SYSTEM CALIBRATION VERSUS PHOTOGRAMMETRIC REFRACTION

Transcription

1 J. BITTENCOURT DE ANDRADE - Ph.D. E. MITISHITA - M.So AERODATA S/A. - ENGENHARIA DE AEROLEVANTAMENTOS CURITIBA -PARANA - BRASIL SYSTEM CALIBRATION VERSUS PHOTOGRAMMETRIC REFRACTION ABSTRACT Nowadays, instruments to measure photo oordinates show a disrimination of half a mirometer. To avoid loss of auray, systemati errors must be orreted at the same level. Among the soures of systemati errors is the photogrammetri refration. The definition of photogrammetri refration permits to distinguish different omponents of the phenomena. Reent studies have been demonstrated that the omponent of refration due to variation on atmospheri density is not the most important. Other soure of refration the disturbed air aused by the airraft displaement - an deviate the optial path ten times more than the atmosphere. If suh systemati errors are important for the task of aerial triangulation, it beomes ruial on the proedures of aerial amera alibration as system alibration. Lens distortion is the non-desirable part of the refration aused by the lenses. As onsequene, both lens distortion and photogrammetri refration are phenomena of the same nature and onsequently physially orrelated. This paper shows how to solve the problem of refration when alibrating the photogrammetri system. The key results of ameras alibrated by the developed tehnique show that the level of auray reahed is ompatible to the auray of the measured photo oordinates - 1 miron. 1. INTRODUCTION The world's photogrammetri ommunity nowadays has instruments apable of measuring photo oordinates within half a mirometer. However, the auray of photogrammetri works doesn't reah the level whih those observations an allow. In the last deade D.C. Brown wrote "Muh very basi work needs to be done in the formulations and testing of error - Models to be used... For example,.. the preise influene of refrative anomalies aused by atmospheri turbulene needs further study". Studies in this area were the objet of the study for the PhD dissertation of J.B. de Andrade, who modelled the photogrammetri refration, taking into onsideration all its auses, inluding the atmospheri turbulene aused by the displaement of the airraft (Andrade, 1977). The proposed model by Andrade was tested by himself on the testing field of Rheidt in Germany showing results for the photogrammetri refration totally different from those obtained through the standard atmosphere; but, the results of the aerotriangulation show drastially redued errors, mainly on altitude where it diminished by an average of 60% and in many ases the error was ten times smaller. By the other side, in the field of photogrammetry we may ount with amera alibration. treatment of systemati errors 1n the tools offerred by the methods of In the area of instrument alibration and the standards Eisenhart, (1963) gives a olaboration whih drastially hanges the onepts aepted until this time. Searhing for a method of alibration whih was apable to

2 attend the onept of system alibration, established by Eisenhart. Merhant, (1972) reated the "Method of Mixed Ranges" whih revolutionized the subjet. This method was improved by Andrade, (1981) in the following aspets: a) to use only 1 test range. b) use either distane between targets or geometri levellinge These improvements were possible due to the studies realized on rank deffiieny of the desin matrix. The rank defiieny was found to be 8. Therefore, it would be neessary to apply 7 onstraints for the realization of the referenial frame and 1 to break the orrelation between the amera onstant and the flying height as a minimum. This method of alibration assumes that all the methods of alibration, exept those onerning the lens system, are appropriately treated either by pre-orretion or by modelling and parametrization. Conerning photogrammetri refration, the parametrization (as developed by Andrade, 1977) was impratiable due to orrelations between photogrammetri refration and the symmetrial radial distortion. The adapted way was the "a priori" orretion of the photogrammetri refration using a standard atmosfere, suh proedure made the proess viable in spite of the fat that the residuals remained disagreable. A new way must be followed in order to make the size of the residuals onord with the preision of the observations involved in the proess. Two approahes were tried to solve the problem: the first was to regroup the parameters of the mathematial model for photogrammetri refration - aiming to break the orrelation between parameters of refration and symetrial radial distortion; the seond was to "a priori"proessing the normal equations augmented by the photogrammetri refration model. In spite of the fat that one of the approahes was unsuessful, it will also be disussed to make lear important points for the omplete understanding of this problem. 2. PHOTOGRAMMETRIC REFRACTION The angle between the ollinear and the target to the atual ray measured at the exterior node, and taken as positive for outward radial displaement of the image, aused by the atmospheri refration, reeives the name of the photogrammetri refration. We onsider only the variation of atmospheri refration index between the terrain and airraft (amera), the photogrammetri refration would always be positive exept when there are large inversions in the refration index. Its maximum value of photogrammetri refration would be 28 mirometers at sea level and flying heights of 16 km. Howewer, other fators must be taken into onsideration: the perturbed air aused by the displaement of the airraft; the differene of pressure on the filter faes aused by the turbulene of the air in front of the amera; and the differene in air pressure inside and outside when the airraft 1S pressurized. Andrade (1977) developed a mathematial model apable of desribing the global angular deviation of a ray of light before entering the amera lens system. z - s n In ~ n p (2. 1 )

3 where: e = photogrammetri refration ~ = height of the mass enter of the air olumn between the terrain and the airraft Z = flying height s Z = terrain height p a e n s n p e nadiral angle angle between the boundary layer and the airraft = refrative index at the amera level refrative index at the ground level null, for subsonis veloities This paper treats only subsoni veloities: * r2 Ox = (1 + ~ ) x * r2 oy (1 + ~ ) y For supersoni veloities the above orretions would be: r2 *;', r2 ox * ( 1 + )x - ~ 1 ( 1 + ~) (2.2) (2.3) (2.4) oy r2 -;'( r2 * ( 1 + )y - ~ 2 ( 1 + ~ ) (2.5) Where: -/( photogrammetri refration for a = 45 and subsoni veloities = photogrammetri refration oeffiients (null veloities) for subsoni Note that for supersoni flights an additive term appears, whih is independent of the image oordinates. We are going to analyze only the ase of subsoni flights. The atmosphere perturbed by the airraft adds. refrative effets on the light rays entering the amera. These phenomena are strongly related to the type of airraft in use. The Boundary-Layer Theory (Shlihting, 1968), shows that in a flat plate at zero inidene in parallel flow, the speed of the air varies from zero to a onstant value equal to the speed of the flow. The harges of the speed an be desribed by Shlihting (1068). /J.p = p l M2 2 ' 1n the worst situation (2.6) Where: /J.p p M v hange in density, density, ~ = "mah number" veloity of the flow, veloity of the sound Based on the above relat due to the flow speed:, we an find the in the refration index

4 n - -L po + IIp IIp 1 = = ::: ::: 1 + M2 no- po po p 2 or (2.7) n == 1 + ( 1 + M2) (n - 1 ) 0 (2.8) Where: n == refration index at V == 0 o The onditions in whih (2.6) is valid are ideals (laminar flux paralell to a flat plate). Below is an example: == 335,28 mls - sound veloity v 400 Km/h - airraft veloity Z == 3.000m - flying height s n 1, refration index at the flying height 0 n == 1, * In , "k ==- 45 3,3998 x 10-5 radions r J, Or == 45 2 x 150 x (-3,3998 x 10-5 ) Or == - 0,010 rnrn In pratie this value an be ten times greater, beause the veloity of flow an be substantially augmented by the ation of the propellers; the surfae below the airraft is not flat and the angle of attak is not zero; the flow may not be parallel and ertainly isn't below the amera window. Typial values found by Andra~~'s tests realized in the Rheidt Test Field are in the order of 45 == -4xl0 radians, whih orresponds to rnrn of image displaement for a nadiral angle of SYSTEM CALIBRATION Quoting Eisenhart: Calibration of instruments and standards is a refined form of measurement. Measurement of some property of a thing is an operation that yields as an end result a number that indiates how muh of the property the thing has. Measurement is ordinarily a repeatable operation, so that it is appropriate to regard measurement as a prodution proess, the "produt" being the members, i.e., the measurements, that it yields; and to apply to measurements proesses in the laboratory the onepts and tehniques of statistial proess ontrol that have proved so useful 1n the quality ontrol of industrial prodution. "Viewed thus, it beomes evident that a partiular measurement operation annot be regarded as onstituting a measurement proess unless statistial stability of the type known as a state of statistial ontrol has been attained. In order to determine whether a partiular measurement operation 1S, or is not, in a state of statistial ontrol it is neessary to be

5 definite on what variations of proedure apparatus, environmental onditions, observors, operators, et., are allowable in "repeated appliations" of what will be onsidered to be the same measurement proess applied to the measurement of the same quantity under the same onditions. To be realisti, the "allowable variations" must be of suffiient sope to braket the irumstanes likely to be met in pratie. Furthermore, any experimental program that aims to determine the standard deviation of a measurement proess as an indiation of its preision, must be based on appropriate random sampling of this likely range of irumstanes. Ordinarily, the auray of a measurement proess may be haraterized by giving (a) the standard deviation of the proess and (b) redible bounds to its likely overall systemati error. Determination of redible bounds to the ombined effet of reognized potential soures of systemati error always involves some arbitrariness, not only in the plaing of reasonable bounds on the systemati error likely to be ontributed by eah partiular assignable ause, but also in the manner in whih these individal ontributions are ombined. Consequently the "inauray" of end results of measurement annot be expressed by 'onfidene limits' orresponding to a definite numerial 'onfidene level', exept in those rare instanes in whih the possible overall systemati error of a final result is negligible in omparison with its impreision. Speifiation of the apparatus and auxiliary equipment to be used, the operations to be performed, the sequene in whih they are to be exeuted, and the onditions under whih they are respetively to be arried out these instrutions olletively serve to define a method of measurement. A measurement proess is the realization of a method of measurement in terms of partiular apparatus and equipment of the presribed kinds, partiular onditions that as best only aproximate the onditions presribed, and partiular persons as operators and observers. It has long been reognized that, in undertaking to apply a partiular method of measurement, a degree of onsisteny among repeated measurements of a single quantity needs to be attained before the method of measurement onerned an be regarded as meaningfully realized, i.e., before a measurement proess an be said to have been established that is a realization of the method of measurement onerned. Indeed, onsisteny or statistial stability of a very speial kind is required: to qualify as a measurement proess a measurement operation must have attained what is known in industrial quality ontrol language as a state of statistial ontrol. Until a measurement operation has been 'debugged' to the extent that it has attained a state of statistial ontrol, it annot be regarded in any logial sense as measuring anything at all. And when it has attained a state of statistial ontrol there may still remain the question of whether it is faithful to the method of measurement of whih it is intended to be a realization". The method of mixed ranges, oneived by Merhant (1972) is the unique appliable to airphotogrammetry whih attends to the modern priniples established by Eisenhart. The amera is alibrated on the same onditions 1n whih it is used and all the parameters to define the metrial properties of the system of measurement (amera, airraft, et.) and the quality of its performane are determined. The method of mixed ranges requires two test fields for alibration: one montaneous, with few signalized 3D-ontrol points and another,flat to failitate the installation of a great number of targets also with three oordinates arefully determined

6 The aurate determination of 3D oordinates on a montaneous terrain (and even on flat terrain) makes the method diffiult to be applied in pratie. Those diffiulties were overomed by the studies that Andrade started in 1977 at The Ohio State University, where was verified that the rank defiieny of the normal equation is 8 for this method - seven to materialize the referene system and one to provide vertial sale. Those studies were followed at the Federal University of Parana (Brazil). A test field (Andrade, 1981) was established in Sao Luiz do Puruna with 36 points 30 on the higher level and 6 on the lower level of a sarped terrain, showing a differene in level of 200m in average. The following goals were reahed: 1) The number of test fields were redued from 2 to 1. 2) The number of photo-images of targeted points were multiplied by taking the photographs in four strips: (N-S); (S-N); (E-W) and (W-E), resulting 12 photographs times 36 ground points, or 432 image-points. 3) The 3-D oordinates for eah targeted point were substituted by a first order levelling. 4) The referene system was arbitrary materialize it. only elevation were used to In this method the models for radial symmetri and deentering distortion were added to the projetive equation (Bundle Method). 4. PHOTOGRAMMETRIC REFRACTION VERSUS CALIBRATION To alibrate the Photogrammetri System it beomes neessary to "a priori" orret the other systemati errors or to add adequated parameters to the projetive equations. Conerning Photogrammetri Refration, fla priori" orretion was done using standard atmosphere due to the impossibility of separation of parameters to define refration from those of symmetrial radial distortion. Refration and lens distortion are physially the same phenomena and so orrelated quantities. Conrady's model of symmetrial radial distortion reads: 6r = Kl r3 + K2 r 5 + K3 r 7 + The model for refration reads: (4. 1 ) 6r = E: * r + E:.,..._ 2 (4.2) Comparing (4.2) and (4.1) one an see, that they differ by a onstant fator. Trying to break this orrelation, a different agrupment of parameters tried: From Andrade (1972): were -;'( ~ 1 ns E: = n- (4.3) Z - Z np s p Q = 1 ns ~ n-- np 6r = Q r + r3 (4.4) Zs - Zp

7 This model was expeted to work beause Zs-Zp varies up to 200 meters on Sao Luiz do Puruna Test Field - about 10% of the flying height. However, beause ~ approahes Zs - Zp due to the high air pressure below the airraft, the orrelation annot be broken. The orrelation oefiient beomes only 5% smaller with this approah. So it was abandoned. The seond approah tried was that of omputing an "a priori" value by adding it to the projetive equation and solving only for it elements of exterior orientation. *he results were amazing. The founded by Andrade (1977) for were repeated and the residuals the most optimisti levels. to 10 DO 40 GO 80 '0 eo eo loci 110 lito GO too 150 and the values drop to MICROMETERS Figure 1: Photogrammetri Refration and radial symetri distortion urves RESIDUAL IN PHOTOCOORDINATES RESIDUAL IN GROUND COORDINATES POINT Vx Vy POINT Vx Vy POINT Vz POINT Vz 3-0,0026 0, ,0002-0, , , ,0034 0, ,0034-0, , , ,0007 0, ,0020-0, , ,0015 0, ,0002-0, , , ,0011 0, ,0027 0, , , ,0007 0, ,0041 0, , , ,0013-0, ,0010-0, , , ,0004 0, ,0012-0, , , ,0001 0, ,0004-0, , , ,0023 0, ,0046 0, , , ,0021-0, ,0012-0, , , ,0016 0, ,0015 0, , , ,0033-0, ,0016-0, , , ,0014-0, ,0036 0, , , ,0026-0, ,0009-0, , , ,0009 0, ,0021-0, , , ,0009 0, ,0009-0, , , ,0032 0, ,0032-0, , , ,0020-0, ,0019 0, , ,0003 Table 1: Residuals obtained through this method. The relative auray reahed was 1: of the flying height theoretial limit obtainable. the

8 The values of photogrammetri refration in four different experiments were: ( (45 Camera Sale Airraft Veloity -4,160. JO- 4 rad 4, rad 2 Zeiss RMK-AR J : , rad 6,7814.IO- 10 rad 2 Wi Id RC-1O J: ,199.1O- 4 rad B, Navajo 300Km/h rad 2 Wi Id RC-5/RC-B I: ,068.IO- 4 rad 5,542.1O- 9 Senea 300Km/h rad 2 Wild RC-5 1: Senea 300Km/h 5. CONCLUSIONS The results of this researh show the possibility of performing aerial triangulation with a very high level of auray. Naturally, one annot expet that in aerial triangulation the same level of auray as a test field may offer, an be reahed. However, there is no doubt that the proper treatment of the systemati errors allowed ompensatory good results. It is important to remember that the total effet of photogrammetri refration (inluding air disturbed by airraft motion) is on the order of 10 times greater than the one omputed through a standard atmosphere and with opposite sign (negative). 6. REFERENCES Andrade, J.B. University (1977), "Photogrammetri Refration" Dissertation. Columbus, Ohio, U.S.A The Ohio State Andrade, J.B. and Olivas, MeA.A. (1981), IICalibra;ao de Cameras Aerofotogrametrias". Boletim da Universidade Federal do Parana. Curso de P6s-Gradua;ao em Cienias Geodesias. Geodesia nq 26. Curitiba. Brown, Duane C. (1974), "Evolution, Appliation'and Potential of The Bundle Method of Photogrammetri Triangulation", Paper presented to Commission III, Symposium in Stuttgart, International Soiety for Photogrammetry. Eisenhart, Churhill (1963), "Realisti Evaluation of The Preision and Auray of Instrument Calibration System", Journal of Researh of the National Bureau of Standards, Vol. 676, nq 2. Merhant, Dean C. (1972) "Metri Calibration of The Aerial Photogrammetri System by The Method of Mixed Ranger", AFSC (RADC) TR , July, Shlihting, H. (1968) Boundary-Layer Theory, Sixth Ed., Translated by Kestin, J. MGraw-Hill Book Company, New York.