Accuracy Aspects of Non-Metric Imageries*

H. M. KARARA Y. I. ABDEL-AZIZ~ University of Illinois Urbana, Illinois 61801 Accuracy Aspects of Non-Metric Imageries* Opening the door to the use of non-metric cameras should enable many engineers and scientists to make full use of the technical and economical advantages of photogrammetry. mations. In view of the relatively large lens ESEARCH IN close-range photogramnletry at distortions and film deformations generally Rthe Urlive~sityof Illinois and elsewhere associated with non-metric cameras, the has clearly shown thatg"3, for numerous areas analytic approach has been almost exclu- of applications and potential applications, sively used so far in photogrammetric data fully acceptable accuracy can often be reduction from non-metric imageries. achieved with better non-metric cameras, In 1971 an analytic data reduction method, such as Hasselblad, Robot, Linhof Technika, particularly suitable for non-metric im- etc., provided that appropriate measures are ageries (Direct Linear Transformation-DLT) ABSTRACTAn updated version of the Direct Linear Transformation (DLT)emphasizes the mathematical modeling of lens distortions and film deformations. Experimental results indicate the levels of accuracy attainable at close range withfour readily available non-metric cumerus (Hasselblad 500 C, Honeywell Pentax Spotinatic, Crown Graphic, Kodak Instainatic 154) and the Hasselblad MK 70 inetric camera. A technique compares the photograinmetric worthiness of iv~easuringsysteins involving any camera (metric or non-metric).Ex- pressions enable one to estimate tlae theoretically expected accuracies in object-space coordinates. The optimum number of object-space control points for the DLT solution is derived. taken in data acquisition and data reduction. was developed by the authors1. This method Essentially, one has to: (a) choose a suitable has since been used in numerous applica- configuration for data acquisition (b) provide tions (e.g., by Williamson14 and Faig6" and the necessary object-space control, (c) coun- others) and has proven its practical merits. teract possible internal instability of the In this paper, an updated version ofthe DLT camera by combining calibration procedures approach is discussed, with emphasis on the with the measuring process, and (d) choose a mathematical modeling of lens distortions suitable mathematical model to correct for and film deformations. A number of the effect of lens distortions and film defor- mathematical models have been investigated and the one deemed most suitable (on the *Presented at the ASPIACSM 1973 Fall Na- basis of the experiments conducted and the tional Convention, Orlando, Florida. Report on statistical analysis undertaken) is recom- studies supported in part by the National Science mended. Foundation (grant GK-11655). To the accuracy-conscious user of non- t Currently a Research Associate, Biostereomet- metric cameras, the experimental investiga- rics Laboratory, Baylor College of Medicine, tion (summarized later in the section, "The Houston, Texas 77025. Photogrammetric Potential of Any Camera") PHOTOCRAMMETRIC ENGINEERING, 1974 may be of some interest. These results indi- nonlinear coinponents of lens distortions and cate the levels of accuracy attainable with film deformation are not taken into account. four readily available non-metric cameras (Hasselblad 500 C, Honeywell Pentax Spot- LENS DISTORTIONS matic, Kodak Instamatic 154, Crown In an ideal lens with perfectly centered Graphic) as well as Hasselblad MK70. elements, lens distortion is strictly symmetrical about the optical axis. Errors in center- BASICDLT EQUATIONS ing lens elements lead to asymmetrical lens As outlined in Reference 9, the basic equa- distortion. tions for the Direct Linear Transformation Sy~nmetricalLens Distortion. A generally method are: accepted mathematical model for symmetrical lens distortion is an odd-powered polyno~nial: where Ar is radial lens distortion, r is the length of the radial vector from the point of symmetry to the point under consideration, r2 where x,y are the comparator coordinates of = (x - x,)~+ (y - yJ2, x,y are image coordi- an image point X,Y,Z are the object-space nates of the point under consideration, x,,y, coordinates of that point, Z1,Z2, . ,Il1 are the are image coordinates of the point of sym- transformation coefficients, and k,Ay are metry, and (2n + 1) is the degree of the odd- image refinement components in x and y to powered polynomial. account for the nonlinear components of lens Investigations have shown that, for the rel- distortions and film deformations. atively simple lenses often used in non- In the linearized form, Equation 1 takes the metric cameras, k1 is the only signifihant coef- following form: ficient in Equation 3. Equation 3 can thus be reduced to: Asyininetrical Lens Distortion. A generally accepted mathematical model for asyin- metrical lens distortion is the following one which reflects the distortion caused by de- centering of lens elements and accounts for where 11, 12, . ,Zll are transformation coeffi- the selection of a point other than the point of cients, A = l9 X + ll0 Y + Zl1 Z + 1, v,, vy are symmetry as reference: residual errors in image coordinates after refinement. It should be pointed out that the linear components of lens distortion and film deformation are taken into account by the elev- en transformation coefficients (11 through where &,by are asymmetrical lens distortion Zll) in Equation 2 in the process of transform- components, r is the length of the radial vec- ing comparator coordinates into object-space tor from the point of symmetry to the point coordinates. These linear components ac- under consideration, x,y are image coordi- count for different scale factors along the x nates of the point under consideration, re- and y directions and for the nonperpendicu- ferred to the point of symmetry, and pl,pz are larity of the comparator axes. coefficients of asymmetrical lens distortion. FILM DEFORMATIONS The incorporation of provisions for image Numerous sources contribute to film de- refinement (to account for the linear and the formations in non-metric imageries includ- nonlinear components of lens distortions and ing irregularities in film material, handling film deformations) in data reduction is highly during processing, unflatness of the film in- recommended if one wishes to obtain side the camera, tension exerted on the film reasonably accurate results. In applications inside the camera (between one photograph of low accuracy requirements, one may dis- and another) and outside the camera (during regard Ax and hy in Equation 2 where the processing), and temperature and relative ACCURACY ASPECTS OF NON-METRIC IMAGERIES humidity during storage of the film and dur- 11. In this instance, the DLT solution involves ing its processing. 12 unknowns. Several mathematical nlodels can be used Model 111. An odd radial polynomial of to represent film deformations. To estimate the seventh degree is added to Model I to the parameters of such models, one has to account for symmetrical lens distortion, have some calibrated references (fiducial marks or reseau). In non-metric cameras, hx = al + a2x + a3y + ;(klr2 + k2r4 + k3r6) however, such references are generally not available and one can only use object-space Ay = a4 + up + a6y + i(klr2 + k2r4 + k3r6). control to determine the combined effect of film defor~nationand lens distortion. The es- (8) timates RMS values of the residual errors The Solution in this instance involves 14 after image refinement, thus reflect, in this unknowns, instance, unrepresented film deformations, Model IV. Model 111 is combined with unrepresented lens distortions and the ran- Conrady,s for asymlnetrical lens dom errors in measurement. distortion, MATHEMATICAL MODELING OF IMAGE REFINEMENT Models Tested. Based on the mathematical models used for fi11n deformation and lens distortions in aerial cameras, and on results of experiments conducted by the authors2, the following six mathematical models for image refinement in non-metric photography were The DLT solution here involves 16 unknowns. selected for an experimental investigation. Model V. The same as Model IV, except Model I. Linear polynomial in x and y, the radial polynomial accounting for lens distortion is a full polynomial of the seventh degree, In this model, only the linear components of lens distortion and film deformation are taken into 'consideration. The nonlinear components of image refinement are neg- lected. Equation 6 accounts for the lack of: perpendicularity of the x and y axes and al- lows for different scale factors along the x and y directions. Here the DLT solution involves 18 unknowns. Incorporating Model I in Equation 2 does Model VI. Same as Model V, except the not change the form of the equations. In other polynomial in x and y ii of the second words, the unknowns in the DLT solution degree, accounting for the nonlinear com- . using Model I remain as 11 unknowns. This ponent of film deformation, should not be surprising as Equation 2, as stated earlier, takes into account the linear components of image refinement, in the process of transforming comparator coordinates into object-space coordinates. In combining the two sets of Equations 6 and 2, the six coefficients in Equation 6 thus absorbed by the 11 coefficients in Equation 2. Model 11. One more unknown (kl) is added to Model I to account for symmetrical lens distortion, In this model, the DLT solution involves 22 unknowns. In all the above models, x,y are the image coordinates of the point under consideration, XS,~JS are the-image coqdinates of the point of symmetry, x = x - xs, y = y - y~,r = length of in which the terms are defined after Equation the vector from the point of symmetry to the Camera Lens Focal Image Approx. Length Format Price (mm) (mm) 6) Kodak Instamatic 154 Kodak 43 12x 12 15 Crown Graphic Graphex 04.7 Honeywell Pentax Spotmatic Super Takumar 01.4 Hasselblad 500 C Planar U2.8 Hasselblad MK 70 Biogon fl5.6 MK 60 55x55 4500 image point under consideration, Ax,Ay = image refinement components, al,az, .

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