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The Precise Evaluation of Lens Distortion*

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The Precise Evaluation of Lens Distortion* The Precise Evaluation of Lens Distortion* FRA CIS E. WASHER, National Bureau of Standards, Washington, D. C. ABSTRACT: Four methods of radial distortion measurement are described. Com­ parison of results obtained on the same lens using each of the four methods is used to locate systematic errors and to evaluate the reliability of each method. Several sources of error are discussed. It is concluded that a precision of ±2 microns can be achieved using anyone of the four methods provided adequate attention is given to the reduction of all potential errors to negligible proportions: that is, the PE 8 of D{3 does not exceed 2 microns where D{3 is the value of distor­ tion obtained at angle fJ and PE 8 is the probable error of a single determination. 1. INTRODUCTION which is used to considerable extent In Europe. HE measurement of radial distortion in T the focal-plane of photographic objec­ Because of the diversity of methods used in tives has been the subject of intensive study evaluation of distortion, it seemed worth­ since the advent of mapping from aerial while to investigate the results of measure­ photographs. This particular aberration is of ments made in a single laboratory on a single prime interest to photogrammetrists as its lens by a variety of methods, to determine magnitude determines the accuracy with whether or not the values so obtained varied which the final negative maintains the correct appreciably \\"ith the method. This has been relationships among the array of point images done using four different methods and a com­ making up the photography of the cor­ parison of the results is reported in this paper. responding array of points in the area photo­ 2.0 METHODS OF MEASUREMENT graphed. I t was early realized that the relia­ bility of the quantitative information ob­ The methods are as follows: tained from a photograph is increased as the 1. Photographic-Precision lens testing distortion in the camera lens is decreased. camera. Consequently, the development of improved 2. Visual-Nodal slide optical bench. lenses was encouraged with the result that 3. Visual-Inverse nodal slide on T-bench succeeding series of new lenses were character­ with small aperture telescope. ized ever lower values of distortion. 4. Visual-Modified goniometric using During this period of change, diverse Wild theodolite. methods of evaluating the distortion of lenses Methods 1, 2, and 4 have been described at came into use at various laboratories. The some length in the literature,1,2,3 while reason for such diversity was primarily the Method 34 has been recently developed at this availability of given types of measuring in­ Bureau and is a modification of a very early struments in various laboratories. There are methodS of measuring lens distortion. In the now three principal methods of measuring following sections, a brief description of each distortion, plus numerous additional methods method is given. that are either the inverse of one of the prin­ cipal methods or a variation thereof. The 2.1 PRECISION LENS TESTING CAMERA. nodal slide bench is one of the oldest methods; METHOD 1 this is a visual method capable of high accu­ The precision lens testing camera,! shown racy and is perhaps the most widely used. The in Figure 1, was developed at this Bureau by photographic method is more recent and arose 1. C. Gardner and F. A. Case. I t is one of the out of the desire to make measurements under earliest successful devices developed to meas­ conditions approximating those of use. The ure the performance of lenses by photographic third principal method is the goniometric, means. I t consists of a bank of collimators * This paper was presented at the 1962 St. Louis ACSM-ASP Convention. 327 328 PHOTOGRAMMETRIC ENGINEERING FIG.!. Precision lens testing camera (Method 1). spaced at 5° intervals having resolution test in evaluating the distortion at a given angle charts as reticles. The lens under test is f3 is mounted at the center of convergence of the D{3 = Y{3 - f tan {j, (1) colli mator fan, and can be aimed at anyone of where Y{3 is the measured distance separating them by rotation of a carriage which carries the images produced by the lens under test in the lens holder and camera back whereon the the 0° collimator and that in the collimator photographic recording plate is mounted. As inclined at angle f3, and f is the value of the presently constituted, the lens testing camera equivalent focal-length based on the separa­ has 10 collimators covering a total angle of tion of the 0° and 5° images and that of the 45°. When used to test wide-angle lenses, the 0° and 10° images. camera is aimed at one of the extreme collim­ ators (Position I) and the test made. In order 2.2 VISUAL OPTICAL BENCH; DIRECT NODAL to cover a complete diameter, a second test is SLIDE. METHOD 2 made with the camera aimed at the collimator The visual optical bench has long been the at the opposite extreme (Position II). It has basic tool for evaluating the constants of been found by experience that the results lenses. The one used at this Bureau2 is shown obtained from two negatives made in this in Figure 2. For measuring distortion, a col­ manner are quite reliable. limator, nodal slide lens holder, and microm­ From the measured separation of the eter microscope are used. The lens is carefully images recorded on the test negative made aligned in the holder and the axial-image with a given lens, it is possible to determine formed by the lens under test of the illumi­ both the equivalent focal-length f and the nated reticle of the target is brought into co­ distortion D{3. The determining equation used incidence with the object-plane of the viewing FTG. 2. Visual optical bench. Direct nodal-slide (Method 2). PRECISE EVALUATION OF LENS DISTORTION 329 FIG. 3. Inverse nodal-slide (Method 3). microscope. By a series of successive adjust­ incidence with the center of rotation of the ments, a condition is found for which a smal1 nodal-slide. The nodal-slide bearing the lens rotation of the lens about a vertical axis does under test is then rotated by amount fJ about not produce a displacement of the axial­ the vertical axis, the target reticle is moved image viewed. The rear nodal-point of the away from the zero position by amount lens is then considered to coincide with the f(secfJ-l), the poin tingoftheviewingtelescope center of vertical rotation of the nodal-slide. is adjusted to compensate for change in direc­ Assuming the equivalent focal-length f to tion in the collimated beam emergent from be known, the nodal-slide is rotated by the lens under test, and the setting of the amount fJ about the vertical axis using the transverse micrometer is recorded. This read­ calibrated circle of the nodal-slide to position ing is 4fj. The process is repeated with the it exactly. The entire saddle carrying the lens rotated to position -fJ, and the reading nodal-slide and lens is then moved away from LfJ is recorded. The distortion Dfj is obtained the microscope toward the col1i mator by an from the relation amountf(sec fJ-l). The viewing microscope is t::.EfJ shifted laterally to the new position of the Dfj = -fsec2 {3 (3) 2 image and its dial read and recorded as read­ ing RfJ. The nodal-slide is then rotated to where LHfJ=f+fJ-LfJ, and f is the equivalent posi tion -fJ and a second setting of the micro­ focal-length of the lens. scope, R-fJ , is made. These displacemen ts are measured in a plane normal to the chief rays 2.4 MODIFIED GONIOMETER. METHOD 4 inclined to the axis at angles +fJ and -fJ, The goniometer method, il1ustrated in respectively. The distortion DfJ is obtained Figure 4, used in this laboratory does not from the relation employ a specially constructed camera goni­ t::.Rfj ometer such as is used in Europe, but more DfJ = -- sec (3 (2) 2 nearly approximates the method used by Merritt3• A calibrated linear-scale is placed in where t:J.RfJ = ·RtfJ - R_fj. the focal-plane of the lens under test and the scale is viewed through the front of the lens 2.3 INVERSE NODAL SLIDE. METHOD 3 with a precision theodolite. The angular dis­ placements fJ of selected points on the scale In the inverse nodal-slide method4, shown in Figure 3, an illuminated target, preferably distant rfJ from the center line on the cali­ a point source, is placed at the rear focal-point brated scale are measured using the precision of the lens under test. Light from the point­ circle of the theodolite. The distortion DfJ is source emerges from the lens in a collimated obtained from the relation beam which enters the viewing telescope. By DfJ = rfj - f tan {3. (4) a series of adjustments, a condition is found for which a smal1 rotation of the lens about a 3.0 RESULTS OF MEASUREMENT vertical axis does not produce an angular dis­ An extensive series of measurements in a placement of the collimated beam observable single wide-angle lens using each of these four through the viewing telescope. The rear methods was made and values of distortion nodal-point of the lens is then in close co- determined with each method. The results of 330 PHOTOGRAMMETRIC ENGINEERING In addition, the referral of the values of D{3 to the calibrated focal-length removes from consideration errors in distortion arising from errors in equivalent focal-length.
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