AIRBORNE LINEAR ARRAY IMAGE GEOMETRIC RECTIFICATION METHOD BASED ON UNEQUAL SEGMENTATION

As the linear array sensor such as multispectral and hyperspectral sensor has great potential in disaster monitoring and geological survey, the quality of the image geometric rectification should be guaranteed. Different from the geometric rectification of airborne planar array images or multi linear array images, exterior orientation elements need to be determined for each scan line of single linear array images. Internal distortion persists after applying GPS/IMU data directly to geometrical rectification. Straight lines may be curving and jagged. Straight line feature -based geometrical rectification algorithm was applied to solve this problem, whereby the exterior orientation elements were fitted by piecewise polynomial and evaluated with the straight line feature as constraint. However, atmospheric turbulence during the flight is unstable, equal piecewise can hardly provide good fitting, resulting in limited precision improvement of geometric rectification or, in a worse case, the iteration cannot converge. To solve this problem, drawing on dynamic programming ideas, unequal segmentation of line feature-based geometric rectification method is developed. The angle elements fitting error is minimized to determine the optimum boundary. Then the exterior orientation elements of each segment are fitted and evaluated with the straight line feature as constraint. The result indicates that the algorithm is effective in improving the precision of geometric rectification. * Corresponding author


INTRODUCTION
Airborne linear array pushbroom sensors are becoming increasingly of interest for multispectral and hyperspectral applications.The pushbroom images are obtained line by line with the time sequence.Each line is shot at the same moment with a central projection, so each line has its own exterior orientation elements.The geometric rectification of linear images is different from airborne planar array images or multi linear array images, because there are no geometric constraints among each scan line, and each line is independently acquired with a different exterior orientation, so a classic bundle adjustment is not realistic, owing to the huge number of unknowns.Usually, with the position and attitude data from the POS system (GPS/IMU) carried on the airborne platform, the geometric correction of linear array images can be achieved by the traditional collinearity equations (Li, 2014, Mostafa et al., 2001, Wang et al., 2011).In order to eliminate POS data errors, (Tuo et al., 2005) used POS data to coarse correct first, and then applied polynomial model based on the reference images for further correct.(Wang et al., 2013) presented a geometric correction method which can be used to correct the linear array image based on the bias matrix.These methods improve the absolute accuracy to some extent, but internal distortion is difficult to eliminate, especially for high-resolution images with high scanning frequency.The line feature along the direction of flight may still be some curved and serrated after geometric rectification.
Straight line feature-based geometrical rectification algorithm was applied to solve this problem (Lee et al., 2000), whereby the exterior orientation elements are fitted by piecewise polynomial and evaluated with the straight line feature as constraint.However, atmospheric turbulence during the flight is unstable, equal piecewise can hardly provide good fitting, resulting in limited precision improvement of geometric rectification or, in a worse case, the iteration cannot converge.To solve this problem, drawing on dynamic programming ideas, unequal segmentation of line feature-based geometric rectification method is developed.

Exterior orientation elements polynomial model
Each pushbroom linear array image line is central projection.The rigorous imaging model of the line array image is where

Airborne linear image unequal segmentation
The airborne linear image was divided into N segments.And the angle elements of each image segment were fitted, the fitting error is   , while i is the number of segments.So the total error of all the segments is computed as follows: (3) The geometric correction result will be best when total e is minimum.Based on dynamic programming ideas, using the polynomial model fitting error i e of each segment as the cost function, and the objective function can be simplified as follows:  For example，the image is divided into N segment as Figure 1.
The optimal solution of each segment can be written as: The right boundary of the first segment can be determined by 1 S , then the right boundary of the second segment can be determined by 2 S , and so on.The right boundary of the ith segment can be determined by i S , and the right boundary of the last segment is the last line of the image.

Processing Flow
In this paper, we use a frame orthophoto as reference image to get the control points and lines.The processing flow is as follows: (1) Extract corresponding points and lines from the linear array image and reference image.Determine the maximum segment number that can be divided by the number of feature points and feature lines.
(2) According to the maximum segment number and the image length, divide the image into equal segments, and the boundary is used as the initial value.The right boundary of the last segment is the last line of the image.
( (5) The objective function is minimized by multistage searching, and the right boundary of each segment is determined by multi iteration, ultimately leading to the optimum segmentation.

GEOMETRIC RECTIFICATION BASED ON POINT AND LINE FEATURE
Using a frame orthophoto as reference image, the linear array image is corrected based on point and straight line feature method.First, corresponding points and straight lines are extracted from the linear array image and reference image.Then, the image is divided into several unequal segments using the method mentioned above.Then the exterior orientation elements of each segments are fitted by polynomial models.
With the smoothing condition between adjacent segments as the constraint, a combined error equation based on point and straight line features can be formulated.At last, taking the straight line equation parameters, image coordinate of points in the straight line, image and object coordinate of feature points as the known numbers, and taking the parameters of the exterior orientation elements polynomial fitting model, the coordinate parameter of points in the straight line as the unknown numbers, solve the equation by adjustment, and get the exterior orientation elements of all the segments.
In this paper, the object space line can be determined by the reference image.We use the following form to express the line: So, all the points belonging to the feature straight line of linear array image are in line with this formula.For each point in the straight line, we only need to estimate the parameter u.
When the linear array image is divided into segments, exterior orientation elements of each segment use different polynomial fit models.The smoothing condition is used to avoid alignment errors between the adjacent segments after geometric rectification.Since the exterior orientation is continuous, the value of the function computed from the polynomial in each of two neighbouring segment is equal at their boundary, and smooth.
The exterior orientation elements may dramatically change when flying in bad circumstances.In order to fit the elements accurately, the segment should have a very short time interval.
Strong relativity in computing the exterior orientation elements is caused by the small image pieces.The method that line elements and angle elements are computed separately is used in this paper for the solution of exterior orientation.

EXPERIMENTAL RESULTS AND ANALYSIS
To validate efficiency of the method proposed, an experiment was carried out on the images acquired by a linear array sensor developed by the Academy of Opto-Electronics (AOE) on 2013, the flight area cover Neimenggu province of China.The image is 1825*1000 pixels with the GSD 0.2.The raw data is shown in Figure 2.  Relative precision evaluation is also taken by check lines.
Compute the coordinates (X, Y) of the point on the line which is closest to the point, the residuals for an estimated point along a check line are computed as relative precision.The evaluation results are shown in From Table 1 and Table 2, we can see that compared with the rectification result only using POS data, the RMS of check points drops from 0.46m to 0.23m on x direction, and 0.46m to 0.21m on y direction.And the RMS of the distance by which the point deviates from the straight line drops from 0.44m to 0.2m.
Three images chosen from different strips were used to verify the effect of unequal segmentation.The precision evaluation comparison of equal segments to unequal segments is shown in Table 3.

Figure 1 .
Figure 1.An example of segmentation ) Calculate the right boundary S i of the ith segment.At this time, the left boundary 1 S i  of the ith segment is determined by the previous stage calculation, and the right boundary 1 S i  of the segment No.n+1 is fixed by the initial value.Search the optimal solution within the range of the initial position.When the right boundary of the ith segment locates in line k belonging to the search range, use quadratic polynomials to fit the angle elements of the segment No.i and No.i+1, and calculate the fitting error of all the points in the segments as follow:  ,  ,  fitting error of point jThe overall fitting error of the segment No.i is: Determine the right boundary of all the segments using step (3), and calculate the overall fitting error of all the segments.
Figure 2. Raw images

Table 1 .
Absolute precision evaluation by check points(Unit: m)

Table 2 .
Relative precision evaluation by check lines (Unit: m)