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The geometry of binocular stereo is shown in Figure 111 The simplest model is two identical cameras separated only in the x direction by a baseline distance b The image planes are coplanar in this model A feature in the scene is viewed by the two cameras at different positions in the image plane The displacement between the locations of the two features in the image plane is called the disparity The plane passing through the camera centers and
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4::: ~i Epipolar lines Left image plane
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, , ,
! Left image ~ xl plane i
Right image plane Pr
Left camera Stereo lens center baseline Right camera lens center
Left camera lens center
Right camera lens center
Figure 111: Any point in the scene that is visible in both cameras will be projected to a pair of image points in the two images, called a conjugate pair The displacement between the positions of the two points is called the disparity
the feature point in the scene is called the epipolar plane The intersection of the epipolar plane with the image plane defines the epipolar line For the model shown in the figure, every feature in one image will lie on the same row in the second image In practice, there may be a vertical disparity due to misregistration of the epipolar lines Many formulations of binocular stereo algorithms assume zero vertical disparity 12 describes the relative orientation problem for calibrating stereo cameras Definition 111 A conjugate pair is two points in different images that are the projections of the same point in the scene Definition 112 Disparity is the distance between points of a conjugate pair when the two images are superimposed In Figure 111 the scene point P is observed at points PI and Pr in the left and right image planes, respectively Without loss of generality, let us
111 STEREO IMAGING
assume that the origin of the coordinate system coincides with the left lens center Comparing the similar triangles P MCI and PILCI, we get (111) Similarly, from the similar triangles P NCr and PrRCr, we get x - b x' = -'!: z f Combining these two equations, we get z= bf (x~ - x~)
Thus, the depth at various scene points may be recovered by knowing the disparities of corresponding image points Note that due to the discrete nature of the digital images, the disparity values are integers unless special algorithms are used to compute disparities to sub pixel accuracy Thus, for a given set of camera parameters, the accuracy of depth computation for a given scene point is enhanced by increasing the baseline distance b so that the corresponding disparity is large Such wideangle stereopsis methods introduce other problems, however For instance, when the baseline distance is increased, the fraction of all scene points that are seen by both cameras decreases Furthermore, even those regions that are seen by both cameras are likely to appear different in one image compared to the corresponding regions in the other image due to distortions introduced by perspective projection, making it difficult to identify conjugate pairs Before we discuss the problem of detecting and matching features in image pairs to facilitate stereopsis, we now briefly consider imaging systems in which the cameras are in any general position and orientation
Even when the two cameras are in any general position and orientation, the image points corresponding to a scene point lie along the lines of intersection between the image planes and the epipolar plane containing the scene point and the two lens centers as shown in Figure 112 It is clear from this figure that the epipolar lines are no longer required to correspond to image rows
Camera 2 focal point
Figure 112: Two cameras in arbitrary position and orientation The image points corresponding to a scene point must still lie on the epipolar lines
In certain systems, the cameras are oriented such that their optical axes intersect at a point in space In this case, the disparity is relative to the vergence angle For any angle there is a surface in space corresponding to zero disparity as shown in Figure 113 Objects that are farther than this surface have disparity greater than zero, and objects that are closer have disparity less than zero Within a region the disparities are grouped into three pools: