoor light and their dynamical changes at different times and under different atmospheric conditions is of interest to computer vision. In this paper, we proposed a simple method for computing SPDs of sunlight and skylight for a given zenith angle and turbidity. In the computer vision community, researchers usually apply the blackbody radiation or the eigenvectors of daylight [16] to approximate real SPDs. Compared with these two kinds of methods, our method has two advantages. First of all, SPDs calculated by our method can approximate those calculated by meteorology ones. The second is that our method can simultaneously calculate SPDs of daylight, sunlight, and skylight. The advantages especially the second one are important for our applications. These applications are difficult to accomplish by the blackbody or the eigenvector model since they cannot simultaneously calculate the corresponding SPDs of daylight, sunlight, and skylight. Therefore, it is difficult to employ these models to simultaneously recover the SPDs of daylight and skylight in Sec. 4.1, estimate K_H in Sec. 4.2 and 4.3, and relight the ColorChecker image from skylight to daylight under the same sun angle in Sec. 4.4.

Our method is much simpler than the existing atmospheric methods and shows its advantages in a number of computer vision applications. Different from other models that need many parameters such as ozone, nitrogen dioxide, mixed gases, and water vapor that have no close relation to computer vision and are not easily obtained in computer vision, our model only requires two parameters that are most related to physically-based computer vision: sun angle (geometry) and turbidity (weather). Our model establishes a bridge between image and physical environmental information, e.g., time, location, and weather conditions. Since we focus on computer vision applications rather than atmosphere physics or meteorology, the basic calculations in atmosphere physics, e.g., transmission functions of scattering, are still based on previous atmospheric sciences literature.

Appendix

Table 4. Our proposed total absorption coefficient.

View Table | View all tables in this article

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61473280 and 61102116.

References and links

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	SMARTS2	Bird	Our
Absorption	T_uλ = exp[−1.41a_uλ m / (1 + 118.93a_uλm)^0.45] m_u = [cosθ+4.57θ^0.07(96−θ)^−1.7]⁻¹	T_uλ = exp[− (ma_uλ u_g)^0.56] m = [cos(θ) + 0.15(93.885 − θ)⁻¹²⁵³]⁻¹	T_λ = exp(−τ′(λ)·m) $τ^{'} (λ) = 0.35 a_{o λ} + 0.003 a_{u λ} + 0.1 {a^{'}}_{w λ}^{0.78}$ ${a^{'}}_{w λ} = a_{w} + 3.8 a_{u λ}$ m = sec(θ)
	T_oλ = exp(−u_oa_oλm_o) m_o =[cosθ +268θ^0.5(115 − θ)^−3.3]⁻¹	T_oλ = exp(−u_oa_oλm_o) m_o = (1 + h_o / 6370) / [cos² (θ) + 2h_o / 6370]^0.5
	T_Nλ = exp(—u_na_nλm_o) m_o = [cosθ+602θ^0.5 (117 − θ)^−3.5]⁻¹	Not mentioned
	T_wλ = exp{−[(m_w)^1.05 f_wⁿ B_wa_wλ]^c} f_w = [0.39 − 0.27λ + (0.46 + 0.24λ)P] n = 0.88 + 0.025λ − 3.59 exp(−4.54λ) c = 0.54+ 0.0033λ + 1.52 exp(−4.29λ) B_w = h(m_ww) exp(0.19 − 0.079m_w + 4.71E − 4m_w²) h (m_ww) = 0.62m_ww^0.4⁶ if a_wλ < 0.01 h(m_ww) = (0.53+ 0.25m_ww)^0.45 otherwise m_w =[cosθ+3.11θ^0.1(92 − θ)^−1.38]⁻¹	T_wλ = exp[−0.2385a_wλ Wm/ (1 + 20.07a_wλ Wm)^0.45] m = [cos(θ) + 0.15(93.885 − θ)⁻¹²⁵³]⁻¹ W = 1.6
Diffuse Irradiance	I_sλ =I_rλ + I_aλ I_rλ = cos(θ)·E_oλ · D · T_aaλ · T_oλ · T_nλ · T_wλ (1 − T_rλ^0.9)0.5F_R₂ I_aλ = cos(θ) · E_oλ · D · T_aaλ · T_oλ · T_nλ · T_uλ · T_wλ · T_rλ (1 –T_asλ)F_a I_gλ =(I_dλ cosθ + I_rλ + I_aλ) r_sλ r_gλ/(1 − r_sλ r_gλ) F_R₂ = 1, if τ_Rλ < τ_Rm, and $F_{R 2} = \exp [- {(\frac{τ_{R λ} - τ_{R m}}{σ_{R}})}^{0.72 + \cos θ}]$ σ_R =3.65−2.3 exp(−4 cosθ) τ_Rm=0.17[1−exp(−8cosθ)] $T_{a a}_{λ} = \exp [- m_{a} (1 - ϖ_{o}) τ_{a λ}]$ $ϖ_{0} = 0.94 - 0.09 \exp (1 - 3.38 λ)$ T_asλ = T_aλ / T_aaλ F_a₁ = 1 − 0.5 exp[(a_s₀ + a_s₁ cosθ) cosθ] a_s₀ = [1.5 + (0.16 + 0.41F_g)F_g]F_g a_s₁ = [0.08+ (0.38+0.58F_g)F_g]F_g F_g = ln(1−g) F_a₂ = 1 for T_asλ ≤ 2, or else $F_{a}_{2} = \exp [- {(\frac{τ_{a s λ} - 2}{σ_{a}})}^{ξ}]$ ζ = −0.5 + exp[0.24(cos θ₀)^−1.24] cosθ₀ = Max(0.05, cosθ) σ_a=Max{1,3.5−(4.53−0.82τ_asλ)cosθ + (8.26−6.02τ_asλ)cos²θ}	I_sλ = I_rλ + I_aλ + I_gλ I_rλ = cos(θ) · E_oλ · D · T_aaλ · T_oλ · T_wλ (1 –T_rλ^0.95)0.5 I_aλ = cos(θ) · E_oλ · D · T_aaλ · T_oλ · T_wλ · T_rλ^1.5 (1 –T_asλ)F_s I_gλ =(I_dλ cosθ + I_rλ + I_aλ) r_sλ r_gλ/(1 − r_sλ r_gλ) T_asλ =exp(−ω_−rβλ^−αm) T_asλ = exp(− (1− ω_r) βλ^−α m) ω_r =0.945exp{−0.095[ln(λ/0.4)]²} F_s = 1−0.5exp[(AFS+cos(θ)BFS)cos((θ)] r_sλ = T′_oλ · T′_wλ · T′_aaλ[0.5 (1 − T′ _rλ) + (1 − F_s) T′ _rλ (1 − T′_asλ)] AFS=ALG[1.459+ALG(0.1595+ALG0.4129)] BFS=ALG[0.0783+ALG(−0.3824−ALG0.5874)] ALG = ln(1 − 0.65) F′_s = 1 − 0.5 exp[(AFS+BFS/1.8)/1.8] $C_{s} = {\begin{matrix} {(λ + 0.55)}^{1.8} & for λ \leq 0.45 μ m \\ 1.0 & for λ > 0.45 μ m \end{matrix}$	I_sλ = E_in·cos(θ)·(1−T_rλ·T_aλ)·κ E = E − T
Parameters	Earth-sun distance, Pressure, ozone, nitrogen dioxide, mixed gases, water vapor, carbon dioxide, albedo, tilt albedo, zenith angle, turbidity.	Earth-sun distance, Pressure, ozone, water vapor, albedo, aerosol optical depth, zenith angle, turbidity.	zenith angle, turbidity.

Turbidity	0.05			0.1			0.15			0.2			0.25			0.3
K_R/K_G/K_B
Sun Angle
20	19.91	13.86	8.20	12.55	9.38	6.14	9.25	7.17	5.00	7.39	5.87	4.26	6.21	5.02	3.76	5.38	4.12	3.40
30	18.40	12.82	7.61	11.61	8.70	5.72	8.58	6.67	4.67	6.87	5.48	4.00	5.78	4.69	3.54	5.02	4.14	3.21
40	17.04	11.86	7.04	10.74	8.04	5.30	7.94	6.17	4.33	6.36	5.07	3.72	5.36	4.36	3.31	4.67	3.86	3.02
50	14.38	10.04	6.02	9.11	6.85	4.58	6.77	5.29	3.78	5.45	4.38	3.28	4.62	3.80	2.95	4.05	3.39	2.72
60	11.62	8.13	4.94	7.40	5.60	3.81	5.53	4.37	3.20	4.50	3.67	2.83	3.85	3.21	2.58	3.40	2.90	2.42
70	8.34	5.88	3.70	5.39	4.24	2.96	4.10	3.32	2.57	3.40	2.86	2.36	2.80	2.58	2.23	2.70	2.40	2.14
80	4.86	3.52	2.49	3.29	2.68	2.20	2.67	2.33	2.08	2.37	2.17	2.03	2.21	2.09	2.01	2.12	2.04	2.01

λ	τ′	λ	τ′	λ	τ′	λ	τ′	λ	τ′	λ	τ′
400	0.0270	450	0.0200	500	0.0190	550	0.0362	600	0.0572	650	0.0370
401	0.0258	451	0.0201	501	0.0212	551	0.0357	601	0.0537	651	0.0334
402	0.0244	452	0.0205	502	0.0255	552	0.0358	602	0.0524	652	0.0506
403	0.0249	453	0.0194	503	0.0285	553	0.0361	603	0.0507	653	0.0333
404	0.0259	454	0.0194	504	0.0398	554	0.0356	604	0.0498	654	0.0320
405	0.0251	455	0.0190	505	0.0296	555	0.0355	605	0.0486	655	0.0387
406	0.0236	456	0.0185	506	0.0307	556	0.0359	606	0.0482	656	0.0348
407	0.0248	457	0.0207	507	0.0311	557	0.0366	607	0.0475	657	0.0342
408	0.0267	458	0.0206	508	0.0301	558	0.0373	608	0.0469	658	0.0296
409	0.0264	459	0.0206	509	0.0287	559	0.0382	609	0.0462	659	0.0268
410	0.0260	460	0.0212	510	0.0271	560	0.0390	610	0.0452	660	0.0255
411	0.0246	461	0.0202	511	0.0248	561	0.0402	611	0.0448	661	0.0222
412	0.0260	462	0.0216	512	0.0234	562	0.0412	612	0.0440	662	0.0202
413	0.0265	463	0.0219	513	0.0225	563	0.0432	613	0.0432	663	0.0204
414	0.0247	464	0.0195	514	0.0213	564	0.0441	614	0.0424	664	0.0195
415	0.0230	465	0.0173	515	0.0211	565	0.0443	615	0.0417	665	0.0189
416	0.0230	466	0.0185	516	0.0211	566	0.0441	616	0.0411	666	0.0182
417	0.0240	467	0.0195	517	0.0216	567	0.0455	617	0.0403	667	0.0177
418	0.0245	468	0.0181	518	0.0226	568	0.0494	618	0.0397	668	0.0167
419	0.0253	469	0.0180	519	0.0231	569	0.0571	619	0.0390	669	0.0164
420	0.0255	470	0.0191	520	0.0231	570	0.0550	620	0.0387	670	0.0159
421	0.0254	471	0.0186	521	0.0231	571	0.0502	621	0.0383	671	0.0155
422	0.0253	472	0.0192	522	0.0236	572	0.0543	622	0.0375	672	0.0152
423	0.0258	473	0.0216	523	0.0244	573	0.0530	623	0.0372	673	0.0148
424	0.0251	474	0.0215	524	0.0271	574	0.0519	624	0.0368	674	0.0145
425	0.0226	475	0.0194	525	0.0289	575	0.0519	625	0.0363	675	0.0142
426	0.0223	476	0.0183	526	0.0285	576	0.0503	626	0.0370	676	0.0139
427	0.0242	477	0.0182	527	0.0297	577	0.0492	627	0.0396	677	0.0135
428	0.0230	478	0.0191	528	0.0294	578	0.0475	628	0.0566	678	0.0133
429	0.0218	479	0.0207	529	0.0302	579	0.0464	629	0.0434	679	0.0130
430	0.0238	480	0.0207	530	0.0308	580	0.0457	630	0.0436	680	0.0129
431	0.0229	481	0.0201	531	0.0313	581	0.0448	631	0.0373	681	0.0129
432	0.0219	482	0.0187	532	0.0312	582	0.0446	632	0.0395	682	0.0130
433	0.0236	483	0.0179	533	0.0309	583	0.0443	633	0.0364	683	0.0129
434	0.0253	484	0.0182	534	0.0303	584	0.0448	634	0.0344	684	0.0132
435	0.0247	485	0.0194	535	0.0299	585	0.0451	635	0.0345	685	0.0127
436	0.0220	486	0.0188	536	0.0297	586	0.0466	636	0.0333	686	0.0127
437	0.0200	487	0.0192	537	0.0299	587	0.0499	637	0.0328	687	0.1496
438	0.0231	488	0.0215	538	0.0311	588	0.0608	638	0.0308	688	0.1041
439	0.0240	489	0.0213	539	0.0315	589	0.0792	639	0.0305	689	0.0931
440	0.0206	490	0.0188	540	0.0330	590	0.0787	640	0.0298	690	0.0768
441	0.0194	491	0.0181	541	0.0372	591	0.0700	641	0.0295	691	0.0566
442	0.0213	492	0.0180	542	0.0387	592	0.0774	642	0.0287	692	0.0382
443	0.0260	493	0.0191	543	0.0361	593	0.0690	643	0.0292	693	0.0420
444	0.0260	494	0.0200	544	0.0387	594	0.0713	644	0.0288	694	0.0509
445	0.0250	495	0.0200	545	0.0380	595	0.0770	645	0.0300	695	0.0420
446	0.0236	496	0.0195	546	0.0402	596	0.0642	646	0.0365	696	0.0447
447	0.0245	497	0.0180	547	0.0384	597	0.0667	647	0.0447	697	0.0213
448	0.0231	498	0.0173	548	0.0373	598	0.0636	648	0.0507	698	0.0213
449	0.0201	499	0.0174	549	0.0365	599	0.0615	649	0.0476	699	0.0363
										700	0.0372

	SMARTS2	Bird	Our
Absorption	T_uλ = exp[−1.41a_uλ m / (1 + 118.93a_uλm)^0.45] m_u = [cosθ+4.57θ^0.07(96−θ)^−1.7]⁻¹	T_uλ = exp[− (ma_uλ u_g)^0.56] m = [cos(θ) + 0.15(93.885 − θ)⁻¹²⁵³]⁻¹	T_λ = exp(−τ′(λ)·m) $τ^{'} (λ) = 0.35 a_{o λ} + 0.003 a_{u λ} + 0.1 {a^{'}}_{w λ}^{0.78}$ ${a^{'}}_{w λ} = a_{w} + 3.8 a_{u λ}$ m = sec(θ)
	T_oλ = exp(−u_oa_oλm_o) m_o =[cosθ +268θ^0.5(115 − θ)^−3.3]⁻¹	T_oλ = exp(−u_oa_oλm_o) m_o = (1 + h_o / 6370) / [cos² (θ) + 2h_o / 6370]^0.5
	T_Nλ = exp(—u_na_nλm_o) m_o = [cosθ+602θ^0.5 (117 − θ)^−3.5]⁻¹	Not mentioned
	T_wλ = exp{−[(m_w)^1.05 f_wⁿ B_wa_wλ]^c} f_w = [0.39 − 0.27λ + (0.46 + 0.24λ)P] n = 0.88 + 0.025λ − 3.59 exp(−4.54λ) c = 0.54+ 0.0033λ + 1.52 exp(−4.29λ) B_w = h(m_ww) exp(0.19 − 0.079m_w + 4.71E − 4m_w²) h (m_ww) = 0.62m_ww^0.4⁶ if a_wλ < 0.01 h(m_ww) = (0.53+ 0.25m_ww)^0.45 otherwise m_w =[cosθ+3.11θ^0.1(92 − θ)^−1.38]⁻¹	T_wλ = exp[−0.2385a_wλ Wm/ (1 + 20.07a_wλ Wm)^0.45] m = [cos(θ) + 0.15(93.885 − θ)⁻¹²⁵³]⁻¹ W = 1.6
Diffuse Irradiance	I_sλ =I_rλ + I_aλ I_rλ = cos(θ)·E_oλ · D · T_aaλ · T_oλ · T_nλ · T_wλ (1 − T_rλ^0.9)0.5F_R₂ I_aλ = cos(θ) · E_oλ · D · T_aaλ · T_oλ · T_nλ · T_uλ · T_wλ · T_rλ (1 –T_asλ)F_a I_gλ =(I_dλ cosθ + I_rλ + I_aλ) r_sλ r_gλ/(1 − r_sλ r_gλ) F_R₂ = 1, if τ_Rλ < τ_Rm, and $F_{R 2} = \exp [- {(\frac{τ_{R λ} - τ_{R m}}{σ_{R}})}^{0.72 + \cos θ}]$ σ_R =3.65−2.3 exp(−4 cosθ) τ_Rm=0.17[1−exp(−8cosθ)] $T_{a a}_{λ} = \exp [- m_{a} (1 - ϖ_{o}) τ_{a λ}]$ $ϖ_{0} = 0.94 - 0.09 \exp (1 - 3.38 λ)$ T_asλ = T_aλ / T_aaλ F_a₁ = 1 − 0.5 exp[(a_s₀ + a_s₁ cosθ) cosθ] a_s₀ = [1.5 + (0.16 + 0.41F_g)F_g]F_g a_s₁ = [0.08+ (0.38+0.58F_g)F_g]F_g F_g = ln(1−g) F_a₂ = 1 for T_asλ ≤ 2, or else $F_{a}_{2} = \exp [- {(\frac{τ_{a s λ} - 2}{σ_{a}})}^{ξ}]$ ζ = −0.5 + exp[0.24(cos θ₀)^−1.24] cosθ₀ = Max(0.05, cosθ) σ_a=Max{1,3.5−(4.53−0.82τ_asλ)cosθ + (8.26−6.02τ_asλ)cos²θ}	I_sλ = I_rλ + I_aλ + I_gλ I_rλ = cos(θ) · E_oλ · D · T_aaλ · T_oλ · T_wλ (1 –T_rλ^0.95)0.5 I_aλ = cos(θ) · E_oλ · D · T_aaλ · T_oλ · T_wλ · T_rλ^1.5 (1 –T_asλ)F_s I_gλ =(I_dλ cosθ + I_rλ + I_aλ) r_sλ r_gλ/(1 − r_sλ r_gλ) T_asλ =exp(−ω_−rβλ^−αm) T_asλ = exp(− (1− ω_r) βλ^−α m) ω_r =0.945exp{−0.095[ln(λ/0.4)]²} F_s = 1−0.5exp[(AFS+cos(θ)BFS)cos((θ)] r_sλ = T′_oλ · T′_wλ · T′_aaλ[0.5 (1 − T′ _rλ) + (1 − F_s) T′ _rλ (1 − T′_asλ)] AFS=ALG[1.459+ALG(0.1595+ALG0.4129)] BFS=ALG[0.0783+ALG(−0.3824−ALG0.5874)] ALG = ln(1 − 0.65) F′_s = 1 − 0.5 exp[(AFS+BFS/1.8)/1.8] $C_{s} = {\begin{matrix} {(λ + 0.55)}^{1.8} & for λ \leq 0.45 μ m \\ 1.0 & for λ > 0.45 μ m \end{matrix}$	I_sλ = E_in·cos(θ)·(1−T_rλ·T_aλ)·κ E = E − T
Parameters	Earth-sun distance, Pressure, ozone, nitrogen dioxide, mixed gases, water vapor, carbon dioxide, albedo, tilt albedo, zenith angle, turbidity.	Earth-sun distance, Pressure, ozone, water vapor, albedo, aerosol optical depth, zenith angle, turbidity.	zenith angle, turbidity.

Turbidity	0.05			0.1			0.15			0.2			0.25			0.3
K_R/K_G/K_B
Sun Angle
20	19.91	13.86	8.20	12.55	9.38	6.14	9.25	7.17	5.00	7.39	5.87	4.26	6.21	5.02	3.76	5.38	4.12	3.40
30	18.40	12.82	7.61	11.61	8.70	5.72	8.58	6.67	4.67	6.87	5.48	4.00	5.78	4.69	3.54	5.02	4.14	3.21
40	17.04	11.86	7.04	10.74	8.04	5.30	7.94	6.17	4.33	6.36	5.07	3.72	5.36	4.36	3.31	4.67	3.86	3.02
50	14.38	10.04	6.02	9.11	6.85	4.58	6.77	5.29	3.78	5.45	4.38	3.28	4.62	3.80	2.95	4.05	3.39	2.72
60	11.62	8.13	4.94	7.40	5.60	3.81	5.53	4.37	3.20	4.50	3.67	2.83	3.85	3.21	2.58	3.40	2.90	2.42
70	8.34	5.88	3.70	5.39	4.24	2.96	4.10	3.32	2.57	3.40	2.86	2.36	2.80	2.58	2.23	2.70	2.40	2.14
80	4.86	3.52	2.49	3.29	2.68	2.20	2.67	2.33	2.08	2.37	2.17	2.03	2.21	2.09	2.01	2.12	2.04	2.01

Simple and effective calculations about spectral power distributions of outdoor light sources for computer vision

Abstract

1. Introduction

2. A simple method for computing the spectral power distribution of sunlight and skylight

2.1. Light and image

2.2. Absorption of extraterrestrial irradiance passing through the atmosphere

2.3. Computing direct sunlight

2.4. Computing diffuse skylight

3. Experiments and comparisons

3.1. Comparison with blackbody radiation

3.2. Comparison with Preetham sky model

4. Applications

4.1. The recovery of camera capture information

4.2. Shadow features for shadow detection

4.3. Deriving intrinsic images

4.4. Lighting conversion

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (19)

Tables (4)

Equations (36)

Optics Express