iccv iccv2013 iccv2013-348 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Anne Jordt-Sedlazeck, Reinhard Koch
Abstract: In underwater environments, cameras need to be confined in an underwater housing, viewing the scene through a piece of glass. In case of flat port underwater housings, light rays entering the camera housing are refracted twice, due to different medium densities of water, glass, and air. This causes the usually linear rays of light to bend and the commonly used pinhole camera model to be invalid. When using the pinhole camera model without explicitly modeling refraction in Structure-from-Motion (SfM) methods, a systematic model error occurs. Therefore, in this paper, we propose a system for computing camera path and 3D points with explicit incorporation of refraction using new methods for pose estimation. Additionally, a new error function is introduced for non-linear optimization, especially bundle adjustment. The proposed method allows to increase reconstruction accuracy and is evaluated in a set of experiments, where the proposed method’s performance is compared to SfM with the perspective camera model.
Reference: text
sentIndex sentText sentNum sentScore
1 Refractive Structure-from-Motion on Underwater Images Anne Jordt-Sedlazeck and Reinhard Koch Institute of Compute Science, Kiel University, Germany { s edl a z e ck ,rk} @mip . [sent-1, score-0.029]
2 de Abstract In underwater environments, cameras need to be confined in an underwater housing, viewing the scene through a piece of glass. [sent-4, score-0.896]
3 In case of flat port underwater housings, light rays entering the camera housing are refracted twice, due to different medium densities of water, glass, and air. [sent-5, score-1.347]
4 This causes the usually linear rays of light to bend and the commonly used pinhole camera model to be invalid. [sent-6, score-0.481]
5 When using the pinhole camera model without explicitly modeling refraction in Structure-from-Motion (SfM) methods, a systematic model error occurs. [sent-7, score-0.644]
6 Therefore, in this paper, we propose a system for computing camera path and 3D points with explicit incorporation of refraction using new methods for pose estimation. [sent-8, score-0.543]
7 Additionally, a new error function is introduced for non-linear optimization, especially bundle adjustment. [sent-9, score-0.107]
8 The proposed method allows to increase reconstruction accuracy and is evaluated in a set of experiments, where the proposed method’s performance is compared to SfM with the perspective camera model. [sent-10, score-0.267]
9 Introduction In the last decade, many applications for images captured underwater arose. [sent-12, score-0.386]
10 They include scientific exploration of geological or archaeological structures on the sea floor [2], maintenance of offshore oil rigs, inspection of ship hulls, and measurements of ships and other fisheries [6]. [sent-13, score-0.153]
11 Due to the need of gaining measurements in the above described scenarios, the geometry ofimage formation is often utilized. [sent-14, score-0.024]
12 However, cameras used in an underwater environment are usually confined in an underwater housing filled with air, viewing the scene through a piece of glass. [sent-15, score-1.174]
13 In case of this glass being a flat port, the light rays entering the camera housing are refracted twice, once at the water-glass interface and again at the glass-air interface. [sent-16, score-1.354]
14 Many of the above described applications require the camera to be lowered into the deep sea, sometimes to water depths ofthousands ofmeters. [sent-17, score-0.446]
15 Therefore, the underwater housing needs to be strong enough to withstand immense water pressures, requiring the glass interface to be several centimeters thick. [sent-18, score-1.505]
16 The double 57 refraction causes the usually straight rays of light to bend and change direction depending on the interface incidence angles. [sent-19, score-0.857]
17 When following the ray in water in Figure 1 without refraction (dashed line), it does not intersect the camera center. [sent-20, score-1.016]
18 [28] showed that the perspective camera model is invalid below water due to the rays not intersecting in one common center of projection. [sent-22, score-0.778]
19 Despite that, the perspective camera model is often used for underwater images, approximating the refractive effect to some extent. [sent-23, score-0.898]
20 [18] showed that a camera calibrated below water approximates refraction with focal length and radial distortion and Sedlazeck and Koch [25] showed that principal point and camera pose absorb some of this model error in addition to focal length and radial distortion. [sent-25, score-1.442]
21 Due to the perspective model being invalid, a systematic model error is introduced, when applying perspective algorithms utilizing imaging geometry like mosaicing or Structure-from-Motion (SfM) [9, 27] to underwater images. [sent-26, score-0.665]
22 Even though, several works can be found in the literature, where the perspective camera model is used to reconstruct 3D scenes in underwater environments (e. [sent-27, score-0.653]
23 In contrast to using the perspective camera model in order to approximate refraction, refraction can also be modeled explicitly, where first a parametrization of the glass port of the housing needs to be found and calibrated. [sent-30, score-1.346]
24 [19] coming from the area of photogrammetry where the housing of a stereo rig can be calibrated. [sent-32, score-0.323]
25 [28] assumes a flat port interface with very thin glass and parallelism between glass and imaging sensor. [sent-34, score-1.082]
26 [1] showed how a more general camera with thick glass and a possible inclination angle between glass interface and imaging sensor can be calibrated, and Jordt-Sedlazeck et al. [sent-36, score-1.074]
27 Building upon a valid calibration of an underwater camera, meaning the intrinsics and a housing parametrization are known, several approaches to refractive SfM exist. [sent-38, score-0.986]
28 [4] proposed a method for refractive SfM, where the camera views a scene at the bottom of a pool through the water surface and the camera’s yaw and pitch with respect to the water surface are assumed to be known. [sent-42, score-0.958]
29 [16] showed results for 3D reconstruction with relative pose between two images with explicit incorporation of refraction. [sent-44, score-0.074]
30 They rely on outlierfree correspondences, which have to be selected manually and glass thickness is not modeled explicitly. [sent-45, score-0.329]
31 The system cannot handle image sequences and because of the use of the reprojection error during bundle adjustment, it cannot be extended easily. [sent-46, score-0.107]
32 Our Contribution: In this paper, we propose a more general method for refractive SfM that can evaluate video sequences with more general patterns of movement compared to [4]. [sent-47, score-0.245]
33 The main problem to overcome is that due to refraction, the computation of the refractive re-projection error is infeasible in large non-linear optimization problems like bundle adjustment [29]. [sent-48, score-0.414]
34 Therefore we propose a new error function that can be computed efficiently and even enables the analytic derivation of the necessary Jacobian matrices of the error function. [sent-49, score-0.088]
35 Finally, a refractive plane sweep proposed in [13] is used to estimate dense depth maps for each view, which are then used to create the final 3D model. [sent-51, score-0.306]
36 Controlled experiments show that the proposed method performs better than a comparable perspective method, where the refractive effect is only approximated. [sent-52, score-0.333]
37 Refractive Camera Model and Non-linear Error Function The camera model is the standard pinhole camera model with distortion [9, 27]. [sent-54, score-0.461]
38 Hence the camera’s intrinsics are defined in the camera matrix K containing focal length f, aspect ratio a, and a principal point (cx , cy), complemented by two coefficients for radial distortion r1 and r2 and two coefficients for tangential distortion t1 and t2. [sent-55, score-0.498]
39 The camera’s extrinsics are the rotation matrix R and the translation vector C, resulting in the projection matrix P = K[RT| − RTC]. [sent-56, score-0.088]
40 Refraction at the underwater housing is described by Snell’s law [11] and depends on the different medias’ indexes of refraction na for air, ng for glass, and nw for water. [sent-58, score-1.048]
41 As seen in Figure 1, the rays coming from the water do not intersect in the camera’s center of projection. [sent-59, score-0.529]
42 However, [1] determined that a camera behind a flat port underwater housing is an axial camera, i. [sent-60, score-1.119]
43 all rays coming from the water intersect a common axis defined by the camera center and the interface normal (blue glass rg, and air ra. [sent-62, score-1.457]
44 All ray segments together with the interface normal lie in a common plane, the Plane of Refraction (POR). [sent-63, score-0.515]
45 The blue line depicts the interface normal passing through the center of projection which is intersected by all rays rw [1] (dashed line). [sent-64, score-0.721]
46 The virtual camera’s center Cv is located at the intersection of the un-refracted ray (dashed line) and the interface normal (blue) and its focal length is fv = d. [sent-66, score-0.819]
47 Moreover, all segments of the light ray ra in air, rg in glass, and rw in water, and the interface normal n lie in one common plane, the Plane of Refraction POR (pale blue plane in Fig. [sent-69, score-0.896]
48 In order to back-project a ray from a 2D image point, the ray in air ra is determined using the perspective parameters explained above. [sent-71, score-0.612]
49 Then, the ray direction in glass rg is computed by [1]: rg = × (1) nngara+? [sent-72, score-0.699]
50 Using rg, ng, and nw, the ray in water rw is computed respectively. [sent-76, score-0.594]
51 Along with the interface distance d and the interface thickness dg, ra and rg allow to determine a starting point p of the ray rw on the outer glass plane (Fig. [sent-77, score-1.429]
52 (2) Hence for each pixel, a raxel [8] can be computed using the proposed parameters, instead of calibrating each raxel independently, which is often difficult. [sent-79, score-0.13]
53 Using the proposed parameter set, [1] derived two constraints for the flat port underwater camera. [sent-80, score-0.612]
54 The first one is called the Flat Refractive Constraint (FRC) and states that if a 3D point X has been transformed into the local camera coordinate system, its direction should be the same as the ray in water rw, hence: (RTX − RTC − p) From the POR follows that: (RTX 58 − RTC)T(n rw = 0 (FRC). [sent-81, score-0.867]
55 Virtual Camera Error Function When projecting a 3D point into a camera confined in an underwater housing with explicit refraction computation, Agrawal et al. [sent-85, score-1.257]
56 [1] determined that a 12th degree polynomial needs to be solved. [sent-86, score-0.025]
57 While this insight allows solving the projection problem much more efficiently than previous approaches, where usually the projection was determined by an optimization using the back-projection function [17], it is still infeasible in classic SfM, especially bundle adjustment. [sent-87, score-0.182]
58 It builds upon the idea in [24], where a virtual camera is defined for each 2D point into which the corresponding 3D point can be projected perspectively (Fig. [sent-89, score-0.402]
59 Note that a similar idea has been expressed in [23], however, the proposed method is adapted to the refractive case and is exact for each pixel. [sent-91, score-0.245]
60 The virtual camera error is computed using the ray in water rw and its starting point p as described above to define a virtual perspective camera. [sent-92, score-1.218]
61 The virtual rotation Rv is defined through its rotation axis, which is the cross product between interface normal and optical axis and its rotation angle, which is the scalar product between interface normal and optical axis. [sent-94, score-0.93]
62 Thus, a 3D point X in the global coordinate system is first transformed into a point in the local camera coordinate system Xl and then into the virtual camera coordinate system Xv by: Xl = RTX − RTC (5) Xv = RvTXl − RvTCv. [sent-96, score-0.699]
63 (6) The starting point on the outer interface is also transformed into the virtual camera: pv = RvTp − RvTCv. [sent-97, score-0.532]
64 (7) The error is then computed from the 2D projections of Xv and pv onto the virtual image plane: gv=? [sent-98, score-0.233]
65 (8) gv can be used as a non-linear error function for optimization with different parametrizations. [sent-101, score-0.092]
66 For example considering one camera and a set of n 2D-3D correspondences, when only extrinsic parameters and 3D points are unknown, the known ray in water and the virtual camera center are used: ? [sent-102, score-0.994]
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