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We propose a novel light field acquisition method based on a planar catadioptric system. Only one camera with a large field-of-view (FOV) is required, and multiple virtual cameras can be created via an array of planar mirrors to acquire the light field from different views. The spatial distribution of the virtual cameras can be configured flexibly to acquire particular light rays, which can be controlled by simply changing the positions of the mirrors. Compared with previous systems, no aberration or reduction in light transmittance is introduced by the planar mirrors, which ensures image quality. In this study, the design method of the planar catadioptric system is provided and a calibration procedure of its computational model is analyzed in detail. The method is verified by a prototype system, with which the correct digital refocusing results are achieved using the acquired, calibrated light field.
© 2015 Optical Society of America
1. Introduction
In recent years, light field techniques have been used in many applications, including three-dimensional (3D) displays [1–3 ], depth estimation [4,5 ], and scene reconstruction [6], with particular interest in acquiring the light field of 3D scenes. In conventional cameras, a sensor pixel averages the radiance of the light rays impinging over the full hemisphere of incident angles. Only two-dimensional (2D) projections of the light field from a 3D scene are recorded by conventional photographs.
An array of image sensors can be used to capture a set of photographs at different positions to estimate the light field. These sensors can be located uniformly along a planar surface or at flexible positions. In practice, accurate calibration and synchronization of multiple sensors must be ensured to obtain an accurate light field. Yang et al [7] created a light field capture system using 64 cameras. A similar system was developed by Wilburn et al [8] with as many as 125 digital video cameras. However, the spatial resolution of the acquired light field is limited by the physical shape of each sensor; additionally, the cost and engineering complexity of such systems prohibits their use in consumer applications. Many computer vision and imaging processing methods [9] have been proposed to reduce the number of sensors while maintaining the resolution of the light field; however, these limitations still exist in light field capture systems that use multiple sensors.
Temporal multiplexing methods using a single image sensor have also been developed. A mechanical gantry [10] can be used to translate a single camera over specific positions. Liang et al [11] demonstrated a novel temporal multiplexing method using a dynamic programmable aperture. The cost and complexity are reduced dramatically by temporal multiplexing; however, these systems cannot capture dynamic light field scenes.
Lippmann [12] and Ives [13] introduced parallax barriers and integral photography to spatially multiplex the light field onto a 2D digital-image sensor; this approach has received much attention since its introduction [14,15 ]. Commercial light field cameras extend the integral photography concept to capture the light field passing through the entrance aperture of a conventional camera (e.g., Lytro [16,17 ] and Raytrix [18] models). Georgeiv et al [19,20 ] developed a hand-held light field camera using a custom lens and an array of prisms and lenses; Perwaß et al [21] analyzed this system in detail. Veeraraghavan et al [22] and Lanman et al [23] introduced frequency multiplexing to encode the light field as a 2D sensor image, to make full use of the 2D sensor. These hand-held light field cameras offer the advantages of an extension of the depth-of-field and a wide range of applications (e.g., 3D depth reconstruction and digital refocusing); however, because they are based on Gaussian optics, the loss of light and optical aberrations were not taken into account.
The goal of optical design for main lens is to correct optical aberrations and to increase light transmittance, thus, an image with high quality can be obtained. Pinhole arrays and other coded masks reduce the light transmittance and increase exposure times. Prism and lens arrays introduce new optical aberrations that will affect imaging performance. Many algorithms have been proposed to enhance the imaging performance [24–26 ] and reduce the introduced aberrations. However, light transmittance reduction and optical aberration can only be alleviated, rather than being completely eliminated. In fact, image with better performance must be obtained when these algorithms are applied to light field acquired with minimum aberration and light transmittance reduction. Moreover, only the light field passing through the entrance pupil of the optical system can be captured by these handheld light field cameras, and it is difficult to enlarge the size of the entrance pupil. The catadioptric capture method can be used to resolve this issue. Lanman et al [27] used spherical catadioptric arrays to acquire multiple views of a 3D scene. Taguchi et al [28] presented a geometric ray model to calibrate the light field acquired by a spherical catadioptric system. However, as mentioned above, convex mirrors introduce new optical aberrations that cannot be corrected via calibration and the placement of the mirrors is not flexible. Planar mirrors do not affect the imaging performance nor reduce light transmittance; therefore, these mirrors have been used for two-view stereo imaging [29] and panoramic imaging [30]. The relationship between the mirrors and camera can be flexible, and the acquired light field is not confined by an entrance pupil.
In this study, we have developed a new light field acquisition method based on a catadioptric system using multiple planar mirrors. Multiple virtual cameras are created by a single image sensor and planar mirrors. No optical aberrations or reduction in light transmittance are introduced by this capture system. The final system is compact and easy to fabricate. Moreover, the system is sufficiently flexible to allow adjustment of the virtual camera distribution; the required light field is obtained by simply controlling the positions of the sensor and mirrors.
2. Overview of the proposed method
Figure 1 shows a schematic diagram of the proposed light field acquisition method using a planar catadioptric system. Multiple mirrors are fabricated, precisely cut, and aligned on a custom-designed mechanical structure. A main camera is placed to face the multiple mirror structure; its image is split into a number of subimages, which form the same number of virtual cameras in space. The position of the each virtual camera is the reflective image of the optical center of the main camera. No new aberrations are introduced, and the size of entrance pupil stays the same thanks to the planar mirrors employed in our method. Thus, light transmittance and the image performance will not be affected by the multiple mirrors. The angular magnification of mirrors is one, so the angular resolution is also the same as that of the main camera. In our proposed method, the field-of-view (FOV) and the pose of a virtual camera is determined by the shape of its corresponding mirror and its position relative to the main camera. Multiple virtual cameras with flexible spatial distribution can be achieved by simply controlling the mirror positions. All of the virtual cameras must share the FOV of the main camera, thus, the FOV of our method is often smaller than that of other methods, but the angular resolution stays the same as that in a traditional camera, and the poses of virtual cameras (the direction of optical axis) can be controlled flexibly. Many applications, including 3D reconstruction, depth estimation, and dense light field acquisition can be achieved based on the light field captured by the proposed system.
3. Design method
3.1 Position of virtual cameras and placement of mirrors
Using a Cartesian coordinate system, the origin OC is defined as the optical center of the main camera. The FOV of the camera is Ψ × Φ, and m × n virtual cameras are reconstructed by the developed planar catadioptric system. The FOV of the camera can be divided into m × n parts with each φi × πj (i = 0, 1, …, m−1; j = 0, 1, …, n−1); the following equality should be satisfied:
For the (i, j)th virtual camera, the central ray from the optical center of the real camera can be expressed as
Let r̂ij be the normalized vector rij. The unit vector of the optical axis of the (i, j)th virtual camera is l̂ij. According to Snell’s Law, the normal vector of the (i, j)th mirror is given by
From the law of reflection, the (i, j)th mirror is the perpendicular bisector plane of line OCVij, in which Vij is the optical center of the (i, j)th virtual camera. Thus, the coordinate of the virtual optical center is given by λijn̂ij, where λij is the distance between the origin and the virtual point. The homogeneous coordinate of the (i, j)th mirror plane is given by
The vertices of the (i, j)th mirror (M1ij, M2ij, M3ij, and M4ij, shown in Fig. 2(a) ) can be calculated by the intersection between the mirror plane and the edge rays from the optical center of the main camera (the origin). Therefore, the position of the virtual optical center and placement of the corresponding mirror can be determined by the optical axis of the virtual camera and a given scale factor.
Fig. 2 Schematic diagrams of (a) the position of virtual cameras & the placement of mirrors and (b) the constraints of mirror placements.
3.2 Virtual camera position and mirror placement
As mentioned above, all of the virtual cameras must share the FOV of the main camera; this requires that two structural constraints be satisfied. The first constraint is that the mechanical structure of the main camera cannot be viewed by any of the virtual cameras. The second constraint is that the placement of the mirrors must be controlled such that none of the mirrors can be viewed by adjacent virtual cameras.
After the mirror placement for the (i, j)th virtual camera with a FOV of φi × πj is calculated, based on the description in Section 3.1, it is required to check that both constraints are satisfied. The main camera is located inside the cuboid A1A2A3A4A5A6A7A8, in which A1 –A8 correspond to the vertices of the cuboid, as shown in Fig. 2(b). The cuboid should be larger than the overall external dimension of the main camera during the design process, due to the machining errors of the multiple mirrors. The (h, k)th virtual camera with mirror M1hkM2hkM3hkM4hk corresponds to one of the virtual cameras already designed, with M1hk–M4hk corresponding to its mirror vertices. If any of the vertices (including the cuboid and all of the designed mirrors) is outside the FOV of the (i, j)th virtual camera, then the two constraints are satisfied; conversely, the parameters of this virtual camera should be changed until all vertices are outside of its FOV.
3.3 Design of a planar catadioptric system with visual camera positions on one plane
To demonstrate the development of a planar catadioptric system, we examine light field acquisition with visual cameras in one plane, as an example. According to the description above, the design method can be summarized in the following steps:
- (1) Choose the number of virtual cameras, the FOV and optical axis of each virtual camera, and the plane upon which the optical centers of the visual cameras are placed. We assume that the FOV of the main camera is Ψ × Φ, and m × n virtual cameras are generated with a FOV of φi × πj. Let the center of the Cartesian coordinate system correspond to the optical center of the main camera. The homogeneous coordinate of the virtual camera plane is given as P = (p̂x, p̂y, p̂z, η)T. Here, p̂ = (p̂x, p̂y, p̂z)T is a unit vector; thus, three variables can be given as input parameters for the virtual camera plane. The optical axis of each virtual camera is given as l̂ij = (l̂ijx, l̂ijy, l̂ijz)T, which is also a unit vector.
- (2) Calculate the central ray r̂ij for each virtual camera according to Eq. (2).
- (3) Calculate the homogeneous coordinate of each mirror. The homogeneous coordinate of the (i, j)th mirror is given by variable λij, according to Eq. (4). Because the virtual optical center λij n̂ij is on plane P, λij for the (i, j)th mirror plane should be satisfied by the following:
- (4) Calculate the intersections between the mirror plane and the marginal light rays of the (i, j)th virtual camera, which are the vertices of the (i, j)th mirror.
- (5) Determine whether or not the (i, j)th structure satisfies the constraints.
The positions of 5 × 4 mirrors have been calculated using Matlab according the method described above. In the process, the optical axis of each virtual camera is all parallel, and the FOV of each virtual camera is the same. The structure (in blue), main camera (in red) and the positions of virtual cameras (in green) have been shown in Fig. 3 .
Fig. 3 Multiple mirrors in a planar catadioptric system with virtual sensors on a plane (take 5 × 4 virtual cameras system as an example).
4. Calibration procedure of an actual system
When m × n mirrors are manufactured, precisely cut and assembled with a camera based on the calculated results mentioned in Section 3, light field of the real scene from m × n virtual cameras can be acquired at one time. The capture image can be divided into m × n subimages; each subimage contains the image from one virtual camera (Vij shown in Fig. 3). However, in actual system, difficulties arise during mirror alignment, due to the precision limit of the mechanical system. Misalignment of the virtual cameras introduces inconsistencies with respect to the designed model. To obtain the light field in the actual planar catadioptric system, one must calculate the extrinsic and intrinsic parameters of each virtual camera, based on a calibration procedure.
In terms of the intrinsic parameters, the image distortion coefficients (radial and tangential distortions) must be taken into account to optimize the FOV of the camera. In the method described, one main camera is used. No optical aberration is introduced by the use of multiple mirrors; i.e., the intrinsic parameters of each virtual camera are the same. Many types of camera computational models have been proposed to represent the relationship between the normalized pixel projection [xn, yn]T in the camera coordinate system and the pixel coordinates [xp, yp]T in the captured image. Despite the merits of the various calculation models, in this study we used that in the Camera Calibration Toolbox for Matlab [31]. The relationship between the [xn, yn]T and [xp, yp]T can be expressed as
where r2 = xn2 + yn2. Here, 10 coefficients must be calibrated to define the intrinsic parameters of the system. f1 and f2 are the focal distances expressed in units of horizontal and vertical pixels. The pixel coordinate of the central point in the image is [x0, y0]T. The coefficient α is the angle between the x and y sensor axes. The coefficients k1–k5 contain both radial and tangential distortion coefficients.Concerning the extrinsic parameters, the mirrors can be placed in any configuration. The relative orientation between the (i, j)th virtual camera coordinate and the world coordinate can be described by the rotation matrix Rij and the translation vector Tij. For each correspondence point from the (i, j)th virtual camera Vij, the relationship between the normalized pixel coordinates [xn, yn]T in the camera coordinate system and [X, Y, Z]T in the world coordinate system can be given as.
where zc is the z-coordinate in the camera reference frame. zc can be eliminated to get two linear equations for each mapping point.The 3 × 3 matrix Rij can be described by three coefficients, θij1, θij2, and θij3 (for example, Rodrigues' rotation formula). The translation vector is also described by three coefficients tij1, tij2, and tij3. Therefore, there are in total (6 × m × n + 10) coefficients (including intrinsic and extrinsic parameters) for the light field capture system with m × n virtual cameras. Many approaches have been developed to obtain a correspondence map between the pixel coordinates on the captured image and the points in the world coordinate system (camera–world map). The key of the calibration procedure is to solve a set of nonlinear equations formed by Eqs. (6) and (7) based on the correspondence points in the obtained camera-world map. Therefore, an iterative method, such as the Levenberg–Marquardt algorithm, can be used to obtain a final solution, given a set of initial values.
Given that the FOV of the main camera is Ψ × Φ, and the resolution as a whole is Ux × Uy, the initial intrinsic parameters are given as
For each mapping point in the camera–world map from the (i, j)th virtual camera, two following linear equations can be given using Eq. (7) by eliminating zc,Therefore, the initial extrinsic parameters can be obtained by solving a set of linear equations if a sufficient number of correspondence points are found. Iterative method can be employed to refine all parameters using the obtained initial values, and thus, the (6 × m × n + 10) coefficients can be given, which includes the intrinsic parameters of the main camera, the rotation matrix Rij and the translation vector Tij for the (i, j)th virtual camera.Light field within a three-dimensional volume can be parameterized by a 7D function [32]. In terms of the light field for a particular moment and a particular wavelength of light, it can be parameterized by 5D function (including three for the coordinate of one point on the ray, and two for the elevation and azimuth of the direction) [10]. In our developed system, the calibrated light field can be defined by the optical center of a virtual camera and the unit direction vector (the elevation and azimuth can be easily obtained by this vector). Thus, for each pixel from the (i, j)th virtual camera, the correspondence light field in the world system can be parameterized as,
where inv(∙) means the inverse of a matrix.The calibration procedure can be summarized as follows:
- a) Obtain a camera–world map. This can be achieved using various methods (e.g., by finding the corner points or decoding the structured light images).
- b) Determine the initial intrinsic parameters of the system based on the number of pixels and the estimated FOV, according to Eqs. (8-10) .
- c) Given the initial intrinsic parameters, calculate the extrinsic parameters of each virtual camera based on Eq. (11), using the least squares method.
- d) Refine all of the parameters using Eqs. (7-9) , according to the Levenberg–Marquardt algorithm [33].
- e) Obtain a correspondence map between the pixel coordinate on the captured image and the accurate light field in the world coordinate using Eqs. (12-13) .
5. Experimental setup and results
To verify the presented method, a prototype was developed. In the experiment, a Logitech HD Pro Webcam C920 (FOV: 66° × 52°; resolution: 5168 × 2907 pixels) was used as the main camera. Virtual cameras (5 × 4) were reconstructed by 5 × 4 mirrors, based on the method described in Section 3. In the design results, the optical axes of the virtual cameras were parallel to each other, and perpendicular with respect to the optical axis of the main camera. Multiple mirrors were constructed from acrylic glass. A 3D print technique was used to manufacture the mechanical structure to hold the mirrors in the required position. Figure 4 shows the module of the developed planar catadioptric system.
To calibrate the positions of the virtual cameras in the prototype, a structured light technique [34] was used to find the camera world map, as given by the method described in [35]. In the experiment, we used sinusoidal phase shift patterns and multi-frequency heterodyne phase unwrapping. Specifically, a three-frequency, four-step phase-shift implementation was used to achieve satisfactory results. A Samsung SyncMaster B2230 liquid-crystal display (LCD) screen (resolution: 1920 × 1080 pixels) provided the sinusoidal phase patterns. Twenty-four images were used to encode the LCD pixels in both the horizontal (X) and vertical (Y) directions.
Figure 5(a) shows an example of one of the structured light images; the display performance was worse around the seam between adjacent subimages than in other parts of the display area. These areas cannot be treated as an effective area in the calibration procedure; also, the application is based on the acquired light field. This problem did not influence the capture performance of the light field and can be alleviated by improving the techniques used in mirror cutting or cementation. Figure 5(b) shows the LCD-camera map for the effective areas of the LCD’s X-direction, in which the color-coded value with a certain camera pixel is the X coordinate of the LCD pixel. Similarly, Fig. 5(c) shows the LCD-camera map for the effective areas of the LCD’s Y-direction, in which the color-coded value is the Y coordinate of the LCD pixel. Let the world coordinates be defined on the LCD plane; thus, the world-camera map is given by the LCD-camera map and the physical size of the pixels on the LCD panel (0.24825 mm in this experiment). The positions of the virtual cameras can be calculated using the camera-world map, based on the description in Section 4. Figure 6 gives the calibrated results for the virtual cameras in the world coordinate system; the position and effective FOV of each virtual camera can be represented by a green pyramid.
Fig. 5 (a) One of the structured light images. (b) Structured light pattern decoding results in horizontal (X) direction. (c) Structured light pattern decoding results in vertical (Y) direction.
Fig. 6 (a) Calibrated results of the virtual cameras. (b) Enlarged image of the calibrated results. (unit: mm)
To verify that the calibration results are accurate and reliable, digital refocusing at different depths was implemented, which is a popular application based on light field techniques. Each pixel in the digital refocusing image was obtained by averaging the light field rays projected at a given depth from all of the virtual cameras, which has been proposed in [36]. Figure 7(a) shows the experimental setup for the digital refocusing application. A toy model of magic cube, an toy model, and a book were used as the reference objects; the reference objects were positioned at different depths (around 800 to 1,300 mm far away from the camera). Figure 7(b) shows a captured light field image of the three objects by the developed system. Reconstructed images refocusing at different depths were achieved based on the captured light field image and the calibration result. Figure 8 shows the reconstructed images of the scene focused at different depths: −140 mm, 40 mm, and + 260 mm in the world coordinate system. The locations and the sizes of the three reconstructed planes are also shown in Fig. 6(a), together with the calibrated virtual cameras. The image of the reference object at the focusing depth was sharp and clear, while the other reference objects away from the focusing depth were blurred. Digital refocused images are also shown in Visualization 1 with focusing depths ranging from −350 mm to 400 mm in the world coordinate system. It can be inferred from the experimental results that the calibrated light field provided an accurate representation of the objects, suggesting its use in practical applications.
Fig. 8 Digital refocusing images using the captured image (Fig. 7(b)) by our developed system (see Visualization 1).
6. Conclusions
We have presented a new type of light field acquisition method using a large FOV camera and multiple planar mirrors. In this study, the design method and calibration procedure for the developed system have been described in detail. The distribution of generated virtual cameras is flexible, and the acquired light field is not limited by the entrance aperture. Moreover, the display performance is not affected by the introduction of multiple planar mirrors. The proposed method was demonstrated by a compact prototype. Our experimental results indicated that the proposed system is capable of capturing the dense light field from a 3D scene. Only one-time capture is required for light field acquisition; additionally, ray energy loss was minimal, with little to no optical aberration. Thus, the proposed planar catadioptric system shows great potential for light field acquisition. In future work, we plan to improve the focusing component of the proposed system to enhance its performance, and hope to apply this approach to human-machine interactive systems.
Acknowledgments
This work was partially funded by the National Natural Science Foundation of China (NSFC) (61235002, 61370134) and the National High Technology Research and Development Program of China (2013AA013904).
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