 RESEARCH PAPER
 Open Access
4D light field reconstruction by irradiance decomposition
 Takahito Aoto^{1}Email authorView ORCID ID profile,
 Tomokazu Sato^{1},
 Yasuhiro Mukaigawa^{1} and
 Naokazu Yokoya^{1}
https://doi.org/10.1186/s410740160014z
© The Author(s) 2017
 Received: 12 July 2016
 Accepted: 29 December 2016
 Published: 8 April 2017
Abstract
Common light sources such as an ordinary flashlight with lenses and/or reflectors make complex 4D light field that cannot be represented by conventional isotropic distribution model nor point light source model. This paper describes a new approach to estimate 4D light field using an illuminated diffuser. Unlike conventional works that capture a 4D light field directly, our method decomposes observed intensities on the diffuser into intensities of 4D light rays based on inverse rendering technique with prior knowledge. We formulate 4D light field reconstruction problem as a nonsmooth convex optimization problem for mathematically finding the global minimum.
Keywords
 Light field reconstruction
 Illumination
 Inverse lighting
 Convex optimization
1 Introduction
As a simplest model of a radiant intensity distribution for a light source, an isotropic point light source has conventionally been used (Fig. 1 a). This type of model has only one intensity parameter. Due to its simplicity, this model established a standard in the field of photometric computer vision [1, 2]. This simplest model is extended to two directions for representing the directivity and the spatial distribution of light sources. For the directivity, an angular radiance distribution is considered (Fig. 1 b) by assigning different intensity parameters for different directions. This model can handle an anisotropic point light source, and is essential for modeling a light with hard directivity like an LED [3]. The other extension is for the spatial distribution for representing the volume of lighting environment (Fig. 1 c). By simply arranging multiple isotropic pointlightsources in a space, the model can handle the spatial distribution of lights. Although these extensions increase the accuracy of lighting environment modeling, they cannot be used to model an actual complex light field, e.g., generated by LEDs or bulbs with reflectors and/or lenses. Differences between illuminating effects of actual lights and modeled ones by (a), (b), (c), become bigger when lights are placed near from the objects, and it prevents photo realistic rendering and high accurate inverse rendering in this situation. 4D light field (Fig. 1 d), which presents light field by 2D directivity × 2D spatial distribution of light sources, is essential for modeling actual lighting environments.
In this study, we focus on reconstructing 4D light field from images on an illuminated object. In order to estimate 4D light field, most of conventional works directly capture a huge number of images for all the directions from all 3D positions and resultantly, they suffer from the cost problem for measuring all rays. Instead of direct capturing, our method estimates unknown parameters of the 4D light field so that the images rendered with reconstructed lighting would become as similar as possible with captured original images based on an inverse lighting techniques [4]. To achieve the goal of both accurate and robust estimation, we have developed a new inverse lighting method based on a convex optimization technique. Our method introduces the range of possible radiant intensities from a physical constraint and it actualizes the reconstruction of the 4D light field from a few images.
Remainder of this paper is organized as follows. Section 2 discusses related work and highlights our contributions. Section 3 expresses a basic idea for 4D light field reconstruction. Section 4 describes an efficient solution for nonsmooth convex optimization problem. Section 5 shows experimental results in real scenes. Finally, Section 6 concludes the present study.
2 Related work and contributions
2.1 Direct method for 4D light field acquisition
Direct methods directly capture the 4D light field by backtracing the rays from the camera. Measured 4D light field is represented by a form of 4D light ray space based on a set of 2D images of a scene captured from different view points.
As straightforward methods for measuring the 4D light field, some methods that capture 2D images for all directions from all the 3D positions using camera mounted on a robot arm have been used [5–8]. These methods are generally expensive in both measuring cost and time. In order to reduce them, Unger et al. [9, 10] used an array of mirrored spheres and a moving mirrored sphere that travels across the plane. Goesele et al. [11] and Nakamura et al. [12] used various kinds of optical filters that spatially limit the incident light rays to the camera. Cossairt et al. [13] used a lens array and created augmented scenes relighting synthetic objects using captured light field. Although these methods are related to our problem, the direct methods still require a comparatively large amount of images.
2.2 Indirect method for light field reconstruction
Indirect methods, also known as inverse lighting, reconstruct the lighting parameters in a scene by minimizing the difference of observed and computed intensities that can be simulated using CG rendering techniques with known scene geometry, surface property, and lighting environment.
Conventionally, many researches utilized a specular reflection or diffuse reflection components to estimate a lighting environment [14, 15]. These approaches solve a linear system. Ramamoorthi and Hanrahan [16] have shown the reason that inverse lighting problem for global illumination is illposed or numerically illconditioned, based on the theoretical analysis in the frequency domain using spherical harmonics. On the other hand, Park et al. [3] estimate the 2D light field emitted from a point light source, which rigidly attached to a camera, using a illuminated plane.
Shadows are areas where direct lights from a light source cannot reach due to the occlusion by other object and thus can provide useful information for estimating the lighting environment. By using the cast shadow information, Sato et al. [17] proposed a method to recover positions of a set of point light sources. Okabe et al. [18] used a Haar wavelet basis to approximate the lighting effect by a small number of basis functions. Takai et al. [4] proposed a skeleton cube as a reference object that creates a self cast shadows from a point light source of arbitrary position.
2.3 Our contributions
As described above, the direct method suffers from the cost problem for data capturing and there is no method that can estimate 4D light field with the indirect method, as far as we know. In this paper, we propose a method for estimating 4D light field in an indirect manner that can estimate light field from a few images. In order to achieve this, the problem of estimating 4D light field is formed as energy minimization problem with several constraints and the problem is solved by convex optimization. It should be noted that this paper is an extension of our conference paper [19]. In this paper, for stable and accurate estimation, we have newly introduced physical constraint with ℓ _{1}norm regularization for energy function and we have also conducted completely new experiments with new solution.
3 Basic idea for 4D light field reconstruction
In this section, we first formulate the problem of basic 4D light field reconstruction as a linear system.

The relative positions and postures of a camera, a diffuser and light sources are known.

The radiance distribution of light sources is static.

The sensor response is linear.

The light is not attenuated by scattering or absorption

The diffuser’s property (transmission model) is given.
In the followings, we first review the relationship between 4D light field and observed intensities, and then discuss how to model the inverse problem of estimating 4D light field from observed intensities.
3.1 Relationship between light field and observed intensities
where, \({\boldsymbol {A} :\mathbb {R}^{M} \mapsto \mathbb {R}^{N}} \) is a matrix of known parameters that represents a rendering process. Equation (4) can be solved by linear least squares in principle if we have sufficient observations.
3.2 Size reduction using spherical harmonics function
In practice, the size of A in Eq. (4) is too large to be solved due to the large number of unknown parameters.
where y _{ f,g } is the basis of realspherical harmonics, and c _{ i,f,g } are H=(F+1)^{2} unknown parameters. Representable distribution of radiant intensities depends on H.
where b _{ i,j } is an element of matrix \({\boldsymbol {B}} \in \mathbb {R}^{\hat {H}} \mapsto \mathbb {R}^{N}\), R(ω) is a transmission distribution function that is determined by diffuser’s property, ω is determined by the angle between a normal of the diffuser and the ray, and D(·) represents the distance between (u,v) and x _{ i } on the diffuser \({\mathcal {B}}\).
4 Efficient solution under insufficient observations
The solution of the linear equation given in Eq. (7) is sensitive to observation noises and errors in pose estimation of the diffuser and often outputs negative intensities due to the lack of valid observations, as shown in the latter experiment. Possible approaches to overcome this problem are gaining more observations or introducing constraints on the parameters.
In this work, in order to achieve stable estimation from a limited number of inputs, we introduce a physical constraint and ℓ _{1}norm regularization into a light field reconstruction algorithm formulated as a convex optimization problem. In the following, we first give the formulation of the problem, and then describe the details of each constraint.
4.1 Formulation of light field reconstruction problem
where the first term represents the squared error between observed intensities and rendered intensities, and the second term represents the ℓ _{1}norm of spherical harmonics coefficients and the λ is a weight parameter for ℓ _{1}norm. The third term represents the physical constraint that limits the numerical range of light ray intensities.
Since each term is convex in Eq. (8), whole function is also convex. Hence, this function has a unique solution.
The convex optimization problem of Eq. (8) can be solved by alternating direction method of multipliers [20], which effectively minimizes the function with iterative manner. By solving this problem, we can get the 4D light field Y c.
4.2 ℓ _{1}norm regularization
In general, regularization is introduced to prevent over fitting when the number of observations is not sufficiently larger than that of unknown parameters. In this research, we employ ℓ _{1}norm of the unknown parameters c as a regularization term for preventing this problem. This term makes most elements in c become zero and it selects important bases on the matrix B. The weight parameter λ, which is empirically determined in the experiment, adjusts the number of selected bases.
4.3 Physical constraint based on nonnegative constraint for light ray intensity
where \(\mathcal {X}_{j}\) is a set of light rays j which hits to local region x _{ i } on the diffuser \({\mathcal {B}}\).
5 Experiments
In this section, we verify the effectiveness of the proposed method using a real data set. We first compare our method with several kinds of approaches under different conditions. In these experiments, since the groundtruth lightfield map is not available, we quantitatively verify the correctness of our algorithm by computing the photometric errors that is the difference between the captured image and the corresponding relit image using reconstructed light field in real scene. In the following, we call the least square as LS, ℓ _{1}norm regularization as L1 and physical constraint as PC.
5.1 Setup
It should be noted that, some reconstructed light fields have negative radiant intensities. In this experiment, we permit a negative intensity for generating relit images.
5.2 Quantitative evaluation
 (I)
4D light field reconstruction with L1 + PC,
 (II)
4D light field reconstruction with L1,
 (III)
4D light field reconstruction with PC,
 (IV)
4D light field reconstruction by LS without L1 and PC,
 (V)
2D light field reconstruction assuming an anisotropic point light source, and
 (VI)
2D light field reconstruction assuming a set of isotropic point light sources.
Here, (I) ∼(III) are the variations of the proposed method and (IV) is the baseline method^{3}. For the method (V), middle of given three light positions is used as a position of a point light source since (V) does not have the spatial dimension. For the method (VI), boxstyle region of the diffuser shown in Fig. 5 c that contains 91×64 points is used as the spatial distribution of lights on \(\mathcal {L}\).
5.2.1 Comparison of 4D and 2D light field reconstruction methods
5.2.2 Effect of constraints
As shown in Figs. 6 and 8, there are very small quantitative and subjective differences in the results for middle to far range d=[60,165] among compared 4D based methods. However, for near range d=[30,45], considerable differences are exposed. First, we can see unnatural ripples around the lights for the methods (I) to (IV). The ripples for (IV) are harder than others.
As shown in Fig. 7 (IV), LS gives large negative intensities in estimated light field and it has caused over parameter fitting problem. By comparing the pairs of {(I), (II)} and {(III), (IV)} in this figure, we can confirm that negative intensity on the estimated light field is successfully suppressed by using PC. Although we cannot see any subjective differences among relit images for (I), (II), and (III), quantitatively, the errors become smaller when we employed L1 and the method (I) which uses both PC and L1 gives the best scores for near range.
5.2.3 Effect of dimension
As shown in Figs. 9 and 10, We can confirm that the ripples artifacts appeared especially for the near rage of [3045], become weaker when we can give more resolution for angle direction, i.e., F become higher. In the case, the model does not have enough resolution for angle direction, the model cannot represent both the details of shapes of lights and background region behind the lights. In this situation, from the characteristic of the spherical harmonic function, repetitive patterns easily appear in the image. On the other hand, the reason, why the ripples for (IV) become harder than others, is considered as an overfitting problem of standard linear programming. Our method successfully reduced this error by convex optimization. Ripples around lights, mentioned in the previous section, are almost disappeared when F=89. From these results, we can say that we need high dimensional parameters for sphericalharmonics function for accurate reconstruction of 4D light field.
5.2.4 Effect of spacial resolution and arrangement of virtual light sources
However, in the range d = [30, 45], different images are generated except for the images (a) and (e). For (a), we arranged virtual light sources at the center positions of highlights and for (e) number of point light sources are increased as shown in Fig. 12. From this comparison, we can say that as far as we can put virtual light sources in front of the true light positions (i.e., centers of highlights), good results will be given with minimum number of virtual light sources. By comparing (e) with F = 69 in Fig. 10, which have the same number of parameters with (e), we can see that latter result is better than (e)’s. It means the angle resolution is more important than the special resolution, as long as we can put virtual lights to appropriate positions. When we cannot arrange them for centers of highlights (case (d)), the relit image becomes different shapes from ideal ones. On the other hand, when we gave more virtual light sources to different positions from highlights, as shown in (b) and (c), undesired highlights appear on near range images. This is considered as the effect of an overfitting problem for these unnecessary positions. It should be noted that except for near range images, good results are obtained even for (b) and (c), and intensities of undesired highlights on near range images are also darker than those of true highlights. It is because the L1 norm suppress the value of coefficients by selecting an important basis of a spherical harmonic function.
5.2.5 Computational cost
In order to reconstruct 4D light field with the method (I), it takes 31 h for F=34 in this experiment using a PC (i n t e l ^{ ®;} c o r e ^{ T M } i73970 3.50GHz × 12, Memory 32 GB, C++ implementation). The core time for computation spent for solving a convex optimization problem in Eq. (8). In this experiment, 5.3 GB memory was required for our implementation. When we set F=89, it takes more than 1 week for 4D light field reconstruction. In order to reduce the cost, we should find more efficient bases for representing light field and efficient way for solving the problem.
5.3 Results for more complex lights
The light source in Fig. 14 a has three LEDs triangularly arranged. We can see that projected shape of each light looks like torus, and part of three shapes are overlapped as shown in Fig. 15. In the relit images, both the methods (I) and (IV) could reproduce good results for the position where input images were captured (red boxed images). However, the LS method (IV) gave completely different shapes for near range d=[30,45] due to the over fitting. In contrast, the proposed method (I) could reproduce much better relit images even for near range.
The light source in Fig. 14 b is a flash light with lens and reflectors. For this light source, although the method (I) recovers higher frequency component in the relit images compared with the method (IV), relit results does not reach a satisfactory level as shown in Fig. 16. It is considered that the poor results are due to the lack of parameters to model the 4D light field for this complex light, and more input images are also necessary to acquire stable results. At this moment, we need more computational resources to estimate this kind of complex 4D light field in which virtual light source positions, arose by reflectors in the light, are spatially distributed.
6 Conclusion
In this paper, we have presented a novel 4D light field reconstruction technique utilizing a physical constraint and a regularization. We have formulated the light field reconstruction problem as a convex optimization problem. This optimization problem was designed to decompose the observed intensities on the measurement plane into light ray intensities. Unlike conventional works, the proposed method can estimate the 4D light field from a few images without special optics such as a mirrorarray, a lens array, and filters. As shown in experiments, we could confirm the effectiveness of both the physical constraint and ℓ _{1}norm regularization. A remaining weakness in the current implementation of the proposed method is the difficulty for increasing dimensions of parameters due to its high computational cost which prevents to handle more complex lighting environments. In order to relax this problem, we should find more efficient bases for representing the light model to reduce the computation cost, in future work.
Although we assumed the board has Lambertiantransmission property and scattering effect is ignored, which did not give obvious effects in the results in the experiment, for more precise reconstruction, calibration method for the diffuser board should be considered. In addition, we should confirm the sensitivity of the proposed method by using images with artificially added noises.
7 Endnotes
^{1} In this equation, the intensity o _{ i } is represented by a continuous system. The effect of the attenuation and incident angle are considered by the integral of ray j.
^{2} In this paper, we regard ρ as constant by assuming the transmission property of the diffuser is Lambertian.
^{3} For LS, we iteratively minimize the function from zero vector using conjugate gradient method, since B in Eq. (7) has small singular values due to the insufficient observations in this experiment. In this case, LS cannot give a unique solution or a stable solution by linear solvers.
Declarations
Acknowledgements
This research was supported in part by JSPS KAKENHI Grant No. 23240024 and GrantinAid for Exploratory Research Grant No. 25540086.
Authors’ contributions
TA designed the study, performed the experiments, and drafted the manuscript. TS participated in the design of the study and helped to draft the manuscript. YM conceived of the study and participated in its design. NY gave technical support and conceptual advice. All authors discussed the results and implications and commented on the manuscript at all stages. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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