Figure 1. Epipolar curve generation in the right image from an image point in the left. TABLE 1. STRAIGHT LINE FITTING ORTHOGONAL DIRECTION RESIDUAL OF LOCAL EPIPOLAR CURVES Figure 3 shows the experimental results for RPC data. The curve at Figure 3b is the generated epipolar curve on the right image for the center point p of the left image. Two points q, and q, along the epipolar curve in Figure 3b are selected considering ground heights of 0 m and +1,000 m. The conjugate point of p should be somewhere between q, and q, along the epipolar curve and should be determined by its true ground height. Thus, epipolar curves for q, and q, are gener- ated using the ground height range (—15,000m to +15,000m) and plotted in Figure 3a. It should be noted that each curve is not a straight line, and the two curves are not identical even though they are not distinguishable in Figure 3a because of the scale. The difference will be shown later. Figure 2. Epipolar curve shape from RPC: (a) hyperbolic shaped curve, (b) magnified curve, and (c) shape changed after RPC update. Figure 4 shows the discrepancy between the two curves in the column direction. This epipolar curve set is generated in the same way that is explained above except that another two points q, and q, are selected using the ground height interval 0 m and +2,000 m for comparison. As seen in Figure 4a, there is not much discrepancy between two epipolar curves around the start point p, but the gap between the epipolar curves tends to grow as the distance from the image center increases. The discrepancy in the case of the larger ground height range set (0 m to 2,000 m) increases faster than in the case of the smaller height range (0 m to 1,000 m). The nonexistence of the global epipolar curve pair can also be identified. In Figure 4b the discrepancy is plotted only for the area corresponding to the ground height ranges, i.e., Om to 1,000 m or 0 m to 2,000 m. The difference is bounded to less than 0.1 pixels, which is sufficiently small to allow the two epipolar curves to be approximated as a single epipolar curve. In other words, the two epipolar curves on the left image shown in Figure 3a can be approxi- mated as identical within the given ground height range. The test results showing the epipolar pairs existing locally is also consistent with (Kim, 2000). Another important concern about the epipolar curve is that the epipolar curve pairs do not exist for the entire scene. To illustrate this property, the two image points q, and q2 on the right image, which correspond to the left image point p depending on the ground height range, are selected along the epipolar curve as can be seen in Figure 1. Then each right image point is used to generate an epipolar curve on the left image. Figure 3. Test result on the epipolar curve pair from RPC: (a) the left image showing the image center and the epipolar curves corresponding to the two selected points in the right image, and (b) the right image showing the epipolar curve from the image center of the left image and two selected points from the ground height (O m, 1,000 m). Figure 4. Discrepancy between two epipolar curves in the left image: (a) discrepancy for the global scene, and (b) discrepancy for the local area determined by the ground height range. Figure 6. Piecewise epipolar curve generation and image resampling from RPC. Epipolar Resampling Figure 5. Discrepancies in column direction around piecewise generated curve points. the y-parallax can be achieved by assigning a constant row coordinate value to each epipolar curve pair in both images. The linear relationship between the x-parallax and the ground height can be attained if the interval between the epipolar curve points is fixed in both images. It is because the epipolar curve points are computed from the fixed height interval in the proposed piecewise method. The constant interval can be obtained from the mean interpoint distance (between neigh- boring points along the epipolar curve) in both images. The point rearrangement yields many conjugate points between the original imagery and the epipolar resampled imagery for both left and right images. In addition, they are uniformly distributed, as a grid over the entire image. Therefore, using the conjugate points, the image resampling equation can be formed. In the study, high-order polynomial transformations, as Equation 2, are used because simple models, such as the affine model cannot account for the curve-to-straight-line transform: Figure 7. Piecewise epipolar curve pair generation from RPC. Figure 8. Piecewise epipolar curve point generation for epipolar resampling: (a) left image, and (b) right image. Figure 10. Example of interpolation for the epipolar resampling, i.e., from original image to epipolar resampled image. Following the resampling, RPCs for the epipolar resampled images have to be recomputed for stereo mapping. The new RPC can be estimated using the virtual GcPs by least squares Figure 9. Proposed epipolar resampling method from the generated epipolar curve image points: (a) left epipolar image, and (b) right epipolar image. TABLE 3. RPC ERRORS IN THE LEFT AND RIGHT IMAGE BEFORE AND AFTER THE UPDATE by correcting only the shift terms from four GcPs, as Toutin (2006) showed that the Ikonos RPC gave similar satisfactory results, which are comparable to Toutin’s 3D physical model when the RFM is adjusted with a shift value for a given height range. In contrast, QuickBird imagery needs to be refined with linear functions due to its relief dependency. Table 3 shows the RPC residuals before and after the compensation; note that RMS decreased significantly to the subpixel level. Figure 12. Piecewise generated epipolar curve points: (a) left image, and (b) right image (triangles indicate the starting point locations and dots are the piecewise generated epipolar curve points). ——e v/s rut ek eee ee Figure 13 shows the line fitting residual and the slope of the fitted line. It can be seen that the piecewise generated epipolar curve is not straight yet, and the higher-order polynomial fitting is expected to attain sub-pixel accuracy. Note that the global epipolar curve, created by the proposed method still cannot be approximated by a straight line for the entire image. On the slope graph, the epipolar curve index starts from the leftmost line in Figure 13b. It should be noted that the angle has a negative value because the row axis has ee To improve the georeferencing accuracy of the images, the test procedure begins with updating the RPC parameters using GCPs; note that the RPC accuracy is relatively poor at Ikonos product level 2. Dial and Grodecki (2002) and Fraser and Hanley (2005) described a model to compensate for and adjust RPC errors. It should be noted that the left sides in the model are bias- compensated image point coordinates. Since there are six unknowns, at least three GcPs are required for full parameter estimation. But for the case of estimating only shift terms, a single GcP is enough. In this study, the Ikonos RPCs are updated TABLE 4. RMS oF EACH POLYNOMIAL EPIPOLAR RESAMPLING EQUATION TABLE 5. Y-PARALLAX FOR TIE POINTS BY THE PROPOSED METHOD TABLE 7. GROUND RESTITUTION RMS BY THE PROPOSED METHOD TABLE 6. Y-PARALLAX FOR 16 MEASURED TIE POINTS BY THE PARALLEL PROJECTION Figure 13. Straight line fitting of the piecewise generated epipolar curve: (a) line fitting residuals for each epipolar curve, and (b) the line slope for each curve. Plate 1. Anaglyph image generated by the proposed method. Plate 2. Sub-images at corners of the anaglyph image.
