ON PROPERTIES OF AUTOMATICALLY MEASURED TIE POINT OBSERVATIONS

Transcription

1 ON PROPERTIES OF AUTOMATICALLY MEASURED TIE POINT OBSERVATIONS Eija Honkavaara and Juha Jaakkola Finnish Geodetic Institute Geodeetinrinne 2 FIN-243 Masala Finland Eija.Honkavaara@fgi.fi, Juha.Jaakkola@fgi.fi Commission III, Working Group III/1 KEY WORDS: Automatic tie point extraction, Aerial triangulation, Block adjustment, Quality, Accuracy, Reliability, Weight ABSTRACT In this article results of an investigation concerning usage of automatically measured tie point observations in aerial triangulation are given. Especially the effect of amount and accuracy of tie point observations on the block quality has been concerned. Simulation and empirical techniques have been used. Simulation study proved that increase in the amount of tie points improve the quality of aerial triangulation, provided that correct functional and stochastic models are used in the block adjustment. The quality improvement was especially seen as increasing accuracy of orientation parameters and decreasing sensitivity of orientation parameters on the tie point measurement errors. Based on simulation, suggestion for sufficient amount of tie points in typical conventional photogrammetric cases was derived. Empirical investigation showed that when using large amount of automatically measured tie points a high accuracy can be achieved. However, results of the empirical test blocks gave an indication that proper handling of systematic image deformations and the stochastic model of adjustment may get more importance when more observations are used. 1 INTRODUCTION Automatic tie point measurement has been accepted as a part of digital photogrammetric production line during the past few years. Examples of commercial systems are MATCH-AT (Krzystek et al., 1996) and PHODIS AT (Tang et al., 1997). In these systems tie point extraction is typically performed as a separate process and followed by the block adjustment. In this article the automatic tie point measurement process is simply considered as a process which measures a certain amount of homologous points in interesting positions of the overlapping images, using a strategy and an image matching method. Good reviews of the concept of automatic tie point measurement have been given by (Förstner 1995 and Schenk 1997). Investigations about the subject have been concentrated mostly on the methods to solve the task. When concerning the quality the basic idea is that a huge amount of tie points is measured to achieve high quality. A winged sentence concerning the amount and quality of tie points is From quality to quantity (Schenk 1997). Hypothesis is that the method which is used to provide the tie points does not affect on the quality of aerial triangulation. The same quality can always be achieved by measuring sufficient amount of tie points with good enough geometry and handling them properly in the block adjustment. Another question is, of course, are all the matching methods capable to generate the sufficient amount of tie points. Some empirical investigations have already shown that by automatic tie point measurement good accuracy can be achieved (Krzystek et al., 1996, Tang et al., 1997). Theoretical expectation is that automation improves the quality of aerial triangulation. In this article effect of using automatically measured tie point observations on aerial triangulation is investigated. Focus is on the amount and accuracy of tie points. Basic theory about quality of aerial triangulation and properties of automatic tie point measurement are shortly discussed in chapter 2. Quality of aerial triangulation when using automatic tie point observation is investigated by simulation and empirically. Test set up is described in chapter 3. Results of simulation study are given in chapter 4 and of empirical study in chapter 5. 2 EFFECT OF AMOUNT AND ACCURACY OF TIE POINTS ON THE QUALITY OF AERIAL TRIANGULATION 2.1 Factors affecting on the quality of aerial triangulation Aerial triangulation is an indirect method to solve orientations of block of aerial images and ground coordinates of unknown points. The task is solved using the non-linear weighted least squares method, as well known. The basic functional model used is the collinearity equation, completed by additional parameters to handle distortions. Stochastic model consists of weights for the observations, ensuring the LS estimators to be BLUE. There are several important factors affecting on the quality of aerial triangulation, including 1. geometry of the block 2. quality of the images 3. accuracy, amount, geometry and distribution of tie point observations 4. accuracy, amount, geometry and distribution of auxillary observations (including ground, GPS and INS control) 5. block adjustment method, including functional and stochastic models and error detection methods

2 Tie point measurement interacts, more or less, with each one of these factors. This article deals with topics 3 and Measures of block quality Quality of photogrammetric blocks has been thoroughly discussed by Förstner (Förstner 1985, 1987). The quality concept consists of two components, i. e. precision (accuracy) and reliability. Estimates for the precision of the estimates of unknowns can be computed by error propagation from the variance covariance matrix of the unknowns, i. e., ( ) V ( x) = Σ A P A = σ = σ 2 Q, (1) xx T 2 1 ll xx where P ll is the weight matrix of the observations l, A is the design matrix and σ is the estimate for the standard error of unit weight. Standard deviation of the parameter estimate is σ = σ q V ( x = ) (2) x x x i i i i Reliability can be divided to two parts, internal and external reliability. The internal reliability concerns the ability to detect gross errors. The external reliability concerns the influence of non-detected gross errors on the parameter estimates. (Förstner 1985, 1987). Measures for the reliability are: Internal reliability, the redundancy number r i of an observation l i. The redundancy number tells about the contribution of observation l i to the total redundancy. Redundancy number of observation l i is computed by where ( ) r = Q P, (3) i vv ll ii ( ) Q = Q A A P A A vv ll T 1 T ll is the weight coefficient matrix of the residuals v. Förstner (1985) has ranked the redundancy numbers as good ( r i >.5), acceptable (.1 r i.5), bad (.4< r i <.1) and not acceptable ( r i <.4). Internal reliability, the controllability factor δ i. Controllability factor gives the lower bound for detectable gross error, meaning that gross errors in l i smaller than δ i -fold standard deviation can not be detected by a statistical test. Computational formula for controllability factor is δ i 1 (4) = δ / r. (5) δ is so called non-centrality parameter, defining the significance and the power of the statistical test used in gross error detection. Good value of δ i 6 δ i <12. ii is <6 and acceptable External reliability, the sensitivity factor δ i. Sensitivity factor gives the upper bound for the effect of non-detectable gross error on the parameters. The adjusted parameter is not contaminated more than δ i times its standard deviation if statistical test has been used. δ i can be computed separately for different parameter sets, like for unknown orientation parameters. Effect of an gross error in the observation l i on the unknown orientation parameters is computed by where δ, i i = δ u, (6) t i ( ) ut = B B P B B P i 1 T ll T ll ii is the contribution of an observation l i to the determination of the orientation parameters. B is the part of the design matrix referring to the transformation parameters. Correspondingly, the effect on the unknown co-ordinates can be computed by replacing u ti by u = 1 r u or on the all unknowns by ki i ti 1 r i. Good values for δ i are <4 and acceptable 4 δ i <1. When concerning the quality of aerial triangulation with automatically measured tie points, the primary interest is the quality of orientation parameters, i. e. standard deviation of orientation parameters (equation 2) and sensitivity of orientation parameters on the tie point measurement errors (equation 6). 2.3 On the effect of amount and quality of tie points Amount of tie points In traditional photogrammetry the number of tie points has been minimised so that the result has still been accurate and reliable enough. In practise typically 12-2 points/photograph have been measured as accurately as possible. When automatic tie point measurement is concerned, amount of tie points is not an economic question anymore. So, quality of aerial triangulation can be improved by increasing the amount of tie point observations, as has been realised by several authors (Förstner 1995, Schenk 1997, Tang 1997). Examples of amount of tie points in commercial systems are 1-6 (Krzystek et al., 1996) and 2-5 (Tang et al., 1997). The factor for accuracy improvement, when making n times more measurements, is n. For instance, the same accuracy is achieved when measuring 25 tie points with positioning error of 5 µm and 1 points with positioning error of 1 µm. The increasing number of tie points affects on the reliability of the block so that the sensitivity of orientation parameters on the tie point measurement errors decreases, as the contribution of single observation on the determination of orientation parameters decreases Quality of tie point observations When concerning quality of tie point observations, again concepts of accuracy and reliability have to be considered see (Förstner 1985). Here only accuracy is further elaborated. From the previous it can be concluded the accuracy of tie point observations is no more a problem because lower accuracy of the observations can compensated by increasing the amount of observations. However, it is important to know the accuracy of the measurements in order to give them correct weights in the block adjustment. Important questions are 1) what is the standard deviation of observation errors, 2) is there (7)

3 heteroscedasticity, i. e. observations with different standard deviations and 3) are there systematic measurement errors. As it is known form the theory of the LS-method, heteroscedasticity disturbs its performance. When there are equally weighted observations which in reality have different variance the LS method will give more weight on the observations with large variances, resulting the best fit on the large variance portion. Anyhow, the LS method gives BLUE estimates also in presence of heteroscedasticity, if the observations are weighted correctly. Effect of incorrect weighting is that although the solved parameters are unbiased and consistent, they are not the minimum variance estimates. In addition, the estimated variances of the estimated parameters will be biased estimators of the true variances resulting the statistical tests and confidence intervals to be incorrect. There are several factors affecting hetroscedasticity on tie point observations. It has been considered that two of them have practical importance (Sarjakoski, 1988): 1. Varying quality of measured objects. 2. Location of the image point. Varying quality of object points has been treated in photogrammetry with interactive observations so that the operator has measured as good points as possible, resulting a low level of heteroscedasticity, and given quality figures for different type of objects for weighting purposes. In automatic matching both selection of good objects and giving quality measures may be difficult. Matching methods can give quality figures for goodness of match but usage of them in practice has not been reported so far in literature. Heteroscedasticity due to location of the image point has been treated in traditional photogrammetry by using so called variance component estimation. Distance of the image point from the principal point of the image has been used to define location dependent variance (Sarjakoski, 1988). This kind of approach could be applicable also in automatic case, at least when concerning conventional aerial images. In our believe usage of variance components has not been very popular in practice. However, in automatic case when huge amount of points is measured it may be necessary to treat location dependent variability correctly. As a result of previous discussion it can be assumed that the accuracy of automatically measured tie points varies more than accuracy of interactively measured and reconsideration of stochastic model of adjustment may be necessary. 3.1 Objectives 3 TEST SET UP Objectives of the investigation are to see how the amount and accuracy of the tie points affect the quality of aerial triangulation. In the simulation study the variables are amount, accuracy and heteroscedasticity of the tie point observations. In the empirical investigation the variables are amount of tie points and the functional and stochastic model of block adjustment. 3.2 Blocks Simulated block Geometry of the simulated block is like the geometry the empirical test block Lahti: Block name: Simulation Focal length: 15 mm Flying direction: North-South Images: 4 strips with 13 images Photo scale: 1:3 Flying height: 4 5 m Overlaps (end lap,side lap): 6%, 3% Terrain type: Plane, ground height m 16 uniformly distributed points were used as ground control points (GCP) and 85 uniformly distributed points as check points (CP). Generated errors were N (, σ ) distributed random errors. Systematic errors were not considered. Results of simulation are averages of ten simulation rounds performed by independent data sets. Two types of tie point distributions were used: optimised and uniform. In the optimised distribution 8% of the tie points are located on the overlap areas of the strips in the direction perpendicular to the flying direction. In the flying direction distribution is uniform. This optimisation results in a very stable block geometry. In the uniform distribution uniform grid of points is generated on the object space and image co-ordinates are computed Empirical blocks In the empirical investigation two test blocks are used: OEEPE test block Forssa, see also (Jaakkola et al., 1996) and test block Lahti, see also (Bilker et al., 1998). Technical details of the blocks are as follows: Image information: Block name Forssa Lahti Date of flight: May 3 rd, 1989 July 2 th, 1997 Camera: Wild RC 2/23 Leica RC3+PAV3 Focal length: mm mm Flying direction: East-West North-South Strips: 4 4 Images/strip: 7 13 Photo scale: 1:4 1:31 Overlap (fw,side) 6%, 2-4% 6%, 3% Pixel size: 3 µm 25 µm Terrain type: Rural area Varying (city, lake, agricultural, forest) Ground control information: Block name Forssa Lahti Type: Targeted, cross Targeted, square Accuracy: - ground - image σ GCP _ XYZ = 15. cm σ GCP_ xy = 6µ m σ GCP _ XYZ = 5cm σ GCP_ xy = 5µ m GCP: 14 XYZ on the block borders CP: 5 XY + 41 Z 31 XYZ 16 XYZ with uniform distribution

4 3.3 Co-ordinate system Right handed coordinate system is used. In the ground coordinate system X goes to east and Y to north. In the image coordinate system x points to the flying direction and y to the perpendicular direction. 3.4 Quality estimates Following quality estimates have been derived: Theoretical accuracy. Standard deviations of unknown parameter estimates are derived from the variance covariance matrix of unknowns, formulas 1-2. Reliability. Redundancy number and controllability and sensitivity factors of observations, formulas 3-7. δ was given a value of 4, giving significance level of 99% and power of 93%. Empirical accuracy. After the block adjustment object coordinates of check points are computed by forward intersection using the solved orientation parameters. RMS value of the differences to the correct ground co-ordinates is computed. In the simulations the value is average of 1 simulation rounds with independent data sets. 3.5 Block adjustment program In the block adjustments ESPA bundle block adjustment program is used (Sarjakoski 1988). The block adjustment is performed using weighted least squares method. Weight, w i of image observation l i is determined from the a priori standard deviation σ li error of unit weight σ as follows w i = σ σ l i 2 of l i and standard Weights can be assigned to different groups of observations. 3.6 Program for automatic tie point extraction Automatic tie point extraction system has been under development at the FGI since 1994 (Honkavaara et al., 1996). The current version of the system is able to generate any kind of reasonable tie point distribution. The system uses information on orientations and ground heights in generating approximate locations for tie point measurement. If the approximate values are not good enough they can be improved automatically by the system. The point transfer and actual tie point measurement is performed using point based multiple image feature based matching and LSM matching between image pairs. Both of the (8) methods assume a locally planar surface, so the transformation between the images is the affine transformation. In LSM two radiometric transformation parameters are used to adjust radiometric differences between image windows (shift and linear term). LSM is used in final matching. A predefined amount of tie points is measured. Then all the observations are taken to a bundle block adjustment. Because the current version of the system does not give any information about quality of matching, weight of 1 is assigned to all tie point observations. Weights of GCP image and ground observations are determined based on a priori information. As robust adjustment for outlier detection is not available in the system, bad observations are rejected based on residuals. The rejection limit is gradually decreased until no residuals bigger than rejection limit exists. Typical sequence of rejection limits is 1, 9, 8, 7, 6, 5, 4, 35, 3, 25 and 2 µm. This is a coarse strategy to reject outliers, but in our believe acceptable, because there are huge amount of observations. After block adjustment the quality of the block is checked. Each image is divided into a 3x3 grid. Requirement is that there are enough links both in strip direction and between the strips in each of the squares. Problem areas are checked interactively. Typical requirement is minimum of 1 links per square. 4 RESULTS OF THE SIMULATION STUDY 4.1 Effect of accuracy and amount of tie point observations on the accuracy of the block Effect of amount and accuracy of tie points was investigated by generating a varying number of tie points with different standard deviations. Distributions of tie points were 5x5, 1x1, 15x15 and 2x2, resulting in 25, 1, 225 and 4 points per image. In most of the cases the optimised distribution was used, but in case 2x2 also the uniform distribution was generated. Standard deviations of generated errors correspond to a quite typical photogrammetric case: GCP: σ GCP_ XYZ = scale * 5µ m = 15 cm, σ GCP_ xy = 5µ m CP: σ _ = 5µ m CP xy Tie points: case s5: σ = 5 µ m, and case s1: σ = 1µ m tie Minimum for the theoretical error of orientation parameters and empirical error of check points was computed using 25 errorless tie points per image. Values were.25,.2 and.25 m for X, Y, Z and.15,.12 and.27 m for check points X, Y, Z. Thus the empirical accuracy of check points will show the quality improvement only to a limited extent. But this is the situation also in practise. Difference of accuracy in X and Y direction is resulted by the block geometry. tie

5 .6.5 a) X Y Z.6.5 a) Error (m) Number of tie points Minimum 1, s5 2 4, s5, uni 4, s5 25, s1 1, s1 225, s1 4, s b) X Y Z b) Number of tie points Figure 2. Combined effect of GCP error and amount of tie points on theoretical accuracy of a) tie point X coordinate and b) perspective centre X coordinate. σ = 1 µ m. tie , s5 1, s5 225, s5 4, s5, uni c) 4, s5 25, s1 1, s1 225, s1 4, s1 X Y Z in the cases with 1 points of accuracy 5 µm and 4 points of accuracy 1 µm are practically the same. Case with 5x5 tie points differs form the other cases with optimal distribution. Obviously geometry is slightly different. The accuracy of the uniform case is slightly worse than of the optimised case. The case with 25 tie points with error of 5 µm comes closest to the traditional case. Empirical accuracy of check point coordinates in X, Y and Z is about.19,.16 and.36 m, giving a difference about 3% to the minimum error. When tie point error is 5 µm, empirical accuracy of check points in the case with 225 tie points differs about 1% and the case with 4 tie points less than 1% from the limiting accuracy. However, the accuracy of orientations is far from the minimum value in all of the simulated cases..1 Minimum 1, s5 2 4, s5, uni Figure 1. Effect of amount and accuracy of tie points on a) empirical accuracy of check points, b) theoretical accuracy of tie point ground co-ordinates and c) theoretical accuracy of perspective centres. Notation: : 25 tie points with σ = 5 µ m per image; uni: uniform distribution. tie Effect of amount and accuracy of tie points on empirical accuracy of check points as well as on theoretical accuracy of tie point ground co-ordinates and perspective centre coordinates is shown in figure 1. The block behave according to theory. For instance, accuracy of orientations and check points 4, s5 25, s1 1, s1 225, s1 4, s1 Combined effect of GCP error and amount of tie points on the theoretical accuracy of the tie point X coordinate and the perspective centre X coordinate is shown in figure 2. GCP standard deviation is.5-1 m and the tie point standard deviation is 1 µm. Accuracy of the other unknowns behave on a similar way and is not shown here. It can be seen that accuracy improvement when increasing number of tie points is independent of the GCP accuracy. Naturally the relative improvement of the accuracy is much bigger when GCP accuracy is better. From these results a conclusion can be drawn that in typical photogrammetric applications with geometry similar to the simulated block and random measurement errors, 1 tie points per image give already a good accuracy and 2-4 tie points per image gives close to a best achievable object accuracy, when the tie point measurement accuracy of 5 µm. If standard

6 .6 (a) 7. (b) r, tiep x r, tiep y In, tiep x In, tiep y , s1 1, s5 1, s , s1 4, s5, uni 4, s5 4, s1 25, s1 1, s5 1, s , s1 4, s5, uni 4, s5 4, s1 2.5 (c) 4.5 (d) Ext, ori x Ext, ori y Ext, XYZ x Ext, XYZ y , s1 1, s5 1, s , s1 4, s5, uni 4, s5 4, s1 25, s1 1, s5 1, s , s1 4, s5, uni 4, s5 4, s1 Figure 3. Reliability of tie points. Effect of amount and accuracy of tie points on the average a) redundancy number, b) controllability factor, and c) sensitivity factor of perspective centres and d) tie point ground co-ordinates for the tie points. Redundancy r, GCP X r, GCP Y r, GCP Z r, GCP x r, GCP y deviation σ tie of tie points is different, this amount should be adjusted by the factor of ( σ tie / 5) 2. 1, s5 1, s1 4, s5, uni 4, s5 4, s1 Figure 4. Effect of amount and accuracy of tie points on the GCP ground and image observation redundancy. 4.2 Effect of accuracy and amount of tie point observations on the reliability of the block Same test set up as in the first part of section 4.1 was used in investigating the effect of amount of tie points on the reliability of the block. Results are given in figure 3. These numbers are mean values for tie points over the whole block area. Shown are the redundancy number ( r), controllability factor (In) as well as sensitivity factors of orientations (Ext, ori) and tie point ground coordinates (Ext, XYZ). Block geometry is good. In the blocks with optimised tie point distribution mean values of redundancy of tie points come in x and y close to.5 and.6. In the block with uniform tie point distribution redundancy in x is slightly worse, about.4. Behaviour of block with 25 points differs slightly from the other blocks with optimised tie point distribution. Obviously the geometry is different. The amount of tie points does not affect on redundancy, controllability or sensitivity of tie point ground co-ordinates, but only on the sensitivity factor of orientation parameters, provided that geometry of the block does not change. Mean sensitivity factor of orientations gets smaller than.5 when there are more than 4 tie points per image. Effect of increasing the number of tie points on the redundancy of ground control ground and image observations is shown in figure 4. Result is that if number of tie points increases and weights do not change, ground control redundancy increases. However, if weights are decreased in relation to amount of tie points, redundancy does not change. For example, ground control redundancy is same for cases 4 tie points of accuracy 1 µm and 1 tie points of accuracy 5 µm. Because controllability and sensitivity are dependent on the redundancy, they improve respectively.

7 Important conclusion of the investigation of reliability is that when more tie points are used, geometry of the photogrammetric block improves which is especially seen on decreasing sensitivity of orientation parameters. Also reliability of GCP observations increases (a) X, emp X, theor r, tiep x r, tiep y r, GCP X r, GCP Y r, GCP Z r, GCP x r, GCP y.5. 4, s5 4, s1 4, corr 1, s5 1, s1 1, corr (b) 5x5 1x1 Interactive Figure 5. Effect of heteroscedastisity on X. The empirical and theoretical accuracy estimates are shown. Variables are the amount of tie points and weighting Effect of heteroscedasticity Effect of heteroscedasticity was investigated by generating errors with different standard deviations on the tie point coordinates. Uniform 1x1 and 5x5 distributions of tie point areas were considered. To each tie point area were generated two points with 5 µm and other two points with 1 and 2 µm standard deviation. Resulting proportions of distributions of errors were: 5% N (, 5µ m), 25% N (, 1µ m) and 25% N (, 2µ m). The average error is thus 1 µm. In total 1 and 4 points were generated per image. In the block adjustment following settings were used: Case s5: σ tie = σ =5µm Case s1: σ tie =σ =1µm Case Corr: σ =5µm, σ tie = σ corret (correct weights). Empirical accuracy of check points and empirical accuracy of orientations (RMS value of difference of the true and the adjusted orientations) was computed. Empirical accuracy of check points was better when correct weights were used (difference about 5%). It was remarkable that better accuracy was achieved by using 1 points/image and correct weights than 4 points/image and incorrect weights. The bias of accuracy estimate of orientation parameter X is shown in figure 5. Both the theoretical estimate and empirical estimate is shown. Theoretical and empirical estimates are practically the same in the case with correct weights, showing that the system performs correctly. In the cases with wrong weights, especially case s5, theoretical error estimates are clearly biased. The accuracy is clearly better if correct weights are used. For instance, difference between the accuracy of the case s5 and case with correct weights is about 5% of size of the error. And again, the case with correct weights and 1 points per image outperforms the case with wrong weights and 4 points per image. In, tiep x In, tiep y Ext, ori x Ext, ori y 5 RESULTS OF THE EMPIRICAL INVESTIGATION 5.1 Tie point measurement and block adjustment Ext, XYZ x Figure 6. Block Lahti. a) Redundancy numbers of tie point and GCP observations and b) average controllability and sensitivity factors of tie points (orientation, tie point ground co-ordinate and the total sensitivity, see chapter 4.2). Uniform 5x5 and 1x1 tie point area distributions were generated on both of the blocks. In each of the tie point areas clusters with 1 (5x5 distribution) and 5 (1x1 distribution) points were measured. The tie point observations with residuals >2 µm were rejected in the block adjustment. Interactive refinement was not necessary. In the automatic case a priori settings for tie point standard deviations were σ tie = σ =pixelsize/2. The a priori standard deviations of ground control observations have been given in chapter Ebner s additional parameters were used in the computations. These settings hold throughout the rest of the text unless mentioned otherwise. For comparison results of interactive measurements are also given. The block Lahti has been measured by manual monomeasurement by Geodata Ltd using the ESPA digital photogrammetric software. The interactive results for block Forssa have been taken from the report of OEEPE test (Jaakkola et a., 1996). They are one of the best results for the data with 3 µm pixel size reported. Block adjustment results are given in tables 1 and 2. Both the empirical and theoretical accuracy estimates show high accuracy and the results correspond quite well the simulated case with 1 µm observation error. Ext, XYZ y Ext, tot x Ext, tot y

8 Table 1. Adjustment result of the block Lahti. Case Interactive 5x5 1x1 σ (µm) Amount of rejected - 2.5% 1.9% Tiep observations/image Tiep observations total Empirical accuracy of CP: X (cm) Y (cm) Z (cm) Theoretical accuracy of tie points X (cm) Y (cm) Z (cm) Theoretical accuracy of orientations X (cm) Y (cm) X (cm) Z omega (mgon) phi (mgon) kappa (mgon) Simu 1x1, s5, eb 1x1, s12.5, eb 1x1, s25, eb (a) 1x1, s5, eb Interactive, eb (b) 1x1, s5 1x1, s12.5 1x1, 25 1x1, s5 Interactive X Y Z X Y Z Table 2. Adjustment result of the block Forssa. Case 5x5 1x1 σ (µm) Amount of rejected 4.7% 4.% Observations/image 25 5 Observations total Empirical accuracy of CP: X (cm) Y (cm) Z (cm) Theoretical accuracy of tie points: X (cm) Y (cm) Z (cm) Theoretical accuracy of orientations: X (cm) Y (cm) X (cm) Z omega (mgon) phi (mgon) kappa (mgon) 5.2 Reliability Reliability numbers (mean values of redundancy of tie point x, y; GCP X, Y, Z and GCP x,y; and internal and external reliability of tie point x and y) of block Lahti are given in figure 6. All the values are clearly better in the automatic cases than in the interactive case. Redundancy of 1x1 block is better than of 5x5 block, especially in the flying direction. Redundancy numbers of both GCP image and ground observations increase when more observations are used. Results of block Forssa are Simu 1x1, s6, eb 1x1, s15, eb 1x1, s3, eb similar to block Lahti and are not shown. Differences are that the reliability of block Forssa is slightly lower (tie point redundancy in x and y:.35 and.55) and there are practically no difference between tie point redundancies of cases 5x5 and 1x1 tie point distribution. 5.3 Effect of weighting 1x1, s6, eb Figure 7. Effect of weighting on the empirical accuracy of a) block Lahti and b) block Forssa. Interactive, eb Effect of weighting was investigated using 1x1 data set (5 points per image). Cases with and without Ebner s additional parameters were computed. Cases with Ebners parameters are expressed by eb. The tie point observation a priori standard deviation and standard error of unit weight were given the same values, σ tie = σ =1/5, 1/3, ½, 1 and 2 of the pixel size. The 1x1, s6 1x1, s15 used standard deviation is expressed by s σ tie. 1x1, s3 1x1, s6 Interactive

9 (a) (b) x5 simu, s1 Figure 8. Block Lahti, effect of amount of observations on the empirical accuracy in a) X, b) Y and c) Z. (c) x1 Interactive.3.25 (a).3.25 (b).6.5 (c) Empirical accuracy of blocks Lahti and Forssa are given in figure 7. Also the simulation result with 2x2 tie points with σ = 1 µ m is shown. First some general notes concerning tie both of the cases are made. Weights have a remarkable effect on the result. Usage of the highest weights ( σ tie = 1 / 5 of the pixel size) always gives the worst result. Based on the empirical accuracy, it seems that weight around one pixel is optimal. Additional parameters seem to have more effect in the automatic cases than in manual cases. And weighting seem to have smaller effect when additional parameters are used. Block Lahti. The empirical accuracy of the block Lahti is very close to the simulated case when additional parameters are used. The weighting affects about 2-3% to the empirical accuracy. When additional parameters are not used the empirical accuracy gets much worse and the block does not behave like expected based on the simulations, for instance, relation between the X and Y accuracy is totally wrong. The weighting affects 25-4% in X and Y and the height error is almost doubled between the worst and the best case. When using additional parameters the best accuracy of X and Y corresponds approximately to 5 µm on the image and.7 of the flying height. Block Forssa. The behaviour of the block Forssa is not as good as the block Lahti. Usage of additional parameters weakens the height accuracy. Despite of this a good accuracy is achieved. In the best cases the empirical error on X and Y is less than 5 µm and on height about.6 of the flying height. With additional parameters best height accuracy is about.8 of the flying height Figure 9. Block Forssa, effect of amount of observations on the empirical accuracy in a) X, b) Y and c) Z A possible explanation for this behaviour is systematic errors whose role may get more importance when huge amount of observations is used. 5.4 Effect of amount of observations Subsets of measured tie points were selected from each tie point cluster to be able to see effect of amount of points. Following selections were made: 1x1 tie point area distribution: 1, 3 and 5 points/cluster 5x5 tie point area distribution: 1, 4, 7 and 1 points/cluster Selection criterion was to select points with maximum number of intersecting rays. Results are shown in the figures 8 (Lahti) and 9 (Forssa) as a function of number of points. Empirical accuracy of interactive measurement and the result of simulation (standard deviations: σ GCP_ XYZ = scale * 5 µ m, σ GCP_ xy = 5 µ m and σ tie = 1 µ m ) are also shown. General impression from the empirical curves is that better accuracy is achieved when more points is used. Accuracy improvement is remarkable when going from 25 to 1 points per image. Improvement is also clear when going from 1 points to 2 or 3 points. However, when going from 3 to 5 points per image (1x1 distribution) no improvement can be seen. In the block Lahti it seems that the 1x1 distribution gives better results than the 5x5 distribution. In the block Forssa the difference between the distributions is a lot smaller. One explanation onto this behaviour could be the redundancy of tie point observations. In the block Lahti (figure 6) redundancy is in the case of 1x1 distribution 5% better than in the case of 5x5 1x1 simu, s1 Interactive

10 5x5 distribution. Instead in the block Forssa the redundancy of both of the cases is about the same. When comparing the results with interactive measurements the result is that if the same amount of tie points is used the interactive measurement gives better accuracy, meaning that pointing accuracy of interactive measurement is better. When using a lot of observations automatic measurement gives better results in the case of block Lahti. In the case of block Forssa automatic measurement gives clearly better accuracy in X and Y, but instead worse accuracy in Z. Possible reasons for the difference between the simulated and empirical result are that the stochastic model used in the block adjustment is not correct, the standard deviations of the observations may not correspond each other and the variance and non-ideality of the empirical case. 6 CONCLUSIONS In this article effect of amount and accuracy of tie points on the quality of aerial triangulation has been investigated. The simulation study showed that in theory block quality improves significantly when using huge amount of points. Requirement however is that in the adjustment correct functional and stochastic models are used. When the standard deviation of tie points was 5 µm, 1 points per image gave already a good accuracy and reliability, and close to the best achievable object accuracy was achieved when using 2-4 tie points per image. Corresponding amount of observations can be computed for different tie point standard deviations, σ tie, by multiplying the amount by the factor of ( σ tie / 5) 2. In the empirical investigation automatic tie point measurement gave very good results. The empirical accuracy was in the X and Y on the level of 5 µm on the image and.7 of the flying height. Standard error of unit weight was in the test blocks ¼ and 1/5 of the pixel size. Förstner W., The Reliability of Block Triangulation. Photogrammetric Engineering and Remote Sensing, Vol. 51, No. 6, pp Förstner W., Reliability Analysis of Parameter Estimation in Linear Models with Applications to Mensuration Problems in Computer Vision. Computer Vision, Graphics and Image Processing 4, pp Förstner W., Matching Strategies for Point Transfer. 45 th Photogrammetric Week, Stuttgart, pp Honkavaara E., Høgholen A., Automatic Tie Point Extraction in Aerial Triangulation. International Archives of Photogrammetry and Remote Sensing, Vol. 31, Part B3, pp Jaakkola, J., Sarjakoski, T., Experimental Test on Digital Aerial Triangulation. OEEPE, Official Publication No 31. Krzystek P., Heuchel T., Hirt U., Petran, F., An Integral Approach to Automatic Aerial Triangulation and Automatic DEM Generation. International Archives of Photogrammetry and Remote Sensing, Vol. 31, Part B3, pp Sarjakoski T., Automation in Photogrammetric Block Adjustment Systems - On the Role of Heuristic Information and Methods, Acta Polytechnica Scandinavica, Ci 88. Schenk, T., Towards Automatic Aerial Triangulation. ISPRS Journal of Photogrammetry and Remote Sensing, Vol. 52, pp Tang, L., Braun, J., Debitsch, R., Automatic Aerotriangulation - Concept, Realization and Results. ISPRS Journal of Photogrammetry and Remote Sensing, Vol. 52, pp According to the empirical investigation there is reason to believe that when using huge amount of tie points functional and stochastic model of adjustment gets more importance compared to the traditional case. Results indicating this are that weighting and usage of additional parameters had a remarkable effect on the empirical accuracy of the block. 7 ACKNOWLEDGEMENTS This investigation has been financially supported by The Technology Development Centre of Finland and the Finnish mapping companies FM-Kartta Ltd and Geodata Ltd. The practical work for the project (production of imagery, field work and interactive image measurements) has been carried out by FM-Kartta Ltd and Geodata Ltd. Janne Ylönen has been of great help in making improvements to the ESPA bundle block adjustment program. They all are gratefully acknowledged. REFERENCES Bilker M., Honkavaara, E., Jaakkola, J., GPS Supported Aerial Triangulation Using Untargeted Ground Control. International Archives of Photogrammetry and Remote Sensing, Vol, 32.