Multiview Reconstruction Using Phase Retrieval of Light Field Signals

Light field rendering is an effective multi-view reconstruction technology. The current multi-view reconstruction methods still have the problem of low view synthesis quality in complex scenes, such as occlusion and non Lambertian surfaces. In this article, we consider a undistorted multiview reconstruction and develop a phase retrieval method for light field (PRLF) signals to improve the rendering quality. Our approach utilizes the phase spectrum in both the time domain and frequency domain of the discrete light field because the phase of the light field carries more scene information. The phase corrections are “light field truncation” in the time domain and “phase replacement” in the frequency domain. The iterative algorithm used provides high-quality estimated results for the relative phase of the light field. By the estimated phase results, we derive a multiview reconstruction method to improve the rendering quality. Finally, to verify the claimed performance, we also compare the PRLF with the most advanced light field reconstruction algorithms. The experimental results show that the proposed PRLF outperforms other known reconstruction schemes.


I. INTRODUCTION
Light field rendering (LFR) [1], [2] is a powerful technique for visualization and human-computer interaction. LFR can ensure good stereo experience and visual immersion by improving the reconstruction quality of a novel view. The reconstruction problem of a novel view given only a set of captured multiview images, as shown in Fig. 1, appears in a wide range of computational imaging and virtual reality applications [3]- [6]. Nonetheless, for the reconstruction of LFR technology [7], the solutions described previously still have some limitations. For instance, reconstruction methods have difficulty generating sufficiently good 3D geometry for foregrounds with complex object shapes or sharp objects with discontinuous depths [8].
Modern novel view quality optimization methods of light fields can be divided into two fundamentally different The associate editor coordinating the review of this manuscript and approving it for publication was Eduardo Rosa-Molinar . approaches. The first approach improves the novel view quality by optimizing the capturing of multiview images. By optimizing the capture position and direction of cameras, the most abundant scene information can be obtained to VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ improve the rendering quality of novel views, such as active rearranged capturing (ARC) [9], correspondence field (CF) capturing [10], and noncoverage field (NCF) capturing [11]. In this case, the captured position and direction of the camera directly determine the reconstruction quality of novel views. Similar to the capturing of multiview images, the rendering quality of novel views can also be improved by depth information optimization [12], [13]. The second approach improves the quality of novel view reconstruction by optimizing the reconstruction algorithms of the light field. The bandwidth of the light field can be obtained through spectrum analysis, and a reasonable reconstruction filter can be designed to reconstruct the novel views [14]- [17]. Furthermore, learning methods are used to optimize the reconstruction quality of novel views [18]- [21]. In this case, the novel view is used to improve the rendering quality by optimizing the filter bandwidth or learning methods and networks.
The optical detection devices of cameras can only capture the intensity information of light rays but cannot measure the phase information of light rays [22], [23]. Therefore, in the process of light field sampling, the captured image sets contain the light intensity values but lack phase information. The traditional light field reconstruction algorithm only considers the reconstruction and optimization of the amplitude information of the light field. Inspired by the theory of signal phase retrieval [22], [23], along with the development of novel view reconstruction algorithms, this paper considers the fundamental problem of rendering some general novel views. First, we mathematically explore phase retrieval methods for light field (PRLF) signals from a set of captured multiview images. The phase spectral support of a light field signal is bounded by the frequencies along the camera plane and image plane. The phase of the light field is then recovered using a least squares method based on the previous phase retrieval theory of the signal. Second, for the application of PRLF signals to novel view reconstruction from a set of captured multiview images, we try to explore the reconstruction quality optimization method from the phase retrieval direction of the light field signal to improve the rendering quality. Some preliminary results of this paper were presented at a conference [24]. To summarize, our main contributions are as follows: • Phase recovery is the basis of applied physics and engineering. It is required to determine the phase of a complex-valued function from modulus data and additional a priori information. We apply the theory of phase retrieval to the light field signal and explore the phase retrieval of the light field signal.
• The problem of recovering a signal from the Fourier transform amplitude is of great significance in various fields of engineering, which has existed for more than 100 years. The relationship between novel view reconstruction and phase retrieval of light field signals is derived in detail to improve the viewpoint quality based on phase information.
• The phase of the light field signal contains more scene information. We develop a new novel view reconstruction framework based on light field phase retrieval.

II. RELATED WORK
Since the emergence of light field rendering technology 25 years ago [1], [2], a large number of light field reconstruction technologies have been proposed. We roughly divide these reconstruction technologies into traditional signal processing methods using Fourier transform theory and recent emerging deep learning methods that track and explain the development process of light field reconstruction technology.

A. RECONSTRUCTION USING SPECTRAL ANALYSIS
Traditional research on light field reconstruction using the Fourier transform mainly uses the method of spectrum analysis to estimate the bandwidth of the light field. Then, the novel view rendering quality is designed and optimized according to the light field bandwidth [25]. For example, a linear and spatially invariant reconstruction filter was used in [17], and an appropriate prefilter was used in [16]. Vagharshakyan et al. [26] proposed using an adapted discrete shearlet transform to sparse the light field and derived a method of light field reconstruction in the directionally sensitive transform domain. Similar methods and results are also introduced in reference [27]. In [28], Shi et al. applied the continuous frequency domain of a light field to sparse light field signals and reconstruct novel views that typically work in the discrete Fourier domain. In [6], a flexible connection scheme of a multilight field system was proposed, and it makes use of the constant radiation of light. Additionally, light field reconstruction by a Fourier disparity layer representation [29] and a concentric multispectral light field [30] can also process the light field through different angles and perform Fourier transforms. Then, the method of light field reconstruction quality optimization is derived by spectrum analysis of the light field for non-Lambertian surface scenes. In [5], Koniaris et al. proposed animated light fields with real-time view correlation reconstruction by converting offline rendered movie content into a novel immersive representation. In [31], Marquez et al. proposed a compressed spectrum analysis method and a light field imaging architecture based on the principle of a compressed imaging framework to capture the representation of multiplexed multidimensional information. This framework can realize real-time and high-quality view reconstruction.
The above solutions mainly apply the amplitude spectrum of the light field to design the reconstruction filter for novel view reconstruction. However, for complex foreground objects, novel view reconstruction is usually not improved, and more redundant captured images will be generated.

B. RECONSTRUCTION USING LEARNING METHODS
In recent years, with the development of deep learning technology, deep learning has been used to train the captured multiview images by the light field, and the learning method has been used to study novel multiview reconstruction. For example, in [18] and [32]- [34], Wu et al. used a convolutional neural network to train the electronic portal image (EPI) texture structure and then reconstructed the novel views according to the trained EPI texture structure model from a sparse set of captured images. In [20], Farrugia and Guillemot proposed a learning-based superresolution method for deep CNN spatial light fields. The deep CNN spatial light field algorithm allows the restoration of a consistent entire light field in all angular views. In [35], Lamba et al. proposed a deep neural network, called L3Fnet, using a two-stage architecture that considers all the light field views to encode the light field geometry and reconstruct each light field view. The proposed L3Fnet can be used to enhance the light field images for low-light light field imaging. In [35], Lamba et al. presented a spatio-angular dense network (SADenseNet), which contains two components: correlation blocks and spatio-angular dense skip connections. SADenseNet convolutes the camera plane and the imaging plane using a spatial filter and a angular filter, respectively. There are also a large number of light field-related deep learning methods, such as the multiangular light field angular reconstruction network (MALFRNet) [37], unified learning framework for light field reconstruction [38], Pseudo 4DCNN for depth estimation and view warping of light fields [39], a new learning-based sparse sampled light field with irregular structures [40], and a generation model based on the central view of the light field [41].
The solutions described above mainly use the learning method with the amplitude of the light field to iteratively optimize the reconstruction quality. However, for some complex scenes, such as complex geometry and mutation, and even scenes with occlusion, the reconstructed novel views obtained by deep learning methods are still distorted.

C. PHASE RETRIEVAL FOR SIGNAL
For the captured images of the light field rendering technology, the optical sensor for captured images collects only the amplitude information of the scene, but cannot measure the phase information. Currently, in other areas, there have been many research results on the problem of estimating phase information according to the amplitude information of a signal [42]- [46]. Two detailed surveys of phase retrieval algorithms were described in [22] and [23]. For example, in 1981, Taylor studied in detail the phase recovery problem in the application of electromagnetic theory [47]. In [48], Fienup investigated and compared various solutions to the phase recovery problem. In [49], Klibanov et al. investigated various results and explained the only situation in which a signal can be recovered from its amplitude information and supplementary information. In [50], Waldspurger et al. solved the phase recovery problem using a provably convergent block coordinate descent algorithm. Candes et al. applied matrix completion to study the phase recovery problem [51]. Candes et al. also presented a method for phase retrieval using Wirtinger flow in [52]. Sun et al. [53] provided a complete geometric characterization of the nonconvex formulation for the generalized phase retrieval problem. This algorithm allows multiple iterations of phase information to find a global minimum without special initialization.

III. LIGHT FIELD SAMPLING MODEL AND FORMULATION A. LIGHT FIELD REPRESENTATION
We apply 4D light field p (t, s, v, u) [1], [2] quantization to describe the imaging signal of 3D scene, and include camera position and direction, and scene information. (t, s) denotes the camera plane, and (u, v) denotes the image plane. To follow the method of light field representation in [14], [15], and [54]- [57], the 4D light field function p (t, s, v, u) can be simplified to the 2D light field p (t, v) as shown in Fig. 1. The properties and results of the light field signal in the 2D light field (s, u) are the same as those in the 2D light field (t, v). Therefore, we only need to study the results of 2D light field p (t, v). The 2D light field is also named EPI [58], which is the mapping between the scene object and the camera plane and image plane.

B. LIGHT FIELD SAMPLING REQUIREMENTS
For light field sampling, we need to reasonably determine the spacing between two captured cameras. According to the results in [14], [15], and [54]- [57], the spectral support of the light field is with respect to the minimum and maximum depths, the scene texture distribution, the resolution of the sampling camera ( v), the resolution of the rendering camera, and the scene attribute ψ (t, v). Using the spectral support of the light field, we next define the light field sampling requirements.
Definition 1: Given a scene, the spacing between two captured cameras is determined by the spectral support of the light field [14], [15], [54]- [57]. The spacing between two cameras is calculated as where z opt is the optimal depth, and ψ (t, v) affects the light field sampling, such as occlusion, non-Lambertian reflection, texture color change, geometric shapes, and whether the sample is smooth [14], [15], [54]- [57]. Using (1), we can determine the sampling rate of the light field according to specific scene conditions. Special scene conditions include the maximum and minimum depths, object surface geometry and other information. Then, a set of multi-views images for the light field can be captured based on the sampling rate. Using the captured multi-views images, we mathematically derive the light field phase spectrum to reconstruct novel multi-views of the light field in the next section.

C. PROBLEM FORMULATION
According to Fourier series decomposition theory, we can decompose a light field signal into the amplitude and phase.  To analyze the phase of the light field, the 2D light field p (t, v) may be written in terms of its magnitude and phase as where |p (t, v)| denotes the amplitude of the light field, and φ t (t) and φ v (v) are light field phases for the t-axis and v-axis, respectively. Generally, the Fourier phase of the light field contains more scene information than the Fourier magnitude [43]. For the captured multi-view image, the optical detection devices of cameras cannot measure the phase of a light wave. The reconstruction of the novel views only uses the amplitude information captured (i.e., |p (t, v)|) by the camera. Thus, the phase information is included in the novel view reconstruction, which will improve the reconstruction quality. Therefore, the phase retrieval for light field signals is important. We next define the phase retrieval problem of the light field.
Definition 2: Given a set of captured multi-view images, the phase retrieval problem of the light field can be formulated as where φ t ,φ v are the phase retrieval of light field.
We mathematically explore the phase retrieval method of light field signals for application to novel view reconstruction. The phase spectral support of a light field signal is bounded by the frequencies along the camera plane and image plane. The phase of the light field is then recovered using a least squares method based on the previous phase retrieval theory of the signal.

IV. PHASE SPECTRUM ANALYSIS OF THE LIGHT FIELD A. LIGHT FIELD PHASE SPECTRUM
To analyze the phase characteristics of the light field, we sample the continuous light field by (n t , m v ) to obtain the discrete light field, denoted by p (n t , m v ). The z-transform of p (n t , m v ), denoted by P (z t , z v ), is defined by The Fourier transform is the z-transform on the unit circle. Thus, the Fourier transform of the discrete light field exists and is given by The Fourier transform result of the light field can also be regarded as a function of the amplitude and phase where t (ω t ), and v (ω v ) are the Fourier transforms of φ t (t) and φ v (v), respectively. Based on (2) and (6), we now state the light field phase spectrum in the following theorem. Theorem 1: The light field phase is calculated bŷ The light field phase spectrum is calculated bŷ where P R (ω t , ω v ) is the real part of P (ω t , ω v ), and P I (ω t , ω v ) is the imaginary part of P (ω t , ω v ).
Proof: See Appendix A. By (8), we can calculate the phase spectrum of the light field. We use a sine function to construct a light field with the texture information as shown in Fig. 2(a), and the corresponding phase spectrum of the light field is presented in Fig. 2(b). The phase spectrum consists of six parallel oblique lines. These six parallel phase spectra are important information for constructing the light field.

B. LIGHT FIELD PHASE OPTIMIZATION
As noted before, the optical phase of the light field signal cannot be directly measured by an electronic detector. The electronic detector of the camera only measures the magnitude of the light field. By (7) and (8), the phase spectrum of the light field can be derived from the frequency along camera axis ω t and image axis ω v . However, we are not able to obtain the mathematical relationship between the phase and amplitude from (7) and (8) because the camera can only measure the amplitude of the light field for the captured multi-view images. Based on the results in [54]- [57], the spatial frequency ω t and imaging frequency ω v of light field can be derived using the captured multi-view images. Thus, we optimize the phase spectrum estimation of the light field in (7) and (8) using the captured multi-view images.
In light field sampling, we often have access only to the Fourier magnitudes of a light field signal p (t, v), i.e., |P (ω t , ω v )|. The phase information of the light field is hard or infeasible to record due to physical constraints. The problem of optimizing the light field signal p (t, v) from its Fourier magnitudes |P (ω t , ω v )| is naturally termed (Fourier) phase optimization. Given N captured multi-view images, the captured multi-view images are described by the 2D light field |p k (t, v)| , k = 1, · · · , N (Only including the amplitude information) and multi-view image acquisition vector p 1 (n t , m v ) , · · · , p n (n t , m v ) ∈ C n . In discrete 2D Fourier phase retrieval of the light field, the measurement vectors of the light field correspond to p k (n t , m v ) = e −j2π(k(n t ,m v )) / N .
The optimized phase spectrum estimation of the light field can be considered a distortion free novel view reconstruction from the captured multi-view images. According to Fourier theory, the distortion free novel view reconstruction of the light field p (t, v) from the measurement of P (ω t , ω v ) can be achieved by simply applying the inverse-DFT operator, i.e., P (ω t , ω v ) = |P (ω t , ω v )| .e jφ(ω t ,ω v ) . The Fourier phase-optimized problem is to optimize p (t, v) when only the magnitude of P (ω t , ω v ) is measured, i.e., to optimize p (n t , m v ) given |P (ω t , ω v )|. Since the DFT operator is bijective, this is equivalent to optimizing the phase of P (ω t , ω v ), i.e., φ (ω t , ω v )-hence the term phase is optimized. We next define the phase optimization of light field.
Definition 3: Generalized phase optimized problem of the light field: Is it possible to efficiently recover an unknown light field p (t, v) from Fourier magnitudes |P (ω t , ω v )| of discrete light field signal p k (n t , m v ) , k = 1, · · · , N , up to a global phase factor e jφ(ω t ,ω v ) ?
The Fourier magnitudes |P (ω t , ω v )| of light field projections onto a set of captured multi-view images, i.e., p 1 (n t , m v ) , · · · , p n (n t , m v ) ∈ C n . The Fourier magnitudes can be written as Obviously, we hope to recover p (t, v) up to a global phase, as p (t, v) = |p (t, v)| e jφ(ω t ,ω v ) for all φ (ω t , ω v ) ∈ [0, 2π) provides exactly the same set of captured multi-view images. The Fourier phase-optimization problem is as a special case of the more general phase-optimization problem, where we are given a set of captured multi-view images.
As opposed to these strategies, we solve the phase optimization problem by explicitly separating the amplitude and phase variables, and by only optimizing the values of the phase variables. Given a set of captured multi-view images, the phase optimization problem of the light field can thus be written as where diag (·) denotes a diagonal matrix. We optimize both variables φ k (ω t , ω v ) ∈ C n and p (t, v) ∈ C p . Specifically, we define an N × N matrix P = P i (ω t , ω v ) · P * j (ω t , ω v ) , i, j = 1, 2, · · · , N . Based on (10), we now state the phase optimized on a novel view of the light field in the following theorem.
Theorem 2: For a light field signal, given N captured multi-view images p k (n t , m v ), the phase of the light field can be balanced and optimized using the amplitude spectrum of the light field aŝ where the Hermitian matrix M i,j = diag P i,j I − A i,j diag P i,j is positive semidefinite, and A = p k (n t , m v ) · p * k (n t , m v ) , k = 1, 2, · · · , N . For each i = 1, · · · , N , when φ (ω t , ω v ) is the optimum solution to the phase recovery problem of the light field signal,φ (ω t , ω v ) amounts to solving Proof: The full proof can be found in Appendix B. Using Theorem 2, we can estimate and optimize the phase spectrum of the light field by using the amplitude information of the captured multi-view images. The optimized phase spectrum carries more abundant scene information.

V. RECONSTRUCTION MODEL OF NOVEL VIEWS A. RECONSTRUCTION MODEL
Based on the above phase spectrum of the light field, we derive a reconstruction model. It is always possible to find another discrete light field that has a Fourier transform with the same phase by simply convolving p (n t , m v ) with a zero phase sequence g (n t , m v ); i.e., where * denotes the convolution operation. In the frequency domain, the form for multiplying two functions iŝ Thus, Ifφ For the zero phase function, the estimation of the phase function is The PRLF is presented for reconstructing a discrete light field from samples of the phase of its Fourier transform. Therefore, a heuristic approach for reconstructingp (n t , m v ) is an iterative algorithm. This algorithm is characterized by the repeated transformation between the time and frequency domains where, in each domain, the known information about the desired sequence is imposed on the current estimate light field.

B. RECONSTRUCTION METHOD BY A PHASE OF THE LIGHT FIELD
Based on the above phase spectrum analysis and Theorem 1, we derive the light field reconstruction method using the phase information. For the reconstruction method of the light field, given a discrete light field p (n t , m v ), whose Fourier transform is expressed as P (ω t , ω v ), the light field reconstructed with the above estimated phase is expressed aŝ The reconstructed light field can be obtained by the inverse Fourier transform of (19) as follows: Based on (19) and (20), we simulate two EPI signals as shown in Fig. 3. One EPI signal is simulated without complex texture information. The other EPI signal contains complex texture information. Then, the PRLF is used to reconstruct the EPI, and the results are shown in Fig. 3. From the results, we find that the two EPIs cannot be well reconstructed. The reason is that the phase estimation of the EPI is inaccurate. Thus, the phase estimation of the light field needs to be further optimized for novel view reconstruction.

C. ITERATIVE RECONSTRUCTION ALGORITHM
To optimize the quality of novel view reconstruction, we can optimize the reconstruction method in (20) based on time-frequency conversion theory [42]. Based on (19) and (20), we develop an iterative algorithm to optimize the quality of novel view reconstruction, which can start from the phase spectrum reconstruction of the captured multi-view images that meet certain conditions. Each iteration of the algorithm includes two transformations from the spatial domain to the frequency domain and from the frequency domain to the spatial domain of the light field signal, and includes two modifications in the time domain and the frequency domain of the light field signal. Thus, the quality of the reconstructed novel views is improved with each iteration.
The PRLF is presented for reconstructing a discrete light field from samples of the phase of its Fourier transform. Therefore, a heuristic approach for reconstructing p (n t , m v ) is an iterative algorithm. This algorithm is characterized by the repeated transformation between the time and frequency domains where, in each domain, the known information about the desired sequence is imposed on the current estimate light field. Additionally, the iteration may be described as follows. Input: ∈-Error threshold. Output: p out (n t , m v )a vector that conforms with both magnitude constraints, i.e., : Initialization: Choose initial p 0 (n t , m v ) = |p (n t , m v )| exp (φ (ω t , ω v )) (e.g., with a random phase φ (ω t , ω v )).
•Step 1: Beginning with the spectrum of the initial light field p 0 (n t , m v ), i.e., P 0 (ω t , ω v ) = DFT [p 0 (n t , m v )], an initial guess of the unknown DFT magnitude, the first estimated light field P 1 (ω t , ω v ) of the initial light field spectrum P 0 (ω t , ω v ) is formed by combining P 0 (ω t , ω v ) with the random phase φ (ω t , ω v ), i.e., Computing the inverse DFT of P 1 (ω n , ω m ) provides the first estimate light field, p 1 (n t , m v ), of p 0 (n t , m v ). Because the light field signal has zero phase in the frequency domain, p 1 (n t , m v ) is equal to p 0 (n t , m v ) within a scale factor.
•Step 2: From p 0 (n t , m v ), another discrete light field, p 1 (n t , m v ), is formed as follows: •Step 3: The magnitude P 1 (ω t , ω v ) of the DFT of p 1 (n t , m v ) is then used as the new estimated light field of P 2 (ω t , ω v ) and the next estimated light field of P 2 (ω t , ω v ) is formed by A new estimated light field, p 2 (n t , m v ), is then obtained by taking the inverse DFT of P 2 (ω t , ω v ). The repeated application of steps two and three defines the iteration. Until: ≤∈. The flowchart of the above light field reconstruction steps is also shown in Fig. 4.

A. LIGHT FIELD PHASE SPECTRUM ANALYSIS
We tested our light field phase spectrum algorithm on two real datasets: ragdoll and orchid, where the scenes are shown in the top of Fig. 5. Each scene includes 200 images uniformly captured at different positions, with each image having a pixel values of 320*240. Then, the 200 captured images compose EPI, and a 2D Fourier transform is performed on the EPI to obtain the phase spectrum presented in Figs. 5(a) and (b). It can be found that the series of parallel phase lines are not visible in the ω t , ω v -plane. The reason for this phenomenon is the complexity of the scene, which makes the light field signal not just a periodic signal. There are other noise and interference signals that make the phase parallel lines inconspicuous. The more complicated the scene, the more cluttered the phase spectrum, this effect is shown by the differences in Figs. 5(a) and (b). Therefore, we can use the phase spectrum to iteratively modify the phase of the light field to obtain the light field information and improve the rendering quality.

B. COMPARISON OF LIGHT FIELD RECONSTRUCTION
To evaluate the performance of the PRLF in improving the quality of LFR, we use synthetic scenes and actual scenes to reconstruct the light field. Each scene captures a set of images  (the sampling rate is calculated by [14]) uniformly at different positions, and then these captured images are used for rendering novel views. Additionally, 200 novel multiviews are reconstructed for each scene. The reconstruction method considers three models: the PRLF, light field reconstruction model using a convolutional network on the EPI (CNNE) by Wu et al. [18] and light field reconstruction using sparsity in the continuous Fourier domain (SCFD) by Shi et al. [28].

1) SYNTHETIC SCENES
To verify that our PRLF can improve the rendering quality of the light field, we capture images from datasets and reconstruct views using two synthetic scenes (buddha [59] and ring by 3ds Max) depicted in Fig. 6. The light field reconstruction results in Fig. 6 show that the novel views reconstructed using the PRLF, SCFD and CNNE methods have similar effects. The reconstruction results have no obvious distortion. For example, for the Buddha scene, the reconstructed results VOLUME 10, 2022  do not show distortion. A similar phenomenon can be seen from either the reconstructed EPI or a single rendered view. The quality of the rendered views is measured by the peak signal-to-noise ratios (PSNRs) against the ground truth image as shown in Fig. 7. Furthermore, similar phenomena can be seen from the reconstructed EPI and novel views of the ring scene. From these experimental results, we find that the PRLF has good light field reconstruction.

2) INDOOR ACTUAL SCENES
We evaluate the proposed approach using two ragdoll and orchid indoor scenes, as depicted in Fig. 5. For the rendering results, we find that the quality of the reconstructed novel views using these three methods is better and the distortion is less. Only a few reconstructed viewpoints still have distortions. Fig. 8 shows that the reconstructed views have some obvious ghosting and aliasing in either the ragdoll scene or the orchid scene. Apparently, the rendering quality of the PRLF is better than that of the CNNE and SCFD on the ragdoll scene. Ghosting occurs because scene complexities, such as leafage and irregular shape, are challenging to reconstruct accurately. However, when using the PRLF, ghosting and aliasing in the rendered views decrease. There are identical phenomena that occur in the orchid scene. These results also suggest that the PRLF can be applied to analyze complex scenes for light field reconstruction.

3) OUTDOOR ACTUAL SCENES
To evaluate the performance of our method, 30 novel views of two actual outdoor scenes are reconstructed using the collected data according to the following methods: the PRLF, CNNE and SCFD. The rendering results are shown in Fig. 9. We find that the quality of the reconstructed novel views using these three methods is better and the distortion is less. This novel view reconstruction comparison experiment, also shows that PRLF has good novel view reconstruction performance. The main reason is that PRLF uses scene phase information, which improves the quality of the rendered views.

C. COMPARISON ON PUBLICLY AVAILABLE DATASETS
To further compare and evaluate the performance of the PRLF method with that of other methods, we select three published datasets form the Stanford light field archive, i.e., Bunny, Crystal Ball, and Jelly Beans [59], for comparison experiments. Seventeen novel views are reconstructed by the PRLF, learning-based spatial light field (LSLF) superresolution method [20], and adapted discrete shearlet transform (ADST) [26].
The rendering results for the EPIs of the three scenes using the three different methods are presented in Fig. 10. From the Bunny scene, the rendered views by the three methods do not experience distortion. However, the rendered EPI using the ADST method has some distortion. When the PRLF and LSLF methods are used, the reconstructed EPIs have no distortion and have better reconstruction quality. For the Crystal Ball scene, the views rendered by the three methods do not experience distortion, and there are good reconstruction effects. However, the rendered EPIs with ADST and LSLF have a small amount of distortion and blur. Similar phenomena can also be seen in the Jelly Beans scene. Therefore, the views rendered by the PRLF method have better display results. These results also have the same change trend as the PNSR statistical results in Fig. Fig. 11. The excellent performance of the PRLF reconstruction method is due to the use of optimized light field phase information, which depends on the large amount of scene information contained in the phase spectrum.

VII. CONCLUSION
In this paper, we proposed a PRLF model for LFR using phase spectrum theory. This reconstruction model has been studied, and its spectrum is presented. We present a light field reconstruction algorithm based on phase iterative correction. Based on the phase spectrum correction, novel views of the light field can be reconstructed. Furthermore, the rendering quality of the light field can be improved for complex scenes.
For the multi-view synthesis of light field technology, the quality of rendered novel views will decline for complex scenes. We increase the captured information of scene from the phase recovery of light field signal, so as to improve the rendering quality. Based on the traditional signal phase recovery works, our work provides a study method of LFR technology. However, there are still some shortcomings in our work on the method of light field phase recovery. The main reason is that light field phase recovery is related to scene information. In our our future work, we will study phase recovery of light field associated with scene information.

VIII. PROOF OF THEOREM 1
Proof: Since P (ω t , ω v ) is, in general, a complex-valued function of the light field, it may be expressed in terms of its real and imaginary parts as follows: where and tan [ t (ω t )] = P I (ω t , ·) P R (ω t , ·) , tan [ v (ω v )] = P I (·, ω v ) P R (·, ω v ) .
By (27), we can calculate the phase spectrum of the light field. We use a sine function to construct a light field with the texture information as shown in Fig. 2(a), and the corresponding phase spectrum of the light field is presented in Fig. 2(b). The phase spectrum consists of six parallel oblique lines. These six parallel phase spectra are important information for constructing the light field. t (ω t ) = arctan φ (ω t , ω v ) = arc tan P I (ω t , ω v ) P R (ω t , ω v ) .

IX. PROOF OF THEOREM 2
Proof: By (10), the inner minimization problem in p (t, v) is a standard least squares and can be solved explicitly by setting where A = p k (n t , m v ) · p * k (n t , m v ) , k = 1, 2, · · · , N . In (30), light field view reconstruction is equivalent to the reduced problem The objective of this last problem can be rewritten as follows where the Hermitian matrix M = diag (P) (I − A) diag (P) is positive semidefinite.
Having transformed the phase recovery problem (30) in the quadratic minimization problem (31), suppose that we are given an initial vector diag (P)φ (ω t , ω v ), and focus on optimizing over a single componentφ (ω t , ω v ). The problem is equivalent to solving min  φ * (ω t , ω v ) Mφ (ω t , ω v ) For (34), the problem of view reconstruction based on phase optimization this then amounts to solving which meanŝ for each i = 1, · · · , N , when φ (ω t , ω v ) is the optimum solution to problem (31).