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Camera Center Estimation Using Vanishing Points

Abstract:

This paper presents a new method for 3D camera calibration, to calculate the optical center of the camera. The proposed technique requires a single image yielding two vanishing points. A rectangular prism is employed as the calibration target to generate vanishing points. The special arrangement of the calibration object adds more accuracy in finding the intrinsic parameters. Based on the simple geometry of the perspective distortion of the edges of the prisms in the image, a relation has been formulated for the calculation of the coordinates of the optical center. Experimental results show the goodness of the proposed approach.

Fig. 1: Depiction of vanishing point in a perspective image.

Fig. 1: Depiction of vanishing point in a perspective image.

Key-Words: Camera calibration, vanishing points, horizon line, picture plane, focal length, center of focus.

(Note: This Paper was presented in ICSIP Please Download the paper for proper formatting, images and symbols)

Introduction:

The purpose of camera calibration is to establish the relationship between 3D world co-ordinates and their corresponding 2D image co-ordinates seen by the camera. Once this relationship is established, 3D information can be inferred from the geometric properties of the object. Importance of applications like, scene mosaicing and depth estimation is found from camera calibration. Approaches to camera calibration distinguish two steps in the calibration procedure, viz.

1. Intrinsic calibration - The determination of the pixel-to-ray mapping with respect to a co-ordinate system fixed to the camera. In other words, finding the principal point, focal length etc.

2. Extrinsic calibration - The determination of the rotation and translation that link the camera co-ordinate system to the world co-ordinate system. In other words, finding the pose of the camera.

Both intrinsic and extrinsic calibration methods have been examined by several authors. The classical approach [1] that originates from the field of photogrammetry solves the problem by minimizing a non linear error function. Due to slowness and computational burden of this technique, closed-form solutions have been also suggested [2][3]. However, these methods are based on certain simplification in the camera model, and therefore, they do not provide as good results as nonlinear minimization. There are various methods that use geometric objects whose images have some characteristics that are invariant to the actual position of the object in space and can be used to calibrate some of the camera parameters [4][5][6][7]. The use of vanishing points of parallel lines drawn on the faces of a cube is one of those and can be used to obtain the pose of the camera and some of the intrinsic parameters. A variety of methods based on this principle is explored in [4][8][9][10][11][12]. There are atleast 15 definitions for image center, which have been summarized in literature by Willson and Shafer [13]. These definitions include Center of radial lens distortion, center of field of view, center of perspective projection and center of expansion for focus and zoom. In this paper, we will be estimating the center of focus based on the known geometry of the rectangular prism which is considered as the calibration object and finding vanishing points with the help of this object itself.

Projective geometry is a geometry unifying Euclidean and non-Euclidean geometry and was developed by mathematicians like Lobachevski, Rieman, Gauss and axiomatized by Hilbert. Compared to the work done by Benosman et. al. [14] our approach differs by the fact that our work is based on computation of vanishing points and on the other hand, we use a single camera to know the center of focus of the camera, where as in the other approaches including the work of Benosman et. al., calibration is based on the theory of projective vectors but uses simple geometry based on colinearity and orthogonality of vectors existing in the scene.

It is necessary to fix up the camera position or in other words it is required to know the extrinsic parameters of the camera. If the camera center is known, the orientation of the objects with respect to the camera center and image plane, it is easy to find out the dimensions of the object, based on analytical calculations. This is the importance of finding the camera center in the intrinsic parameter estimation of a camera.

This paper is divided into following sections. The section that follows this introduces the basic setup of the experimentation. Followed by the section presenting the calibration method and introduces the mathematical preliminaries. Subsequent sections describes about the results and concluding remarks.

Experimental setup:

The following basic setup is considered to be made in order to carry out our experimentation.

1. Images of rectangular prisms are considered as calibration objects taken for camera calibration.

2. The calibration objects are so aligned while image acquisition that the image yields two vanishing points for this experiment. This is for the reason that, we get a clearer horizon line which acts are the eye height from the ground level. Another reason being, the horizon line always passes through the center of vision [17]. Ultimately it is required to find out this point on the horizon line.

3. Known angle orientation of the objects to the picture plane is considered. The calibration objects that is the faces of the rectangular prism facing the camera make certain angle to the picture plane. This angle is supposed to be known in order to proceed with the experimentation.

With the above settings, the estimation of the optical center of the camera is analytically calculated in the manner explained in the section that follows.

Method:

The picture plane [15] is also called as the image plane. Pixel representation of an image is considered with the matrix convention of reading the index of any co-ordinates. Where as in the regular convention of the co-ordinate system the abscissa with increasing values from left to right and ordinate values increasing from bottom to top is considered. Since both the indexing are in opposite direction in the y-axis, it is required to bring the image plane into the regular convention before further calculations. For this, (r, c) being the point of consideration in an image of size (h, w) should have (h-r, c) in the new space and there on (h-r, c) shall be considered instead of (r,c).

Fig. 1: Depiction of vanishing point in a perspective image.

In this work, manual vanishing point detection is considered [16]. Since the vanishing point detection technique is a manual procedure, the points to be picked up for the calculation are selected carefully. They are the end points of the parallel lines of the object which are actually perspectively distorted in the image are identified as (x₁, y₁), (x₂, y₂), (x₃, y₃) & (x₄, y₄) as shown in the figure. Due to the reason of perspective distortion, the lines shall be meeting at the vanishing point (X,Y) as shown in fig. 1. The vanishing point is given by the relation for X and Y as follows.

(1)

(2)

Where m₁ is the slope of the line joining points (x₁, y₁) & (x₂, y₂) and m₂ is the slope of the line joining points (x₃, y₃) & (x₄, y₄).

It is required to find all the vanishing points in the image. Images which yield only two vanishing points are considered and hence we get two points which can be joined to form a line. This line joining the vanishing points is called the horizon line. A line parallel to the horizon line is constructed which is at a convenient distance for clarity purpose. This shall be called as the Picture plane (image plane) and is the top view. Thus the vanishing points plotted on the image are also projected back on to the picture plane [17]. The station point ‘S’ is the eye position when viewed from top.

Fig. 2: Depiction of Station point from the top view.

Fig. 3: Points depicted on the top view and their notations.

Objects beyond the picture plane are oriented in some angles. i.e. the faces of the rectangular prisms that are considered makes angle ? and ?. When this is viewed from top, it is visible as shown in figure 2. Station points are fixed up based on the equation given.

(3)

(4)

Where (x_t,y_t) are the co-ordinates of the station point, (x₁,y₁) & (x₂,y₂) are the co-ordinates of the vanishing points on the picture plane. ‘m’ is the slope of the line joining these two vanishing points. This is given by,

(5)

(6)

Equation of line passing through (x₁,y₁) with an inclination of ? to the picture plane is given by

(7)

(8)

Similarly the equation of line passing through (x₂,y₂) with an inclination of ? to the picture plane is given by

(9)

(10)

From the station point the shortest distance to the picture plane, is the focal length of the camera. This is determined by the co-ordinates of the foot of the perpendicular of station point over the picture plane. This can be determined by the vector property of orthogonality.

Thus, distance formula of points P(X,Y) to S(x_t,y_t) is given by the relation

(11)

At the intersection of the two lines, viz. the line joining the points P(X,Y) and S(x_t,y_t) and the horizon line is calculated. This is the point of Center of Focus ‘C’. This is again given by the equation for intersection of lines, viz. equations 1 and 2. But here in this case, (x₁,y₁), (x₂,y₂) are replaced by P(X,Y), S(x_t,y_t) and (x₃,y₃), (x₄,y₄) are replaced by VP1(x,y),VP2(x,y) respectively. Center of focus is also the optical center of the camera.

Results:

In the previous sections, relationship for the coordinates for the camera center was derived based on the analytical geometry based calculations. Upon the subjecting to the three different objects shown in figure 4, 5 & 6 with different orientations in 30 degrees, 45 degrees and 60 degrees to the picture plane, the following outputs are obtained as shown in table 1 and 2. The resolution of the image captured is 640 X 480. The camera used to capture these images is Sony Handycam DCR-HC40E.

Focal Length	30	45	60
Prism 1	695	700	705
Prism 2	699	701	704
Prism 3	709	701	705
Table 1: Depicting focal length of the camera in pixel units.

Camera Center	30	45	60
Prism 1	(336, 227)	(337, 228)	(337, 229)
Prism 2	(337, 227)	(337, 227)	(337, 229)
Prism 3	(339, 230)	(337, 227)	(337, 229)
Table 2: Camera center estimation results obtained.

The median among the measures of central tendency, (337,228) is the most valued average among the set of points obtained. In figure 7 the center of the crosshair depicts the center of vision or the center of focus or the optical center of the camera.

Fig. 7: Depiction of Center of Focus which is not exactly the center of the image.

Conclusion:

A new approach to camera calibration based on the use of the vanishing point is proposed in this paper. A rectangular prism is adequate for the calibration purpose. The focal length along with the optical center of the camera is found out in this approach. The two vanishing points results in the horizon line. With the help of this horizon line, a top view of the image plane is constructed. Onto this plane, all the visual rays are projected from the station point. This results in the calculation of the center of focus, by projecting the station point over the horizon line. The computation is analytical and thus no iteration is necessary. This speeds up the calibration work.

The proposed method is appropriate for indoor computer vision applications like robot navigation and location because of its simplicity of environmental setup.

References:

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(Note: This Paper was presented in ICSIP Please Download the paper for proper formatting, images and symbols)

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