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Evaluation of the Viewpoint Shift for a Fisheye Lens based on Stereo Geometry Nobuyuki Kita Intelligent Systems Research Institute National Institute of Advanced Industrial Science and Technology (AIST ) Tsukuba, Japan [email protected] Abstract— Many studies have developed methods of making three-dimensional measurements from fisheye images. Most of these methods assume a single-viewpoint model. However, the viewpoint of a fisheye lens shifts along the optical axis according to the incident angle between the optical axis and the incident light ray. The author previously analyzed the error resulting from using the single-viewpoint model for three-dimensional measurements from fisheye images and clarified that the error is not negligible under some conditions. In particular, the errors in stereo measurements were remarkable. This implies that some knowledge about the viewpoint shift can be obtained from the stereo matching coordinates on stereo images. The present paper obtains an interesting relation of the shift magnitude at a couple of incident angles from a pair of coordinates. Moreover, the relations of shift magnitudes are integrated to obtain an important hint for the calibration of the viewpoint-shifting model. Keywords—fisheye lens; stereo geometry; viewpoint shift; calibration; I. INTRODUCTION Many methods of making three-dimensional (3D) measurements from images have been proposed. Most use the single-viewpoint (SVP) camera model where light rays traveling towards a fixed point make images, such as in the case of the pinhole camera. Recently, the usefulness of an image having a wide field of view has again been recognized and 3D measurement methods have been applied to such wide images [1-7]. In the case that the viewing field is not so large (e.g., 120 degrees in the horizontal), methods use the perspective projection model (f tan ) with radial distortion [8]. In the case that the viewing field is large (e.g., larger than 180 degrees in the horizontal), methods use different projection models, such as fθ and f sin [4,5,9,10]. Whichever projection model is used, the viewpoint is assumed to be fixed. Although the viewpoint is not fixed and shifts along the optical axis for actual lenses, the assumption is usually reasonable because the shift of the viewpoint is negligible against the distance to the target objects. The author previously analysed the error resulting from using the SVP model for 3D measurements from images that are captured by viewpoint-shifting cameras through simulation [11]. Such error is referred to as SVP approximation error. When targets are close to the camera and the viewpoint shift exceeds several tens of millimeters, as in the case of some fisheye lenses, the simulation clarified that SVP approximation error is not negligible. To eliminate SVP approximation error, Gennery [12] proposed the use of viewpoint-shifting camera models. He also proposed a calibration method for the viewpoint-shifting camera models. The accuracy of 3D measurements improved but the calibration method needed many calibration targets whose 3D positions have been accurately measured. Kumar [13] proposed a new viewpoint-shifting camera model not for eliminating SVP approximation error but for modeling the radial distortion of ordinary (non-fisheye) lens. The method also needed calibration targets whose 3D positions have been accurately measured. In [11], SVP approximation errors were analysed also for stereo measurements and they were remarkably larger than those for other 3D measurements. This implies that some knowledge about the viewpoint shift can be obtained from the stereo matching coordinates on stereo images. The present paper investigates stereo geometry considering a viewpoint shift. Light rays emanating from an object point enter left and right cameras with different shifts and according to the different incident angles and . From the projected coordinates in the left and right images, an interesting relation between and is introduced. By introducing properties of the relation between the incident angle and shift position such that the viewpoint position smoothly shifts forward according to the incident angle, the possible range of shift at a particular incident angle can be confined and an important hint for the calibration of the viewpoint shift curve is obtained. The remainder of the paper is organized as follows. Section II explains a viewpoint-shifting camera model while Section III investigates stereo geometry considering a viewpoint shift. Section IV explains how to confine the possible ranges of shift and presents results. Section V summarizes the findings of the paper. II. CAMERA MODEL Fig. 1 shows conceptual diagrams of an SVP camera model and a viewpoint-shifting camera model. While light rays traveling towards a fixed position on the optical axis form images for the former model, the light rays traveling towards This work was supported by a Grant-in-Aid for Scientific Research KAKENHI (16H02885)
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Evaluation of the Viewpoint Shift for a Fisheye Lens … of the Viewpoint Shift for a Fisheye Lens based on Stereo Geometry Nobuyuki Kita Intelligent Systems Research Institute National

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Page 1: Evaluation of the Viewpoint Shift for a Fisheye Lens … of the Viewpoint Shift for a Fisheye Lens based on Stereo Geometry Nobuyuki Kita Intelligent Systems Research Institute National

Evaluation of the Viewpoint Shift for a Fisheye Lens based on Stereo Geometry

Nobuyuki Kita Intelligent Systems Research Institute

National Institute of Advanced Industrial Science and Technology (AIST ) Tsukuba, Japan

[email protected]

Abstract— Many studies have developed methods of making

three-dimensional measurements from fisheye images. Most of these methods assume a single-viewpoint model. However, the viewpoint of a fisheye lens shifts along the optical axis according to the incident angle between the optical axis and the incident light ray. The author previously analyzed the error resulting from using the single-viewpoint model for three-dimensional measurements from fisheye images and clarified that the error is not negligible under some conditions. In particular, the errors in stereo measurements were remarkable. This implies that some knowledge about the viewpoint shift can be obtained from the stereo matching coordinates on stereo images. The present paper obtains an interesting relation of the shift magnitude at a couple of incident angles from a pair of coordinates. Moreover, the relations of shift magnitudes are integrated to obtain an important hint for the calibration of the viewpoint-shifting model.

Keywords—fisheye lens; stereo geometry; viewpoint shift; calibration;

I. INTRODUCTION Many methods of making three-dimensional (3D)

measurements from images have been proposed. Most use the single-viewpoint (SVP) camera model where light rays traveling towards a fixed point make images, such as in the case of the pinhole camera. Recently, the usefulness of an image having a wide field of view has again been recognized and 3D measurement methods have been applied to such wide images [1-7]. In the case that the viewing field is not so large (e.g., 120 degrees in the horizontal), methods use the perspective projection model (f tan𝜃𝜃) with radial distortion [8]. In the case that the viewing field is large (e.g., larger than 180 degrees in the horizontal), methods use different projection models, such as fθ and f sin𝜃𝜃 [4,5,9,10]. Whichever projection model is used, the viewpoint is assumed to be fixed. Although the viewpoint is not fixed and shifts along the optical axis for actual lenses, the assumption is usually reasonable because the shift of the viewpoint is negligible against the distance to the target objects.

The author previously analysed the error resulting from using the SVP model for 3D measurements from images that are captured by viewpoint-shifting cameras through simulation [11]. Such error is referred to as SVP approximation error. When targets are close to the camera and the viewpoint shift

exceeds several tens of millimeters, as in the case of some fisheye lenses, the simulation clarified that SVP approximation error is not negligible.

To eliminate SVP approximation error, Gennery [12] proposed the use of viewpoint-shifting camera models. He also proposed a calibration method for the viewpoint-shifting camera models. The accuracy of 3D measurements improved but the calibration method needed many calibration targets whose 3D positions have been accurately measured. Kumar [13] proposed a new viewpoint-shifting camera model not for eliminating SVP approximation error but for modeling the radial distortion of ordinary (non-fisheye) lens. The method also needed calibration targets whose 3D positions have been accurately measured.

In [11], SVP approximation errors were analysed also for stereo measurements and they were remarkably larger than those for other 3D measurements. This implies that some knowledge about the viewpoint shift can be obtained from the stereo matching coordinates on stereo images. The present paper investigates stereo geometry considering a viewpoint shift. Light rays emanating from an object point enter left and right cameras with different shifts 𝑠𝑠𝑙𝑙 and 𝑠𝑠𝑟𝑟 according to the different incident angles 𝜃𝜃𝑙𝑙 and 𝜃𝜃𝑟𝑟 . From the projected coordinates in the left and right images, an interesting relation between 𝑠𝑠𝑙𝑙 and 𝑠𝑠𝑟𝑟 is introduced. By introducing properties of the relation between the incident angle and shift position such that the viewpoint position smoothly shifts forward according to the incident angle, the possible range of shift at a particular incident angle can be confined and an important hint for the calibration of the viewpoint shift curve is obtained.

The remainder of the paper is organized as follows. Section II explains a viewpoint-shifting camera model while Section III investigates stereo geometry considering a viewpoint shift. Section IV explains how to confine the possible ranges of shift and presents results. Section V summarizes the findings of the paper.

II. CAMERA MODEL Fig. 1 shows conceptual diagrams of an SVP camera

model and a viewpoint-shifting camera model. While light rays traveling towards a fixed position on the optical axis form images for the former model, the light rays traveling towards

This work was supported by a Grant-in-Aid for Scientific Research KAKENHI (16H02885)

Page 2: Evaluation of the Viewpoint Shift for a Fisheye Lens … of the Viewpoint Shift for a Fisheye Lens based on Stereo Geometry Nobuyuki Kita Intelligent Systems Research Institute National

SVP camera model

Viewpoint-shifting camera model

Fig. 1. An SVP camera model and a viewpoint-shifting camera model.

Fig. 2. Examples of an image height and viewpoint shift curves.

shifted positions on the optical axis form images for the latter model. Both the image heights, which are the distances from optical centres on images, and the viewpoint shifts, which are the distances from reference viewpoints, follow curves according to incident angles.

Fig. 2 shows the two curves for an off the shelf fisheye camera whose viewing angle is 214 degree and image height curve is based on fθ and the view point shifts from about -6 mm to +18 mm. For the curve of the viewpoint shift, the reference viewpoint is the viewpoint for the light ray whose incident angle is 45 degrees. The curves were obtained by the calibration using precicely located calibration targets. We use these curves for simulation in the latter part of the paper.

III. STEREO GEOMETRY WITH A VIEWPOINT SHIFT Two viewpoint-shifting cameras that have the same image

height and viewpoint shift curves are set as shown in Fig. 3. The figure shows geometrical relations between the corresponding image points 𝑝𝑝𝑙𝑙 and 𝑝𝑝𝑟𝑟 on the left and right images and 3D light rays for the image points. 𝑂𝑂𝑙𝑙𝑋𝑋𝑙𝑙𝑌𝑌𝑙𝑙𝑍𝑍𝑙𝑙 and 𝑂𝑂𝑟𝑟𝑋𝑋𝑟𝑟𝑌𝑌𝑟𝑟𝑍𝑍𝑟𝑟 are the left and right camera frames whose origins

are at reference viewpoints and Z axes are the optical axes. 𝑜𝑜𝑙𝑙𝑥𝑥𝑙𝑙𝑦𝑦𝑙𝑙 and 𝑜𝑜𝑟𝑟𝑥𝑥𝑟𝑟𝑦𝑦𝑟𝑟 are the left and right image frames. Here we assume that the image planes are perpendicular to the optical axes for simplicity. The origins are at the position where the light rays incoming along the optical axes make images and the x and y axes are parallel to the X and Y axes.

Fig.3. Stereo geometry with a viewpoint shift.

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The light ray back-projected from the image point 𝑝𝑝𝑟𝑟 is represented as 𝑆𝑆𝑟𝑟 + 𝑡𝑡𝑟𝑟 ∙ 𝐿𝐿𝑟𝑟 . 𝐿𝐿𝑟𝑟 is a unit vector whose direction is decided by the angle 𝜑𝜑𝑟𝑟 , which is the angle between the 𝑥𝑥𝑟𝑟 axis and 𝑜𝑜𝑟𝑟𝑝𝑝𝑟𝑟������ , and the incident angle 𝜃𝜃𝑟𝑟 , which is determined from the image height ℎ𝑟𝑟. 𝑆𝑆𝑟𝑟 is a vector whose direction is the 𝑍𝑍𝑟𝑟 axis and magnitude is the viewpoint shift 𝑠𝑠𝑟𝑟 , which is determined from the incident angle 𝜃𝜃𝑟𝑟. 𝑡𝑡𝑟𝑟 is any real number larger than zero. The situation is the same for the left-image point 𝑝𝑝𝑙𝑙 . If the correspondence and all parameters are correct, the two light rays cross at a 3D point where the object point is located.

Let’s consider the geometrical relation when the same corresponding image points 𝑝𝑝𝑙𝑙 and 𝑝𝑝𝑟𝑟 are given and only the viewpoint shifts are unknown. The light ray back-projected from the image point 𝑝𝑝𝑙𝑙 is unknown but on the plane 𝜋𝜋𝑙𝑙 , which includes 𝐿𝐿𝑙𝑙 and 𝑍𝑍𝑙𝑙 as shown in Fig. 4. Assuming 𝑅𝑅𝑟𝑟 is the light ray when 𝑠𝑠𝑟𝑟 = 𝑠𝑠𝑟𝑟′ , which is a signed real unknown number, 𝑅𝑅𝑟𝑟 crosses with 𝜋𝜋𝑙𝑙 at a 3D point that is denoted P. The light ray for 𝑝𝑝𝑙𝑙 passes through the point P and makes an angle 𝜃𝜃𝑙𝑙 with 𝑍𝑍𝑙𝑙 as shown by a thick dotted arrow in Fig. 4. The shift for the incident angle 𝜃𝜃𝑙𝑙 is 𝑠𝑠𝑙𝑙′ in Fig. 4 for the left camera when the shift for the incident angle 𝜃𝜃𝑟𝑟 is 𝑠𝑠𝑟𝑟′ for the right camera. Because all parameters other than the viewpoint shifts are known, the 3D coordinates of the point P in the left camera frame can be calculated and the viewpoint shift 𝑠𝑠𝑙𝑙′ can be straightforwardly calculated from the coordinates. Because the viewpoint shift curves of the left and right cameras are the same, a pair of (𝑠𝑠𝑟𝑟′ , 𝜃𝜃𝑟𝑟) and (𝑠𝑠𝑙𝑙′, 𝜃𝜃𝑙𝑙) is drawn as a thick line on a graph as shown in Fig. 5. The graph shows such lines for the calculated pairs when 𝑠𝑠𝑟𝑟′ is sampled at the same interval

though it is actually continuous. We refer to the graph as the angle–shift relation graph in the latter part of the paper.

Fig. 6 shows the simulation results. First, twenty-four object points are arbitrarily located in the world frame and 24 pairs of corresponding image points 𝑝𝑝𝑙𝑙 and 𝑝𝑝𝑟𝑟 are calculated by using the image height and viewpoint shift curves in Fig. 2 for the three different stereo configurations, which are parallel, vergent and divergent as depicted in Fig. 7. Next, the angle–shift relation graphs for the 24 pairs of corresponding image points are drawn as explained above at 2-mm intervals between −20 and +40 mm for 𝑠𝑠𝑟𝑟′ though it is actually continuous and the range is between −∞ and +∞.

IV. CONFINING THE RANGE OF THE VIEWPOINT SHIFT This section confines the possible ranges of viewpoint shift

by defining and using properties of a viewpoint shift curve.

Fig. 4. The relation between 𝑠𝑠𝑟𝑟′ and 𝑠𝑠𝑙𝑙′.

Fig.5. An angle–shift relation graph.

Fig. 6. The angle–shift relation graphs by simulations.

Fig. 7. The three different stereo configurations.

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A. Properties of a Viewpoint Shift Curve Fig. 2 gave an example of a viewpoint shift. Viewpoint

shift curves for actual lenses can be found on the website [14] of the photographer Toscani, where -∆LPP is equivalent to our shift amount. He introduces one lens that has a completely opposite property of its viewpoint shift curve; see Fig. 16 of [14]. However, he also mentions that the viewpoint for general fisheye lenses moves forward according to the incident angle at just below of Fig. 7 of [14]. Here we assume that the viewpoint shift curve is monotonically increasing and convex downward like the curve in Fig. 2 and Fig. 9 of [14]. It is also possible to infer the forward and backward limits for the viewpoint shift position from the structure of a lens system.

B. Confining the Viewpoint Shift for Each Angle–shift Relation Graph The possible range of viewpoint shift is confined first

independently for each angle–shift relation graph. For the incident angles 𝜃𝜃𝑟𝑟 and 𝜃𝜃𝑙𝑙, right or left has no more meaning and whether the angles are small or large is important. Then we sort them as follows.

if 𝜃𝜃𝑙𝑙 > 𝜃𝜃𝑟𝑟 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝜃𝜃𝑏𝑏 = 𝜃𝜃𝑙𝑙 , 𝑠𝑠𝑏𝑏 = 𝑠𝑠𝑙𝑙′,𝜃𝜃𝑠𝑠 = 𝜃𝜃𝑟𝑟 , 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑟𝑟′

𝑒𝑒𝑒𝑒𝑠𝑠𝑒𝑒 𝜃𝜃𝑏𝑏 = 𝜃𝜃𝑟𝑟 , 𝑠𝑠𝑏𝑏 = 𝑠𝑠𝑟𝑟′ ,𝜃𝜃𝑠𝑠 = 𝜃𝜃𝑙𝑙 , 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑙𝑙′. (1)

As inferred from Fig. 6, for any angle-shift relation graph, 𝑠𝑠𝑏𝑏 and 𝑠𝑠𝑠𝑠 are linearly related to each other:

𝑠𝑠𝑏𝑏 = 𝑓𝑓𝑏𝑏(𝑠𝑠𝑠𝑠) = 𝑎𝑎𝑏𝑏 ∙ 𝑠𝑠𝑠𝑠 + 𝑏𝑏𝑏𝑏

𝑠𝑠𝑠𝑠 = 𝑓𝑓𝑠𝑠(𝑠𝑠𝑏𝑏) = 𝑎𝑎𝑠𝑠 ∙ 𝑠𝑠𝑏𝑏 + 𝑏𝑏𝑠𝑠 (2)

This can be simply proved and the demonstration is omitted here for brevity. 𝑎𝑎𝑏𝑏, 𝑏𝑏𝑏𝑏, 𝑎𝑎𝑠𝑠 and 𝑏𝑏𝑠𝑠 are obtained numerically.

Since the viewpoint shift position must monotonically increase according to the incident angle and stay between the forward and backward limits, 𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚, at any incident angle, the following condition must be satisfied.

𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑠𝑠𝑠𝑠 ≤ 𝑠𝑠𝑏𝑏 ≤ 𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚 (3)

Let’s represent maximum and minimum bound of the possible range of 𝑠𝑠𝑠𝑠 and 𝑠𝑠𝑏𝑏 by 𝑚𝑚𝑎𝑎𝑥𝑥(𝑠𝑠𝑠𝑠) , 𝑚𝑚𝑚𝑚𝑒𝑒(𝑠𝑠𝑠𝑠) , 𝑚𝑚𝑎𝑎𝑥𝑥(𝑠𝑠𝑏𝑏) and 𝑚𝑚𝑚𝑚𝑒𝑒(𝑠𝑠𝑏𝑏) . The values of 𝑚𝑚𝑎𝑎𝑥𝑥(𝑠𝑠𝑠𝑠) and 𝑚𝑚𝑎𝑎𝑥𝑥(𝑠𝑠𝑏𝑏) are initially set as +∞ and those of 𝑚𝑚𝑚𝑚𝑒𝑒(𝑠𝑠𝑠𝑠) and 𝑚𝑚𝑚𝑚𝑒𝑒(𝑠𝑠𝑏𝑏) are initially set as −∞, and those are confined by using condition (3). Fig. 8 shows examples of the confining process using condition (3). 𝑠𝑠𝑒𝑒 is the shift position when 𝑠𝑠𝑏𝑏 = 𝑠𝑠𝑠𝑠. For other cases, the confining processes are similarly deduced and omitted here.

C. Confining the Viewpoint Shift for a Pair of Angle–shift Relation Graphs For any two angle–shift relation graphs, graphs 1 and 2,

the possible ranges of the viewpoint shift are confined as follows. When a member of graph 1, (𝑠𝑠𝑠𝑠1,𝜃𝜃𝑠𝑠1), (𝑠𝑠𝑏𝑏1,𝜃𝜃𝑏𝑏1) , is

inside the possible range, there must exist at least a member for graph 2, (𝑠𝑠𝑠𝑠2,𝜃𝜃𝑠𝑠2), (𝑠𝑠𝑏𝑏2,𝜃𝜃𝑏𝑏2), which satisfies the conditions

𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑠𝑠1 ≤ 𝑠𝑠2 ≤ 𝑠𝑠3 ≤ 𝑠𝑠4 ≤ 𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚

∆𝑠𝑠1𝑠𝑠2����� ≤ ∆𝑠𝑠2𝑠𝑠3������ ≤ ∆𝑠𝑠3𝑠𝑠4����� (4)

Fig. 9 depicts the conditions. 𝜃𝜃𝑚𝑚(𝑒𝑒 = 1, 2, 3, 4) are the incident angles in increasing order of 𝜃𝜃𝑠𝑠1,𝜃𝜃𝑏𝑏1,𝜃𝜃𝑠𝑠2 𝑎𝑎𝑒𝑒𝑎𝑎 𝜃𝜃𝑏𝑏2, and 𝑠𝑠𝑚𝑚(𝑒𝑒 = 1, 2, 3, 4) denotes the corresponding shift positions. ∆𝑠𝑠𝑚𝑚𝑠𝑠𝑚𝑚+1��������� is the inclination of the line connecting 𝑠𝑠𝑚𝑚 and 𝑠𝑠𝑚𝑚+1.

Fig.8. An example of the confining process using condition (1).

Fig. 9. The condition (2).

Fig. 10. An example of the confining process using condition (2).

Page 5: Evaluation of the Viewpoint Shift for a Fisheye Lens … of the Viewpoint Shift for a Fisheye Lens based on Stereo Geometry Nobuyuki Kita Intelligent Systems Research Institute National

One example of the confining process using condition (4) is shown in Fig. 10. In that case, 𝑚𝑚𝑎𝑎𝑥𝑥(𝑠𝑠1) and 𝑚𝑚𝑎𝑎𝑥𝑥(𝑠𝑠2) are decreased and 𝑚𝑚𝑚𝑚𝑒𝑒(𝑠𝑠3) and 𝑚𝑚𝑚𝑚𝑒𝑒(𝑠𝑠4) are increased to satisfy condition (4) as shown by red dotted arrows.

After applying condition (3) to each graph, the possible ranges are updated by applying condition (4) to every combination of any two angle–shift relation graphs repeatedly until there are no further updates possible. Fig. 11 shows the final results after confining the graphs that were introduced in Fig. 6. Comparing with the viewpoint shift curve in Fig. 2 which was used for the simulation, the possible ranges of shifts are properly confined.

V. SUMMARY For a fisheye lens, the viewpoint shifts along the optical

axis. We examined the stereo geometry while considering the viewpoint shift and showed that a hint about the magnitude of the viewpoint shift can be obtained from corresponding image coordinates in left and right images.

Calibration methods that consider the viewpoint shift are now being developed along the lines of the present paper. Although several calibration methods consider a viewpoint shift [12][13], they all need calibration targets whose positions have been accurately measured. Our method will need only the image coordinates of corresponding points for two images instead of 3D coordinates of the targets. This is expected to drastically reduce the effort needed for calibration.

ACKNOWLEDGMENT The author would like to thank Dr. K. Yokoi, Dr. F.

Kanehiro and Dr. Y. Kita for their support in this research.

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[14] P. Toscani. Fisheyes. Available: http://www.pierretoscani.com/echo_fisheyes_english.html

Fig.11. The confined angle–shift relation graphs.